/
Автор: Kleinert H. Jantzen R.T.
Теги: physics gravity field theory theory of relativity relativistic mechanics
ISBN: 978-981-283-426-3
Год: 2008
Текст
THE ELEVENTH
MARCEL GROSSMANN MEETING
On Recent Developments in Theoretical and Experimental
General Relativity, Gravitation and Relativistic Field Theories
Also published by World Scientific:
PROCEEDINGS OF THE SIXTH
MARCEL GROSSMANN MEETING ON GENERAL RELATIVITY
PART A & PART B
Eds. Humitaka Sato and Takashi Nakamura
Series Ed. Remo Ruffini
PROCEEDINGS OF THE SEVENTH
MARCEL GROSSMANN MEETING ON GENERAL RELATIVITY
PART A & PART B
Eds. Robert T. Jantzen and G. Mac Keiser
Series Ed. Remo Ruffini
PROCEEDINGS OF THE EIGHTH
MARCEL GROSSMANN MEETING ON GENERAL RELATIVITY
PART A & PART B
Ed. Tsvi Piran
Series Ed. Remo Ruffini
PROCEEDINGS OF THE NINTH
MARCEL GROSSMANN MEETING ON GENERAL RELATIVITY
PART A, PART B & PART C
Eds. Vahe G. Gurzadyan and Robert T. Jantzen
Series Ed. Remo Ruffini
PROCEEDINGS OF THE TENTH
MARCEL GROSSMANN MEETING ON GENERAL RELATIVITY
PART A, PART B & PART C
Eds. Mario Novello and Santiago Perez Bergliaffa
Series Ed. Remo Ruffini
PARTC
THE ELEVENTH
MARCEL GROSSMANN MEETING
On Recent Developments in Theoretical and Experimental
General Relativity, Gravitation and Relativistic Field Theories
Proceedings of the MG11 Meeting
on General Relativity
Berlin, Germany 23-29 July 2006
Editors
Hagen Kleinert
Freie Universitat Berlin, Germany
Robert T Jantzen
Villanova University, USA
Series Editor
Remo Ruffini
University of Rome "La Sapienza"
Rome, Italy
\fc World Scientific
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THE ELEVENTH MARCEL GROSSMANN MEETING (In 3 Volumes)
On Recent Developments in Theoretical and Experimental General Relativity,
Gravitation and Relativistic Field Theories
Copyright © 2008 by World Scientific Publishing Co. Pte. Ltd.
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ISBN-13 978-981-283-426-3 (Set)
ISBN-10 981-283-426-5 (Set)
ISBN-13 978-981-283-427-0 (Vol. 1)
ISBN-10 981-283-427-3 (Vol. 1)
ISBN-13 978-981-283-428-7 (Vol.2)
ISBN-10 981-283-428-1 (Vol.2)
ISBN-13 978-981-283-429-4 (Vol.3)
ISBN-10 981 -283-429-X (Vol. 3)
Printed in Singapore by Mainland Press Pte Ltd
THE MARCEL GROSSMANN MEETINGS
The Marcel Grossmann Meetings were conceived with the aim of reviewing recent
developments in gravitation and general relativity, with major emphasis on
mathematical foundations and physical predictions. Their main objective is to bring
together scientists from diverse backgrounds in order to deepen our
understanding of spacetime structure and review the status of experiments testing Einstein's
theory of gravitation.
Publications in the Series of Proceedings
Proceedings of the Eleventh Marcel Grossmann Meeting on General Relativity
(these volumes)
(Berlin. Germany, 2006)
Edited by H. Kleinert, R.T. Jantzen, R. Ruffini
World Scientific, 2008
Proceedings of the Tenth Marcel Grossmann Meeting on General Relativity
(Rio de Janiero, Brazil, 2003)
Edited by M. Novello, S. Perez-Bergliaffa, R. Ruffini
World Scientific, 2005
Proceedings of the Ninth Marcel Grossmann Meeting on General Relativity
(Rome, Italy, 2000)
Edited by V.G. Gurzadyan, R.T. Jantzen, R. Ruffini
World Scientific, 2002
Proceedings of the Eighth Marcel Grossmann Meeting on General Relativity
(Jerusalem, Israel, 1997)
Edited by T. Piran
World Scientific, 1998
Proceedings of the Seventh Marcel Grossmann Meeting on General Relativity
(Stanford, USA, 1994)
Edited by R.T. Jantzen and G.M. Keiser
World Scientific, 1996
Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity
(Kyoto, Japan, 1991)
Edited by H. Sato and T. Nakamura
World Scientific, 1992
v
VI
Proceedings of the Fifth Marcel Grossmann Meeting on General Relativity
(Perth, Australia, 1988)
Edited by D.G. Blair and M.J. Buckingham
World Scientific, 1989
Proceedings of the Fourth Marcel Grossmann Meeting on General Relativity
(Rome, Italy, 1985)
Edited by R. Ruffini
World Scientific, 1986
Proceedings of the Third Marcel Grossmann Meeting on General Relativity
(Shanghai, People's Republic of China, 1982)
Edited by Hu Ning
Science Press - Beijing and North-Holland Publishing Company, 1983
Proceedings of the Second Marcel Grossmann Meeting on General Relativity
(Trieste, Italy, 1979)
Edited by R. Ruffini
North-Holland Publishing Company, 1982
Proceedings of the First Marcel Grossmann Meeting on General Relativity
(Trieste, Italy, 1976)
Edited by R. Ruffini
North-Holland Publishing Company, 1977
Series Editor: REMO RUFFINI
SPONSORS
Free University Berlin (FUB)
The German Research Foundation (DFG)
The German Academic Exchange Service (DAAD)
The Frankfurt Institute for Advanced Studies (FIAS)
The Frankfurt Institute for Advanced Studies (FIAS)
The International Union of Pure and Applied Physics (IUPAP)
The Center of Applied Space Technology and Microgravity (ZARM)
The Wilhelm and Else Heraus Foundation
International Center for Theoretical Physics (ICTP)
MPI for Gravitational Physics (Albert Einstein Institute)
MPI for Extraterrestrial Physics
ORGANIZING BODIES OF THE
ELEVENTH MARCEL GROSSMANN MEETING:
INTERNATIONAL ORGANIZING COMMITTEE
David Blair, Yvonne Choquet-Bruhat, Demetrios Christodoulou, Thibault Damour,
Jurgen Ehlers, Francis Everitt, Fang Li-Zhi, Stephen Hawking, Yuval Ne'eman
Remo RufHni (Chair), Huinitaka Sato, Rashid Sunayev, Steven Weinberg
LOCAL ORGANIZING COMMITTEE
Bernd Briigmann, A. Chervyakov, Hansjorg Dittus, W. Janke, Hagen Kleinert
(chair), Jutta Kunz, Claus Laemmerzahl, Flavio Nogueira, Axel Pelster, Luciano
Rezzolla, Erwin Sedlmayr, Stefan Theisen, Thomas Thiemann
INTERNATIONAL COORDINATING COMMITTEE
Bob Jantzen (chair)
ALBANIA: Hafizi M., ARGENTINA: Jakubi A.S., Mirabel F., Nunez C.A.,
ARMENIA: Gurzadyan V., AUSTRALIA: Lun A., Manchester D., Scott S.M., Steele
J.D., Veitch P., AUSTRIA: Aichelburg P.C., Schindler S., BELGIUM: Henneaux
M., Surdej J., BELORUSSIA: Minkevich A.V., BOLIVIA: Aguirre C.B., BRAZIL:
Aguiar O., Aldrovandi R., Novello M., Opher R., Perez Bergliaffa S.E., Villela T.,
CANADA: Cooperstock F., Page D.N., Papini G, Smolin L., CHILE: Bunster
Weitzman C, CHINA (Beijing): Feng L.-L., Gao J.-G, Lee D.-S, Lee W.OL., Li
M., Ni W.-T., Wu X.-P., Yipeng J., CHINA (Taepei): Lee D.S., Lee W.L., Ni W.T.,
COLOMBIA: Sepulveda H.A., Torres S., CROATIA: Milekovic M., CUBA: Quiros
I., CZECK REPUBLIC: Bicak J., DENMARK: Novikov I., EGYPT: Wanas M.I.,
ESTONIA: Einasto J., FRANCE: Brillet A., Chardonnet P., Coullet P., de Fre-
itas Pacheco J.A., Deruelle N., Iliopoulos J., Mignard, F., GEORGIA: Lavrelashvili
G, GERMANY: Biermann P., Danzmann K., Fritzsch H., Genzel R., Greiner W.,
Hasinger G, Hehl F., Kiefer C, Neugebauer G, Nicolai H., Renn J., Ringwald
A., Ruediger A., Schutz B., GREECE: Batakis N., Cotsakis S., HUNGARY: Fodor
G, ICELAND: Bjornsson G, INDIA: Narlikar J., Sahni V., Vishveshwara C.V.,
IRAN: Mansouri R., Sobouti, Y., IRELAND: O'Murchada N., ISRAEL: Piran T.,
Sobouti, Y., ITALY: Belinsky V., Bianchi M., Ciufolini I., Menotti P., Regge T.,
Stella L., Treves A., JAPAN: Fujimoto M.K., Makino J., Nakamura T., Sasaki M.,
Sato K., Tomimatsu A., KAZACHSTAN: Abdildin A.M., Mychelkin E.G., KOREA
(Pyeongyang): Kim J.S., Kim Y.G, KOREA (Seoul): Lee Chul H., Lee Hyung W.,
Song Jong D., KYRGYZSTAN: Gurovich V.Ts., LIBYA: Gadri M., LITVA: Piragas
K.A., MEXICO: Garcia-Diaz A.A., Macias-Alvarez A., Mielke E.W., Rosenbaum
M., Ryan M.P., NETHERLANDS: 't Hooft G, NEW ZEALAND: Visser M.,
Wiltshire D., NORWAY: Knutsen H., POLAND: Demianski M., Nurowski P., Sokolowski
VII
VIII
L., PORTUGAL: Costa M., Vargas Moniz R, ROMANIA: Visinescu M.,
RUSSIA: Bisnovatyi-Kogan G.S., Blinnikov, S., Chechetikin V.M., Cherepaschuk A.M.,
Khriplovich I.B., Kotov Y., Lipunov V.M., Lukash V., Melnikov V., Rudenko V.,
Starobinsky A.A., Tchetchetkine V. M., SERBIA: Sijacki D., SLOVENIA: Cadez
A., SOUTH AFRICA: Maharaj S., SPAIN: Ibanez J., Perez Mercader J., Verda-
guer E., SWEDEN: Marklund M., Rosquist K., SWITZERLAND: Durrer R., Jet-
zer P., TURKEY: Nutku Y., UK: Barrow J., Cruise A.M., Green M., Kibble T.,
Maartens R., USA: Ashtekar A., Bardeen J., Barish B., Chen P., Cornish N., Der-
mer C, DeWitt-Morette C, Drever R., Finkelstein D., Halpern L., Hellings R.W.,
Jantzen R.T., Klauder J., Kolb R., Lousto C, Mashhoon B., Matzner R., Melia F.,
Nordtvedt K., Parker L., Pullin J. Schwarz J., Shapiro I., Shoemaker D., Smoot
G., Thorne K.S., van Nieuwenhuizen P., York J.W. Jr., UZBEKISTAN: Zalaletdi-
nov R.M., VATICAN CITY: Stoeger W., VENEZUELA: Herrera L., Percoco U.,
VIETNAM: van Hieu N.
ACKNOWLEDGMENTS
We acknowledge the help of the following individuals before, during and after the
actual meeting itself: Michael Kleinert (meeting webmaster and local IT organizer),
Anneinarie Kleinert (chief local organizer and finance manager), Flavio Nogueira
(local meeting point man), and the staff, students and postdocs of Hagen Kleinert's
research group, and the ICRANet/ICRA secretarial support: Federica Di Berardino,
Veronica D'Angelo, Gilda Massa, Cesare Corsetti. We also acknowledge the
generous assistance of the Italian Foreign Ministry and in particular of the Science and
Technology Attache of the Italian Embassy in Berlin Prof. Vincenzo Tovi.
In an age of increasing technological sophistication, this meeting could not have
functioned without the tireless dedication of ICRA system manager Vittorio
Vanning nor could these proceedings have been possible without his patient
management of the email and web communication and data handling necessary to produce
them. We also recognize the past contributions of the late system ICRA co-system
manager Maurizio Cosma whose friendship and valuable contributions to past MG
Meetings should not go unrecognized.
Finally we acknowledge the loss of our friend Leopold Halpern, a physicist,
humanitarian, environmentalist, naturalist, world traveler and participant in every
MG Meeting whose advice to a young physicist (Remo Ruffini) at a key moment
influenced his choice to enter the field of general relativity and later cofound this
Meeting series.
MARCEL GROSSMANN AWARDS
ELEVENTH MARCEL GROSSMANN MEETING
Institutional Award
Freie Universitat Berlin
"for the successful endeavour of re-establishing
— in the spirit of the Humboldt tradition - -
freedom of thinking and teaching within a democratic society
in a rapidly evolving cosmos"
—presented to Dr. Dieter Lenzen, President of FUB
Individual Awards
Ro}' Kerr
"for his fundamental contribution to Einstein's theory of
general relativity: The gravitational field of a
spinning mass as an example of algebraically special metrics"
George Coyne
"for his committed support for the international development of
relativistic astrophysics and for his dedication to fostering
an enlightened relationship between science and religion"
Joachim Triknper
"for his outstanding scientific contributions to the physics of compact
astrophysical objects and for leading the highly successful ROSAT mission
which discovered more then 200,000 galactic and extragalactic X-ray sources:
a major step in the observational capabilities of X-ray astronomy
and in the knowledge of our universe"
Each recipient is presented with a silver casting of the TEST sculpture by the
artist A. Pierelli. The original casting was presented to His Holiness Pope John Paul
II on the first occasion of the Marcel Grossmann Awards.
IX
X
TENTH MARCEL GROSSMANN MEETING
Institutional Award
CBPF (BRAZILIAN CENTER FOR RESEARCH IN PHYSICS)
Individual Awards
YVONNE CHOQUET-BRUHAT, JAMES W. YORK, JR., YUVAL NE'EMAN
NINTH MARCEL GROSSMANN MEETING
Institutional Award
THE SOLVAY INSTITUTES
Individual Awards
RICCARDO GIACCONI, ROGER PENROSE
EIGHTH MARCEL GROSSMANN MEETING
Institutional Award
THE HEBREW UNIVERSITY OF JERUSALEM
Individual Awards
TULLIO REGGE, FRANCIS EVERITT
SEVENTH MARCEL GROSSMANN MEETING
Institutional Award
THE HUBBLE SPACE TELESCOPE INSTITUTE
Individual Awards
SUBRAHMANYAN CHANDRASEKHAR, JIM WILSON
SIXTH MARCEL GROSSMANN MEETING
Institutional Award
RESEARCH INSTITUTE FOR THEORETICAL PHYSICS (Hiroshima)
Individual Awards
MINORU ODA, STEPHEN HAWKING
FIFTH MARCEL GROSSMANN MEETING
Institutional Award
THE UNIVERSITY OF WESTERN AUSTRALIA
Individual Awards
SATIO HAYAKAWA, JOHN ARCHIBALD WHEELER
FOURTH MARCEL GROSSMANN MEETING
Institutional Award
THE VATICAN OBSERVATORY
Individual Awards
WILLIAM FAIRBANK, ABDUS SALAM
XII
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V V--Si?**''**'. --■.'.:-.',;
i'.•>*■".«fififs • ?' ■* 1V1
.^
■m.
*
■/*;
TEST; sculpture by A. Pierelli.
PREFACE
The Eleventh Marcel Grossmann Meeting on General Relativity (MGll) took place
during July 23-29, 2006 on the Campus of the Freie Universitat Berlin, an attractive
location for both practical and historical reasons. It is situated in the park-like
district of Berlin-Dahleni, where many famous German researchers of the early
20th century lived and worked, among them Planck and Einstein (Fig. 1). The
conference site lies close to the former Kaiser-Wilhelm-Tnstitute of Physics where
Hahn, Meitner, and Strassmann discovered the fission of uranium in 1938 (Fig. 2).
Fig. 1 Fig. 2
Kg. 3
Otto Halm's house is just around the corner. So is Einstein's apartment in Ehren-
bergstrasse 33 where he lived after moving from Zrich in 1914 (Fig. 3, with zoomed
bronze memorial plate at the entrance). Around 800 participants and accompany-
XIII
XIV
J' J*
Fig. 4
Fig. 5
ing persons were present during a week of exceptionally warm summer weather in
Berlin. The meeting began with the Marcel Grossmann Awards ceremony on July
23. The institutional award went to Freie Universitt (FU) Berlin (Fig. 4) "for the
successful endeavor of re-establishing — in the spirit of the Humboldt tradition —
freedom of thinking and teaching within a democratic society in a rapidly evolving
cosmos". Remo Ruffini handed the award to Dieter Lenzen, president of the FU
Berlin (Figs. 4 and 5).
Three individual awards were presented to Roy Kerr "for his fundamental
contribution to Einstein's theory of general relativity: The gravitational field of a spinning
mass as an example of algebraically special metrics".
Three individual awards were presented to
Roy Kerr "for his fundamental contribution to Einstein's
theory of general relativity: The gravitational field of a
spinning mass as an example of algebraically special
metrics" .
George Coyne "for his committed support for the
international development of relativistic astrophysics and for his
dedication to fostering an enlightened relationship between
science and religion".
XV
Joachim Triimper "for his outstanding scientific
contributions to the physics of compact astrophysical objects and
for the leading successful ROSAT mission which
discovered more than 200,000 galactic and extragalactic X-ray
sources: a major step in the observational capabilities of
X-ray astronomy and in the knowledge of our universe".
Each laureate received a silver casting of the TEST sculpture by the artist
A. Pierelli. The original casting was presented on the first occasion of the Marcel
Grossrnann Award to His Holiness Pope John Paul II.
After the prize ceremony the plenary program started with lectures by:
Thibault Damour (IHES, Bures-sur-Yvette) "Cosmology and string theory"
Sasha Polyakov (Princeton University) "The structure beyond spacetime"
Hermann Nicolai (Albert-Einstein-Inst. Potsdam) "Hidden symmetries and cos-
mological singularities"
They were continued each morning from Tuesday to Saturday with the following
speakers:
Claes Uggla (Karlstaads University) "The nature of generic cosmological
singularities"
Eva Silverstein (Stanford University) "Cosmological singularities in string theory"
Igor Klebanov (Princeton University) "Gauge theories, strings and cosmology"
Joe Polchinski (UC Santa Barbara) "Cosmic superstrings"
Abhay Ashtekar (Pennsylvania State University) "Loop quantum gravity"
Dieter Luest (Humboldt Univ., Berlin) "String theory and the standard model of
particle physics"
Karsten Danzmann (Univ. Hannover) "LISA"
Marie Anne Bizouard (Univ. Paris XI) "VIRGO"
David Shoemaker (MIT) "LIGO: Status of instruments and observations"
Alessandra Buonanno (Univ. of Maryland) "Analytical approach to coalescing
binary black holes"
Francois Mignard (Observatoire C6te d'Azur) "Relativistic effects from HIPPAR-
COS and GAIA missions"
Michael Kramer (Univ. of Manchester) "Binary pulsars and general relativistic
effects"
Josh Grindlay (Harvard Univ.) "Globular clusters and millisecond pulsars"
tE-is
XVI
Richard Mushotzky (NASA Goddard SFC) "Intermediate mass black holes and
X-ray sources"
Rashid Sunyaev (MPA Garching) "The sky in the hard x-ray spectrum"
Reinhard Genzel (MPE Garching) "The black hole in our galactic center"
George Djorgovski (CALTECH), "The origins of massive black holes and quasars
at high redshifts"
Remo Ruffini (ICRA, Roma) "Gamma ray bursts"
Francis Halzen (University of Wisconsin-Madison) "ICE CUBE"
Peter Biermann (MPI for Radioastronomy, Bonn) "Sterile neutrinos in
astrophysics and cosmology"
Volker Springel (MPI for Astrophysics Garching) "Simulations of the formation,
evolution and clustering of galaxies and quasars"
Paolo De Bernardis (Univ. Roma La Sapienza) "CMB science from Boomerang
to PLANCK"
David Spergel (Princeton Center for Theoretical Physics) "WMAP and its cos-
rnological implications"
Ethan J. Schreier (AUI, Washington, DC) "ALMA"
John Mester (Stanford University) "Equivalence principle from space"
Francis Everitt (Stanford University) "The NASA Gravity Probe B Mission:
technical report"
Guy Monnet (Europ. South. Observatory, Garching) "Science and technology of
the European ELT"
Michael Garcia (Harvard-Smithsonian Ctr. for Astroph.) "Science from Chandra
to Constellation-X"
Nicholas White (HEASARC) "Beyond Einstein: from the big bang to black holes"
Theodor Haensch (Ludwig-Maximilian Univ. Miinchen) "Precise clocks"
Juergen Renn (MPI for the History of Science, Berlin) "The genesis of general
relativity"
On Monday, Tuesday, Thursday, and Friday, public lectures were presented by
Hanns Ruder (University Tubingen) "Visualizations of relativistic effects"
Giinter Hasinger (MPE Garching) "The fate of the universe — new clues from
cosmology"
Bruno Leibundgut (Eur. Southern. Obs., Garching) "Das nene Weltbild der Kos-
mologie — Was ist Dunkle Energie?"
Christian Spiering (DESY Zeuthen) "Neutrinoastronomie — ein neues Fenster
zum Kosmos"
These lectures were well attended by Berlin citizens and conference participants
and found broad resonance in the media. Parallel sessions were held on the
afternoons in 20 lecture halls. Some 850 scientific papers were presented during 82
parallel sessions over four afternoons. A typical setting in front of one of the lecture
halls is shown below.
XVII
Many speakers at MGll were accommodated in the famous Harnack House, a
place where much of the "Dahlem Legend" happened. The house was built during
the Weimar republic by the theologian Adolf von Harnack, the first president of
the Kaiser-Wilhelm Society. Many German Nobel Prize winners and their students
met here for social interaction and academic discussion. Here they held lectures and
eolloquia, took lunch together, read the new international press, drank coffee in the
garden, engaged in sports, and played music. The list of former guests and lecturers
reads like a "Who's Who of Science": Albert Einstein, Peter Debye, Werner Heisen-
berg, Fritz Haber, Adolf Butenandt, Otto Hahn, Lise Meitner, Otto Meyerhof, Max
Planck, Max von Laue and Otto Warburg. One Nobel Prize winner, the biologist
Hans Fischer, even received the news of his award during his stay at the Harnack
■"■-Vs.
XVIII
House. Also great non-scientists stayed at this house, for instance Ricarda Huch,
the Swiss art historian Heinrich Wolfflin, and the Indian philosopher Rabindranath
Tagore. In 1935, in direct opposition to the government, Max Planck led an
impressive commemoration of Fritz Haber here. The Kaiser-Wilhelm Institutes were later
re-organized and renamed as the Max Planck Institutes.
During MG11, a big beer tent was set up in the courtyard of the physics
department in the style of the famous Munich Oktoberfest, which was well frequented by
all participants since its informal atmosphere was very beneficial for social
interactions and the exchange of ideas.
A video stream exchange was set up with the Einstein Institute in Potsdam
so that its members were able to follow the Marcel Grossmann lectures and the
participants in Berlin could listen to lectures at the Einstein Institute if desired.
The opulent MG11 conference banquet dinner was held at the Ritz Carlton Hotel
next to Potsdamer Platz. A Prussian 19th century type brass orchestra was there
to play music from the emperor's time.
On July 29 Remo Rumni closed the meeting thanking all the speakers and
participants and sponsoring institutions.
These three volumes represent the proceedings of the meeting. The first
volume contains articles by many of the plenary speakers together with some of the
review articles from the parallel sessions. The second and third volumes contain
the remaining contributions from the parallel sessions. The participant list and the
author index complete the third volume.
INAUGURAL ADDRESS
Dear Mr. Ruffini,
Dear Mr. Sreenivasan,
Dear Mr. Umbach,
My Dear Colleague Mr. Kleinert,
Ladies and Gentlemen,
Honored Guests,
On the occasion of this year's Marcel Grossmann Conference at the Freie
Universitat Berlin, it is a special honor for me to welcome all of you here to Dahlem,
one of Berlin's largest and most important centers of science and scholarship.
To honor the epochal achievements of Albert Einstein, who worked in Berlin-
Dahlem for nearly two decades as director of the Kaiser Wilhelm Institute for
Physics, is a central concern of the Freie Universitat Berlin. Albert Einsteins time in
Berlin saw the emergence of contributions to physics that were so outstanding that
they have continued to be a source of fascination in the field of physics and beyond.
This morning, exceptional scientists received the distinction of the Marcel
Grossmann Prize. The Freie Universitat Berlin too will be distinguished as an institution
that has rendered extraordinary services to unfettered science and scholarship ever
since its foundation. It is with great pleasure that I receive this honor in the name of
the Freie Universitat Berlin, for like virtually no other German institution of higher
learning, our university is closely associated with the concept of "freedom".
The establishment of this university in 1948 can be traced back to the struggle
for academic freedom. The impetus for its foundation emanated primarily from
students who—after bitter experiences with the National Socialist dictatorship—were
committed to freedom and democracy, and who rejected the relegation of Students
of Berlins Humboldt University in accordance with the worldview prevailing in the
East.
Through international material assistance, the university's members and
numerous sponsors among Berlin's citizens saw to it that students from the surrounding
regions and from Berlins eastern sector who had been refused the opportunity to
study for political reasons were able to complete their training at the Freie
Universitat. All of this transpired against a backdrop of escalating conflict between the
Western allies and the Soviet Union concerning Europe's future political
organization. The founding of the Freie Universitat proceeded in the middle of the Soviet
blockade of West Berlin, which sealed the city off from the outside world from June
1948 to May 1949. Freie Universitat survived even the Berlin Blockade, because
international aid arrived, especially from the United States. Until 1961, when the
partition of the city was cemented by the erection of the Berlin Wall, the
university's founders had succeeded in establishing firm and supportive international
networks that won Freie Universitat a recognised position among German and in-
XIX
XX
ternational universities. We continue to benefit from these international networks
today. The crimes of German fascism only began to be examined in earnest in 1968.
The student movement, which largely originated at Freie Universitat, was also a
response to this need for unfettered scrutiny of our past. In the following years, the
traditional university under professorial governance was replaced by more
accountable structures, in which all members of the university are represented in university
governance. Our university continued to play a vital role after the breakdown of
the communist GDR and in the course of German reunification. The key challenge
then was to rebuild the universities that had lived under communist dictatorship
by giving them both financial and intellectual support. In the course of this internal
reform, our university was even able to raise its performance by some ten per cent
per annum since 2000. Now that this reform has been completed, we look forward
to addressing the universitys strategic globalization, based on its long tradition of
international networking. It is no coincidence that the Freie Universitat is delighted
to host a series of events such as the Marcel Grossinann Conference, for intensive
exchange between the sciences has been a key priority of the Freie Universitat Berlin
since its foundation here in Dahleni in 1948. And here in particular, in Dahlem in
the south of Berlin, the Freie Universitat Berlin perpetuates a scientific tradition
that provides ideal preconditions for an indispensable ingredient of contemporary
scientific and scholarly work: networked, interdisciplinary activities that transcend
subject areas and disciplinary boundaries. In light of this "Dahlem Myth" and of
the tradition of interdisciplinary exchange that is bound up with it, we can only
regard our own times—which demand so much readiness for change and reform in the
sphere of education and elsewhere as representing a new departure, one we must
use to our advantage, since for all of us, the future lies in science and education.
In this spirit, honored guests, I wish all of you a stimulating time at this year's
Marcel Grossmann Conference. Thanks to the organizing committee, to you, my
dear colleague Mr. Kleinert, and to the Department of Physics. And to all of you,
a warm welcome to the Freie Universitat Berlin!
Dieter Lenzen
President of Freie Universitat Berlin
MARCEL GROSSMANN AWARD ESSAY
George Coyne, S.J.
Director' Emeritus of the Vatican Observatory
I was deeply honored to have received a Marcel Grossman Award at the July
2006 meeting in Berlin, a city so rife with memories of discoveries in physics. The
citation noted my interest in the relationship between science and religion. Even
these few years since that meeting have seen many interesting developments in that
relationship. In fact, most recently some have even arrived at seriously posing the
question: Is God a mathematician? The background to that question harks back to
Albert Einstein's comment: "The most incomprehensible thing about the universe
is that it is comprehensible." But, in what way is it comprehensible? Here enters
the question as to God and mathematics. This question is, I think, at the core of
the intersection of the two cultures of science and religion in today's world.
Let us begin by marveling, as many others, including Einstein, did, that the
universe is comprehensible. In fact, I have co-authored with Michael Heller a book
entitled: "A Comprehensible Universe" (Springer Vcrlag, in press). We see the com-
prehensibility of the universe as due to its mathematical structure. One can
challenge the notion that physics is limited to the investigation of matter. In fact, in
much of the research in physics emphasis is placed on the fact that physics
constructs mathematical models of the world and then confronts them with empirical
results. And such an approach has had an astonishing success because, indeed, the
world has a mathematical structure to it. And who set up that structure?
Science itself cannot find the WHO? But, that mathematical structure can serve
as an enticement, an invitation to go beyond the strict methodology of science to
the ultimate question: WHO? But let us look more closely at the concept of the
mathematical structures of the universe, which provide its comprehensibility and,
ultimately, the invitation to approach the WHO. At the birth of modern science
there was the persistent idea, as there had been for the Pythagoreans, that
physicists were discovering some grand transcendental design incarnate in the universe.
As to religious insights, the concept in St. John's Gospel of the logos becoming
incarnate was particularly appropriate and hailed back in some way to Platonic and
Pythagorean concepts of the world of eternal ideas and of the transcendental
character of mathematics. Indeed, Newton, Descartes, Kepler and others can be cited
as viewing physics and mathematics in this way. Kepler for instance, saw geometry
as providing God with a model for creation. He went so far as to see the circle as
transcendentally perfect, the straight line as the totally created and incarnate and
the ellipse as a combination of the two, an incarnation in this world of what would
have been the perfect geometry for the motion of the heavenly bodies in an ideal
world. The simple equations in which Newton expressed the law of gravity and the
laws of motion redirected for future centuries the role of mathematics in physics.
No longer was mathematics simply a description of what was observed; it was a
XXI
XXII
probe of the very nature of what was observed. This role of mathematics was only
enhanced as relativity theory, quantum mechanics and then quantum cosmology
came on the scene.
Leibniz once claimed that "When God calculates and thinks things through, the
world is made." Things thought through by God might be identified with
mathematical structures interpreted as structures of the visible universe. For God to plan
is the same as to implement the plan and thus to create. God has planned and,
thereby, created a structured world which participates, through the subtle random
events intrinsic to the structure, in the very creativity of God.
Will we eventually understand comprehensively the structure of the universe
and, therefore, the mind of the mathematician God? I suggest a definitive no. God
is mystery and the source of all that is mysterious in the universe. The search for
the ultimate mathematical structure is unending and that is what makes the search
being carried on by many scholars such a passionate adventure.
CONTENTS
The Marcel Grossmann Meetings — Publications in this Series and Sponsors . v
Organizing Committees vii
Marcel Grossmann Awards ix
Preface xiii
Inaugural Address xix
Marcel Grossmann Award Essay xxi
PART A
PLENARY AND REVIEW TALKS
A Brief History of X-Ray Astronomy in Germany
Truemper, Joachim E 3
On the Discovery of the Kerr metric
Kerr, Roy Patrick 9
Chaos and Symmetry in String Cosmology
Damour, Thibault 39
Hidden Symmetries, Cosmological Singularities and the
EW/K{EW) Sigma Model
Kleinschmidt, Axel; Nicolai, Herman 49
The Nature of Generic Cosmological Singularities
Uggla, Claes 73
QCD and String Theory
Klebanov, Igor R 90
The Cosmic String Inverse Problem
Polchinski, Joseph 105
Loop Quantum Gravity: Four Recent Advances and a Dozen
Frequently Asked Questions
Ashtekar, Abhay 126
String Theory Landscape and the Standard Model of Particle
Physics
Lust, Dieter 148
The Status of the Virgo Gravitational Wave Detector
Bizouard, Marie-Anne; for the Virgo collaboration 177
XXIII
XXIV
Analytical Modeling of Binary Black Holes Coalescence
Buonanno, Alessandra 197
Binary Pulsars and General Relativistic Effects
Kramer, Michael 225
Space Astronometry and Relativity
Mignard, Frangois; Klioner, Sergei A 245
Neutrino Astronomy 2006
Halzen, Francis 272
Dark Matter and Sterile Neutrinos
Biermann, Peter L.; Munyaneza, Faustin 291
Supercomputer Simulations of the Joint Formation and Evolution
of Galaxies and Quasars
Springel, Volker 309
CMB Observations: From BOOMERanG to Planck ... and Beyond
De Bernardis, Paolo 326
The Origins and the Early Evolution of Quasars and Supermassive
Black Holes
Djorgovski, S. George; Volonteri, Marta; Springel, Volker; Bromm,
Volker; Meylan, Georges 340
On Gamma Ray Bursts
Ruffini, Remo; Bernardini, Maria Grazia; Bianco, Carlo Luciano;
Caito, Letizia; Chardonnet, Pascal; Cherubini, Christian; Dainotti,
Maria Giovanna; Fraschetti, Federico; Geralico, Andrea; Guida,
Roberto; Patricelli, Barbara; Rotondo, Michael; Rueda Hernandez,
Jorge Armando; Vereshchagin, Gregory; Xue, She-Sheng 368
Passion for Precision
Theodor, Hansch W 506
The Genesis of General Relativity
Renn, Jiiergen 532
Superposition of Fields of Two Reissner-Nordstrom Sources
Alekseev, George A.; Belinski, V.A 543
Quasiperiodic Oscillations due to Axisyminetric and Non-
axisymmetric Shock Oscillations in Black Hole Accretion
Chakrabarti, Sandip K.; Debnath, D,; Pal, P.S-; Nandi, A.; Sarkar,
R.; Samanta, M.M.; Wiita, P. J.; Ghosh, H.; Som, D 569
Power Spectra of Black Holes (BH) and Neutron Stars (NS) as
a Probe of Hydrodynamical Structure of the Source: Diffusion
Theory and its Application to X-ray Observations of NS and
XXV
BH Sources
Titarchuk, Lev; Shaposhnikov, Nikolai; Arefiev, Vadim 589
Quark Matter in Compact Stars: Astrophysical Implications and
Possible Signatures
Bombaci, Ignazio 605
Gauge Gravity and Electroweak Theory
Hestenes, David 629
Black Holes in Higher Dimensions (Black Strings and Black Rings)
Kunz, Jutta 648
Some Remarks on Microlensing Towards LMC and M31
Jetzer, Philippe 663
OGLE-2005-BLG-390Lb — Gravity Reveals First Cool Rocky /Icy
Exoplanet
Dominik, Martin 670
Theoretical Gravitational Lensing — Beyond the Weak-Field
Small-Angle Approximation
Perlick, Volker 680
Nonsingular Collapse of Spherically Symmetric Charged Dust
Krasinski, Andrzej; Bolejko, Krzysztof 700
Quantum Cosmology Standpoint
Vargas Moniz, Paulo 708
Gamma Ray Burst Host Galaxies and the Link to Star-Formation
Fynbo, Johan P. U.; Hjorth, Jens; Malesani, Daniele; Sollerman,
Jesper; Watson, Darach J.; Jakobsson, Pall; Gorosabel, Javier;
Jaunsen, Andreas 0 726
Gamma-Ray Bursts with and without Supernova Fireworks
Delia Valle, Massimo 736
Talking about Singularities
Cotsakis, Spiros 758
Time Machines and Quantum Theory
Hadley, Mark J 778
Slowly and Rigidly Rotating Perfect Fluid Balls of Petrov Type D
Bradley, Michael; Eriksson, Daniel; Fodor, Gyula; Rdcz, Istvdn 795
Numerical Wave Optics and the Lensing of Gravitational Waves
by Globular Clusters
Moylan, Andrew J.; McClelland, David E.; Scott, Susan M.;
Searle, Antony C; Bicknell, Geoff V. 807
XXVI
Inflation, Bifurcations of Nonlinear Curvature Lagrangians and
Dark Energy
Mielke, Eckehard W.; Kusmartsev, Fjodor V.; Schunck, Franz E 824
Virgo Data Analysis for C6 and C7 Engineering Runs
Cuoco, Elena et al 844
Leopold Ernst Halpern and the Generalization of General
Relativity
Overduin, James M.; Plendl, Hans S 870
Post-Newtonian Approximations, Compact Binaries, and Strong
Field Tests of Gravity
Blanchet, Luc; Grishchuk, L.P.; Schdfer, Gerhard 881
Tests of Lorentz Symmetry in the Photon Sector
Herrmann, Sven; Senger, Alexander; Moehle, Katharina;
Kovalchuk, Evgeny; Peters, Achim,; 895
OPTIS - High Precision Test of Special and General Relativity in
Space
Ldemmerzahl, Glaus; Dittus, Hansjorg; Hackmann, Eva;
Scheithauer, Silvia; Peters, Achim; Schiller, Stephan 905
Testing Special and General Relativity: Clocks and Trajectories
Dittus, Hansjorg; Ldrnmerzahl, Claus; Peters, Achim;
Salomon, Christophe 916
Laboratory Limits for Temporal Variations of Fundamental
Constants: An Update
Peik, Ekkehard; Lipphardt, Burghard; Schnatz, Harold; Tamm,
Christian; Weyers, Stefan; Wynands, Robert 941
Some Old and Some New Opportunities for Quantum Gravity
P henomenology
Amelino-Camelia, Giovanni 952
Visualization of Relativistic Effects
Ruder, Hanns; Nollert, Hans-Peter; Muller, Thomas; Borchers, Marc . . . 972
PART B
PARALLEL SESSIONS
• Dark Matter
Chairperson: Biermann, Peter
Impact of Dark Matter on Reionization and Heating
Mapelli, Michela; Ripamonti, Emanuele 979
xxvii
Impact of Dark Matter Decays and Annihilations on Structure
Formation
Ripamonti, Emanuele; Mapelli, Michela 982
Thermal and Chemical Evolution of the Primordial Clouds in
Warm Dark Matter Models with keV Sterile Neutrinos in One-
Zone Approximation
Stasielak, Jaroslaw; Biermann, Peter L.; Kusenko, Alexander 985
Restrictions on Sterile Neutrino Parameters from Astrophysical
Observations
Ruchayskiy, Oleg 988
Upper Limits on Density of Dark Matter in Solar System
Khriplovich, Iosif; Pitjeva, Elena 991
The Observed Properties of Dark Matter on Small Astrophysical
Scales
Gilmore, Gerard 994
Is Dark Matter Futile on the Brane?
Gergely, Ldszlo A 997
Direct X-Ray Constraints on Sterile Neutrino Warm Dark Matter
Watson, Casey R 1000
Limits on the Dark Matter Particle Mass from Black Hole Growth
in Galaxies
Munyaneza, Faustin 1003
Dark Matter: The Case of Sterile Neutrino
Shaposhnikov, Mikhail 1006
• Neutrino Masses: Experimental
Chairperson: Drexlin, Guido
Neutrino Background, Diffuse Backgrounds and CMB: Is the
Picture Consistent?
Popa, Lucia Aurelia; Vasile, Ana 1019
Constraining the Cosmological Lepton Asymmetry through Cosmic
Microwave Background Observations
Lattanzi, Massimiliano; Ruffini, Remo; Vereshchagin, Gregory V. 1022
Possible Neutrino-Antineutrino Oscillation under Gravity and its
Consequences
Mukhopadhyay, Banibrata 1025
How Gravity Can Distinguish Between Dirac and Majorana
Neutrinos
Singh, Dinesh; Mobed, Nader; Papini, Giorgio 1028
XXVIII
• Cosmic Rays
Chairperson: Schlickeiser, Reinhard
Anisotropics of Ultra-High Energy Cosmic Rays
Serpico, Pasquale D 1033
Recent Progress in Describing Cosmic Ray Transport
Tautz, Robert C 1036
Propagation of Ultra-High Energy Cosmic Rays: Towards a New
Astronomy
Mattel, Alvise; Chardonnet, P 1039
• Astrophysics of Neutron Stars and Black Holes: Observations
Chairperson: Pian, Elena
Extragalactic X-Ray Jets
Worrall, Diana M 1045
Initial Results from the Suzaku Satellite
Dotani, Tadayasu; The Suzaku Team 1048
Soft Gamma Repeaters and Magnetars
Hurley, Kevin C 1051
• Theoretical Models of Observations from Black Hole Candidates
Chairperson: Chakrabarti, Sandip K.
Epicyclic Frequencies and Resonant Phenomena Near Black Holes:
The Current Status
Aliev, Alikram N 1057
Humpy LNRF-Velocity Profiles in Accretion Discs Orbiting
Rapidly Rotating Kerr Black Holes
Stuchlik, Zdenek; Slany, Petr; Torbk, Gabriel 1060
Standing Shocks in Pseudo-Kerr Geometry
Mondal, Soumen; Chakrabarti, Sandip K. 1063
Properties of Accretion Shock Waves in Viscous Flows with
Cooling Effects
Das, Santabrata; Chakrabarti, Sandip K 1066
Model of Radiating Annuli near Black Holes for Iron Ka Line
Profile Interpretations
Zakharov, Alexander F 1069
QPOs due to Centrifugally Supported Shocks around Stellar-Mass
and Supermassive Black Holes
Okuda, Toru; Teresi, Vincenzo; Molteni, Diego 1072
XXIX
Observing the Flares of Sgr A* with the Very Large Telescope
Interferometer
Paumard, Thibaut; Miiller, Thomas; Genzel, Reinhard; Eisen-
hauser, Frank; Gillessen, Stefan 1075
Simulating VLBI Images of Sgr A*
Noble, Scott C.; Leung, Po Kin; Gammie, Charles F.; Book, Laura G. . . . 1078
• Astrophysical Black Holes
Chairperson: Chakrabarti, Sandip K.
Astrophysical Black Holes — Do They Have Boundary Layers?
Chakrabarti, Sandip K.; Ghosh, Himadri; Som, Debopam 1085
Secondary Perturbation Effects in Keplerian Accretion Disks:
Elliptical Instability
Mukhopadhyay, Banibrata 1098
Gravitational Collapse of Population III Stars
Suwa, Yudai; Takiwaki, Tomoya; Kotake, Kei; Sato, Katsuhiko 1101
Near-Infrared Observations of Sagittarius A*
Trippe, Sascha; Paumard, Thibaut; Gillessen, Stefan; Otto,
Thomas; Eisenhauser, Frank; Martins, Fabrice; Genzel, Reinhard 1104
Long-Term Monitoring of the Hard X-Ray/Gamma-Ray Emission
from Galactic Black Hole with BATSE
Case, Gary L.; Anzalone, Evan; Cherry, Michael L.; Rodi, James
C; Ling, Jam.es C; Radocinski, Robert G; Wells, Derek; Wheaton,
William A 1107
Marginally Stable Thick Discs Orbiting Kerr-de Sitter Black Holes
Slany, Petr; Stuchlik, Zdenek 1110
Black Holes in Scalar Field or Quintessential Cosmology
Harada, Tomohiro 1113
A New Solution for Einstein Field Equation in General Relativity
Mousavi, Sadegh 1116
Pseudo-Kerr Geometry
Mondal, Soumen; Chakrabarti, Sandip K 1119
Extreme Gravitational Lensing by Supermassive Black Holes
Bozza, Valeria 1122
XXX
• Spectral and Timing Appearances of the Galactic and
Extragalactic Black Holes
Chairperson: Titarchuk, Lev
Physical Characteristics of XTE J1650-500 and GRS 1915+105
with BeppoSAX
Montanari, Enrico; Titarchuk, Lev; Frontera, Filippo 1127
Spectral and Timing Properties of Magnetized Advective Flows
with Standing Shocks
Mandal, Sarnir; Chakrabarti, Sandip K. 1130
Estimating Black Hole Masses in ULXs
Soria, Roberto 1133
• Extreme Properties of Neutron Stars: Observations and Theory
Chairperson: Mendez, Mariano
Equation of State of Dense Matter in Neutron Stars
Cerny, Slavomir; Stone, Jifina, Rikovskd; Stuchlik, Zdenek;
Hledik, Stanislav 1139
Detectability of Gravitational Waves from the r-Mode Instability
in Newly-Born Neutron Stars
Sd, Paulo M.; Tome, Brigitte 1142
X-Ray Dim Isolated Neutron Stars: A Review of the Latest Timing
and Spectral Properties
Zane, Silvia 1145
X-Ray Observations of Neutron Stars and the Equation of State
at Very High Densities
Truemper, Joachim E 1148
Eigenmodes of Rapidly Rotating Neutron Stars
Boutloukos, Stratos 1152
Parameter Space Study of Magnetohydrodynamic Flows Around
Magnetized Compact Objects
Das, Santabrata; Chakrabarti, Sandip K 1155
Gravitational Radiation from Accreting Millisecond Pulsars
Vigelius, Matthias; Payne, Donald; Melatos, Andrew 1158
Dynamical Stability of Fluid Spheres in Spacetimes with a Nonzero
Cosmological Constant
Hledik, Stanislav; Stuchlik, Zdenek; Mrdzovd, Kristina 1161
XXXI
• Strange Stars
Chairperson: Usov, Vladimir
Strangelets in Cosmic Rays
Madsen, Jes 1167
Can Strange Stars be Distinguished from Neutron Stars?
Harko, Tiberiu; Cheng, Kwong Sang 1177
Pair Winds in Schwarzschild Spacetime with Application to
Strange Stars
Aksenov, Alexey, G.; Milgrom, Mordehai; Usov, Vladimir, V. 1180
Evidence for White Dwarfs with Strange-Matter Cores
Mathews, Grant J.; Suh, In-Saeng; Lan, Nguyen Quynh; Otsuki,
Kaori; Weber, Fridolin 1183
• Thermal Behavior of Compact Stars
Chairperson: Page, Dany
Magnetars: Internal Heating and Energy Budget
Yakovlev, Dmitry G.; Kaminker, Alexander D.; Potekhin,
Alexander Y.; Shternin, Peter S.; Chabrier, Gilles; Shibazaki, Noriaki 1189
Trapping of Neutrinos in Extremely Compact Neutron Stars
Stuchlik, Zdenek; Urbanec, Martin; Torok, Gabriel;
Hledik, Stanislav; Hladik, Jan 1192
A Self-Consistent Model of the Isolated Neutron Star RX J0720.4-
3125
Miralles, Juan A.; Pons, Jose A.; Perez-Azorin, J. Fernando;
Miniutti, Giovanni 1195
kHz QPO Pairs Expose the Neutron Star of Circinus X-l
Boutloukos, Stratos; van der Klis, Michiel; Altamirano, Diego;
Klein Wolt, Marc; Wijnands, Rudy 1198
Neutron Star Atmospheres and X-Ray Spectra
Kundt, Wolfgang 1201
• Alternative Theories (A)
Chairperson: Schmidt, Hans- Juergen
Anisotropically Inflating Universes
Hervik, Sigbj0rn; Barrow, John D 1207
Thick Brane Solution with Two Scalar Fields
Dzhunushaliev, Vladimir; Schmidt, Hans-Juergen; Myrzakulov,
Kairat; Myrzakulov, Ratbay 1210
Shear Dynamics in Bianchi I Cosmologies with i?™-Gravity
Leach, Jannie A.; Dunsby, Peter K.S.; Carloni, Sante 1213
XXXII
Spontaneous Lorentz Violation, Gravity and Nambu-Goldstone
Modes
Bluhm, Robert; 1217
Spontaneous Lorentz Breaking, Nambu-Goldstone Modes, and
Gravity
Potting, Robertus 1220
The Significance of Matter Coupling in f(R) Gravity
Sotiriou, Thomas P 1223
Constraining Alternative Theories of Gravity with the Energy
Conditions
Perez Bergliaffa, Santiago Esteban 1226
An f(R) Gravitation for Galactic Environments
Sobouti, Yousef 1230
Causality and Superluminal Fields
Bruneton, Jean-Philippe 1233
Gravitation as a Vacuum Nonlinear Electrodynamics Effect
Chernitskii, Alexander A 1236
Asymptotic Flatness and Birkhoff's Theorem in Higher-Derivative
Theories of Gravity
Clifton, Timothy 1239
Cosmological Model with a Born-infield Type Scalar Field
Kerner, Richard; Serie, Emmanuel; Troisi, Antonio 1242
A Teleparallel Representation of the Weyl Lagrangian
Vassiliev, Dmitri 1245
Nonlinear Supersymmetric General Relativity
Shima, Kazunari; Tsuda, Motomu 1248
Black Hole Solutions in N > 4 Gauss-Bonnet Gravity
Alexeyev, Stanislav 0.; Popov, Nikolai 1251
Electrostatics and Confinement in Einstein's Unified Field Theory
Antoci, Salvatore; Liebscher, Dierck-Ekkehard; Mihich, Luigi 1254
Galactic Disks in Theories with Yukawian Gravitational Potential
de Araujo, Jose Carlos N.; Miranda, Oswaldo D 1257
On the Field Theoretic Description of Gravitation
Nieuwenhuizen, Theo M. 1260
De Sitter Stability in Theories with Second Order Curvature Terms
Toporensky, Alexey V.; Tretyakov, Petr V. 1263
Basic Relations of a Unified Theory of Electrodynamics, Quantum
Mechanics, and Gravitation
Ostermann, Peter 1266
Physical Interpretation and Viability of Various Metric Nonlinear
Gravity Theories
Sokolowski, Leszek M 1269
• Alternative Theories (B)
Chairperson: Hammond, Richard
Are Active and Passive Electric Charges Equal?
Ldemmerzahl, Glaus; Macias, Alfredo; Miiller, Holger 1275
Charged Fluid Dynamics in Scalar-Tensor Theories of Gravity
with Torsion
Wang, Chih-Hung 1278
Validation of the Weak Equivalence Principle in a Spatially-VSL
Gravitation Model
Broekaert, Jan 1281
• Higher Dimensional Theories
Chairperson: Coley, Alan
Exact Solution of the 5D Space-Time-Matter Universe and Their
Implications
Fukui, Takao 1287
Hamiltonian Theory of Brane-World Gravity
Kovdcs, Zolton; Gergely, Ldszlo A 1290
Casimir Force Test of a 6D Brane World
Linares, Roman; Morales-Tecotl, Hugo A.; Pedraza, Omar 1293
Electro-Weak Model within a 5-Dimensional Lorentz Group
Theory
Lecian, Orchidea Maria; Montani, Giovanni 1296
Spacetimes with Constant Scalar Invariants
Hervik, Sigbj0rn; Coley, A.A.; Pelavas, Nicos 1299
Higher Dimensional VSI Spacetimes and Supergravity
Fuster, Andrea; Pelavas, Nicos 1302
VSI & VSI; Spacetimes in Higher Dimensions
Pravdovd, Alena 1305
The Electro-Weak Model as a Phenomenological Issue of
Multidimensions
Cianfrani, Francesco; Montani, Giovanni 1308
XXXIV
Hamiltonian Formulation of the 5-D Kaluza-Klein Model and
Test-Particle Motion
Lacquaniti, Valentino; Montani, Giovanni 1311
Electromagnetism and Perfect Fluids Interplay in
Multidimensional Spacetimes
Mitskievich, Nikolai V. 1314
Torsion Induces Gravity
Aros, Rodrigo 1317
Final Fate of Higher-Dimensional Spherical Dust Collapse in
Einstein-Gauss-Bonnet Gravity
Maeda, Hideki 1320
Classification of the Weyl Tensor in Higher Dimensions and its
Applications
Pravda, Vojtech 1323
• Geometric Calculus in Gravity Theory
Chairperson: Hestenes, David
Geometrical and Kinematical Aspects of Rindler Observers
Romero, Carlos; Brasileiro Formiga, Jansen 1329
On the Zeros of Spinor Fields and an Orthonormal Frame Gauge
Condition
Nester, James M 1332
New Special Solutions of the Ricci Flow Equation in Two
Dimensions Using a Linearization Approach
Visinescu, Anca; Visinescu, Mihai 1335
• Black Hole and Pair Creation in Strong Fields
Chairperson: Greiner, Walter
Pair Creation in Inhomogeneous Fields
Schubert, Christian 1341
Monopole Decay in a Variable External Field
Monin, Alexander K.; Zayakin, Audrey V. 1346
World-Making with Extended Gravity Black Holes for Cosmic
Natural selection in the Multiverse Scenario
Barrau, Aurelien 1349
Neutral Nuclear Core vs Super Charged One
Rotondo, Michael; Ruffini, Remo; Xue, She-Sheng 1352
• Black Holes in Higher Dimensions (Black Rings and Black Strings)
Chairperson: Kunz, Jutta
Gravitational Perturbations of Higher Dimensional Rotating Black
Holes
Kunduri, Hari K.; Lucietti, James; Reall, Harvey S. 1359
Gravitating Non-Abelian Solitons and Hairy Black Holes in Higher
Dimensions
Volkov, Mikhail S 1379
Derivation of the Dipole Black Ring Solutions
Yazadjiev, Stoytcho S 1397
Charged Rotating Black Holes in Higher Dimensions
Kunz, Jutta; Navarro-Lerida, Francisco; Viebahn, Jan;
Maison, Dieter 1400
Solitonic Generation of Solutions Including Five-Dimensional
Black Rings and Black Holes
Mishima, Takashi; Iguchi, Hideo 1403
Kaluza-Klein Black Hole with Gravitational Charge in Einstein-
Gauss-Bonnet Gravity
Maeda, Hideki; Dadhich, Naresh K 1406
Higher Dimensional Rotating Charged Black Holes
Aliev, Alikram N 1409
Perturbative Stability and Absorption Cross-Section in String
Corrected Black Holes
Moura, Filipe 1412
Ultrarelativistic Boost of the Black Ring
Ortaggio, Marcello; Krtous, Pavel; Podolsky, Jin 1415
The Results of a New Solution of the Einstein Field Equations in
General Relativity and Black Hole New Movement
Mousavi, Sadegh 1418
Hamiltonian Treatment of Static and Collapsing Spherically
Symmetric Charged Thin Shells in Lovelock Gravity
Dias, Goncalo A.S.; Lemos, Jose P.S.; Gao, Sijie . . . 1421
New Nonuniform Black String Solutions
Kleihaus, Burkhard; Kunz, Jutta; Radu, Eugen 1424
Lovelock Gravity and The Counterterm Method
Bostani, Neda; Dehghani, Mohammad Hossein; Sheykhi, Ahmad 1427
LG(Landau-Ginzburg) in GL(Gregory-Lafiamme)
Kol, Barak; Sorkin, Evgeny 1431
XXXVI
Causal Structure Around Spinning 5-Dimensional Cosmic Strings
Slagter, Reinoud Jan 1434
Short Distances, Black Holes, and TeV Gravity
Agullo, Ivan; Navarro-Salas, Jose; Olmo, Gonzalo J 1437
Black String Solutions with Negative Cosmological Constant
Mann, Robert; Radu, Eugen; Stelea, Cristian 1440
Matched Asymptotic Expansion for Caged Black Holes
Gorbonos, Dan; Kol, Barak 1443
Perturbatively Non-Uniform Charged Black Strings: A New Stable
Phase
Miyamoto, Umpei; Kudoh, Hideaki 1446
• Analog Models of and for General Relativity
Chairperson: Volovik, Grigory
From Quantum Hydrodynamics to Quantum Gravity
Volovik, Grigory 1451
Looking Beyond the Horizon
Babichev, Eugeny; Mukhanov, Viatcheslav; Vikman, Alexander 1471
A Dielectric Analogue Model of the Kerr Equatorial Plane
Rosquist, Kjell 1475
Bose-Einstein Condensates and QFT in Curved Space-Time
Fagnocchi, Serena 1479
Electromagnetic Light Rays in Local Dielectrics
De Lorenci, Vitorio A.; Klippert, Renato 1482
Scattering Problems on Rotating Acoustic Black Holes
Cherubini, Christian; Filippi, Simonetta 1485
• Black Hole Thermodynamics
Chairperson: Khriplovich, Iosif
Thermodynamical Properties of Hairy Black Holes with Cosmo-
logical Constant
Nadalini, Mario; Vanzo, Luciano; Zerbini, Sergio 1491
Radiation of Quantized Black Holes. Is it Observable?
Khriplovich, Iosif 1494
Effects of Quantized Fields on the Spacetime Geometries of Static
Spherically Symmetric Black Holes
Anderson, Paul R.; Binkley, Mathew; Calderon, Hector; Hiscock,
William A.; Mottola, Emil; Vaulin, Ruslan 1497
XXXVII
Thermodynamic Quantities of Kaluza-Klein Black Holes with
Squashed Horizons
Kurita, Yasunari; Ishihara, Hideki 1500
Hawking Radiation and Black Hole Thermodynamics
Page, Don N 1503
Entropy from Conformal Horizon States in D-Dimensional
Spherical, Toroidal, and Hyperbolic Anti-de Sitter Black Holes
Lemos, Jose P.S.; Dias, Goncalo A.S 1508
Thermodynamic Geometries of Black Holes
Aman, Jan E.; Bengtsson, Ingemar; Pidokrajt, Narit; Ward, John . . . .1511
• Alternative Black Hole Models
Chairperson: Mazur, Pawel 0.
On Quantum Effects in the Vicinity of Would-be Horizons
Marecki, Piotr 1517
Stable Dark Energy Stars: An Alternative to Black Holes?
Lobo, Francisco S.N. 1520
Horizon News Function and Quasi-Local Energy-Momentum Flux
Near Black Hole
Wu, Yu-Huei 1523
Black Holes or Eternally Collapsing Objects?
Mitra, Abhas; Glendenning, Norman K 1526
The Proposed Black Holes Around Us
Kundt, Wolfgang 1529
Gravastars and Bifurcation in Quasistationary Accretion
Malec, Edward; Roszkowski, Krzysztof 1537
• Numerical Relativity, Black Hole Collisions, and Algebraic
Computation
Chairperson: Husa, Sascha
Lifetime of Oscillons
Fodor, Gyula; Forgdcs, Peter; Grandclement, Philippe; Racz, Istvan .... 1543
A Virtual Trip to the Schwarzschild-de Sitter Black Hole
Bakala, Pavel; Hledik, Stanislav; Stuchlik, Zdenek; Truparovd
Kamila; Cermdk, Petr 1546
Similarity Solutions Using DESOLV
Vu, Khai T.; Butcher, J.; Carminati, John 1549
xTensor: A Free Fast Abstract Tensor Manipulator
Martin-Garcia, Jose M 1552
XXXVIII
Tensor Computer Algebra
Martin-Garcia, Jose M 1555
• Simulations of Relativistic Flows and Compact Objects
Chairperson: Font, Jose A.
Relativistic MHD Simulations and Synthetic Synchrotron Emission
Maps: A Diagnostic Tool for Pulsar Wind Nebulae
Del Zanna, Luca; Volpi, Delia; Amato, Elena; Bucciantini,
Nicolo 1561
GRMHD Simulations of Jet Formation with RAISHIN
Mizuno, Yosuke; Nishikawa, Ken-Ichi; Koide, Shinji; Hardee,
Philip; Fishman, Gerald J 1564
Standing Shocks in Pseudo-Kerr Geometry
Mondal, Soumen; Chakrabarti, Sandip K. 1567
Evolving Relativistic Fluid Spacetimes Using Pseudospectral
Methods and Finite Differencing
Duez, Matthew D.; Kidder, Lawrence E.; Teukolsky, Saul A 1570
3D Relativistic MHD Simulation of a Tilted Accretion Disk Around
a Rapidly Rotating Black Hole
Fragile, P. Chris; Anninos, Peter; Blaes, Omer M.; Salmonson,
Jay D 1573
Fragmentation of General Relativistic Quasi-Toroidal Polytropes
Zink, Burkhard Sebastian; Stergioulas, Nikolaos; Hawke, Ian; Ott,
Christian D.; Schnetter, Erik; Miiller, Ewald 1576
Adaptive Mesh Refinement and Relativistic MHD
Neilsen, David; Hirschmann, Eric W.; Anderson, Matthew;
Liebling, Steven L 1579
3-D GRMHD and GRPIC Simulations of Disk-Jet Coupling and
Emission
Nishikawa, Ken-Ichi; Mizuno, Yosuke; Watson, Michael; Hardee,
Philip; Fuerst, Steve; Wu, Kinwah; Fishman, Gerald J 1582
Resistive General Relativistic MHD Simulations of Jet Formation
Around Kerr Black Hole
Koide, Shinji; Shibata, Kazunari; Kudoh, Takahiro 1585
Making Up a Short GRB: The Bright Fate of Mergers of Compact
Objects
Aloy, Miguel Angel; Mimica, Petar 1589
Spacetime Modes of Rapidly Rotating Relativistic Stars
Stergioulas, Nikolaos; Kokkotas, Kostas D.; Hawke, Ian 1592
• Dynamics of Compact Binaries
Chairperson: Rezzolla, Luciano
Reducing Orbital Eccentricity in Binary Black Hole Simulations
Pfeiffer, Harold P.; Brown, Duncan A.; Kidder, Lawrence E.;
Lindblom, Lee; Lovelace, Geoffrey; Scheel, Mark A 1597
Negative Komar Masses in Regular Stationary Spacetimes
Ansorg, Marcus; Petroff, David 1600
Constraint Relaxation
Marronetti, Pedro 1603
Relativistic Hydrodynamic Simulations of Multiple Orbits for
Close Neutron Star Binaries
Mathews, Grant J.; Haywood, J. Reese; Wilson, James R 1606
The Final Fate of Binary Neutron Star Systems: What Happens
After the Merger?
Duez, Matthew D.; Liu, Yuk Tung; Shapiro, Stuart L.; Stephens,
Branson C; Shibata, Masaru 1609
Head-On Collisions of Different Initial Data
Sperhake, Ulrich; Briigmann, Bernd; Gonzalez, Jose A.; Hannam,
Mark D.; Husa, Sascha 1612
Tearing instability in Relativistic Magnetically Dominated Plasmas
Barkov, Maxim V.; Komissarov, Serguei S.; Lyutikov, Maxim 1615
• Black Hole Collisions
Chairperson: Lousto, Carlos
Black Hole Bremsstrahlung in the Nonlinear Regime of General
Relativity
Oliveira, Henrique P.; Soares, I. Damiao; Tonini, Eduardo V. 1621
Hyperboloidal Foliations with J^-Fixing in Spherical Symmetry
Zenginoglu, Anil; Husa, Sascha 1624
High-Order Perturbations of a Spherical Spacetime
Brizuela, David; Martin-Garcia, Jose Maria; Mena Marugdn,
Guillermo A 1627
Analytic Solutions of the Linearized Einstein Equations Used to
Test and Develop a Characteristic Code
Bishop, Nigel T 1630
The Kerr Metric in Bondi-Sachs Form
Venter, Liebrecht R.; Bishop, Nigel T 1633
xl
Binary Black Hole Merger Waveforms in the Extreme Mass Ratio
Limit
Damour, Thibault; Nagar, Alessandro 1636
• CMB Theory
Chairperson: Dore, Olivier
CMB Anomalies from Relic Anisotropy
Gumriikgiioglu, A. Emir; Contaldi, Carlo, R.; Peloso, Marco 1641
Can Extragalactic Foregrounds Explain the Large-Angle CMB
Anomalies?
Rakic, Aleksandar; Rdsdnen, Syksy; Schwarz, Dominik J 1647
A New Realization of a Low Quadrupole Universe
Lee, Wo-Lung 1653
Perturbations of Dark Sectors from the CMB
Bashinsky, Sergei 1659
• CMB Experiment Space
Chairperson: Masi, Silvia
Observations of the CMB and Galactic Foregrounds at 11-17 GHz:
The COSMOSOMAS Experiment
Hildebrandt, Sergi R 1667
• CMB Data Analysis
Chairperson: Natoli, Paolo
Probing Cosmic Dark Ages with the CMB Polarization
Measurements
Popa, Lucia Aurelia; Stefanescu, Petruta; Burigana, Carlo 1671
Dark Energy Constraints from Needlets Analysis of WMAP3 and
NVSS Data
Pietrobon, Davide; Balbi, Amedeo; Marinucci, Domenico 1674
The Matter Power Spectrum as a Tool to Discriminate Dark
Matter-Dark Energy Interaction
Olivares, German; Pavon, Diego; Atrio-Barandela, Fernando 1677
The BRAIN Experiment
Polenta, Gianluca; for the BRAIN collaboration 1680
• Observational Gravitational Lensing
Chairperson: Jetzer, Philippe
Microlensing with the Radioastron Space Telescope
Zakharov, Alexander F 1691
Microlensing Towards M31
Calchi Novati, Sebastiano 1694
xli
Does the LMC Halo Contribute Significantly to the MACHO
Events?
Scarpetta, Gaetano 1697
A New Analysis of the MEGA M31 Microlensing Events
Nucita, Achille A.; Ingrosso, Gabriele; De Paolis, Francesco;
Strafella, Francesco; Calchi Novati, Sebastiano; Scarpetta,
Gaetano; Jetzer, Philippe 1700
On the Lens Nature in Microlensing Searches
De Paolis, Francesco; Ingrosso, Gabriele; Nucita, Achille A 1702
• Theoretical Gravitational Lensing
Chairperson: Perlick, Volker
Gravitational Lensing by Braneworld Black Holes
Whisker, Richard 1707
Gravitational Lensing of Stars Surrounding Supermassive Black
Holes
Bozza, Valerio; Mancini, Luigi 1710
Kerr Black Holes Gravitational Lensing in the Strong Deflection
Limit: An Analytical Approach
De Luca, Fabiana 1713
On Gravitational Lensing by a Kerr Black Hole
Sereno, Mauro; De Luca, Fabiana 1716
Testing Theories of Gravity with Black Hole Lensing
Keeton, Charles R.; Fetters, Arlie 0 1719
Gravitational Lensing by Higher Dimensional Black Holes
Majumdar, Archan S.; Mukherjee, Nupur 1722
Iron KQ Line Profiles and Shadow Shapes as Evidences of a
Gravitational Lensing in a Strong Gravitational Field near BHs
Zakharov, Alexander F.; De Paolis, Francesco; Nucita, Achille A.;
Ingrosso, Gabriele 1725
Lensing Effects on Gravitational Waves in a Clumpy Universe
Yoo, Chul-Moon; Nakao, Ken-ichi; Kozaki, Hiroshi; Takahashi, Ryuichi . . 1728
QSO Lensing
Miranda, Marco; Jetzer, Philippe; Maccib, Andrea V 1731
JLenses and XFGLenses
Frutos-Alfaro, Francisco; Solis-Sanchez, Hugo 1734
Wave Fronts in General Relativity Theory
Grave, Frank; Frutos-Alfaro, Francisco; Miiller, Thomas; Adis, Daria . . . 1737
xlii
• Galaxies and the Large-Scale Structure
Chairperson: Sheth, Ravi
Spherical Voids in a Newton-Friedmann Universe
Triay, Roland; Fliche, Henri H 1743
• Dark Energy and Universe Acceleration
Chairperson: Starobinsky, Alexei
Dark Energy and Universe Acceleration of Nonlinear Supersym-
metric General Relativity
Shima, Kazunari; Tsuda, Motomu 1749
Testing the Dark-Energy-Dominated Cosmology by the Solar-
System Experiments
Dumin, Yurii V. 1752
A Darkless Spacetime
Tartaglia, Angelo; Capone, Monica 1755
On Intrinsic Invariance in Gurzadyan-Xue Cosmological Models
Khachatrian, Harutyun; Vereshchagin, Gregory V.; Yegorian, Gegham . . . 1758
Phantom Dark Energy and its Cosmological Consequences.
Dabrowski, Mariusz P 1761
The Generalized Second Law in Dark Energy Dominated Universes
Izquierdo, German; Pavon, Diego 1764
Accelerated Expansion by Non-Minimally Coupled Scalar Fields
Bieli, Roger 1767
Vacuum Energy Generating Mechanisms in Cosmic Expansion
with Natural UV Cutoff
Kempf, Achim 1770
New Kinematical Constraints on Cosmic Acceleration
Rapetti, David; Allen, Steve W.; Amin, Mustafa A.; Blandford,
Roger D 1773
Dark Energy and Decaying Dark Matter
Mathews, Grant J.; Lan, Nguyen Quynh; Wilson, James R 1776
Gravitational Instanton-Solution to Cosmological Constant
Xue, She-Sheng 1779
An Alternative Source for Dark Energy
Wanas, Mamdouh I. 1782
An Awesome Hypothesis for Dark Energy: The Abnormally
Weighting Energy
Fiizfa, Andre; Alimi, Jean-Michel 1785
Xljjj
Perturbations of a Cosmological Constant Dominated Universe
Vasuth, Mdtyds; Czinner, Viktor 1788
On the Gurzadyan-Xue Cosmological Models and the Dynamics
of Density Perturbation
Yegorian, Gegham 1791
Scalar-Tensor Dark Energy Models
Ganouji, Radouane; Polarski, David; Banquet, Andre; Starobinsky,
Alexei A 1794
Reconstruction of Dark Energy Using Supernova and Other
Datasets
Alam, Ujjaini; Sahni, Varun; Starobinsky, Alexei A 1797
f(R) Dark energy: From the Time of Recombination till Present
Day
Gurovich, Viktor; Folomeev, Vladimir; Tokareva, Iya 1800
Probing Dynamical Dark Energy with Press- Schechter Mass
Functions
Le Delliou, Morgan 1803
Broken Scale Invariance and Quintessence (A Quarter of a Century
Ago)
Venturi, Giovanni 1807
• Topology of the Universe
Chairperson: Demianski, Marek
An Axisymmetric Object-Based Search for a Flat Compact
Dimension
Mathews, Grant J.; Menzies, Dylan 1813
Topological Gravitation on Graph Manifolds
Mitskievich, Nikolai V.; Efremov, Vladimir N.; Hernandez Magdaleno . . . 1816
Supernovae Constraints on Cosmological Density Parameters and
Cosmic Topology
Reboucas, Marcelo J 1819
Supernovae Constraints on DGP Model and Cosmic Topology
Reboucas, Marcelo J 1824
• Inhomogeneous Cosmology
Chairperson: Krasinski, Andrzej
Reinterpreting Dark Energy Through Backreaction: The Minimally
Coupled Morphon Field
Larena, Julien; Buchert, Thomas; Alimi, Jean-Michel 1831
xliv
Initial Conditions for Primordial Black Hole Formation
Musco, Ilia; Polnarev, Alexander G 1834
Is the Apparent Acceleration of the Universe Expansion Driven by
a Dark Energy-Like Component or by Inhomogeneities?
Marie-Noelle; Celerier 1837
Evolution of a Void and an Adjacent Galaxy Supercluster in the
Quasispherical Szekeres Model
Bolejko, Krzysztof 1847
Covariant Description of the Inhomogeneous Mixinaster Chaos
Benini, Riccardo; Montani, Giovanni 1857
The Mass and the Geometry of the Cosmos
Hellaby, Charles; Lu, Hui-Ching 1860
• Nonsingular Cosmology — Inflation
Chairperson: Novello, Mario
Emergent Universe with Bulk Viscosity
Mukherjee, Sailo; Paul, Bikash C; Dadhich, Naresh K.; Maharaj,
Sunil D.; Beesham, Aroonkumar 1873
Bulk Viscosity Impact on the Scenario of Warm Inflation
Mimoso, Jose Pedro; Nunes, Ana; Pavon, Diego 1876
• Quantum Cosmology and Quantum Effects in the
Early Universe
Chairperson: Vargas Moniz, Paulo
Scalar Field Phase Dynamics in Preheating
Charters, T.; Nunes, Ana; Mimoso, Jose Pedro 1881
Branch Wave Functions for Quasi-Classical Homogeneous
Universes
Craig, David 1884
Quantum Phantom Cosmology
Sandhoefer, Barbara 1887
Generic Evolutionary Quantum Universe
Battisti, Marco Valerio; Montani, Giovanni 1890
Quantum Cosmology from Three Different Perspectives
Esposito, Giampiero 1893
On the Thermal Boundary Condition of the Wave Function of the
Universe
Bouhmadi-Lopez, Mariam; Vargas Moniz, Paulo 1898
xlv
Dark Energy from Quantum Wave Function Collapse of Dark
Matter
Majumdar, Archan S.; Home, D 1901
Cosmological Perturbations in Quantum Cosmological
Backgrounds
Pinto-Neto, Nelson 1904
Classical and Quantum Aspects of the Inhomogeneous Mixmaster
Chaoticity
Benini, Riccardo; Montani, Giovanni 1909
Cosmological Dynamics with Vacuum Polarization
Toporensky, Alexey V.; Tretyakov, Petr V 1912
Some Cosmological Consequences of Loop Quantum Gravity
Mulryne, David J.; Tavakol, Reza 1915
Semiclassical Supersymmctric Quantum Gravity
Kiefer, Claus; Luck, Tobias; Vargas Moniz, Paulo 1920
Multigravity and Spacetime Foam
Garattini, Remo 1925
Boundary Conditions and Predictions of Quantum Cosmology
Page, Don N 1928
Quantum Cosmology with Nontrivial Topology
Fagundes, Helio V.; Vargas, Teofilo 1933
Non-Singular Solutions in Loop Quantum Cosmology
Vereshchagin, Gregory V 1936
Minimal Energy and Factor Ordering in Quantum Cosmology
Hinterleitner, Franz; Steigl, Roman 1939
On the False Vacuum Bubble Nucleation
Lee, Bum-Hoon; Lee, Chul Hoon; Lee, Wonvioo; Park, Chanyong 1942
PART C
PARALLEL SESSIONS
• The GRB - Supernova Connection
Chairperson: Chardonnet, Pascal
Swift Observations of GRB050712
De Pasquale, Massimiliano; Poole, Tracey; Zane, Silvia; Page,
Mathew; Breeveld, Alice; O 'Mason, Keith; Grupe, Dicke; Burrows,
David; Nousek, John; Roming, Peter; Krimm, Hans; Gehrels,
Neil; Zhang, Bing; Kobayashi, Shiho 1947
xlvi
No Astrophysical Dyadospheres
Page, Don N 1950
Magnetized Hypermassive Neutron Star Collapse: A Candidate
Central Engine for Short-Hard GRBs
Stephens, Branson C; Duez, Matthew D.; Liu, Yuk Tung; Shapiro,
Stuart L.; Shibata, Masaru 1953
Theoretical Interpretation of Luminosity and Spectral Properties
of GRB 031203
Bianco, Carlo Luciano; Bernardini, Maria Grazia; Chardonnet,
Pascal; Fraschetti, Federico; Ruffini, Remo; Xue, She-Sheng 1956
GRB980425 and the Puzzling URCA1 Emission
Bernardini, Maria Grazia; Bianco, Carlo Luciano; Caito, Letizia;
Dainotti, Maria Giovanna; Guida Roberto; Ruffini, Remo J 1959
• The Afterglow, Short and Long GRBs
Chairperson: Arkhangelskaja, Irene
The EPti-EiSO Correlation and the Nature of Sub-Energetic GRBs
Amati, Lorenzo 1965
The GRB Detected by AVS-F Apparatus Onboard CORONAS-F
Satellite in 2001-2005 Years
Arkhangelskaja, Irene V.; Arkhangelsky, Andrey I.; Glyanenko,
Alexander S.; Kotov, Yuri D.; Kuznetsov, Sergey N 1968
Special Relativistic Simulations of Magneto-Driven Jet from
Core-Collapse Supernovae
Takiwaki, Tomoya; Kotake, Kei; Yamada, Shoichi; Sato, Katsuhiko .... 1971
Theoretical Interpretation of "Long" and "Short" GRBs
Bianco, Carlo Luciano; Bernardini, Maria Grazia; Caito, Letizia;
Chardonnet, Pascal; Dainotti, Maria Giovanna; Fraschetti,
Francesca; Guida, Roberto; Ruffini, Remo; Xue, She-Sheng 1974
Theoretical Interpretation of GRB011121
Caito, Letizia; Bernardini, Maria Grazia; Bianco, Carlo Luciano;
Dainotti, Maria Giovanna; Guida, Roberto; Ruffini, Remo 1977
On GRB 060218 and the GRBs Related to Supernovae Ib/c
Dainotti, Maria Giovanna; Bernardini, Maria Grazia; Bianco,
Carlo Luciano; Caito, Letizia; Guida, Roberto; Ruffini, Remo 1981
The "Fireshell" Model in the Swift Era
Bianco, Carlo Luciano; Ruffini, Remo 1989
xlvii
GRB970228 as a Prototype for the Class of GRBs with an Initial
Spikelike Emission
Bernardini, Maria Grazia; Bianco, Carlo Luciano; Caito, Letizia;
Dainotti, Maria Giovanna; Guida, Roberto; Ruffini, Remo 1992
Theoretical Interpretation of GRB060124: Preliminary Results
Guida, Roberto; Bernardini, Maria Grazia; Bianco, Carlo Luciano;
Caito, Letizia; Dainotti Maria Giovanna; Ruffini, Remo 1995
• GRBs and Host Galaxies
Chairperson: Bjornsson, Gunnlaugur
Numerical Counterparts of GRB Host Galaxies
Courty, Stephanie; Bjornsson, Gunnlaugur; Gudmundsson, Einar H. ... 2003
The Host Galaxies of Long Gamma-Ray Bursts: The Mid-Infrared
view from the Spitzer Space Telescope
Le Floe % Emeric 2006
Gamma-Ray Burst Host Galaxy Gas and Dust
Starling, Rhaana; Wijers, Ralph; Wiersema, Klaas 2009
Low Redshift GRBs and their Host Galaxies
Tanvir, Nial R 2012
The Analysis of GRB Redshift Distribution
Arkhangelskaja, Irene V. 2015
Fundamental Properties of GRB-Selected Galaxies: A Swift/VLT
Legacy Survey
Jakobsson, Pall; Hjorth, Jens; Fynbo, Johan P. U.; Gorosabel,
Javier; Jaunsen, Andreas 0 2019
• GRB Observations by SWIFT
Chairperson: Angelini, Lorella
The Swift XRT: Early X-Ray Afterglows
Tagliaferri, Gianpiero 2025
Initial Results from Swift/UVOT
Marshall, Francis E 2030
Investigation of Jet Break Features in Swift Gamma-Ray Bursts
Sato, Garo et al 2033
Recent Results from the Swift Burst Alert Telescope
Krimm, Hans A.; for the Swift/BAT team 2036
Optical Observations of Gamma-Ray Bursts at the First Russian
Robotic Telescope MASTER
Tyurina, Nataly; Lipunov, Vladimir M.; Kornilov, Victor G.;
Gorbovskoy, Evgeniy S.; Kuvshinov, Dmirtiy A 2039
xlviii
• Cosmological Singularities
Chairperson: Cotsakis, Spiros
Flat, Radiation Universes with Quadratic Corrections and
Asymptotic Analysis
Cotsakis, Spiros; Tsokaros, Antonios 2045
The Recollapse Problem of Closed Isotropic Models in Second
Order Gravity Theory
Miritzis, John 2048
Big-Rip, Sudden Future and Other Exotic Singularities in the
Universe
Dqbrowski, Mariusz; Balcerzak, Adam 2051
Braneworld Cosmological Singularities
Antoniadis, Ignatios; Cotsakis, Spiros; Klaoudatou, Ifigeneia 2054
Generalized Puiseux Series Expansion for Cosmological Milestones
Cattoen, Celine; Visser, Matt 2057
• Chaos in General Relativity and Cosmology
Chairperson: Gurzadyan, Vahe
Chaos in the Yang-Mills Theory and Cosmology: Quantum Aspects
Matinyan, Sergei 2063
Chaos, Gravity and Wave Maps with Target SU(2)
Szybka, Sebastian Jan 2078
Chaos in Core-Halo Gravitating Systems
Ghahramanyan, Tigran; Gurzadyan, Vahe G 2081
Transient Chaos in Scalar Field Cosmology on a Brane
Toporensky, Alexey 2084
Toward a Holographic Origin of Cosmological Large Scale Structure
Mureika, Jonas R 2087
Vector Field Induced Chaos in Multi-dimensional Homogeneous
Cosmologies
Benini, Riccardo; Kirillov, A. Alexander; Montani, Giovanni 2090
• Einstein-Maxwell Systems
Chairperson: Lee, Chul Hoon
Dynamo Action on Relativistic Spherical Stars
Nadiezhda, Montelongo-Garcia; Thomas, Zannias 2095
External Electromagnetic Fields of a Slowly Rotating Magnetized
Star with Nonvanishing Gravitomagnetic Charge
Ahmedov, Bobomurat J.; Khugaev, Avas V.; Rakhmatov, Nemat I. .... 2098
xlix
Aligned Electromagnetic Excitations of the Kerr-Schild Solution
Burinskii, Alexander 2101
Static Perturbations of a Reissner-Nordstrom Black Hole by a
Charged Massive Particle
Bird, Donato; Geralico, Andrea; Ruffini, Remo 2104
Charged Black String Solutions of the Einstein-Maxwell Equations
in Higher Dimensions
Lee, Chul Hoon 2107
On the Hypothesis of Gravimagnetism
Abdil'din, Meirkhan M.; Abishev, Medeu E 2110
Static Perturbations of a Reissner-Nordstrom Black Hole by a
Charged Massive Particle
Bini, Donato; Geralico, Andrea; Ruffini, Remo 2113
• Theoretical Issues in GR
Chairperson: Brill, Dieter
A Framework for the Discussion of Singularities in General
Relativity
Whale, Benjamin E.; Scott, Susan M 2119
Axial Symmetric Gravitomagnetic Monopole in Cylindrical
Coordinates
Kagramanova, Valeria G.; Ahmedov, Bobomurat J 2122
Optical Reference Geometry and Inertial Forces in Kerr-de Sitter
Spacetimes
Kovdf, Jin; Stuchlik, Zdenek 2125
On the Construction of Syzgies of the Polynomial Invariants of the
Riemann Tensor
Lim, Allan E.K.; Carminati, John 2128
A General Covariant Stability Theory
Wanas, Mamdouh I.; Bakry, Mohamed A 2131
Relativistic Generalization of the Inertial and Gravitational Masses
Equivalence Principle
Mitskievich, Nikolai V. 2134
Static Perturbations by a Point Mass on a Schwarzschild
Black Hole
Bini, Donato; Geralico, Andrea; Ruffini, Remo 2137
Spatial Noncommutativity in a Rotating Frame
Beciu, Mircea 2140
I
On Energy and Momentum of the Friedman and Some More
General Universes
Garecki, Janusz 2143
Quasilocal Energy for an Unusual Slicing of Static Spherically
Symmetric Metrics
Chen, Chiang-Mei; Nester, James M. 2146
Quasilocal Energy for Cosmological Models
Nester, James M,; Chen, Chiang-Mei; Liu, Jian-Liang 2149
Relative Strains in General Relativity
Bird, Donato; de Felice, Fernando; Geralico, Andrea 2152
Dyonic Kerr-Newman Black Holes, Complex Scalar Field and
Cosmic Censorship
Semiz, Ibrahim 2155
The Ideas of GR, Quantization, Non-equilibrium Theormodynam-
ics and Gravimagnetism in Planetary Cosmogony
Abdil'din, Meirkhan M.; Abishev, Medev E.; Beissen, Nurzada A 2158
• Wormholes, Energy Conditions and Time Machines
Chairperson: Hadley, Mark
A^-Spheres: Regular Black Holes, Static Wormholes and Gravastars
with a Tube-Like Core
Zaslavskii, Oleg B 2169
Averaged Energy Inequalities for Non-Minimally Coupled Classical
Scalar Fields
Osterbrink, Lutz W. 2172
Self-Sustained Traversable Wormholes and the Equation of State
Garattini, Remo 2175
Classical and Quantum Wormholes in a Cosmology with Decaying
Dark Energy
Darabi, Farhad 2178
Nariai-Bertotti-Robinson Spacetimes as a Building Material for
One-Way Wormholes with Horizons, but without Singularity
Mitskievich, Nikolai V.; Medina Guevara, Maria Guadalupe;
Rodriguez, Hector Vargas 2181
Cosmic Time Machines and Gamma Ray Bursts
De Felice, Fernando 2184
Static and Dynamic Traversable Wormholes
Adamiak, Jaroslaw P 2187
Wormholes in the Accelerating Universe
Gonzalez-Diaz, Pedro F.; Martin-Moruno, Prado 2190
Traversable Wormholes Supported by Cosmic Accelerated
Expanding Equations of State
Lobo, Francisco S.N. 2193
On Wormholes of Massless if-Essence
Estevez-Delgado, Joaquin; Zannias, Thomas 2196
Dynamic Wormhole Spacetimes Coupled to Nonlinear
Electrodynamics
Berrocal Arellano, Aaron V.; Lobo, Francisco S.N. 2199
• Exact Solutions (Mathematical Aspects)
Chairperson: Alekseev, George
Robinson-Trautman Spacetimes in Higher Dimensions
Ortaggio, Marcello; Podolsky, Jifi 2205
Solutions of Seiberg-Witten and Einstein-Maxwell-Dirac Equations
in Euclidean Signature
Cihan, Saclioglu 2208
Euler Numbers on Cobordant Hypersurfaces
Harriott, Tina A.; Williams, Jeff G 2211
Symmetries of the Weyl Tensor in Bianchi V Spacetimes
Kashif, Abdul Rehman; Saifullah, Khalid; Shabbir, Ghulam S 2213
Classification of Spacetimes according to Conformal Killing Vectors
Saifullah, Khalid 2216
Exact Solutions for Radiating Relativistic Star Models
Misthry, Suryakumari S.; Maharaj, Sunil D 2219
An EMP Model of Bianchi 1 Cosmology
Williams, Floyd L 2222
Exact Static Solutions for Scalar Fields Coupled to Gravity in
(3+l)-Dimensions
Bilge, Ayse H.; Daghan, Durmus 2225
Thermodynamic Description of Inelastic Collisions in General
Relativity
Neugebauer, Gemot; Hennig, Joerg 2228
Distorted Killing Horizons and Algebraic Classification of
Curvature Tensors
Pravda, Vojtech; Zaslavskii, Oleg B 2231
Quasi-Stationary Routes to the Kerr Black Hole
Meinel, Reinhard 2234
lii
Classification Results on Purely Magnetic Perfect Fluid Models
Wylleman, Lode; Van den Bergh, Norbert 2237
Purely Electric Perfect Fluids of Petrov Type D
Wylleman, Lode 2240
Self-Dual Fields on the Space of a Kerr-Taub-Bolt Instanton
Aliev, Alikram N.; Saclioglu, Cihan 2243
The Kerr Theorem, Multisheeted Twistor Spaces and Multiparticle
Kerr-Schild Solutions.
Btirinskii, Alexander 2246
Electrical Force Lines of a 2-Soli ton Solution of the Einstein-
Maxwell Equations
Pizzi, Marco 2249
Monodromy Transform Approach in the Theory of Integrable
Reductions of Einstein's Field Equations and Some Applications
Alekseev, George 2252
Closed Timelike Curves and Geodesies of Godel-Type Metrics
Sarioglu, Ozgilr 2255
Conformal Symmetries in Spherical Spacetimes
Maharaj, Sunil D.; Moopanar, Selvan 2258
A Theorem of Beltrami and the Integration of the Geodesic
Equations
Boccaletti, Dino; Catoni, Francesco; Cannata, Roberto; Zampetti, Paolo . . 2261
Gravitational Collapse and Horizon Formation in 2 +1 Dimensional
Gravity
Brill, Dieter; Khetarpal, Puneet 2264
Purely Magnetic Silent Universes Do Not Exist
Vu, Khai T.; Carminati, John 2268
• Exact Solutions (Physical Aspects)
Chairperson: Scott, Susan M.
Zeeman-Type Dragging in the Kerr-Newman and NUT Spacetimes
Mitskievich, Nikolai V.; Lopez Benitez, Luis I. 2273
Physical Implications for the Uniqueness of the Value of the
Integration Constant in the Vacuum Schwarzschild Solution
Mitra, Abhas 2276
Singularity Analysis of Generalized Cylindrically Symmetric
Spacetimes
Konkowski, Deborah A.; Helliwell, Thomas M 2279
liii
Some Properties of Kerr Geometry with a Repulsive Cosmological
Constant
Petrdsek, Martin; Hledik, Stanislav 2282
Solution Generating Theorems: Perfect Fluid Spheres and the
TOV Equation
Boonserm, Petarpa; Visser, Matt; Weinfurtner, Silke 2285
Spherically Symmetric Gravitational Collapse of Perfect Fluids
Lasky, Paul; Lun, Anthony 2288
High-Speed Cylindrical Collapse of Two Dust Fluids
Sharif, Muhammad; Ahmad, Zahid 2291
Some Physical Consequences of the Multipole Structure of the
Kerr and Kerr-Newman Solutions
Rosquist, Kjell 2294
Visualising Spacetimes via Embedding Diagrams
Hledik, Stanislav; Stuchlik, Zdenek; Cipko, Alois 2299
Canonical Analysis of Radiating Atmospheres of Stars in
Equilibrium
Kovdcs, Zolton; Gergely, Ldszlo A.; Horvdth, Zsolt 2302
• Self-Gravitating Systems
Chairperson: Mielke, Eckehard W.
Platonic Sphalerons in Einstein-Yang-Mills and Yang-Mills-Dilaton
Theory
Kleihaus, Burkhard; Kunz, Jutta; Myklevoll, Kari 2307
Comment on "General Relativity Resolves Galactic Rotation
without Exotic Dark Matter" by F.I. Cooperstock and S. Tieu
Fuchs, Burkhard; Phleps, Stefanie 2310
Solitonic and Non-Solitonic Q-Stars
Verbin, Yosef 2313
Rotating Monopole-Antimonopole Pairs and Vortex Rings
Neemann, Ulrike; Kunz, Jutta; Kleihaus, Burkhard 2316
Sources of Static Cylindrical Spacetimes
Zofka, Martin 2319
Gravitating Multi-Skyrmions
Kleihaus, Burkhard; Ioannidou, Theodora; Kunz, Jutta 2322
A New Exact Static Thin Disk with a Central Black Hole
Gonzalez, Guillermo A 2325
liv
Bifurcations of Nonlinear Curvature Lagrangians in the Boson
Star Model
Schunck, Franz E 2328
Approximate Dynamics of Dark Matter Ellipsoids
Bisnovatyi-Kogan, Gennadyl S.; Tsupko, Oley Yu 2331
Nonextensive Statistical Theory of Density Distributions in Grav-
itationally Clustered Structures
Leubner, Manfred P 2334
General Relativistic Accretion with Backreaction
Karkowski, Janusz; Kinasiewicz, Bogusz; Mach, Patryk; Make,
Edward; Swierczynski, Zdobyslaw 2337
Non-Homogeneous Axisymmetric Models of Self-Gravitating
Systems
Cherubini, Christian; Filippi, Simonetta; Ruffini, Remo; Sepul-
veda, Alonso; Zuluaga, Jorge I. 2340
Gravitational Wave Damping from a Self-Gravitating Vibrating
Ring of Matter around a Black Hole
Basu, Prasad; Chakrabarti, Sandip K. 2343
Variational Principles and Hamiltonian Formulation of Spherical
Shell Dynamics
Kijowski, Jerzy; Magli, Giulio; Malafarina, Daniele 2346
• Operating GW Detectors
Chairperson: Bassan, Massimo
Virgo Commissioning Progress
Barsuglia, Matteo; for the Virgo Collaboration 2351
Results from LIGO Observations: Stochastic Background and
Continuous Wave Signals
Christensen, Nelson; for the Ligo Scientific Collaboration 2356
Explorer and Nautilus Gravitational Wave Detectors - A Status
Report
Bassan, Massimo; for the ROG Collaboration 2359
AURIGA on the Air: Sensitivity, Calibration, Diagnostics and
Observations
Ortolan, Antonello; for the A URIGA collaboration 2365
• Advanced GW Detectors
Chairperson: Blair, David
Optical Spring at Thermal Equilibrium
Di Virgilio, Angela 2373
Iv
Measurements of Electrical Charge Distribution Variations on
Fused Silica
Prokhorov, Leonid G.; Mitrofanov, Valery P 2376
Developments toward Monolithic Suspensions for Advanced
Gravitational Wave Detectors
Heptonstall, Alastair; Cantley, Caroline; Crooks, David; Cumming,
Alan; Hough, James; Jones, Russell; Martin, Iain; Rowan, Sheila;
Cagnoli, Gianpietro 2379
Concept Study of Yukawa-like Potential Tests Using Dynamic
Gravity-Gradients with Interferometric Gravitational-Wave
Detectors
Raffai, Peter; Mdrka, Szabolcs; Matone, Luca; Mdrka, Zsuzsa 2382
Astrophysical Sources of the Gravitational Waves
Lipunov, Vladimir M 2385
• Space and Third Generation GW Detectors
Chairperson: Hough, Jim
DECIGO: The Japanese Space Gravitational Wave Antenna
Ando, Masaki; et al 2393
Design and Construction of the LISA Technology Package Optical
Bench Interferometer
Killow, Christian J.; Bogenstahl, Johanna; Perruer-Lloyd,
Michael; Ward, Henry; Robertson, David I; Guzman Cervantes,
Felipe; Steier, Frank 2398
Compact Binary Inspiral and the Science Potential of Third-
Generation Ground-Based Gravitational Wave Detectors
Van Den Broeck, Chris; Sengupta, Anand S 2401
Discrete Sampling Variation Measurement Technique for Sub-SQL
Sensitivity Detection of Gravitational Waves
Danilishin, Stefan L.; Khalili, Farid Ya 2404
The Detection of Gravitational Waves with Matter Wave
Interferometers
Delva, Pacome; Angonin, Marie-Christine; Tourrenc, Philippe 2407
• GW Data Analysis
Chairperson: Ricci, Fulvio
Detecting LISA Sources Using Time-Frequency Techniques
Gair, Jonathan R.; Jones, Gareth 2413
Determining the Neutron Star Equation of State using the Narrow-
Band Gravitational Wave Detector Schenberg
de Araujo, Jose Carlos N.; Marranghello, Guilherme F 2416
Ivi
Approximate Waveform Templates for Detection of Extreme Mass
Ratio Inspirals with LISA
Gair, Jonathan R 2419
GW-Detector's Output Processing at the Non-Gaussian Noise
Background
Gusev, Andrei V.; Popov, Serghei M.; Rudenko, Valentin 2422
Detecting a Stochastic Background of Gravitational Waves in the
Presence of Non-Gaussian Noise
Himemoto, Yoshiaki 2426
Coincidences between the Gravitational Wave Detectors
EXPLORER and NAUTILUS in the Years 1998, 2001, 2003 and
2004
Pizzella, Guido 2429
Incoherent Strategies for the Network Detection of Periodic
Gravitational Waves
Astone, Pia; Frasca, Sergio; Palomba, Cristiano 2438
Search for Continuous Gravitational Waves: Simple Criterion for
Optimal Detector Networks
Prix, Reinhard 2441
First Coincidence Search among Periodic Gravitational Wave
Source Candidates Using Virgo Data
Palomba, Cristiano; for the Virgo Collaboration 2444
Primordial Black-Hole Gravitational Wave Background Noise in
the LISA, DECIGO and BBO Frequency Bands
de Araujo, Jose Carlos N.; Aguiar, Odylio D.; Miranda, Oswaldo P 2448
• Recent Advances in the History of General Relativity
Chairperson: Renn, Juergen
The Einstein-Varicak Correspondence on Relativistic Rigid
Rotation
Sauer, Tilman 2453
The History of the So-Called Lense-Thirring Effect
Pfister, Herbert 2456
M.-A. Tonnelat's Research Concerning Unified Field Theory
Goenner, Hubert 2459
Rosenfeld, Bergmann, Dirac and the Invention of Constrained
Hamiltonian Dynamics
Salisbury, Donald C 2467
Ivii
Stellar and Solar Positions in 1701-1703 Observed by Francesco
Bianchini at the Clementine Meridian Line in the Basilica of Santa
Maria degli Angeli in Rome, and its Calibration Curve
Sigismondi, Costantino 2470
• Strong Gravity and Binaries
Chairperson: Blanchet, Luc
The Effacing Principle in the Post-Newtonian Mechanics
Kopeikin, Sergei; Vlasov, Igor 2475
Gravitational Waves of a Lense-Thirring System
Vasuth, Mdtyds; Majdr, Jdnos 2478
York Map, Non-Inertial Frames and the Physical Interpretation of
the Gauge Variables of the Gravitational Field
Lusanna, Luca 2481
• Post-Newtonian Dynamics in Binary Objects
Chairperson: Schaefer, Gerhard
Accurate and Efficient Gravitational Waveforms for Certain
Galactic Compact Binaries
Tessmer, Manuel; Gopakumar, Achamveedu 2487
Dimensional Regularization of the Gravitational Interaction of
Point Masses in the ADM Formalism
Damour, Thibault; Jaranowski, Piotr; Schaefer, Gerhard 2490
New Results at 3PN via an Effective Field Theory of Gravity
Porto, Rafael A 2493
Orbital Phase in Inspiraling Compact Binaries
Vasuth, Mdtyds; Mikoczi, Baldzs; Gergely, Ldszlo A 2497
Gravitational Wave Emission from a Stellar Companion Black
Hole in Presence of an Accretion Disk Around a Kerr Black Hole
Basu, Prasad; Chakrabarti, Sandip K.; Mondal, Soumen; Goswami,
Kushalendu 2500
The Second Post-Newtonian Order Generalized Kepler Equation
Gergely, Ldszlo A.; Keresztes, Zolton; Mikoczi, Baldzs 2503
• Tests of Local Lorentz Invariance
Chairperson: Peters, Achim
The Standard-Model Extension and Tests of Relativity
Russell, Neil 2509
New Measurements of the One-Way Speed of Light and its Relation
to Clock-Comparison Experiments
Unnikrishnan, C.S 2512
Iviii
Test of Time Dilation with a Two-Velocity Atomic Clock
Saathoff, Guido; Karpuk, Sergey, Reinhardt, Sascha; Buhr, Hen-
rik; Hansch, Theodor W.; Holzwarth, Ronald; Huber, Gerhard;
Novotny, Christian; Schwalm, Dirk; Udem, Thomas; Wolf,
Andreas; Zimmermann, Marcus; Gwinner, Gerald 2515
• Laboratory Gravity Tests
Chairperson: Laemmerzahl, Claus
Atom Interferometry for Precision Tests of Gravity: Measurement
of G and Test of Newtonian Law at Micrometric Distances
Bertoldi, Andrea; Cacciapuoti, Luigi; de Angelis, Marella;
Drullinger, Robert E.; Ferrari, Gabriele; Lamporesi, Giacomo;
Poli, Nicola; Prevedelli, Marco; Sorrentino, Fiodor; Tino,
Guglielmo M 2519
Development of Accelerometer Prototype for Testing the
Equivalence Principle in Free Fall
lafolla, Valerio; Lucchesi, David; Milyukov, Vadim; Nozzoli,
Sergio; Santoni, Francesco; Shapiro, Irvin I.; Lorenzini, Enrico
C; Cosmo, Mario L.; Ashenberg, Joshua; Cheimets, Peter N.;
Glashow, Sheldon 2530
Measurement of the Gravitational Constant G
Meyer, Hinrich; Kleinevoss, Ulf; Piel, Helm,ut 2534
Solar Radius at Minimum of Cycle 23
Sigismondi, Costantino 2537
The Newtonian Gravitational Constant: Modern Status and
Perspective of New Determination
Milyukov, Vadim; Luo, Jun 2540
Relativistic Astrometry with Gaia: Advances in the RAMOD
Project
Bucciarelli, Beatrice; Crosta, Maria Teresa; Lattanzi, Mario G.;
Vecchiato, Alberto; Preti, Giovanni; de Felice, Fernando 2543
• Clock and Space Tests of Gravity
Chairperson: Salomon, Christophe
Dynamical Clock Synchronization in Einstein's Theory:
Implications for ACES mission of ESA
Lusanna, Luca 2549
STEP Prototype Development Status
Mehls, Carsten et al 2553
On Stellar System Tests of the Cosmological Constant
Sereno, Mauro; Jetzer, Philippe 2556
lix
The Lense-Thirring Effect and the Pioneer Anomaly: Solar System
Tests
Iorio, Lorenzo 2558
The Equivalence Principle and Its Tests in the Context of Gravity,
Quantum Mechanics and Cosmology
Unnikrishnan, C.S 2561
The Flyby-Anomaly
Ldemmerzahl, Claus; Dittus, Hansjoerg 2564
Gravity Tests and the Pioneer Anomaly
Jaekel, Marc-Thierry; Reynaud, Serge 2567
• Astrometry
Chairperson: Klioner, Sergei
A Nice Tool for Relativistic Astrometry: Synge's World Function
Teyssandier, Pierre; Le Poncin-Lafitte, Christophe 2573
Lunar Laser Ranging: A Space Geodetic Technique to Test
Relativity
Muller, Jiirgen 2576
APOLLO: Next Generation Lunar Laser Ranging
Murphy, Thomas W. Jr.; Michelsen, Eric L.; Orin, Adam E.;
Battat, James B.; Stubbs, Christopher W.; Adelberger, Eric G.;
Hoyle, CD.; Swanson, H. Erik 2579
Metric Extensions of General Relativity and Gravity Tests in the
Solar System
Reynaud, Serge; Jaekel, Marc-Thierry 2582
Measurement of the PPN-7 Parameter with a Space-Born Dyson-
Eddington-like Experiment
Vecchiato, Alberto; Gai, Mario; Lattanzi, Mario G.; Morbidelli, Roberto . . 2585
Relativistic Light Deflection near Giant Planets Using Gaia
Astrometry
Anglada-Escude, Guillem; Klioner, Sergei A.; Torra, Jordi 2588
Astrometrical Microlensing with Radioastron
Zakharov, Alexander F 2591
Asteroidal Occultation of Regulus: Differential Effect of Light
Bending
Sigismondi, Costantino; Troise, Davide 2594
Testing General Relativity by Astrometric Measurements Close to
Jupiter, the Real GAREXPart II
Crosta, Maria Teresa; Gardiol, Daniele; Lattanzi, Mario G.;
Morbidelli, Roberto 2597
Ix
Relativistic Tests from the Motion of Asteroids
Hestroffer, Daniel; Mouret, Serge; Berthier, Jerome; Mignard, Frangois . . 2600
• Quantum Gravity Phenomenology
Chairperson: Amelino-Camelia, Giovanni
Effective Vacuum Refractive Index from Gravity and Present
Ether-Drift Experiments
Consoli, Maurizio . 2605
Quantum Gravity Effects in Rotating Black Holes
Reuter, Martin; Tuiran, Erick 2608
Lorentz Symmetry from Lorentz violation in the Bulk
Bertolami, Orfeu; Carvalho, Carla 2611
Quantum Gravity and Spacetirne Symmetries
Lehnert, Ralf 2615
Lorentz Invariance Violation in Higher Order Electrodynamics
Lorek, Dennis; Ldemmerzahl, Claus 2618
Hubble Meets Planck: A Cosmic Peek at Quantum Foam
Ng, Y. Jack . 2621
Evolutionary Reformulation of Quantum Gravity
Montani, Giovanni 2626
Kerr's Gravity as a Quantum Gravity on the Compton Level
Burinskii, Alexander 2631
A Link between General Relativity and Quantum Mechanics
Rosquist, Kjell 2634
Spacetirne Fluctuations and Inertia
Goklii, Ertan; Ldemmerzahl, Claus; Camacho, Abel; Macias, Alfredo .... 2639
Quantum Gravity in Cyclic (Ekpyrotic) and Multiple (Anthropic)
Universes with Strings and/or Loops
Chung, T.J 2642
• Quantum Fields
Chairperson: Belinski, Vladimir
Quantum Liouville Theory with Heavy Charges
Menotti, Pietro; Tonni, Erik 2647
On the Path Integral for Non-Commutative (NC) QFT
Dehne, Christoph 2650
An Irreducible Form for the Asymptotic Expansion Coefficients of
the Heat Kernel of Fermions
Yajima, Satoshi; Higasida, Yoji; Fukuda, Makoto; Tokuo, Shoshi;
Kubota, Shin-Ichiro; Kamo, Yuki 2653
Ixi
Quantum Anomalies for Generalized Euclidean Taub-Newman-
Unti-Tamburino Metrics
Visinescu, Mihai; Visinescu, Anca 2656
A New Expression for the Transition Rate of an Accelerated
Particle Detector
Louko, Jorma; Satz, Alejandro 2659
On the Geoinetrization of the Electromagnetic Interaction for a
Spinning Particle
Cianfrani, Francesco; Milillo, Irene; Montani, Giovanni 2662
Can EPR Correlations be Driven by an Effective Wormhole?
Santini, Eduardo Sergio 2665
Is Torsion a Fundamental Physical Field?
Lecian, Orchidea Maria; Mercuri, Simone; Montani, Giovanni 2668
Unitary Quantization of the Gowdy T3 Cosmology
Corichi, Alejandro; Cortez, Jeronimo; Mena Marugdn, Guillermo A. ... 2671
On the Interaction of the Gravitational Field of a Cosmic String
with Some Quantum Systems
Marques, Geusa; Bezerra, Valdir B 2674
Einstein-Rosen Waves Coupled to Matter
Barbero Gonzalez, Jesus Fernando; Garay, Inaki; Villasenor,
Eduardo J.S 2677
Electromagnetic Radiation from a Charge Rotating in
Schwarzschild Spacetiine
Castineiras, Jorge; Crispino, Luis C.B.; Murta, Rodrigo; Matsas,
George E.A 2680
Recent Developments in Quantum Energy Inequalities
Fewster, Christopher J 2683
Black Holes as Boundaries in 2D Dilaton Supergravity
Bergamin, Luzi; Grumiller, Daniel 2686
Quasinormal Modes for Arbitrary Spins in the Schwarzschild
Background
Khriplovich, Iosif; Ruban, Gennady 2692
Can Quantum Mechanics Heal Classical Singularities?
Helliwell, Thomas M.; Konkowski, Deborah A 2695
Quantizing Two-Dimensional Dilaton Gravity with Fermions: The
Vienna Way
Meyer, Rene 2698
Ixii
Vacuum Polarization for a Spinor Massive Field in an Einstein-
Maxwell Spacetime
Bezerra, Valdir B.; Khusnutdinov, Nail R 2701
• Casimir Effect and Short-Range Gravity
Chairperson: Mostepanenko, Vladimir
The Casimir Effect in Relativistic Quantum Field Theories
Mostepanenko, Vladimir M 2707
Local and Global Casimir Energies in a Green's Function Approach
Milton, Kimbal A.; Cavero-Peldez, Ines; Kirsten, Klaus 2727
Boundary Induced Quantum Fluctuation Effects: From Moving
Mirror to Electron Coherence
Hsiang, Jen-Tsung; Lee, Da-Shin 2746
A Theory of Electromagnetic Fluctuations for Metallic Surfaces
and van der Waals Interactions between Metallic Bodies
Bimonte, Giuseppe 2749
Theory of the Casimir Effect between Dielectric and Semiconductor
Plates
Klimchitskaya, Galina L.; Geyer, Boro 2752
A Novel Experimental Approach for the Measure of the Casimir
Effect at Large Distances
Antonini, Piergiorgio; Bressi, Giacomo; Carugno, Giovanni;
Galeazzi, Giuseppe; Messineo, Giuseppe; Ruoso, Giuseppe 2755
Measurement of the Casimir Force in the Range above 5 Microns
and Detection of the Finite Temperature Effect
Rajalakshm, Gurumukthy I.; Suresh, Doravari; Cowsik, Ramanath;
Unnikrishnan, CS. 2758
Scalar Casimir Effect with Non-Local Boundary Conditions
Saharian, Aram; Esposito, Giampiero 2761
Sample Dependence of the Casimir Force
Pirozhenko, Irina; Lambrecht, Astrid, Svetovoy, Vitaly B 2764
Casimir Interaction between Absorbing and Meta Materials
Intravaia, Francesco; Henkel, Carsten 2767
Casimir Energy and a Cosmological Bounce
Herdeiro, Carlos A.R 2770
Photon Generation from the Vacuum: An Experiment to Detect
the DCE
Braggio, Caterina; Bressi, Giacomo; Carugno, Giovanni; Ruoso,
Giuseppe; Zanello, Dino 2773
ixiii
• Loop Quantum Gravity, Quantum Geometry, Spin Foams
Chairperson: Lewandowski, Jerzy
The Emergence of AdS2 from Quantum Fluctuations
Ambj0rn, Jan; Janik, Romuald; Westra, Willem; Zohren, Stefan 2779
The Ponzano-Regge Model and Reidemeister Torsion
Barrett, John W.; Naish-Guzman, Ileana 2782
The Proca-Field in Loop Quantum Gravity
Helesfai, Gabor 2785
Ambiguity of Black Hole Entropy in Loop Quantum Gravity
Tamaki, Takashi; Nomura, Hidefumi 2788
Exploring the Diffeomorphism Invariant Hilbert Space of a Scalar
Field
Sahlmann, Hanno 2791
Nieh-Yan Invariant and Fermions in Ashtekar-Barbero-Immirzi
Formalism
Mercuri, Simone 2794
A Generalized Schroedinger Equation for Loop Quantum
Cosmology
Salisbury, Donald C; Schmitz, Allison 2797
Spectral Analysis of the Volume Operator in Loop Quantum
Gravity
Brunnemann, Johannes; Rideout, David 2800
Counting Entropy in Causal Set Quantum Gravity
Zohren, Stefan; Rideout, David 2803
Algebraic Approach to 'Quantum Spacetime Geometry'
Raptis, Ioannis; Wallden, Petros; Zapatrin, Roman 2806
Noncummutative Translations and ^-Product Formalism
Daszkiewicz, Marcin; Lukierski, Jerzy; Woronowicz, Mariusz 2809
• Brane Worlds and String Motivated Cosmology
Chairperson: Galtsov, Dmitry
Black Holes on Cosmological Branes
Gergely, Ldszlo A 2815
Generalized Cosmological Equations for a Thick Brane
Khakshournia, Samad 2818
Cerenkov Radiation from Collisions of Straight Cosmic
(Super)Strings
Melkumova, Elena; Gal'tsov, Dmitri V.; Salehi, Karim 2821
Ixiv
High-Energy Effects on the Spectra of Cosmological Perturbations
in Braneworld Cosmology
Hiramatsu, Takashi; Koyama, Kazuya; Taruya, Atsushi 2824
Braneworlds and Quantum States of Relativistic Shells
Ansoldi, Stefano 2827
Rotating Braneworld Black Holes
Aliev, Alikram N 2830
General Solution for Scalar Perturbations in Bouncing Cosmologies
Bozza, Valerio 2833
Constraints on Accelerating Brane Cosmology with Exchange
between the Bulk and Brane
Mathews, Grant J.; Umezu, Ken-Ichi; Kajino, Toshitaka; Ichiki,
Kiyomoto; Nakamura, Ryoko; Yahiro, Masanobu 2836
Testing DGP Modified Gravity in the Solar System
Iorio, Lorenzo 2839
The Dynamics of Scalar-Tensor Cosmology from RS Two-Brane
Model
Kuusk, Piret; Jdrv, Laur; Saal, Margus 2842
Self-T-Dual Brane Cosmology
Rinaldi, Massimiliano 2845
• Brane Worlds
Chairperson: Bianchi, Massimo
Catching Photons from Extra Dimensions
Dobado, Antorio; Maroto, Antorio L.; Cembranos, Jose A.R 2851
Lorentz Invariance Violation in Braneworld Models
Koroteev, Peter A 2854
The Bazanski Approach in Brane-Worlds: A Brief Introduction
Kahil, Magd Elias 2857
• M-Theory and Dualities
Chairperson: Stelle, Kellogg
M-Theory and Dualities
Mac Conamhna, Oisin 2863
AdS Spacetimes in M-Theory
Gauntlett, Jerome P.; Mac Conamhna, Oisin A.P.; Mateos, Toni;
Waldrarn, Daniel 2875
Global Aspects of Seven-Brane Configurations
Bergshoeff, Eric A.; Hartong, Jelle; Ortin, Tomas; Roest, Diederik .... 2878
Ixv
Duality and Black Hole Partition Functions
Mohaupt, Thomas 2881
M-Theory on Calabi-Yau Five-folds
Haupt, Alexander S.; Stelle, Kellogg S 2884
Hagedorn Transition and Chronology Protection in String Theory
Herdeiro, Carlos A.R 2887
KK-Masses and Dipole Theories
Landsteiner, Karl; Montero, Sergio 2890
List of Participants 2893
Author Index 2911
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The GRB - Supernova
Connection
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SWIFT OBSERVATIONS OF GRB050712
M. De PASQUALE*, T. POOLE, S. ZANE, M. PAGE, A. BREEVELD and K. O'MASON
Milliard Space Science Laboratory, University College London,
Dorking, RH56NT, United Kingdom
* ab-mdp@mssl.ucl.ac.uk
u>u>u>. mssl. ac. uk
D. GRUPE, D. BURROWS, J. NOUSEK and P. ROMING
Department of Astronomy and Astrophysics,
University Park, PA 16802, USA
H. KRIMM and N. GEHRELS
NASA/Goddard Space Science Flight Center Street,
Greenbelt, MD 20771, USA
B. ZHANG
Department of Physics, University of Nevada,
Las Vegas, NV 89154, USA
S. KOBAYASHI
Astrophysics Research Group, John Moore University,
Liverpool, Twelve Quay House, Birkenhead CH4 1LD, United Kingdom
Swift observations of GRB050712 show that the X-ray lightcurve of this burst exhibits
flaring activity in the first 500s. We find that the initial flares can be due to "inner
engine" activity, where the later flare may be explained in terms of the interaction of the
ejecta with the surrounding medium. An optical counterpart was detected in the U and
V band of UVOT up to 15000s after the trigger.
Keywords: Gamma Ray Bursts; ultrarelativistic shocks, high energy sources
1. Introduction
Follow-up observations of GRBs have shown that the initial prompt 7-ray emission
is followed by X-ray and optical afterglows. The rapid response of Swift enables
us to study the GRB from the late prompt emission onwards, thus unveiling the
early burst epoch. This has led to the discovery of interesting features, such as the
X-ray flares in the lightcurve of several Swift GRBs, including GRB050712. These
features are widely interpreted as late " internal emission", moved to energies lower
than those of the prompt. Another possible origin of few X-ray flares is the beginning
of the emission of the forward shock, arising into the circumburst medium once the
burst ejecta plough into it. Swift GRB050712 may be an object where both these
2 different causes are present. In the following we investigate the properties of this
remarkable burst a.
2. Observations
GRB050712 triggered the BAT at 14:00:28UT on July 12 2005 (Grupe et al. GCN
3573). The 7-ray emission started 8s before the BAT trigger time, and the lightcurve
aFor a more complete discussion and for all references we refer to De Pasquale et al. (2006).
1947
1948
shows a broad peak. XRT observations started 160s after the BAT trigger (Grupe et
al. GCN 3579). The background-subtracted lightcurve shows an interesting sequence
of flares (Fig. 1) at 210s, 240s and 480s after the trigger. The X-ray spectrum of
the early afterglow presents evolution, as can be deduced from the bottom panel
of Fig.l, which displays the hardness ratio. The spectrum softens in the first 400s,
changing from a energy index p = —1.1 to p = —1.7 (hereafter, we'll use the
convention F oc tav®, where t is the time from the trigger and v is frequency),
while it becomes harder at t = 450s, in correspondence of the last flare, with
p = 0.96. The spectrum of the late afterglow is consistent with that of the this flare.
UVOT observations started 164s after the BAT trigger (Poole et al. GCN 3598).
The burst was observed in the V and U band until it faded below the detector
limit ~ 15000s after the trigger. It was not detected in the UV filters and there
were no observations in the B band due to a countrate limit violation. The V band
lightcurve shows no significant flares (Fig. 1, upper curve) and is very flat at the
beginning of the observations, but after s«m500s it has a decay consistent with that
of the XRT lightcurve.
3. Discussion
The X-ray lightcurve of GRB050712 shows three large flares during the first 500s. A
hypothesis for the origin of X-ray flares is that they are basically a continuation of
the prompt 7-ray emission at lower energies and later times. In this case, they would
share a common origin. They would be caused by shocks occurring in ultrarelativis-
tic shells emitted by the GRB "inner engine". These shells have different Lorentz
factor and they eventually collide, driving shocks which heat electrons; these finally
radiate in form of synchrotron emission. According to theory, flares due to internal
shocks obey the relation a = p — 2, where a and p are the powerlaw decay and the
energy spectral indexes of the emission. We find that the first two flares agree with
this prediction. Other features of these flares indicating the scenario of production
via internal shocks are the high time variability and the uninterrupted softening
of the spectrum from the prompt till these late times. This interpretation requires
that the central engine does not switch off at the end of the main high-energy event,
but is active for longer.
This explanation does not seem to apply for the third flare, which has a spectrum
that differs from the previous flares and is similar to that observed in the late
afterglow. Furthermore, the flux of this flare can be connected with the late afterglow
lightcurve using a broken powerlaw model if the zero time is rescaled to the onset
of the peak (fig 2). This behaviour is expected in the "thick shell" regime, when
the duration of central engine activity is longer than the deceleration time, defined
as the time the ejecta take to sweep a surrounding medium mass equal to the their
rest-mass. In this case, we can have the peak of the forward shock emission, caused
by shocks running into the circumburst medium, at the end of the central engine
activity. The peak would be followed by powerlaw decay, a mark of self-similar ejecta
expansion.
1949
The optical-to-X-ray energy index pox fluctuates until ~ 500s after the trigger,
then it stabilizes at —0.8 for the rest of the afterglow. The different X-ray and
optical lightcurves suggest that there is still another mechanism responsible for the
optical before 500s. We propose it may be due the "reverse shock", which are shocks
crossing the ejecta inwards and short-lived. Under certain circumstances, the reverse
shocks can be responsible for the flat optical emission seen in the first few hundreds
seconds, while, after ~ 500s, the optical and X-ray emission are produced by the
same mechanism, likely the forward shock.
O 1
O 6
J**\
-h'
Sw
V.
7V
A-r
■-i
500 J OOO
TliTie sine
SOOO
-«L Is]
Fig. 1. The optical (top) and X-ray lightcurve (middle) of GRB050712. Bottom: hardness Ratio
of the X-ray afterglow, defined as (H — S)/(H + S), where H and S are the count/rate in 0.3-1
keV and 1-10 keV bands respectively.
■^t
^-K
1 OOO I O
Time - -140 k
Fig. 2. X-ray lightcurve of 050712 with the zero time rescaled at t = 440s.
References
1. De Pasquale, M., Grupe D., Poole T. S., et al. in MNRAS (370, 1859, 2006)
NO ASTROPHYSICAL DYADOSPHERES *
DON N. PAGE
Theoretical Physics Institute, University of Alberta, Edmonton, Alberta, Canada T6G 2G7,
don@phys.ualberta.ca
Pair production itself prevents the development of dyadospheres, hypothetical
macroscopic regions where the electric field exceeds the critical Schwinger value. Pair
production is a self-regulating process that would discharge a growing electric field, in the
example of a hypothetical collapsing charged stellar core, before it reached 6% of the
minimum dyadosphere value, keeping the pair production rate more than 26 orders of
magnitude below the dyadosphere value.
Ruffini and his group1-16 have proposed a model for gamma ray bursts that
invokes a dyadosphere, a macroscopic region of spacetime with rapid Schwinger pair
production,17 where the electric field exceeds the critical electric field value
Ec = — = I^!«l.32xl016V/cm. (1)
q Hq ' W
(Here m and — q are the mass and charge of the electron, and I am using Planck
units throughout.) The difficulty of producing these large electric fields is a problem
with this model that has not been adequately addressed. Here I shall summarize
calculations18 showing that dyadospheres almost certainly don't develop astrophys-
ically.
The simplest reason for excluding dyadospheres is that if one had an astro-
physical object of mass M, radius R > 2M, and excess positive charge Q in the
form of protons of mass mp and charge q at the surface, the electrostatic repulsion
would overcome the gravitational attraction and eject the excess protons unless
qQ < rripM or
E_ _ qQ mpM mp 2 13 (Mq\
Ec ~ m2R2 S m2R2 4m2M \M J' [ >
where M© is the solar mass. (If the excess charge were negative and in the form of
electrons, the upper limit would be smaller by m/mp.) Then the pair production
would be totally negligible.
However, one might postulate the implausible scenario in which protons are
bound to the object by nuclear forces,2 which in principle are strong enough to
balance the electrostatic repulsion even for dyadosphere electric fields. Therefore,
for the sake of argument, I did a calculation18 of what would happen under the
highly idealized scenario in which the surface of a positively charged stellar core
with initial charge Qq~M (the maximum allowed before the electrostatic repulsion
would exceed the gravitational attraction on the entire core, not just on the excess
protons on its surface) freely fell from rest at radial infinity along radial geodesies
in the external Schwarzschild metric of mass M.
* This research has been partially supported by Natural Sciences and Engineering Research Council
of Canada.
1950
1951
This idealization18 ignores the facts that a realistic charged surface would (a)
not fall from infinity, (b) have one component of outward acceleration, relative to
free fall, from the pressure gradient at the surface, (c) have another component of
outward acceleration from the electrostatic repulsion, and (d) fall in slower in the
Reissner-Nordstrom geometry if the gravitational effects of the electric field with
Q ~ M were included. Because of each one of these effects, the actual surface would
fall in slower at each radius and hence have more time for greater discharge than in
the idealized model. Hence the idealized model gives a conservative upper limit on
the charge and electric field at each radius, even under the implausible assumption
that the excess protons are somehow sufficiently strongly bound to the surface that
they are not electrostatically ejected.
Even in this highly idealized model,18 the self-regulation of the pair production
process itself will discharge any growing electric field well before it reaches dya-
dosphere values. This occurs mainly because astrophysical length scales are much
greater than the electron Cornpton wavelength, which is the scale at which the pair
production becomes significant at the critical electric field value for a dyadosphere.
Therefore, the electric field will discharge astrophysically even when the pair
production rate is much lower than dyadosphere values.
These calculations18 lead to the conclusion that it is likely impossible
astrophysically to achieve, over a macroscopic region, electric field values greater than a
few percent of the minimum value for a dyadosphere, if that. The Schwinger pair
production itself would then never exceed 10-26 times the minimum dyadosphere
value.
Since the idealized model does give pair production at macroscopically
significant rates (though more than 26 orders of magnitude below that of a dyadosphere),
one might revise the definition of a dyadosphere to include any macroscopic electric
field which gives macroscopically significant pair production. Then (assuming that
sufficient charge separation can somehow be achieved by forces necessarily much
stronger than gravitational forces, to evade the limitations discussed above), my
calculations do not exclude the possibility of such a revised concept of a
dyadosphere. However, the much weaker amount of pair production gives an efficiency,
even under the highly idealized conditions of having maximal initial charge at such
large radii that it seems inconceivable that the charge carriers could be sufficiently
bound to such objects so much larger than neutron stars, that is always much
less than unity for collapsing objects with much less mass than three million solar
masses: the efficiency is very conservatively bounded by 2 x W~4y'M/MQ.18
Therefore, even these idealized charged collapsing objects, unless they were enormously
more massive than the sun, would not produce enough energy in outgoing charged
particles to be consistent with the observed gamma ray bursts.
In conclusion, macroscopic dyadospheres (by the original definition) almost
certainly cannot form astrophysically, and the much weaker pair production rates that
might occur, under highly idealized and implausible scenarios, do not seem sufficient
for giving viable models of gamma ray bursts.18
1952
References
1. T. Damour and R. Ruffini, Phys. Rev. Lett. 35, 463-466 (1975).
2. R. RufRni, in Black Holes and High Energy Astrophysics, Proceedings of the Yamada
Conference XLIX on Black Holes and High Energy Astrophysics held on 6-10 April,
1998 in Kyoto, Japan, eds. H. Sato and N. Sugiyama (Frontiers Science Series No.
23, Universal Academic Press, Tokyo, 1998), p. 167; astro-ph/9811232; Astron. and
Astrophys. Supp. 138, 513 (1999); in Proc. 11th Marcel Grossman Meeting on General
Relativity, ed. H. Kleinert, R. T. Jantzen, and R. RufRni (World Scientific, Singapore,
in press).
3. G. Preparata, R. RufRni, and S.-S. Xue, Astron. Astrophys. 338, L87 (1998), astro-
ph/9810182; Nuovo Cim. B115, 915 (2000); J. Korean Phys. Soc. 42, S99 (2003),
astro-ph/0204080.
4. R. RufRni and S.-S. Xue, Abstracts of the 19th Texas Symposium on Relativistic
Astrophysics and Cosmology, held in Paris, France, Dec. 14-18, 1998, eds. J. Paul, T. Mont-
merle, and E. Aubourg (CEA Saclay, 1998).
5. R. RufRni, J. Salmonson, J. Wilson, and S.-S. Xue, Astron. and Astrophys. Supp.
138, 511 (1999), astro-ph/9905021; Astron. and Astrophys. 350, 334 (1999), astro-
ph/9907030; Astron. and Astrophys. 359, 855 (2000), astro-ph/0004257.
6. C. L. Bianco, R. RufRni, and S.-S. Xue, Astron. and Astrophys. 368, 377 (2001),
astro-ph/0102060.
7. R. RufRni, C. L. Bianco, F. Fraschetti, P. Chardonnet, and S.-S. Xue, Nuovo Cim.
B116, 99 (2001), astro-ph/0106535.
8. R. RufRni, C. L. Bianco, F. Fraschetti, S.-S. Xue, and P. Chardonnet, Astrophys.
J. 555, L107-L111 (2001), astro-ph/0106531; L113 (2001), astro-ph/0106532; L117
(2001), astro-ph/0106534.
9. R. RufRni and L. Vitagliano, Phys. Lett. B545, 233(2002), astro-ph/0209072.
10. R. RufRni, S.-S. Xue, C. L. Bianco, F. Fraschetti, and P. Chardonnet, La Recherche
353, 30(2002).
11. R. RufRni, C. L. Bianco, P. Chardonnet, F. Fraschetti, and S.-S. Xue, Astrophys.
J. 581, L19 (2002), astro-ph/0210648; Int. J. Mod. Phys. D12, 173 (2003), astro-
ph/0302141.
12. P. Chardonnet, A. Mattei, R. RufRni, and S.-S. Xue, Nuovo Cim. 118B, 1063 (2003).
13. R. RufRni, L. Vitagliano, and S.-S. Xue, Phys. Lett. B559, 12 (2003), astro-
ph/0302549; Phys. Lett. B573, 33 (2003), astro-ph/0309022.
14. R. RufRni, M. G. Bernardini, C. L. Bianco, P. Chardonnet, F. Fraschetti, and S.-
S. Xue, Advances in Space Research 34, 2715 (2004), astro-ph/0503268.
15. R. RufRni, C. L. Bianco, P. Chardonnet, F. Fraschetti, V. Gurzadyan, and S.-S. Xue,
Int. J. Mod. Phys. D13, 843 (2004), astro-ph/0405284.
16. C. L. Bianco and R. RufRni, Astrophys. J. 605, LI (2004), astro-ph/0403379; 620,
L23 (2005), astro-ph/0501390; 633, L13 (2005), astro-ph/0509621.
17. F. Sauter, Z. Phys. 69, 742 (1931); W. Heisenberg and H. Euler, Z. Phys. 98, 714
(1936); V. Weisskopf, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 14, No. 6 (1936);
J. S. Schwinger, Phys. Rev. 82, 664 (1951); A. I. Nikishov, Nucl. Phys. B21, 346
(1970).
18. D. N. Page, astro-ph/0605432; in Proceedings of the VII Asia-Pacific International
Conference on Gravitation and Astrophysics, Chungli, Taiwan, 23-36 Nov. 2005, ed.
C.-M. Chen (in press), astro-ph/0605432; Astrophys. J., 653, 1400 (2006), astro-
ph/0610340.
MAGNETIZED HYPERMASSIVE NEUTRON STAR COLLAPSE:
A CANDIDATE CENTRAL ENGINE FOR SHORT-HARD GRBs
BRANSON C. STEPHENS, MATTHEW D. DUEZ*, YUK TUNG LIU
and STUART L. SHAPIRO t
Department of Physics,
University of Illinois at Urbana- Champaign,
Urbana, IL 61801, USA
MASARU SHIBATA
Graduate School of Arts and Sciences,
University of Tokyo, Komaba, Meguro,
Tokyo 153-8902, Japan
Hypermassive neutron stars (HMNSs) are equilibrium configurations supported against
collapse by rapid differential rotation and likely form as transient remnants of binary
neutron star mergers. Though HMNSs are dynamically stable, secular effects such as
viscosity or magnetic fields tend to bring HMNSs into uniform rotation and thus lead to
collapse. We simulate the evolution of magnetized HMNSs in axisymmetry using codes
which solve the Einstein-Maxwell-MHD system of equations. We find that magnetic
braking and the magnetorotational instability (MRI) both contribute to the eventual
collapse of HMNSs to rotating black holes surrounded by massive, hot accretion tori and
collimated magnetic fields. Such hot tori radiate strongly in neutrinos, and the resulting
neutrino-antineutrino annihilation could power short-hard GRBs.
1. Introduction and Methods
Short-hard gamma-ray bursts (SGRBs) emit large amounts of energy in gamma
rays1 with durations ~ 10~3-2 s and may originate from binary neutron star
mergers.1,2 If the total mass of the binary is below a certain threshold, a hypermassive
neutron star (HMNS) likely forms as a transient merger remnant.7'8 HMNSs have
masses larger than the maximum allowed mass for rigidly rotating neutron stars
and are supported against collapse mainly by rapid, differential rotation.10
We have performed general relativistic magnetohydrodynamic (GRMHD)
simulations of differentially rotating T-law HMNSs3,4'9 using two new GRMHD codes.5,6
Here, we consider an HMNS collapse model with a more realistic hybrid EOS.
For details of the EOS, the chosen HMNS model, and the simulation, see Shibata
et al.9 and Duez et al.4 We construct a differentially rotating HMNS with mass
M = 2.65M© and angular momentum J = 0.82GM2/c. This HMNS is similar to
one found in a BNS merger simulation.8 A small seed poloidal magnetic field is
introduced into the HMNS as described in Shibata et al.9 These calculations are
the first in general relativity to self-consistently generate a candidate GRB central
engine (i.e., a rotating BH surrounded by a magnetized torus) from non-singular
initial data.
* Current address: Center for Radiophysics and Space Research, Cornell, Ithaca, NY 14853.
t Also at the Department of Astronomy and NCSA, University of Illinois, Urbana, IL 61801.
1953
1954
0 5 10 15 20 25) 5 10 15 20 25
X(km) X(km)
Fig. 1. Upper panels: Density contours (solid curves) and velocity vectors at the initial time and
at a late time. The contours are drawn for p = 1016 g/cm3 X 10_0A% g/cm3 (i = 0-9). In the
second panel, a curve with p = 1011 g/cm3 is also drawn. The (red) circle in the lower left of the
second panel denotes an apparent horizon. The lower panels show the poloidal magnetic field lines
at the same times as the upper panels.
2. Results and Discussion
In Figure 1, we show snapshots of the meridional density contours, velocity
vectors, and poloidal magnetic field lines at the initial time and at a late time. The
differential rotation of the HMNS winds up a toroidal magnetic field, which then
begins to transport angular momentum from the inner to the outer regions of the
star (magnetic braking), inducing quasistationary contraction of the HMNS.3
After the toroidal field growth saturates, the evolution is dominated by the MRI,12
which leads to turbulence, thus contributing to the angular momentum transport.
The star eventually collapses to a BH, while material with high enough specific
angular momentum remains in an accretion torus. The accretion rate M
gradually decreases and eventually settles down to M ~ 10Me/s, giving an accretion
timescale of MtOTUS/M ~ 10 ms.
To explore the properties of the torus, we calculate the surface density E and the
typical thermal energy per nucleon, u. We find u ~ 94 MeV/nucleon, or equivalently,
T ss 1.1 x 1012 K. Because of its high temperature, the torus radiates strongly
in thermal neutrinos.13'14 However, the opacity inside the torus is approximately
k ~ 7 x W~15Ti2 cm2 g_1. The optical depth is then estimated as r ~ kE ~
7200Ei8T122, so that the neutrinos are effectively trapped.14 Here, T12 = T/1012 K,
Ei8 = E/1018 g cm-2. This regime of accretion has been described as a neutrino-
dominated accretion flow (NDAF).15
We note that the properties of the torus are not specific to the chosen initial
data. For example, we consider the model labeled star A in Duez et al.4 This is
also an HMNS model which collapses to a BH surrounded by a hot accretion torus
under the influence of magnetic fields. However, star A has simple F-law EOS (not
a hybrid EOS) and has a different rotation profile and compactness. In this case, we
1955
find that the disk has u « 5 MeV/nucleon, which gives an optical depth of about
70. Thus, the evolution of star A also produces a hyperaccreting NDAF.
Returning to the hybrid EOS HMNS model, we estimate the neutrino
luminosity in the optically-thick diffusion limit.16 We obtain i„ ~ 2 x
1053 erg/s^/lO km)2!^^1, which is comparable to the neutrino Eddington
luminosity.14 A model for the neutrino emission in a similar flow environment with
comparable Lv gives for the luminosity due to vv annihilation Lv„ ~ 1050 ergs/s.14
Since the lifetime of the torus is ~ 10 ms, the total energy, EVy ~ 1048 ergs,
may be sufficient to power SGRBs as long as the emission is somewhat beamed.17
Our numerical results, combined with accretion and jet models,14'17 thus suggest
that magnetized HMNS collapse is a promising candidate for the central engine of
SGRBs.
Acknowledgments
Numerical computations were performed at NAOJ, IS AS, and NCSA. This work
was supported in part by Japanese Monbukagakusho Grants (Nos. 17030004
and 17540232) and NSF Grants PHY-0205155 and PHY-0345151, NASA Grants
NNG04GK54G and NNG046N90H at UIUC.
References
1. B. Zhang and P. Meszaros, Int. J. Mod. Phys. A 19, 2385 (2004); T. Piran, Rev. Mod.
Phys. 76, 1143 (2005).
2. R. Narayan, B. Paczynski, and T. Piran, Astrophys. J. Lett. 395, L83 (1992).
3. M. D. Duez, Y. T. Liu, S. L. Shapiro, M. Shibata, and B. C. Stephens, Phys. Rev.
Lett. 96, 031101 (2006).
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73, 104015 (2006).
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16. See, e.g., Appendix I of S. L. Shapiro and S. A. Teukolsky, Black Holes, White. Dwarfs,
and Neutron Stars, Wiley Interscience (New York, 1983).
17. M. A. Aloy, H.-T. Janka, and E. Miiller, Astron. Astrophys. 436, 273 (2005).
THEORETICAL INTERPRETATION OF LUMINOSITY AND
SPECTRAL PROPERTIES OF GRB 031203
C.L. BIANCO,1'2-* M.G. BERNARDINI,1'2^ P. CHARDONNET,1'4** F. FRASCHETTI,6>§
R. RUFFINI1'2*3^ and S.-S. XUE1*!!
1 ICRANet and ICRA, Piazzale delta Repubblica 10, 1-65122 Pescara, Italy
2 Dip. di Fisica, Universita di Roma "La Sapienza", Piazzale Aldo Moro 5, 1-00185 Roma, Italy
3 ICRANet, Universite de Nice Sophia Antipolis, Grand Chateau, BP 2135, 28, avenue de
Valrose, 06103 NICE CEDEX 2, France
* bianco@icra.it
*rnaria.bernardini@icra.it
f chardon@lapp.in2p3.fr
§ fraschetti@icra.it
^ ruffini@icra.it
"xue@icra.it
We show how an emission endowed with an instantaneous thermal spectrum in the co-
moving frame of the expanding fireshell can reproduce the time-integrated GRB observed
non-thermal spectrum. An explicit example in the case of GRB 031203 is presented.
1. Introduction
One aim of our model (see e.g. Ref. 1 and references therein) is to derive from
first principles both the luminosity in selected energy bands and the time
resolved/integrated spectra of GRBs.2 The luminosity in selected energy bands is
evaluated integrating over the equitemporal surfaces (EQTSs)3'4 the energy
density released in the interaction of the optically thin fireshell with the CircumBurst
Medium (CBM) measured in the co-moving frame, duly boosted in the observer
frame. The radiation viewed in the co-moving frame of the accelerated baryonic
matter is assumed to have a thermal spectrum and to be produced by the
interaction of the CBM with the front of the expanding baryonic shell.2
2. The instantaneous GRB spectra
In Ref. 5 it is shown that, although the instantaneous spectrum in the co-moving
frame of the optically thin fireshell is thermal, the shape of the final instantaneous
spectrum in the laboratory frame is non-thermal. In fact, as explained in Ref. 2,
the temperature of the fireshell is evolving with the co-moving time and, therefore,
each single instantaneous spectrum is the result of an integration of hundreds of
thermal spectra with different temperature over the corresponding EQTS. This
calculation produces a non thermal instantaneous spectrum in the observer frame.5
Another distinguishing feature of the GRBs spectra which is also present in these
instantaneous spectra is the hard to soft transition during the evolution of the
event.6~9 In fact the peak of the energy distributions Ep drift monotonically to
softer frequencies with time.5 This feature explains the change in the power-law
low energy spectral index10 a which at the beginning of the prompt emission of the
1956
1957
burst (if = 2 s) is a = 0.75, and progressively decreases for later times.5 In this
way the link between Ep and a identified in Ref. 6 is explicitly shown.
3. The time-integrated GRB spectra - Application to GRB 031203
The time-integrated observed GRB spectra show a clear power-law behavior. Within
a different framework (see e.g. Ref. 11 and references therein) it has been argued
that it is possible to obtain such power-law spectra from a convolution of many non
power-law instantaneous spectra monotonically evolving in time. This result was
recalled and applied to GRBs12 assuming for the instantaneous spectra a thermal
shape with a temperature changing with time. It was shown that the integration of
such energy distributions over the observation time gives a typical power-law shape
possibly consistent with GRB spectra.
Our specific quantitative model is more complicated than the one considered in
Ref. 12: the instantaneous spectrum here is not a black body. Each instantaneous
10'
10'
10u
$ 10"1
10"
10"'
10"
convolution
INTEGRAL data
0 s < tjj < 5 s
5s<t5j<10s
10s<rf<20s
ciy-,
10
100
Energy (keV)
1000
Fig. 1. Three theoretically predicted time-integrated photon number spectra N(E), computed
for GRB 031203,5 are here represented for 0 < td < 5 s, 5 < tda < 10 s and 10 < tda < 20 s
(dashed and dotted curves), where td is the photon arrival time at the detector.5,13 The hard to
soft behavior is confirmed. Moreover, the theoretically predicted time-integrated photon number
spectrum N(E) corresponding to the first 20 s of the "prompt emission" (black bold curve) is
compared with the data observed by INTEGRAL. This curve is obtained as a convolution of
108 instantaneous spectra, which are enough to get a good agreement with the observed data.
Details in Ref. 5.
1958
spectrum is obtained by an integration over the corresponding EQTS:3'4 it is itself
a convolution, weighted by appropriate Lorentz and Doppler factors, of ~ 106
thermal spectra with variable temperature. Therefore, the time-integrated spectra are
not plain convolutions of thermal spectra: they are convolutions of convolutions of
thermal spectra.2'5
In Fig. 1 we present the photon number spectrum N(E) time-integrated over
the 20 s of the whole duration of the prompt event of GRB 031203 observed by
INTEGRAL:14 in this way we obtain a typical non-thermal power-law spectrum
which results to be in good agreement with the INTEGRAL data5,14 and gives a
clear evidence of the possibility that the observed GRBs spectra are originated from
a thermal emission.5
References
1. R. Ruffini, M. G. Bernardini, C. L. Bianco, L. Caito, P. Chardonnet, M. G. Dainotti,
F. Fraschetti, R. Guida, M. Rotondo, G. Vereshchagin, L. Vitagliano and S.-S. Xue,
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Institute of Physics Conference Series, Vol. 910 (June 2007).
2. R. Ruffini, C. L. Bianco, S.-S. Xue, P. Chardonnet, F. Fraschetti and V. Gurzadyan,
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3. C. L. Bianco and R. Ruffini, Astrophysical Journal 605, LI (April 2004).
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Xue, Astrophysical Journal 634, L29 (November 2005).
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Paciesas, D. L. Band and J. L. Matteson, Astrophysical Journal 479, p. L39 (April
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7. T. Piran, Physics Reports 314, 575 (June 1999).
8. F. Frontera, L. Amati, E. Costa, J. M. Muller, E. Pian, L. Piro, P. Soffitta, M. Ta-
vani, A. Castro-Tirado, D. Dal Fiume, M. Feroci, J. Heise, N. Masetti, L. Nicastro,
M. Orlandini, E. Palazzi and R. Sari, Astrophysical Journal Supplement Series 127,
59 (March 2000).
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M. Briggs, W. Paciesas, G. Pendleton, G. Fishman, C. Kouveliotou, C. Meegan,
R. Wilson and P. Lestrade, Astrophysical Journal 413, 281 (August 1993).
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Journal 555, L107 (July 2001).
14. S. Y. Sazonov, A. A. Lutovinov and R. A. Sunyaev, Nature 430, 646 (August 2004).
GRB980425 AND THE PUZZLING URCA1 EMISSION
M. G. BERNARDINI*, C. L. BIANCO, L. CAITO, M. G. DAINOTTI, R. GUIDA
and R. RUFFINI
Dipartimento di Fisica, Universita di Roma "La Sapienza"
Roma, 1-00185, Italy
* maria. bernardini@icra.it
ICRANet and ICRA, Piazzale delta Repubblica 10
Pescara, 1-65122, Italy
We applied our "fireshell" model to GRB980425 observational data, reproducing very
satisfactory its prompt emission. We use the results of our analysis to provide a possible
interpretation for the X-ray emission of the source SI. The effect on the GRB analysis
of the lack of data in the pre- Swift observations is also outlined.
Keywords: Gamma rays: bursts — black hole physics
1. Theoretical interpretation of GRB980425 prompt emission
GRB980425 triggered the BeppoSAX GRBM (40-700 keV) at 21:49:11 UT and was
simultaneously detected by the BeppoSAX WFC (2-26 keV).1 This GRB received
particular attention because of its spatial and temporal (~ 1 day2) coincidence
with the bright Type Ic Supernova (SN) 1998bw. Since the probability of a chance
coincidence between them was very low, GRB980425 provided the first evidence for
a physical association between GRBs and SNe.3
The follow-up of GRB980425 with BeppoSAX NFI revealed the presence of
two X-ray sources, one (Si) consistent with SN1998bw, and the other (S2) not
consistent.1 The SI X-ray light curve shows a decay much slower than usual X-
ray GRB afterglows.1 This trend would be similar to the X-ray behavior of other
SNe.1 Further observations on 2002 performed by XMM4 confirmed S2 as a sum of
several faint field sources. Si resulted indeed definitely linked to SN1998bw,4 and it
showed a faster temporal decay than the one observed by BeppoSAX. The temporal
behavior of SI was confirmed by a further observation performed by Chandra.5
We applied our "fireshell" model6 to analyze GRB980425 observational data.1
It is based on two independent variables characterizing the source: the total
energy Efot of the e± plasma and the baryon loading B, which for this source are,
respectively, El°± = 1.2 x 1048 erg and B = 7.7 x 10~3. The temporal structure of
the prompt emission has been reproduced assuming a succession of spherical Cir-
cumBurst (CBM) overdense regions. The CBM mean density during this phase is
{ricbm) = 2.18 x 10"2 particles/cm3 and (11) = 1.24 x 10"8.
In Fig. 1 we test our assumptions comparing our theoretically computed light
curves in the 40-700 and 2-26 keV energy bands with the observations by the
BeppoSAX GRBM and WFC.1 The results obtained (see Figs. 1) is very satisfactory.
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Fig. 1. Theoretical light curves of GRB980425 prompt emission in the 40-700 keV and 2-26 keV
energy bands, compared with the observed data respectively from BeppoSAX GRBM and WFC.1
Fig. 2. Theoretical light curves of GRB980425 in the 40-700 keV, 2-26 keV, 2-10 keV energy
bands, represented together with URCAl observational data. All observations are by BeppoSAX^
with the exception of the last two URCAl points, which are observed by XMM and Chandra.4^
2. The GRB980425 late afterglow emission
We turn now to the analysis of the late observations of the source SI performed by
BeppoSAX NFL1 XMM4 and Chandra.5 Since there is a gap of ~ 104 s between the
first observations of SI and the end of GRB980425 prompt emission, it is hardly
1961
possible to determine the real behavior of the CBM parameters in that region.
For this reason we extrapolate the CBM parameters from the value they assume
after the last prompt emission observation. This choice is not unique and different
behaviors produce quite different results. We recently analyzed GRB0602187 which
belongs to the same class of GRB980425 and is the only source in such a class to
have an excellent data coverage without gaps. Since GRB060218 fulfills the Amati
et al.8 relation unlike other sources in its same class,9 we are currently examining
if the missing data in GRB980425 may have a prominent role in its non fulfillment
of the Amati et al. relation.10
Despite all the possible choices for the CBM parameters during the late afterglow
emission, we are able to state that the X-ray emission of the source Si definitely
does not belong to the GRB (see Fig. 2). In fact, due to the very low energy of
the source and of the Lorentz factor at the transparency, we cannot expect such
emission at so late times. This implies that SI must be linked to the SN event
instead of the GRB. In order to emphasize the different origin of this source, we
named it URCAl, and we possibly interpret it as URCA emission from the neutron
star left by the SN explosion.11
Also4 noticed that the late X-ray emission of SN1998bw is compatible with
cooling radiation from the compact remnant, provided the GRB has swept up all the
surrounding material by creating an evacuated cone.12 has shown, in the context of
X-ray afterglows of GRBs, that cooling neutron stars with "external" disturbances
(e.g., a fallback) may radiate in X-rays with a temporal rate faster than a power-law.
References
1. Pian, E., Amati, L., Antonelli, L.A., et al. 2000, ApJ, 536, 778.
2. Iwamoto, K., Mazzali, P.A., Nomoto, K., et al. 1998, Nature, 395, 672.
3. Galama, T., Vreeswijk, P.M., van Paradijs, J., et al. 1998, Nature, 395, 670.
4. Pian, E., et al. 2004, Adv. Sp. Res., 34, 2711.
5. Kouveliotou, C, Woosley, S.E., Patel, S.K., et al. 2004, ApJ, 608, 872.
6. Ruffini, R., Bernardini, M.G., Bianco, C.L., et al. 2005, AIP Con.Proc, 782, 42.
7. Dainotti, M.G., Bernardini, M.G., Bianco, C.L., et al. 2007, A&A , 471, 29.
8. Amati, L., Frontera, F., Tavani, M., et al. 2002, A&A, 390, 81.
9. Amati, L., Delia Valle, M., Frontera, F., et al. 2007, A&A, 463, 913.
10. Ghisellini, G., Ghirlanda, G., Mereghetti, S., et al. 2006, MNRAS, 372, 1699.
11. Ruffini, R., Bernardini, M.G., Bianco, C.L., et al. 2007, ESA Spec.Pub., in press.
arXiv:0705.2456.
12. Tavani, M. 1997, ApJ, 483, L87.
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The Afterglow, Short and
Long GRBs
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THE EpA - Eiso CORRELATION AND THE NATURE OF
SUB-ENERGETIC GRB
LORENZO AMATI
INAF - Istituto di Astrofisica Spaziale e Fisica cosmica,
via P. Gobetti 101, Bologna 40129, Italy
amati@iasfbo. inaf. it
GRB 060218 is a peculiar event which shares several properties with GRB 980425,
the proto-type event of the GRB-SN connection: a very low redshift (0.0331), a
very low isotropic—equivalent radiated energy, E{so, a clear association with a SN
event (SN2006aj). However, while the spectral peak energy, EPii, and E{so values of
GRB 980425 are completely inconsistent with the Eps - Eiao (" Amati") correlation
holding for long GRBs/XRFs (as is possibly true for the other sub-energetic event
GRB 031203), those of GRB 060218 are fully consistent with it. This evidence, togehter
with the " achromatic" behaviour of the GRB 060218 afterglow light curve and some
properties of SN2006aj, challanges the popular explanations for the nature of sub-energetic
GRBs (like, e.g., the very off—axis scenario) and points towards the existence of a (still
mostly uncovered) population of intinsecally faint GRBs.
1. The Ep>i — E[so correlation
In the last decade, thanks to the discovery and study of afterglow emission and host
galaxies, it has been possible to estimate the redshift of several tens of Gamma-
Ray Bursts (GRBs), and thus to derive their distance scale, luminosities and other
intrinsic properties. Among these, the correlation between the cosmological rest-
frame v¥v spectrum peak energy, Ep-U and the isotropic equivalent radiated energy,
£iso, is one of the most intriguing and robust. Indeed, as shown initially by Ref. 1
and more recently by Ref. 2, all long GRBs with known redshift and estimated EPti
are consistent with the relation EPji = K x E[£0 (K ~85 and m ^0.57, with Epj in
keV and E-lso in units of 1052 erg), with the only exception of GRB 980425 (which
is anyway a peculiar event under several other aspects). The EPt{ - E-lso correlation
holds from the brightest GRBs to the weakest and softest ones (X-Ray Flashes,
XRFs) and is characterized by a scatter in log(EPti) of ^0.2 dex (by assuming a
Gaussian distribution of the deviations). The implications and uses of the Ep^ - E-lso
correlation include prompt emission physics, jet geometry and structure, testing of
GRB/XRF synthesis and unification models, pseudo-redshift estimators, cosmology
(when additional observables, like e.g. the break time of the optical afterglow light
curve or the high signal time scale, are included; see Ref. 2 for a review). In the
recent years there has been a debate, mainly based on BATSE GRBs without known
redshift, about the impact of selection effects on the sample of GRBs with known
redshift and thus on the EPt\ - Eiso correlation. Based on the analysis of BATSE
GRBs without known redshift, different conclusions were reported (e.g., Ref. 3,4).
The recent confirmation or the correlation by the Swift satellite5 is a clear evidence,
and a further confirmation, that the Ep<i - Eiso correlation is likely not an artifact
of selection effects.
1965
1966
104B 1049 1050 1051 1052 1053 1054
EiSo (erg)
Fig. 1. Epi - Eiso points of the GRB with firm estimates of redshift and Epi included in the
sample of Amati (2006); the continuous and line is the best fit power-law and the dotted lines
delimitate the la region. The three peculiar sub—energetic GRBs discussed in the text are shown
as big dots.
2. Sub-energetic GRB in the EP:i - Eiso plane
GRB 980425 was not only the first example of the GRB-SN connection, but also
a very peculiar event. Indeed, with a redshift of 0.0085 it was much closer than
the majority of GRBs with known redshift (~ 0.1 < z < 6.3) and its value of
i^iso was very low (^1048 erg), well below the typical range for "standard" bursts
(^1051 - ^1054 erg). Moreover, this event was characterized by values of EPti and
Eiso completely inconsistent with the EP)i - Eiso correlation (Fig. 1). In addition
to GRB 980425, also GRB031203/SN20031w6 was characterized by a value of EPti
which, combined with its low value of £jso, makes it the second (possible) outlier
of the EPti - Eiso correlation (the Ep^ value of this event is still debated). Both
cases may point towards the existence of a class of nearby and intrinsically faint
GRBs with different properties with respect to "standard" GRBs. However, it has
been suggested by several authors that the low measured £\so of these events and
their inconsistency with the Epi - E-iso correlation are due to viewing angle effects
(off-axis scenarios, see, e.g., Ref. 7). Recently, it was also shown8 that the deviation
of GRB 980425 and GRB 031203 from the EP)i - £:iso correlation may be due to
undetected hard to soft spectral evolution.
1967
3. The intriguing case of GRB 060218
The Swift GRB 060218, similarly to GRB 980425 and GRB 031203, was very close
(z = 0.033) and exhibited a very low afterglow kinetic energy (~ 100 times less
than standard GRBs), as inferred from radio observations (9) and a very low value
of £iS0. Thus, it was classified as a sub-energetic GRB. However, as can be seen
in Fig. 1, differently from GRB 980425 and (possibly) GRB 031203, GRB 060218, is
fully consistent with the Ep[ - E-lso correlation. This evidence favors the hypothesis
that this is a truly sub-energetic event rather than a GRB seen off axis. The ratio
between E[so and Lx,io (the X-ray afterglow flux at 10 hours from the GRB onset)
and the radio afterglow properties of this event further support this conclusion. If
this is the case, GRB 060218 can be considered as the prototype of a local sub-
energetic GRB class.
Based on simple considerations on co-rnoving volume and jet solid angle effects
on GRB detection probability as a function of redshift, it is found that the
detection of a close, weak and poorly collimated (as suggested by modeling of radio
data) event like GRB 060218 is consistent with the hypothesis that the rate and
jet opening angle distributions of local GRBs are similar to those of cosmological
GRBs. A correlation between jet opening angle and luminosity can explain the lack
of detection of local bright GRBs and of distant, weakly collimated events. If this is
the case, the occurrence rate of GRBs may be as high as ^1000 GRBs Gpc-3 yr_1,
both in the local Universe and at high redshift. Finally, all GRB/SN events are
consistent with the EPti - Eiso correlation, except for GRB 980425 and GRB 031203.
However, the first event is so close that an off-axis detection is possible, whereas for
the latter there are observational indications that the Ep\ value could be consistent
with the correlation. The consistency of GRB/SN events with the EPi[ - E[so
correlation, combined with energy budget considerations and their location in the -Eiso
~ Lxw diagram, show that the emission properties of long GRBs do not depend
on the properties of the associated SN.10 No clear evidence of correlation is found
between GRB and SN properties; in particular, all GRB/SN events seem to cluster
in the Ep\ - SN peak magnitude plane, with the only exception of GRB 060218.10
References
1. L. Amati, F. Frontera, M. Tavani et al., A&A 390, 81 (2002).
2. L. Amati, MNRAS 372, 233 (2006).
3. D. Band and R. Preece, ApJ 627, 319 (2005).
4. G. Ghirlanda, G. Ghisellini, C. Firmani, A. Celotti, Z. Bosnjak MNRAS 360, L45
(2005).
5. L. Amati, astro-ph/0611189
6. S.Y. Sazonov, A.A. Lutovinov, R.A. Sunyaev, Nature 430, 646 (2004).
7. R. Yamazaki, D. Yonetoku and T. Nakamura, ApJ 594, L79 (2003).
8. G. Ghisellini, G. Ghirlanda, S. Mereghetti, Z. Bosnjak, F. Tavecchio and C. Firmani,
MNRAS 372, 1699 82006).
9. A.M. Soderberg, S.R. Kulkarni, E. Nakar, et al, Nature 442, 1014 (2006).
10. L. Amati, M. Delia Valle, F. Frontera et al, A&A 463, 913 (2007)
THE GRB DETECTED BY AVS-F APPARATUS ONBOARD
CORONAS-F SATELLITE IN 2001-2005 YEARS
IRENE V. ARKHANGELSKAJA, ANDREY I. ARKHANGELSKIY,
ALEXANDER S. GLYANENKO, YURI D. KOTOV.
Moscow Engineering Physics Institute (State University), Kashirskoe shosse, 31
Moscow, 115409, Russia
[SERGEY N. KUZNETSOV|.
Scobeltsyn Institute of Nuclear Physics of Moscow State University, Vorobjevi Gori
Moscow, 119992, Russia
The AVS-F apparatus onboard CORONAS-F satellite operated from 31.07.2001 up to 06.12.2005.
This instrument constitutes the system for data processing from two detectors: SONG-D (CsI(Tl)
detector 0200 mm and 100 mm height, fully surrounded by plastic anticoincidence shield) and XSS-
1 (CdTe detector 4.9 mm x 4.9 mm). Despite of this satellite was Solar-oriented, over 30 GRB
during August 2001 - December 2005 period were registered in the energy band of-0.1-20 MeV by
preliminary data analysis. The characteristics of GRB detected by AVS-F device are discussed.
1 Apparatus
The AVS-F (amplitude-time Sun spectrometry) apparatus [1,2] was installed onboard
CORONAS-F satellite. Instrumentation is intended to study characteristics of fluxes of
hard X-rays, y-rays and neutrons from the Sun and solar flares and to detect other non-
stationary fluxes of cosmic y-rays. CORONAS-F was the second special-purpose
automatic station within frameworks of the CORONAS (Complex ORbiting
ObservatioNs of the Active Sun) international project. NORAD catalog number of this
satellite was 26873 and International Designator was 2001-032A. It had been launched
from Russian kosmodrom Plesetsk at 11:00 UT of 31 July 2001 by Cyclone-3 satellite-
launching rocket into a circular orbit oriented towards the Sun with inclination 82.5°,
altitude ~ 500 km and period ~ 90 min. CORONAS-F finished its operation of 6
December 2005. At the latest period of operation the altitude of orbit was approximately
270 km. Despite of the satellite was Solar-oriented, over 30 GRB during August 2001 -
December 2005 period were registered by AVS-F apparatus by the results of preliminary
data analysis.
The AVS-F apparatus was an electronic system for data treatment using signals
produced by the SONG-D (CsI(Tl) crystal 20 cm in diameter and 10 cm height), XSS-1
(CdTe with size 4,9 mm by 4,9 mm for X-ray analysis in 3-30 keV energy range) detectors
and the anticoincidence signal generated by the plastic scintillation counter of the SONG-
D. We use CsI(Tl) crystal because there are two light-output components in this crystal
with different fluorescence decay times Tfasi ~ 0.5-0.7 fis and rsto„, ~ 7 fjs which allows to
recognize gamma-rays and neutrons in this type detectors by pulse shape discrimination.
The energy resolution of the system was 13.0% for y-quanta from l37Cs with energy
0.662 MeV. There are two energy bands for y-emission registration in SOND-D detector.
1968
1969
The low energy band was 0.1 -11 MeV and high energy band was 4-94 MeV by first year
calibration data. The detector threshold and amplification coefficient were changed
approximately on 1 percent per month. At the last period of apparatus operation they
were -0.1-22 MeV and 2-260 MeV respectively.
2 Results
More than 30 GRB during August 2001 - December 2005 period were registered by
AVS-F apparatus by the preliminary data analysis. One of such bursts is GRB021008.
The temporal profiles of this burst in different energy bands by RHESSI and AVS-F data
are presented at Figure la.
For some GRB AVS-F detected y-emission in high energy band within RHESSI,
HETE and SWIFT t90 intervals. One example of such GRB was GRB050525. It was
detected by RHESSI at 00:49:50 UT. The temporal profiles of this burst according to
RHESSI and AVS-F data in different energy bands are shown at Figure lb.
10 20 30 40
time since 25.05.05 00:49:40.0, s
Figure 1. Temporal profiles of GRB021008 and GRB050525 in different energy bands.
The duration of this GRB was approximately 13 seconds according to RHESSI data,
15 seconds in low energy AVS-F y-band and more than 20 seconds in high energy AVS-
F y-band. As you can see, these profiles are different in high and low energy bands.
During this burst time of maximum in energy bands of 0.1-20 MeV and 25-1500 keV not
correspond to time of maximum in high energy band. There are 4 maxima in temporal
profiles of this burst on RHESSI data. Fourth maximum is the biggest in low energy band
1970
but high energy maximum corresponds to third one. The shifts of maxima times is about
7 seconds, so high energy maximum was registered earlier than low energy one.
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Figure 2. Energy spectra of GRB021008 and GRB050525.
The summarized energy spectra of GRB021008 and GRB050525 by AVS-F data are
presented at Figure 2. Spectra are smooth without any spectral features by preliminary
data analysis which correspond to RHESSI results in 100-700 keV energy band.
Maximum energy of gamma-emission detected by AVS-F apparatus during GRB050525
is approximately 147 MeV. Unfortunately there are no redshift measurements for this
GRB with high energy emission on AVS-F data [4].
3 Conclusion
More than 30 GRB during August 2001 - December 2005 period were registered by
AVS-F apparatus by the preliminary data analysis. For some GRB AVS-F detected y-
emission in high energy band. In some cases the temporal profiles of GRB in low and
high energy bands are similar but in some cases they are different. The moments of
origin of y-emission in low and high energy bands behave in the same manner. Now we
continue the processing of the data from AVS-F apparatus and use the results obtained
for preparation of the next stage of CORONAS project - CORONAS-PHOTON that will
be started at the end of 2008.
References
1. A.I. Arkhangelsky, A.S. Glyanenko, Yu.D. Kotov, et al., Instruments an Experimental
Techniques, Moscow, 1999, vol. 42., No. 5, pp. 596-603.
2. Yu.D. Kotov, S.V. Bogovalov, A.S. Glyanenko, et al., Comprehensive Studies of the
Sun and Solar-Earth Links, Leningrad, LFTI, 1989, pp. 130-159.
3. http://grb.web.psi.ch/grb_list_2005.html
4. http://www.mpe.mpg.de/~jcg/grbgen.html
SPECIAL RELATIVISTIC SIMULATIONS OF MAGNETO-DRIVEN
JET FROM CORE-COLLAPSE SUPERNOVAE*
TOMOYA TAKIWAKI
Department of Physics, School of Science, the University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan, takiwaki@utap.phys.s.u-tokyo.ac.jp
KEI KOTAKE
National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan
SHOICHI YAMADA
Science & Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo, 169-8555, Japan
KATSUHIKO SATO
Department of Physics, School of Science, the University of Tokyo,
7-3-1 Hongo, Bunkyo-ku,Tokyo 113-0033, Japan
We performed 2.5 dimensional numerical simulations of magnetized rotational core-
collapse. From the observational fact, it is probable that gamma-ray bursts and core-
collapse supernovae have same origin. And the very collimated jet is preferable to explain
the light curve of their afterglow. How this collimated jet is produced due to the process
of core-collapse of massive star? There are two main energy-sources for this explosion.
One is neutrino pair annihilations and another is magneto-hydrodynamical process. In
this time we concentrate the magneto-hydrodynamical process. We found a jet-like shock
wave is launched in the direction of the pole. The strength of initial magnetic field
determine the profile of jet. We found common structures of these jets. We continue the
simulation that the jet-like shock wave produced at the center propagate to the surface
of the star. Since the previous works are performed separately at each stage such as core-
collapse and jet-propagation, many initial uncertainty remains in each stage. However we
have performed the simulations from the magnetized core-collapse to the jet-propagation
totally. This study gives us deeper knowledge on the relation between the profile of the
jet and the progenitor.
1. Introduction
Some of long durational GRBs and supernovae have common origin. This is
supported by the observational evidences. For example, the spectrum of GRB030329's
afterglow is similar to the spectrum of SNIQOSbw.1 The study of SNe is
important to understand GRBs. For the theoretical model of GRBs, fire ball model is
promising. In this model, rapidly rotating massive star becomes BH and disk-like
objects and the rotational energy of disk-like objects powers GRBs.relativistic shells
ejected from a compact source radiates GRBs. Therefore rapidly rotating massive
stars are favorable for central compact object. In this time we concentrate the
magneto-hydrodynamical process for the mechanism powers GRBs.
"This research has been partially supported by Grants-in-Aid for the Scientific Research, from the
Ministry of Education, Science and Culture of. Japan (No.S14102004, No.14079202).
1971
1972
2. Method
We use numerical simulation to clarify this topic. The novel point from our previous
work2 is the special relativistic formulation. The simulations of the magneto-driven
jet encounter a problem. In the jet, the magnetic fields are strong and the matter-
density becomes dilute.In that region alfven speed exceeds the speed of light. Total
energy density should be accounted on the inertia. Our computational method is
similar to that of De Villiers et al.3 However we use equation of state by Shen et al.
that is calculated from nuclear physics.4 The use of this realistic equation of state
enables us the computation of neutrino cooling rate.
For the initial models, density, internal energy and electron fraction are imported
from 25M© rotational progenitor.5 For the rotation, we assume cylindrical rotation
following this formula. Inner regions are rapidly rotating and outer region rotate
slowly and the magnetic field we assume poloidal one. It consists of two parts
constant region and dipole region. It smoothly connects at 2000km. In this paper
central angular velocity is fixed to 70 radian per second. And change the strength of
the central magnetic field from 1010 to 1012 to investigate the role of the magnetic
fields.
3. Result
Our result consist of two parts. One is Effect of initial magnetic field on the
generation of the jet and another is how the generated jet penetrate the whole star.
We begin with weak magnetic model. Initial central magnetic fields are 1010G.
In this model, We found oscillation again and again after bounce. After that very
collimated jet is found. On the other hand in the strong magnetic model: 1011 —
1012G, the feature is that the collimated jet is launched just after the bounce.
Our result are shown in Table 1.
Table 1. Comparison on jet-profiles in various magnetic models.
Explosion Energy[1050 erg] and time from core-bounce is measured when jet
reaches 1000km from the center.
Initial Time from Radial Explosion Collimation
Magnetic Fields core-bounce velocity Energy Angle
(G) (ms) (cm/s) (1050ergs) (°)
1010 172 5.5 x 109 0.094 6.7
10u 102 8.0 x 109 0.23 7.1
1012 95 8.0 x 109 1.4 7.1
Strong initial magnetic fields makes prompt explosion. On the other hands Weak
initial magnetic field require long duration for amplify the magnetic field. Strong
magnetic field makes relatively strong shock. The toroidal magnetic fields are very
1973
similar among these models. It reflects the almost same velocity of the shock. In
weak magnetic model strong toroidal magnetic field is confined to the rotational
axis. That reflects the difference on the explosion energy.
Our results above are in the 2000km, we next show the result that jet
penetrate star. The whole star is 5.45MQ, this star does not have Hydrogen envelope.
Therefore it is Wolf-Rayet star.
7 sec after bounce shock reaches surface of the star. Mass ejection rate is 0.03
solar mass per second. And its explosion energy is 0.6 x 1050ergs. The velocity does
not change so much because the very dense nature prevent them accelerating by the
pressures. At surface we found density, p = 100g/cm3 and velocity, v = 5x 109cm/s
Here we show the various pressure in Figure 1. We found the kinetic energy is
dominate in the shock and next magnetic pressure, is strong gas pressure is most
weak. However in the tips of the jet, hot spot, gas pressure and kinetic energy is
comparable.
1e+26
1e+24 -
1e+22 -—-
le+20 •
£. 1e+18
1e+16 •
1e+14 -
1e+12 •
1e+10
-
_
l*-4 •
P B •
kin/3
P
•V "-•■
'
-V-.
V; '
""•
1e+09 2e+09 3e+09 4e+09 5e+09 6e+09 7e+09 8e+09 9e+09 1e+10
radius[cm]
Fig. 1. The matter and magnetic pressures: This figure shows magnetic pressure and matter
pressure and one third of kinetic energy. Three time sequential data arc included in this file and
the data at 472ms, 1150ms and 1540ins after bounce coresspond left to right.
References
1. K. Z. Stanek, T. Matheson, P. M. Garnavich, P. Martini, P. Berlind, N. Caldwell,
P. Challis, W. R. Brown, R. Schild, K. Krisciunas, M. L. Calkins, J. C. Lee, N. Hathi,
R. A. Jansen, R. Windhorst, L. Echevarria, D. J. Eisenstein, B. Pindor, E. W.
Olszewski, P. Harding, S. T. Holland and D. Bersier, ApJ Letter 591, L17(July 2003).
2. T. Takiwaki, K. Kotake, S. Nagataki and K. Sato, ApJ 616, 1086(December 2004).
3. J.-P. De Villiers, J. F. Hawley and J. H. Krolik, ^pJ599, 1238(December 2003).
4. H. Shen, H. Toki, K. Oyainatsu and K. Sumiyoshi, Nuclear Physics A 637, 435(July
1998).
5. A. Heger and N. Langer, ApJ 544, 1016(December 2000).
THEORETICAL INTERPRETATION OF "LONG"
AND "SHORT" GRBs
C.L. BIANCO,1'2 M.G. BERNARDINI,1'2 L.CAITO,1.2 P. CHARDONNET,1*4 M.G.
DAINOTTI,1-2 F. FRASCHETTI,5 R. GUIDA,1-2 R. RUFFINI1'2'3 and S.-S. XUE1
1 ICRANet and ICRA, Piazzale della Repubblica 10, 1-65122 Pescara, Italy
2 Dip. di Fisica, Universita dt Roma "La Sapienza", Piazzale Aldo Moro 5, 1-00185 Roma, Italy
3 ICRANet, Universite de Nice Sophia Antipolis, Grand Chateau, BP 2135, 28, avenue de
Valrose, 06103 NICE CEDEX 2, Prance
4 Universite de Savoie, LAPTH - LAPP, BP 110, F-74941 Annecy-le-Vieux Cedex, France
5 CEA/Saclay, F-91191 Gif-sur-Yvette Cedex, Saclay, France
E-mails: bianco@icra.it, maria.bernardini@icra.it, letizia.caito@icra.it, chardon@lapp.in2p3.fr,
dainotti@icra.it, fraschetti@icra.it, roberto.guida@icra.it, ruffini@icra.it, xue@icra.it.
Within the "fireshell" model we define a "canonical GRB" light curve with two sharply
different components: the Proper-GRB (P-GRB), emitted when the optically thick
fireshell of electron-positron plasma originating the phenomenon reaches transparency,
and the afterglow, emitted due to the collision between the remaining optically thin
fireshell and the CircumBurst Medium (CBM). We here present the consequences of
such a scenario on the theoretical interpretation of the nature of "long" and "short"
GRBs.
1. Introduction
We assume that all GRBs, both "long" and "short", originate from the gravitational
collapse to a black hole.1'2 The e± plasma created in the process of the black hole
formation expands as an optically thick and spherically symmetric "fireshell" with
a constant width in the laboratory frame, i.e. the frame in which the black hole is at
rest.3 We have only two free parameters characterizing the source: the total energy
of the e± plasma E1^ and the e± plasma baryon loading B = MBc2/E1*"*, where Mb
is the total baryons' mass.4 These two parameters fully determine the optically thick
acceleration phase of the fireshell, which lasts until the transparency condition is
reached and the Proper-GRB (P-GRB) is emitted.1'2 The afterglow emission then
starts due to the collision between the remaining optically thin fireshell and the
CircumBurst Medium (CBM).1,2,5-7 It clearly depends on the parameters describing
the effective CBM distribution: its density ncbm and the ratio 1Z = Aeff/Avis
between the effective emitting area of the fireshell Aeff and its total visible area
A • 8~n
2. The "canonical GRB" scenario
Unlike treatments in the current literature,15'16 we define a "canonical GRB" light
curve with two sharply different components (see Fig. 1 and Refs. 1,2,11,14,17):
• The P-GRB, which has the imprint of the black hole formation, an harder
spectrum and no spectral lag;18'19
• the afterglow, which presents a clear hard-to-soft behavior;9'20'21 the peak
of the afterglow contributes to what is usually called the "prompt
emission".111-21
1974
1975
DeteciOf amusllime ll^lis) g
Fig. 1. Left: The "canonical GRB" light curve theoretically computed for GRB 991216. The
prompt emission observed by BATSE is identified with the peak of the afterglow, while the small
precursor is identified with the P-GRB. For this source we have B ~ 3.0 X lo-3.1'8'12,13 Right:
The energy radiated in the P-GRB (solid line) and in the afterglow (dashed line), in units of the
total energy of the plasma (Etc^), are plotted as functions of the B parameter. Also represented
are the values of the B parameter computed for GRB 991216, GRB 030329, GRB 980425, GRB
970228, GRB 050315, GRB 031203, GRB 060218. Remarkably, they are consistently smaller than,
or equal to in the special case of GRB 060218, the absolute upper limit B < 10~2.4 The "genuine"
short GRBs have a P-GRB predominant over the afterglow: they occur for B < 10-5.114
The ratio between the total time-integrated luminosity of the P-GRB (namely, its
total energy) and the corresponding one of the afterglow is the crucial quantity for
the identification of GRBs' nature. Such a ratio, as well as the temporal separation
between the corresponding peaks, is a function of the B parameter (see Fig. 1 and
Ref. 1).
When B < 10~5, the P-GRB is the leading contribution to the emission and the
afterglow is negligible: we have a "genuine" short GRB.1 When 10~4 < B < 10~2,
instead, the afterglow contribution is generally predominant. Still, this case presents
two distinct possibilities: the afterglow peak luminosity can be either larger or
smaller than the P-GRB one.14'17 The simultaneous occurrence of an afterglow
with total time-integrated luminosity larger than the P-GRB one, but with a smaller
peak luminosity, can indeed be explained in terms of a peculiarly small average value
of the CBM density (nc&m ~ 10~3 particles/cm3), compatible with a galactic halo
environment ("fake" short GRBs).14'17
References
1. R. Ruffim, C. L. Bianco, F. Fraschetti, S.-S. Xue and P. Chardonnet, Astrophysical
Journal 555, L113 (July 2001).
1976
2. R. RufRni, M. G. Bernardini, C. L. Bianco, L. Caito, P. Chardonnet, M. G. Dainotti,
F. Fraschetti, R. Guida, M. Rotondo, G. Vereshchagin, L. Vitagliano and S.-S. Xue,
The blackholic energy and the canonical gamma-ray burst, in Xllth Brazilian School
of Cosmology and Gravitation, eds. M. Novello and S. E. Perez Bergliaffa, American
Institute of Physics Conference Series, Vol. 910 (June 20D7).
,3. R. RufRni, J. D. Salmonson, J. R. Wilson and S.-S. Xue, Astronomy & Astrophysics
350, 334 (October 1999).
4. R. RufRni, J. D. Salmonson, J. R. Wilson and S.-S. Xue, Astronomy & Astrophysics
359, 855 (July 2000).
5. C. L. Bianco and R. RufRni, Astrophysical Journal 605, LI (April 2004).
6. C. L. Bianco and R. RufRni, Astrophysical Journal 620, L23 (February 2005).
7. C. L. Bianco and R. RufRni, Astrophysical Journal 633, L13 (November 2005).
8. R. RufRni, C. L. Bianco, P. Chardonnet, F. Fraschetti and S.-S. Xue, Astrophysical
Journal 581, L19 (December 2002).
9. R. RufRni, C. L. Bianco, S.-S. Xue, P. Chardonnet, F. Fraschetti and V. Gurzadyan,
International Journal of Modern Physics D 13, 843 (2004).
10. R. RufRni, C. L. Bianco, S.-S. Xue, P. Chardonnet, F. Fraschetti and V. Gurzadyan,
International Journal of Modern Physics D 14, 97 (2005).
11. M. G. Dainotti, M. G. Bernardini, C. L. Bianco, L. Caito, R. Guida and R. RufRni,
Astronomy & Astrophysics 471, L29 (August 2007).
12. R. RufRni, C. L. Bianco, P. Chardonnet, F. Fraschetti, L. Vitagliano and S.-S. Xue,
New perspectives in physics and astrophysics from the theoretical understanding of
gamma-ray bursts, in Cosmology and Gravitation, eds. M. Novello and S. E. Perez
Bergliaffa, American Institute of Physics Conference Series, Vol. 668 (June 2003).
13. R. RufRni, M. G. Bernardini, C. L. Bianco, P. Chardonnet, F. Fraschetti,
V. Gurzadyan, L. Vitagliano and S.-S. Xue, The blackholic energy: long and short
gamma-ray bursts (new perspectives in physics and astrophysics from the theoretical
understanding of gamma-ray bursts, ii), in Xlth Brazilian School of Cosmology and
Gravitation, eds. M. Novello and S. E. Perez Bergliaffa, American Institute of Physics
Conference Series, Vol. 782 (August 2005).
14. M. G. Bernardini, C. L. Bianco, L. Caito, M. G. Dainotti, R. Guida and R. RufRni,
Astronomy & Astrophysics 474, L13 (October 2007).
15. T. Piran, Reviews of Modern Physics 76, 1143 (January 2005).
16. P. Meszaros, Reports of Progress in Physics 69, 2259 (2006).
17. C. L. Bianco, M. G. Bernardini, L. Caito, M. G. Dainotti, R. Guida and R. RufRni,
The "fireshell" model and the "canonical" grb scenario, in Relativistic Astrophysics,
eds. C. L. Bianco and S. S. Xue, American Institute of Physics Conference Series,
Vol. 966 (2007).
18. C. L. Bianco, R. RufRni and S.-S. Xue, Astronomy & Astrophysics 368, 377 (March
2001).
19. R. RufRni, F. Fraschetti, L. Vitagliano and S.-S. Xue, International Journal of Modern
Physics D 14, 131 (2005).
20. M. G. Bernardini, C. L. Bianco, P. Chardonnet, F. Fraschetti, R. RufRni and S.-S.
Xue, Astrophysical Journal 634, L29 (November 2005).
21. R. RufRni, M. G. Bernardini, C. L. Bianco, P. Chardonnet, F. Fraschetti, R. Guida
and S.-S. Xue, Astrophysical Journal 645, L109 (July 2006).
THEORETICAL INTERPRETATION OF GRB 011121
CAITO LETIZIA*, BERNARDINI MARIA GRAZIA, BIANCO CARLO LUCIANO,
DAINOTTI MARIA GIOVANNA, GUIDA ROBERTO and RUFFINI REMO
Dipartimento di Fisica, Universita di Roma "La Sapienza"
Roma, 1-00185, Italy
* letizia. caito@icra.it
ICRANet and ICRA, Piazzale della Repubblica 10
Pescara, 1-65122, Italy
GRB 011121, detected by the BeppoSAX satellite,1 is studied as a prototype to
understand the presence of flares observed by Swift in the afterglow of many GRB sources.
Detailed theoretical analysis of the GRB 011121 light curves in selected energy bands are
presented and compared with observational data. An interpretation of the flare of this
source is provided by the introduction of the three-dimensional structure of the Circum
Burst Medium(CBM).
Keywords: Gamma-Ray Bursts, Flares
1. Introduction
GRB 011121 is a near, long burst with T90 = 28 s and redshift z = 0.36.2 Its fluence3
is 2.4 x 10~5 erg/cm2 that corresponds, in the hypothesis of isotropic emission at
the observed redshift, to an energy in the band 2 — 700 keV of 2.8 x 1052 erg. This
is the second brightest source detected by BeppoSAX in 7-rays and X-rays. A weak
X-Ray precursor has been observed for this burst about thirty seconds before the
Gamma emission.4 The presence of a red bump in the optical observations, in the
time of the second week after the Prompt-emission, is very probably linked with the
explosion of a supernova almost at the same time of the burst (SN 2001ke5). This
issue is still an open discussion.
At the time t = 240 s, in the X-ray 2 — 26 keV energy band, there is a big
flare.4'6 It lasts about seventy seconds and corresponds to a bump of an order of
magnitude in luminosity. It is however very soft, since its energy is about 3% of the
total amount of the prompt emission4 .
In this work we show how it is possible to reproduce the flare just considering the
distribution of the Circum Burst Medium (CBM) in its three-dimensional structure.
2. The fit of the GRB 011121 observed luminosity and the
interpretation of the flare
In figure 1 we present the observed GRB 011121 light curves in the three different
energy bands we analyzed, together with their theoretical fit in the framework of our
model (see refs.7"12 and references therein): 40 - 700 keV, 2-26 keV, 2-10 keV.
Looking at the observational data we can see that the 40 — 700 keV energy band light
curve presents a temporal profile particularly regular, smooth and homogeneous,
while the 2 — 26 keV light curve has a remarkably irregular profile. This is quite
1977
1978
? 10*
I ,o*
1 104(
I,
r
;
„
: / /
v; /
\/
\
i ..,i
z
s
/
/\
-~V\
\
\.,
-i^J-
fl
•i
-!
1
i ' *i
40
',
1
L
,
QRBM
WFC • /
NFI ( •*-■ . :
-700 keV (QRBM) - —
2-26 keV (WFC -
2-10 keV (NFI) ;
\
■
t 1
I ,
, \* .!..]
i0d 10J 10* 10°
etector arrival time (s)
\ QRBM
\ WFC •,
\ 40-700 keV (QRBM)
2-26 keV (WFC)
1
Detector arrival time (s)
Fig. 1. Left: Theoretical fit of the GRB 011121 light curves in the 40 - 700 keV (BeppoSAX
GRBM), 2 - 26 keV (BeppoSAX WFC), 2/10 keV (BeppoSAX NFI). Bight: Enlarginent of the
Flare.
anomalous, in fact generally the light curves in these energy bands presents just the
opposite trend.
We recall that, in our model, the entire emission phase (from the gamma 'prompt
emission' to the lower energies), with its flares and its peculiar features, is due to the
interaction with the CBM of the shell of baryous accelerated during the optically
thick plasma expansion. This implies a strong dependence of the theoretical curves
to the value of the baryon loading parameter ( B = Mbc2/Etot ). As B increases
1979
the peak energy becomes lower, the spikes in the light curve become sharper and
the emission prolonged. The very similar behavior observed in morphology, duration
and distribution for the X-ray flares13,14 confirms the above hypothesis of the same
origin of the 'prompt emission' and the late afterglow phase. The flare phenomenon
is still an open issue and many hypothesis have been put forward to give an
interpretation of it. The most popular one is the origin of flares from a central engine
activity15-18 , in particular models involving Late Internal Shocks19-21 or Delayed
External Shocks4'22 produced by the long duration of activity or by a re-activation
of it. Other models assert the presence of a short duration central engine activity to
produce Late Internal Shock23 and many different mechanism to produce flares like
refreshed shock24 , Inverse Compton scattering25 or give a clou value to the
curvature effect of the fireball26 . In a different way, an interpretation of flares is provided
also within the hyperaccretion model for GRBs27 and the dust scattering-driven
emission model28 .
In figure 1 there is an enlargement of the flare of this source that shows in detail
the comparison between the theoretical light curve and the observational data.
In the computation of the theoretical light curve for the flare we reproduce it
as due to a spherical cloud of CBM along the line of sight introducing, in this way,
a three-dimensional structure for the Circum Burst Medium. In fact, in the first
approximation, we assume a modeling of thin spherical shells for the distribution
of the CBM. This allows us to consider a purely radial profile in the expansion.7'8
This radial approximation is valid until the visible area of emission of photons
is sufficiently small with respect to the characteristic size of the CBM shell. The
visiblre area of emission is defined by the maximum value of the viewing angle; it
varies with time and is inversely proportional to the Lorentz Gamma Factor. So it
happens that, at the beginning of the expansion, when the Gamma Factor is big
(about 102), the effective distribution of the CBM doesn't matter for the narrowness
of the viewing angle but, at the end of the expansion, the remarkable lessening of
the Gamma Factor produces a strong increase of the viewing angle and a correct
estimation of the CBM by the introduction of the angular coordinate distribution
becomes necessary.
We can see that our results are in very good agreement with the observational
data, also in the late tail of the flare.
Here we performed just a first attempt of interpretation of flares within our
inelastic collisions modeling and we found an encouraging result. Now we plan to
verify our hipothesys by its application to other sources and to produce a detailed
cinematic and dinamic theory concerning this fondamental features of Gamma-Ray
Burst.
References
1. Piro L., GCN Circ, 1147 (2001).
2. Infante L., Garnavich P.M., Stanek K.Z. and Wyrzykokowski L., GCN Circ, 1152
(2001).
1980
3. Price P.A. et al., Astrophys. J. Lett., 572 (2002) L51.
4. Piro L. et al., Astrophys. J., 623 (2005) 314.
5. J. S. Bloom et al., Astrophys. J., 572 (2002) L45-L49
6. Greiner J. et al., Astrophys. J., 599 (2003) 1223.
7. Ruffini R., Bianco C.L., Chardonnet P., Fraschetti F. and Xue S.-S., Astrophys. J.
Lett., 581 (2002) L19.
8. Ruffini R., Bianco C.L., Chardonnet P., Fraschetti F., Vitagliano L. and Xue S.-S.,
in Cosmology and Gravitation: Xth Brazilian School of Cosmology and Gravitation;
25th Anniversary (1977-2002), edited by Novello M. and Perez Bergliaffa S.E., AIP
Conf. Proc, 668 (2003) 16.
9. Ruffini R., Bianco C.L., Chardonnet P., Fraschetti F., Gurzadyan V. and Xue S.-S.,
Int. J. Mod. Phys. D, 13 (2004) 843.
10. Ruffini R., Bianco C.L., Chardonnet P., Fraschetti F., Gurzadyan V. and Xue S.-S.,
Int. J. Mod. Phys. D, 14 (2005) 97.
11. Bianco C.L. and Ruffini R., Astrophys. J. Lett., 620 (2005) L23.
12. Bianco C.L. and Ruffini R., Astrophys. J. Lett., 633 (2005) L13.
13. Chincarini G. et al., Astrophys. J. preprint doi:10.1086/'521591'.
14. Falcone A. D. et al., accepted by Apj (2007).
15. O'Brien P.T. et al., New J. Phys. 8 (2006) 121.
16. Pagani C. et al., Astrophys. J. 645 (2006) 1315-1322.
17. Falcone A.D. et al., to appear in the proceeding of the 16th Annual October
Astrophysics Conference in Maryland '"Gamma Ray Bursts in the Swift Era'" (2006).
18. Zhang B., (2006), to appear in "16th Annual October Astrophysics Conference in
Maryland", AIP Conf.Procs
19. Burrows D.N. et al., (2005a), Science, 309, 1833.
20. Wu X.F. et al., 36th cOSPAR Scientific Assembly, p.731.
21. Galli A. and Guetta D., (2007) A&A, submitted.
22. Galli A. and Piro L., (2007), A&A, in press.
23. Zhang B. et al., (2006), Apj, 642, 354.
24. Guetta G. et al., (2007), A&A, 461, 95-101.
25. Xiang-Yu W. et al., (2006), Astrophys.J., 641, L89-L92.
26. Liang E.W. et al., (2006) Astrophys.J., 646, 351-357.
27. Proga D. and Zhang B., (2006), Mon.Not.Roy.Astron.Soc.Lett., 370, L61-L65.
28. Shao L. and Dai Z.G., (2007), ApJ, in press.
ON GRB 060218 AND THE GRBs RELATED
TO SUPERNOVAE Ib/c
DAINOTTI MARIA GIOVANNA, BERNARDINI MARIA GRAZIA, BIANCO CARLO
LUCIANO, CAITO LETIZIA, GUIDA ROBERTO and RUFFINI REMO
ICRANet and ICRA, Piazzale della Repubblica 10, 1-65100 Pescara, Italy
E-mails: ruffini@icra.it, xue@icra.it, fraschetti@icra.it
Dipartimento di Fisica, Universita di Roma "La Sapienza", Piazzale Aldo Moro 5, 1-00185
Roma, Italy
E-mails: maria.bernardini@icra.it, bianco@icra.it, dainotti@icra.it, roberto.guida@icra.it
Universite de Savoie, LAPTH-LAPP, BP 110, F-74941 Annecy-le-Vieux Cedex, France
E-mail: chardon@lapp.in2p3.fr
We study the Gamma-Ray Burst (GRB) 060218: a particularly close source at z = 0.033
with an extremely long duration, namely Tg0 ~ 2000 s, related to SN 2006aj. This
source appears to be a very soft burst, with a peak in the spectrum at 4.9 keV, therefore
interpreted as an X-Ray Flash (XRF) and it obeys to the Amati relation. We fit the X-
and 7-ray observations by Swift of GRB 060218 in the 0.1-150 keV energy band during
the entire time of observations from 0 all the way to 106 s within a unified theoretical
model. The details of our theoretical analysis have been recently published in a series
of articles. The free parameters of the theory are only three, namely the total energy
E^ of the e^ plasma, its baryon loading B = Mbc2/Ele°^, as well as the CircumBurst
Medium (CBM) distribution. We fit the entire light curve, including the prompt emission
as an essential part of the afterglow. We recall that this value of the B parameter is the
highest among the sources we have analyzed and it is very close to its absolute upper limit
expected. We successfully make definite predictions about the spectral distribution in the
early part of the light curve, exactly we derive the instantaneous photon number spectrum
N(E) and we show that although the spectrum in the co-moving frame of the expanding
pulse is thermal, the shape of the final spectrum in the laboratory frame is clearly non
thermal. In fact each single instantaneous spectrum is the result of an integration of
thousands of thermal spectra over the corresponding EQuiTemporal Surfaces (EQTS).
By our fit we show that there is no basic differences between XRFs and more general
GRBs. They all originate from the collapse process to a black hole and their difference is
due to the variability of the three basic parameters within the range of full applicability
of the theory.
1. Introduction
GRB 060218, discovered by the Swift satellite13 with cosmological redshift z =
0.033,18 is one of the best examples of very long duration GRBs associated with
core collapse Supernovae.9 We present a detailed fit of the X- and 7-ray luminosity
in the entire time and energy band of observation, as well as details on the spectral
distribution during the prompt emission phase. The additional peculiarities of the
source evidence the lowest value of the Circumburst Medium (CBM) number density
as well as the highest value of the baryon loading parameter yet observed in a GRB
source. In this sense this source explores the applicability of GRB models in a yet
untested range of physical parameters.
We present the observational data of the source and the theoretical fit of the light
1981
1982
curves observed by BAT and XRT (15-150 keV and 0.1-10.0 keV respectively. The
latest Chandra observations evidence possible analogies to the class of faint GRBs
at low value of cosmological redshift, see Fig. 1). We show in Fig. 2 the predicted
instantaneous spectrum from 100 s (i.e. during the so called "prompt emission") all
the way up to about 103 s (i.e. until the gamma peak ends).
We emphasize the aspects which makes this source so special: i) the total energy
El± = 2.32 x 1050 erg, ii) the baryon loading parameter B = Mbc2/-E*± = 1.0 x
10"2.
In our approach we assume that all GRBs, short or long, originate from the
gravitational collapse to a black hole.27 We have only two free parameters describing the
source, namely the total energy El± of the e± plasma and its baryon loading B.28
They characterize the optically thick adiabatic acceleration phase of the GRB, which
lasts until the transparency condition is reached. After this acceleration phase, it
starts the afterglow emission, due to the collision between the accelerated baryonic
matter and the CBM. The CBM is described by two additional parameters: the
effective particle number density (ticbm) and the ratio between the effective
emitting area and the total area of the pulse, 1Z = Aeff/AviS23 which both takes into
account the CBM filamentary structure.24
2. GRB 060218-SN 2006aj
GRB 060218 has been triggered and located by the BAT instrument10 on board
of the Swift satellite on 18 February 2006. It has a very long duration with Tgo ~
(2100 ± 100)s. The XRT10 began observations ~ 153 s after the BAT trigger16 and
continued to detect the source for - 12.3 days.30 The source is characterized by a
flat gamma ray light curve and a soft spectrum.2 It has an X-ray light curve with a
long, slow rise and gradual decline and it is considered an X-ray flash since its peak
energy occurs at Ep = 4.9^3 keV.9 The burst fluence in the 15-150 keV band is
(6.8 ± 0.4) x 10~6 erg/cm2.30 The spectroscopic redshift has been found to be z =
0.033.18 At this redshift the isotropic equivalent energy is Eiso = (1.9 ± 0.1) x 1049
erg.30 This faint, low redshift GRB could be a good candidate to be associated
with a Supernova. In fact it has been found an underlying Type Ic Supernova:
SN2006aj.19 This Supernova shows observational features very similar to the other
ones associated with GRBs. In particular it has a very large expansion velocity of
v ~ 0.1c.11'19-31 This source obeys to the Amati relation.
3. The fit of the observed data
In this section we present the fit of our fireshell model to the observed data (see Fig.
1). The fit leads to a total energy of the e± plasma El°± = 2.32 x 1050 erg, with an
initial temperature T = 1.86 McV and a total number of pairs Ne± = 1.79 x 1055.
The second parameter of the theory, B — 1.0 x 10~2, is close to the limit for the
stability of the adiabatic optically thick acceleration phase of the fireshell (for further
details see28). The Lorentz gamma factor obtained solving the fireshell equations of
1983
to46 - Theoretical bolometric luminosity
Detector arrival time (tg) (s)
Fig. 1. GRB 060218 complete light curves: our theoretical fit (blue line) of the 15-150 keV BAT
observations (pink points), our theoretical fit (red line) of the 0.1-10 keV XRT observations (green
points) and the 0.1-10 keV Chandra observations (black points) are represented together with our
theoretically computed bolometric luminosity (black line) (data from9'32).
motion6'7 is 7o = 99.2 at the beginning of the afterglow phase at a distance from
the progenitor r0 = 7.82 x 1012 cm.
In Fig. 1 we show the afterglow light curves fitting the prompt emission both in
the BAT (15-150 keV) and in the XRT (0.3-10 keV) energy ranges, as expected
in our "canonical GRB" scenario (see Dainotti et al., in preparation).
Initially the two luminosities are comparable to each other, but for a detector arrival
time td > 1000 s the XRT curves becomes dominant. The displacement between
the peaks of these two light curves leads to a theoretically estimated spectral lag
greater than 500 s in perfect agreement with the observations.17
We recall that at td ~ 104 s there is a sudden enhancement in the radio
luminosity and there is an optical luminosity dominated by the SN2006aj emission.9'32
Although our analysis addresses only the BAT and XRT observations, for r > 1018
cm corresponding to td > 104 s the fit of the XRT data implies two new features:
1) a sudden increase of the ft factor from ft = 1.0 x lO"11 to ft = 1.6 x 10~6,
corresponding to a significantly more homogeneous effective CBM distribution; 2)
an XRT luminosity much smaller than the bolometric one (see Fig. 1). Therefore,
we identify two different regimes in the afterglow, one for tda < 104 s and the other
for td > 104 s. Nevertheless, there is a unifying feature: the determined effective
CBM density decreases with the distance r monotonically and continuously through
both these two regimes from ncbm = 1 particle/cm3 at r = rQ to ncbm = 10~6
particle/cm3 at r = 6.0 x 1018 cm: ncbm oc r~a, with 1.0 < a < 1.7.
4. The spectrum
A theoretical attempt to identify the physical process responsible for the afterglow
emission of Gamma Ray Bursts (GRBs) is presented, leading to the occurrence of
1984
thermal emission in the comoving frame of the shock wave giving rise to the bursts.
The determination of the luminosity and spectra involves integration of an infinite
number of Planckian spectra, weighted by appropriate relativistic transformations
each one corresponding to a different viewing angle in the last past cone of the
observer. The relativistic transformations have been computed using the equations
of motion of GRBs within our theory, giving special attention to the determination
of the equitemporal surfaces.
4.1. The Equitemporal surdaces (EQTS)
the "equitemporal surfaces" (EQTSs) are surfaces of revolution about the line of
sight for a relativistically expanding spherically symmetric source. The general
expression for their profile, in the form ■& = $(r), corresponding to an arrival time ta
of the photons at the detector, can be obtained from (see e.g.5'6'25 ):
cta = ct(r) — rcostf + r* , (1)
where r* is the initial size of the expanding source, -d is the angle between the radial
expansion velocity of a point on its surface and the line of sight, and t = t(r) is its
equation of motion, expressed in the laboratory frame, obtained by the integration
of Eqs.of the afterglow dynamics. From the definition of the Lorentz gamma factor
7~2 = 1 — (dr/cdt)2, we have in fact:
ct(r)= f [l-7-2(r')r1/2^', (2)
Jo
where j(r) comes from the integration of Eqs. of the afterglow dynamics.
4.2. Thermal spectrum in the comoving frame
We adopt three basic assumptions:23
• the resulting radiation as viewed in the comoving frame has a thermal
spectrum
• the CBM swept up by the front of the baryonic shell is responsible for this
thermal emission.
• the expansion occurs with spherical symmetry.
The choice of thermal spectrum is the only possibility if we rule out the existence
of the strong magnetic fields that produce the synchrotron emission. In fact such
emission requires highly magnetized outflows, but it is not clear how they can be
achieved.
In our case the radiation is produced in the inelastic collision between the
accelerated baryons and the ISM. The structure of the collision is determined by mass,
momentum and energy conservation. The only additional free parameter of our
model to model this emission process is the size of the "effective emitting area" of
the emitting shell: Aeff.
1985
The power emitted in the interaction of the baryonic shell with the CBM inho-
mogeneities measured in the comoving frame is:
^^ = 4Trr21ZaT4 , (3)
At w
where AEint is the internal energy developed in the collision with the CBM in the
co-moving frame, T is the black body temperature in the comoving frame, a is the
Stefan-Boltzmann constant and
^=~L (4)
is the "surface filling factor" which accounts for the fraction of the shell's surface
becoming active, being the ratio between the "effective emitting area" and the total
area Atot- The ratio TZ is a priori a function that varies as the system evolves so
it is evaluated at every given value of the radius r. We are now ready to evaluate
the source luminosity in a given energy band. The source luminosity at a detector
arrival time if, per unit solid angle dVt and in the energy band \v\, v-i\ is given by:25
d£?["i."2] r Ae „ ._, dt
dtidn -JEQTS^VC0Si)A~ ^W(^T-r)dZ, (5)
where Ae = AEint/V is the emitted energy density released in the comoving frame
assuming, for simplicity, that all the shell is emitting, A = 7(1 — (w/c) cost?) is the
Doppler factor, dS is the surface element of the EQTS at detector arrival time tda
on which the integration is performed and Tarr is the observed temperature of the
radiation emitted from dS:
A_1T
Tarr = (TT^y ■ (6)
The "effective weight" W (vi,i>2,Tarr) is given by the ratio of the integral over
the given energy band of a Planckian distribution at a temperature Tarr to the total
integral aT^rr:
W{vl)v2,Tarr) = -7^- [2 p(Tarr,v)d(^) , (7)
alarr •> V\ \ C /
where p(Tarr, v) is the Planckian distribution at temperature Tarr:
p(TQrr,^) = -expW(fc^r)_i (8)
Once we have the luminosity in a given energy band in the same way we can
evaluate the instantaneous and the time-integrated photon number spectrum.
4.3. The istantaneous spectrum
Following the above procedure, already described in,3 we derive the instantaneous
photon number spectrum N(E). In Fig. 2 are shown integrated photon number
spectra for selected time intervals covering the firsts 1000 s of the event, namely
1986
Energy (keV)
Fig. 2. Ten theoretically predicted photon number spectra N(E), each one time-integrated over
100 s, encompassing all the first 1000 s of the prompt emission (colored curves). The top black
and bold curve represents the theoretical spectrum time-integrated from 0s to 1000s.
during the 15-150 keV energy band emission, until the end of the afterglow's peak.
It is clear from this picture that, although the spectrum in the co-moving frame of
the expanding pulse is thermal, the shape of the final spectrum in the laboratory
frame is clearly non thermal. In fact, as explained in,23 each single instantaneous
spectrum is the result of an integration of thousands of thermal spectra over the
corresponding EQuiTemporal Surfaces (EQTS5,6). This calculation produces a non
thermal, instantaneous spectrum in the observer frame3 (see Fig.2). A distinguishing
feature of the GRB spectra which is also present in these instantaneous spectra is the
hard to soft transition during the evolution of the event.8'12,14 In fact, the peak of
the energy distribution Ep drifts monotonically to softer frequencies with time. This
feature explains the change in the power law low energy spectral index a,1 which
at the beginning of the prompt emission of the burst is a = 0.75 and progressively
decreases for later times. So the correlation between a and Ep8 is explicitly shown.
It is interesting that the spectrum corresponding to the 900-1000 s time interval
shows a marked enhancement in the 0.1-3 keV, which is a direct consequence of the
sharp variation of TZ and Ucbm discussed above.
4.4. The Band relation
The GRB observed spectrum appears to be non thermal and it usually varies
strongly from one burst to another. Nevertheless, an excellent phenomenological
fit for the spectrum was introduced using two power-laws joined smoothly at a
break energy (a ~ (3)E0:
(hv)at
7?,, L
hu<(a- l3)E0
N{v) = N0<" ' (9)
{ ({a - f3)E0Ya-P\hisfe{P-a\ hv > (a - f3)E0 .
1987
This function provides an excellent fit to most of the observed spectra but there is no
particular theoretical model that predicts this spectral shape. Within our treatment
it is not necessary to have a power-law spectrum in the comoving frame to obtain
an observed power-law spectrum as we have already explained.
5. Conclusions
GRB060218 presents a variety of peculiarities, including its extremely large Tgo and
its classification as an XRF. The anomalously long Tg0 led us to infer a monotonic
decrease in the CBM effective density. The spectrum from the prompt phase to
the early part of the afterglow varies smoothly and continuously with characteristic
hard to soft transition. What is impressive is that no different scenarios need to be
advocated in order to explain the GRB emission global behavior: both the prompt
and the afterglow emission are just due to the thermal radiation in the comoving
frame produced by inelastic collisions with the CBM duly boosted by the relativistic
transformations over the EQTSs. Our scenario originates from the gravitational
collapse to a black hole and is now confirmed over a 106 range in energy. It is clear
that, although the process of gravitational collapse is unique, there is a large variety
of progenitors which may lead to the formation of black holes, each one with precise
signatures in the energetics. The low energetics of the class of GRBs associated with
SNe, and the necessity of the occurrence of the SN, naturally leads in our model
to identify their progenitors with the formation of the smallest possible black hole
originating from a NS overcoming his critical mass in a binary system. GRB060218
is the first GRB associated with SN with complete coverage of data from the onset
all the way up to ~ 106 s. This fact offers an unprecedented opportunity to verify
theoretical models on such a GRB class.
References
1. Band, D., et al. 1993, apj, 413, 281.
2. Barbier, L., et al. 2006, GCN 4780.
3. Bernardini, M.G., Bianco, C.L., Chardonnet, P., Fraschetti, F., Ruffini, R., & Xue,
S.S. 2005, apjl, 634, L29.
4. Bernardini, M.G., Bianco, C.L., Chardonnet, P., Fraschetti, F., Ruffini, R., & Xue,
S.S. 2006, in "Proceedings of the Xth Marcel Grossmann Meeting", M. Novello, S.E.
Perez-Bergliaffa (eds.), World Scientific, Singapore, 2459.
5. Bianco, C.L., & Ruffini, R. 2004, apjl, 605, LI.
6. Bianco, C.L., & Ruffini, R. 2005a, apjl, 620, L23.
7. Bianco, C.L., & Ruffini, R. 2005b, apjl, 633, L13.
8. Crider, A., et al. 1997, apjl, 479, L39.
9. Campana, S., et al. 2006, nat, 442, 1008.
10. Cusumano, G., et al. 2006, GCN Circ, 4775.
11. Fatkhullin, T.A., et al. 2006, GCN Circ, 4809.
12. Frontera, F., et al. 2000, apjs, 127, 59.
13. Gehrels, N., et al. 2004, apj, 611, 1005.
14. Ghirlanda, G., Celotti, A., & Ghisellini, G. 2002, aap, 393, 409.
1988
15. Kelson, D., & Berger, E. 2005, GCN 3101.
16. Kennea, J.A., et al. 2006, GCN Circ, 4776.
17. Liang, E.W., Zhang, B.-B., Stamatikos, M., et al. 2006a, apjl, 653, L81.
18. Mirabal, N., Halpern, J.P., Thorstensen, J.R., and Terndrup, D.M. 2006, apjl, 643,
L99.
19. Pian, E., et al. 2006, nat, 442, 1011.
20. Ruffini, R., Bernaxdini, M.G., Bianco, C.L., Chardonnet, P., Dainotti, M.G.,
Fraschetti, F., and Xue, S.S. 2006a, apj, submitted to.
21. Ruffini, R., Bernardini, M.G., Bianco, C.L., Chardonnet, P., Fraschetti, F., Guida, R.,
and Xue, S.S. 2006b, apj, 645, L109.
22. Ruffini, R., Bernardini, M.G., Bianco, C.L., Chardonnet, P., Fraschetti, F.,
Gurzadyan, V., Vitagliano, L., and Xue, S.S. 2005a, in "COSMOLOGY AND
GRAVITATION: XIth Brazilian School of Cosmology and Gravitation", M. Novello, S.E.
Perez Bergliaffa (eds.), AIP Conf. Proc. 782, 42.
23. Ruffini, R., Bianco, C.L., Chardonnet, P., Fraschetti, F., Gurzadyan, V., and Xue,
S.S. 2004, Int. Journ. Mod. Phys. D14, 13, 843.
24. Ruffini, R., Bianco, C.L., Chardonnet, P., Fraschetti, F., Gurzadyan, V., and Xue,
S.S. 2005b, Int. J. Mod. Phys. D14, 97 (2005).
25. Ruffini, R., Bianco, C.L., Chaxdonnet, P., Fraschetti, F., Vitagliano, L., and Xue, S.S.
2003, "COSMOLOGY AND GRAVITATION: Xth Brazilian School of Cosmology and
Gravitation; 25th Anniversary (1977-2002)", M. Novello, S.E. Perez Bergliaffa (eds.)
AIP Conf. Proc. 668, 16.
26. Ruffini, R., Bianco, C.L., Chaxdonnet, P., Fraschetti, F., and Xue, S.S. 2001a, apjl,
555, LI07
27. Ruffini, R., Bianco, C.L., Chaxdonnet, P., Fraschetti, F., and Xue, S.S. 2001b, apjl,
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28. Ruffini, R., Salmonson, J.D., Wilson, J.R., and Xue, S.S. 2000, aap, 359, 855.
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30. Sakamoto, T., et al. 2006, GCN Circ, 4822.
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33. Sollermann, J., et al., aap. 454, 503 (2006).
THE "FIRESHELL" MODEL IN THE SWIFT ERA
C.L. BIANCO1-2 and R. RUFFINI,1'2'3
1 ICRANet and ICRA, Piazzale della Repubblica 10, 1-65122 Pescara, Italy
2 Dip. di Fisica, Universita di Roma "La Sapienza", Piazzale Aldo Moro 5, 1-00185 Roma, Italy
3 ICRANet, Universite de Nice Sophia Antipolis, Grand Chateau, BP 2135, 28, avenue de
Valrose, 06103 NICE CEDEX 2, France.
E-mails: bianco@icra.it, ruffini@icra.it.
We here re-examine the validity of the constant-index power-law relation between the
fireshell Lorentz gamma factor and its radial coordinate, usually adopted in the current
Gamma-Ray Burst (GRB) literature on the grounds of an "ultrarelativistic"
approximation. Such expressions are found to be mathematically correct but only approximately
valid in a very limited range of the physical and astrophysical parameters and in an
asymptotic regime which is reached only for a very short time, if any.
1. Introduction
The consensus has been reached that the afterglow emission originates from a rel-
ativistic thin shell of baryonic matter propagating in the CircumBurst Medium
(CBM) and that its description can be obtained from the relativistic conservation
laws of energy and momentum. In both our approach and in the other ones in the
current literature (see e.g. Refs. 1-5) such conservations laws are used. The main
difference is that in the current literature an ultra-relativistic approximation,
following the Blandford & McKee self-similar solution,6 is widely adopted, leading to
a simple constant-index power-law relations between the Lorentz 7 factor of the
optically thin "fireshell" and its radius:
7«r-Q, (1)
with a = 3 in the fully radiative case and a = 3/2 in the adiabatic case.1,4 On the
contrary, we use the exact solutions of the equations of motion of the fireshell. 3~5'7,8
2. Exact vs. approximate solutions in the Swift era
A detailed comparison between the equations used in the two approaches has been
presented in Refs. 3,4,7,8. In particular, in Ref. 4 it is shown that the regime
represented in Eq.(l) is reached only asymptotically when
70 » 7 » ! (2)
in the fully radiative regime and
7o2 » 72 » 1 (3)
in the adiabatic regime, where 70 the initial Lorentz gamma factor of the optically
thin fireshell.
In Fig. 1 we show the differences between the two approaches. In the upper panel
there are plotted the exact solutions for the fireshell dynamics in the fully radiative
1989
1990
10'
5 10' ■
10
3 ■
2.5 ■
2 ■
i 1,5
o
w 0.5
a = 3.0
a =1.5
10'
10'° 10'° 10'
Radial coordinate (r) (cm)
10'
Fig. 1. In the upper panel, the analytic behavior of the Lorentz 7 factor during the afterglow
era is plotted versus the radial coordinate of the expanding optically thin fireshell in the fully
radiative case (solid line) and in the adiabatic case (dotted line) starting from 70 = 102 and the
same initial conditions as GRB 991216.4 In the lower panel are plotted the corresponding values of
the "effective" power-law index ae// (see Eq.(4)), which is clearly not constant, is highly varying
and systematically lower than the constant values 3 and 3/2 purported in the current literature
(horizontal thin dotted lines).
and adiabatic cases. In the lower panel we plot the corresponding "effective" power-
law index aeff, defined as the index of the power-law tangent to the exact solution:4
Xeff
din 7
din 7*
(4)
Such an ''effective" power-law index of the exact solution smoothly varies from 0 to
a maximum value which is always smaller than 3 or 3/2, in the fully radiative and
adiabatic cases respectively, and finally decreases back to 0 (see Fig. 1).
1991
Thanks to the Swift satellite,9 we have now for many GRBs almost gapless
multi-wavelength light curves from the beginning of the prompt emission (which in
our model coincides with the peak of the afterglow, see Refs. 10-14) all the way to
the latest afterglow phases. In the interpretation of such gapless data it is therefore
crucial to use the exact solution for the fireshell dynamics.
References
1. T. Piran, Physics Reports 314, 575 (June 1999).
2. J. Chiang and C. D. Dernier, Astrophysical Journal 512, 699 (February 1999).
3. C. L. Bianco and R. Ruffini, Astrophysical Journal 620, L23 (February 2005).
4. C. L. Bianco and R. Ruffini, Astrophysical Journal 633, L13 (November 2005).
5. R. Ruffini, M. G. Bernardini, C. L. Bianco, L. Caito, P. Chardonnet, M. G. Dainotti,
F. Fraschetti, R. Guida, M. Rotondo, G. Vereshchagin, L. Vitagliano and S.-S. Xue,
The blackholic energy and the canonical gamma-ray burst, in Xllth Brazilian School
of Cosmology and Gravitation, eds. M. Novello and S. E. Perez Bergliaffa, American
Institute of Physics Conference Series, Vol. 910 (June 2007).
6. R. D. Blandford and C. F. McKee, Physics of Fluids 19, 1130 (August 1976).
7. C. L. Bianco and R. Ruffini, Astrophysical Journal 605, LI (April 2004).
8. C. L. Bianco and R. Ruffini, Astrophysical Journal 644, L105 (June 2006).
9. N. Gehrels, G. Chincarini, P. Giommi, K. O. Mason, J. A. Nousek, A. A. Wells, N. E.
White, S. D. Barthelmy, D. N. Burrows, L. R. Cominsky, K. C. Hurley, F. E. Marshall,
P. Meszaros, P. W. A. Roming, L. Angelini, L. M. Barbier, T. Belloni, S. Campana,
P. A. Caraveo, M. M. Chester, O. Citterio, T. L. Cline, M. S. Cropper, J. R. Cummings,
A. J. Dean, E. D. Feigelson, E. E. Fenimore, D. A. Frail, A. S. Fruchter, G. P. Garmire,
K. Gendreau, G. Ghisellini, J. Greiner, J. E. Hill, S. D. Hunsberger, H. A. Krimm,
S. R. Kulkarni, P. Kumar, F. Lebrun, N. M. Lloyd-Ronning, C. B. Markwardt, B. J.
Mattson, R. F. Mushotzky, J. P. Norris, J. Osborne, B. Paczynski, D. M. Palmer,
H.-S. Park, A. M. Parsons, J. Paul, M. J. Rees, C. S. Reynolds, J. E. Rhoads, T. P.
Sasseen, B. E. Schaefer, A. T. Short, A. P. Smale, I. A. Smith, L. Stella, G. Tagliaferri,
T. Takahashi, M. Tashiro, L. K. Townsley, J. Tueller, M. J. L. Turner, M. Vietri,
W. Voges, M. J. Ward, R. Willingale, F. M. Zerbi and W. W. Zhang, Astrophysical
Journal 611, 1005 (August 2004).
10. R. Ruffini, C. L. Bianco, F. Fraschetti, S.-S. Xue and P. Chardonnet, Astrophysical
Journal 555, LI 13 (July 2001).
11. R. Ruffini, M. G. Bernardini, C. L. Bianco, P. Chardonnet, F. Fraschetti, R. Guida
and S.-S. Xue, Astrophysical Journal 645, L109 (July 2006).
12. M. G. Dainotti, M. G. Bernardini, C. L. Bianco, L. Caito, R. Guida and R. Ruffini,
Astronomy & Astrophysics 471, L29 (August 2007).
13. M. G. Bernardini, C. L. Bianco, L. Caito, M. G. Dainotti, R. Guida and R. Ruffini,
Astronomy & Astrophysics 474, L13 (October 2007).
14. C. L. Bianco, M. G. Bernardini, L. Caito, M. G. Dainotti, R. Guida and R. Ruffini,
The "fireshell" model and the "canonical" grb scenario, in Relativistic Astrophysics,
eds. C. L. Bianco and S. S. Xue, American Institute of Physics Conference Series,
Vol. 966 (2007).
GRB970228 AS A PROTOTYPE FOR THE CLASS OF GRBs WITH
AN INITIAL SPIKELIKE EMISSION
M. G. BERNARDINI*, C. L. BIANCO, L. CAITO, M. G. DAINOTTI, R. GUIDA
and R. RUFFINI
Dipartimento di Fisica, Universita di Roma "La Sapienza"
Roma, 1-00185, Italy
* maria.bernardini@icra.it
ICRANet and ICRA, Piazzale della Repubblica 10
Pescara, 1-65122, Italy
We interpret GRB970228 prompt emission within our "canonical" GRB scenario,
identifying the initial spikelike emission with the Proper-GRB (P-GRB) and the following
bumps with the afterglow peak emission. Furthermore, we emphasize the necessity to
consider the "canonical" GRB as a whole due to the highly non-linear nature of the
model we applied.
Keywords: Gamma rays: bursts — black hole physics — galaxies: halos
1. Theoretical interpretation of GRB970228 prompt emission
GRB970228 was detected by the Gamma-Ray Burst Monitor (GRBM, 40-700 keV)
and Wide Field Cameras (WFC, 2-26 keV) on board BeppoSAX on February
28.123620 UT.1 The burst prompt emission is characterized by an initial 5 s strong
pulse followed, after 30 s, by a set of three additional pulses of decreasing
intensity.1 Such observations revealed a discontinuity in the spectral index between the
end of the first pulse and the beginning of the three additional ones.1-3 Moreover,
the spectrum of the last three pulses appear to be consistent with the late X-ray
afterglow.1'3 This was interpreted by Frontera et al.1,3 as if the emission mechanism
producing the X-ray afterglow already took place after the first pulse.
Consistently with BeppoSAX observations, within our "canonical GRB"
scenario4"6 we identify the first main pulse with the Proper-GRB (P-GRB) and the
three additional pulses with the afterglow peak emission. The adopted theoretical
"fireshell" model7 is based on two independent variables characterizing the source:
the total energy Efot of the e^ plasma and the baryon loading B, which for this
source are, respectively, El°± = 1.45 x 1054 erg and B = 5.0 x 10~3.6 The
theoretically estimated total isotropic energy emitted in the P-GRB is Ep-grb =
l.\%Et(± = 1.54 x 1052 erg, in excellent agreement with the one observed in the
first main pulse (Epb^GRB ~ 1.5 x 1052 erg in 2 — 700 keV energy band, see Fig. 1),
as expected due to their identification. The last three pulses have been reproduced
assuming three overdense spherical CircumBurst Medium (CBM) regions. On
average we have (1Z) = 1.5 x 10~7 and (nc{,m) = 9.5 x 10~4 particles/cm3.6 This very
low average value for the CBM density is compatible with the observed occurrence
of GRB970228 in its host galaxy's halo.8-10
1992
1993
6.0x1050
| 5.0x1050
■£
"1 4 0xl050
§ 3.0xlO50
a
s 2.0x1050
I 1.0X1050
0 0x10°
3.5X1049
5 3.0x1049
§ 2.5x1049
~ 2 0xl049
S- t.5x1049
ill
5 1 0x1049
| 5.0x1048
0.0x10°
GRBM observations in 40-700 keV band
Theoretical fil In 40-700 keV band - -
■ I
k
,
WFC observations in 2-26 keV band
Theoretical fit in 2-26 keV band --
:
\ i ^ ■—~
' ;
.
-
■
-
;
-
-
■
3.5x10
30xt06
2 5x10 6 £
2 0x1O'6 J5.
1 5x10"s ■§
t.OxlO'6 g
5.0x10"7
20 40 60
Detector arrival time (t^) (s)
Fig. 1. BeppoSAX GRBM (40-700 keV, above) and WFC (2-26 keV, below) light curves (green
points) compared with the theoretical ones (red lines). The onset of the afterglow coincides with
the end of the P-GRB (represented qualitatively by the blue lines).
2. Discussions on the uniqueness of the fit
Our fit of the observational data has to take into account all the variations in the
luminosity and, what is most stringent, this is done consistently and simultaneously
for every energy band in which observations exist. What is more important, the P-
GRB and the afterglow components are clearly identified, both in their relative
energies and time separation. These extremely stringent requirements do uniquely
determine the free parameters of the model, the total energy El°£ and the baryon
loading B, as well as the details of the CBM distribution.7
This process of data fitting is far from being trivial and must be performed
consistently step after step in view of the highly nonlinear behavior of all the phenomena
involved. We first start from the identification of the P-GRB and then move to the
subsequent part of the fit by consistently determining the CBM inhomogeneity as
well as the values of its filling factor.
In order to exemplify how neglecting any single set of observational data would
drastically affect the result of the fit, we have performed a simulation which neglects
the P-GRB contribution and fits reasonably well the sole afterglow component (see
Fig. 2). It is interesting that the value of El°l = 5.10 x 1052 erg obtained from this
second fit coincide with the usual estimates of Eiso in the current literature.1 The
value of the baryon loading B is also modified in this second fit (B = 4.5 x 10~3) but
1994
6.0x1 o50
5 0x1050
4.0x1O50
3.0x1050
2 0x1050
1.0x1050
0.0x10°
EEL
/V
GRBM observations in 40-700 keV band
=5 10x10^ erg, B=4 5x10 , 40-700 keV band
-
<\ ^.
!0 40 60
Detector arrival time (t^) (s|
100 120
Fig. 2. Theoretical fit of BeppoSAX GRBM observations in 40-700 keV energy band (red line)
neglecting the contribution of the P-GRB, which has been identified with the first sharp pulse.
Even if the resulting afterglow peak emission fits reasonably well the observed data, the predictions
of such a fit for the P-GRB energetic (blue line) is completely wrong.
what is more interesting is that the average CBM density is (ncbm) = 2.25, namely
103 times larger than the one we found in the complete analysis. Despite the good
results obtained for the afterglow peak emission, this analysis fails in reproducing
the P-GRB component. In fact we obtain E^tGRB = 1.8%E^ = 9.18 x 1050
erg, which is two orders of magnitude lower than the energy emitted in the initial
spikelike emission. This dramatically shows that the superposition principle does not
hold for the different components of GRB observational data set. This argument is
more important for those cases in which the P-GRB component is hardly detectable
(see e.g. GRB05031511), hence it is difficult to evaluate precisely its energetic.
References
1. Frontera, F., Costa, E., Piro, L., et al. 1998, ApJ, 493, L67.
2. Costa, E., Frontera, F., Heise, J., et al. 1997, Nature, 387, 783.
3. Frontera, F., Amati, L., Costa, E., et al. 2000, ApJS, 127, 59.
4. Ruffini, R., Bianco, C.L., Chardonnet, P., et al. 2001, ApJ, 555, L113.
5. Ruffini, R., Bernardini, M.G., Bianco, C.L., et al. 2007, AIP Con.Proc. 910, 55.
6. Bernardini, M.G., Bianco, C.L., Caito, L., et al. 2007, A&A, 474, L13.
7. Ruffini, R., Bernardini, M.G., Bianco, C.L., et al, 2005, AIP Con.Proc, 782, 42.
8. Sahu, K.C., Livio, M., Petro, L„ et al. 1997, Nature, 387, 476.
9. Van Paradijs, J., Groot, P.J., Galama, T., et al. 1997, Nature, 386, 686.
10. Panaitescu, A. 2006, MNRAS, 367, L42.
11. Ruffini, R., Bernardini, M.G., Bianco, C.L., et al. 2006, ApJ, 645, L109.
THEORETICAL INTERPRETATION OF GRB060124:
PRELIMINARY RESULTS
R. GUIDA*, M.G. BERNARDINI, C.L. BIANCO, L. CAITO, M.G. DAINOTTI
and R. RUFFINI
Dipartimento di Fisica, Universitd La Sapienza,
Roma, 00185, Italy
* roberto.guida@icra. it
www.icra.it
We show the preliminary results of the application of our "fireshell" model to
GRB060124. This source is very peculiar because it is the first event for which both
the prompt and the afterglow emission were observed simultaneously by the three Swift
instruments: BAT (15 - 350 keV), XRT (0.2 - 10 keV) and UVOT (170 - 650 nm), due
to the presence of a precursor ~ 570 s before the main burst. We analyze GRB060124
within our "canonical" GRB scenario, identifying the precursor with the P-GRB and the
prompt emission with the afterglow peak emission. In this way we reproduce correctly
the energetics of both these two components. We reproduce also the observed time delay
between the precursor (P-GRB) and the main burst. The effect of such a time delay in
our model will be discussed.
Keywords: Gamma rays: bursts — Black hole physics — Radiation mechanisms: thermal
1. GRB060124 observational properties
On 2006-01-24 at 15:54:52 UT, Swift-BAT triggered on the precursor of GRB060124,
that occurred ~ 570 s before the main burst peak.1 This allowed Swift to
immediately re-point the narrow field instruments (NFIs) and acquire a pointing towards
the burst ~ 350 s before the main burst occurred. The burst has a highly structured
profile, comprising three major peaks following the precursor and has the longest
duration (even excluding the precursor) ever recorded.2
GRB060124 also triggered Konus-Wind (10 - 770 keV)3 559.4 s after the BAT
trigger.4 The Konus light curve confirmed the presence of both the precursor and
the three peaks of prompt emission.
The prompt emission of GRB 060124 was observed simultaneously by XRT with
exceptional signal-to-noise (S/N) and was detected by UVOT at V = 16.96 ± 0.08
(T + 183s) and V = 16.79 ± 0.04 (T + 633s).1 This fact makes it an exceptional
test case to study prompt emission models, since this is the very first case that the
burst could be observed with an X-ray CCD with high spatial resolution imaging
down to 0.2 keV.
2. The fit
Within our "canonical GRB" scenario5 we identify the first main pulse with the
P-GRB and the three major peaks following the precursor with the afterglow peak
emission.
We therefore obtain for the two parameters characterizing the source in our
model Elf = 3.73 x 1054 erg and B = 2.3 x 10"3. This implies an initial e± plasma
1995
1996
XRT observations in 0.2-10 keV band i + i :
Theoretical fit in 0.2-10 keV band :
102 103 104 105 106
Detector arrival time (s)
Fig. 1. The XRT light curve (0.2—10 keV, red points) and the preliminary theoretical simulation
in the same energy band (green line). The fit is quite good, but the double peaked structure is
not reproduced, due to the fact that our radial approximation for modeling the CBM is not valid
anymore at the late time of the peaks (see text).
created between the radii r\ = 1.12 x 107 cm and r2 = 4.58 x 108 cm with a total
number of e* pairs Ne± = 1.46 x 1059 and an initial temperature T = 2.23 MeV.
The theoretically estimated total isotropic energy emitted in the P-GRB is
Ep-grb — lAWoE1^ — 5.26 x 1052 erg, in excellent agreement with the one
observed in the first main pulse (E^GRB ~ 6.00 x 1052 erg in 15 — 350 keV
energy band), as expected due to their identification. After the transparency point
at ro = 4.76 x 1014 cm from the progenitor, the initial Lorentz gamma factor of
the fireshell is 70 = 430. The distribution of the CircumBurst medium has been
parametrized assuming an average value for the effective density in the prompt
phase of 10~2 particle per cm3 and in the afterglow phase of 10~4 particle per cm3.
Such a low effective density has been assumed in order to reproduce the ~ 500 s of
quiescence between the P-GRB and the prompt, according to the way in which the
emission is produced within our model, that it will be clarified in the next session.
In Fig. 1 we present the preliminary theoretical fit of the Swift XRT data (0.2-10
keV), while in Fig. 2 of the BAT ones (15-350 keV). The problems of the fit will be
discussed in the next section.
3. The CircumBurst 3D structure
Within our fireshell model all the GRB emission after the transparency is produced
by the interaction of the accelerated baryons with the CBM, and such interaction
1997
2.0x10
BAT observations in 15-350 keV band
BAT observations in 15-350 keV band (precursor)
Theoretical fit in 15-350 keV bahd
5.0x10"'
4.0x10'~
2.0x10"'
1.0x10"'
300 400 500
Detector arrival time (s)
700
800
Fig. 2. The BAT light curve in the 15 — 350 keV band (red points) comprising also the precursor
(green points) and our preliminary theoretical simulation in the same energy band (blue line).
Clearly the energetics is well reproduced, but in order to have a good fit of the peaks, a correct
treatment of the 3-dimensional structure of the CBM is needed (see text).
is modeled as inelastic collisions.6 The number of such collisions, hence, depends on
the CBM density.
The simplest way to model the CBM structure is to assume that ncbm is a
function only of the radial coordinate, nc/,m = ncbm(r) (radial approximation). The
CBM is arranged in spherical shells of width ~ 10l5 cm positioned in such a way
that the modulation of the emitted flux coincides with the observed peaks. It is
important to emphasize that, when the accelerated baryons collide with a shell, the
increase in the flux is almost immediate due to the photons coming from the line
of sight. Then it follows an exponential decrease of the flux due to the contribution
of the photons emitted from different angles. In this way we obtain the observed
FRED structure for each peak, together with all the other observed peculiarities
(hard to soft transition, spectral lag).
Clearly our radial approximation is valid until the visible area of the incoming
baryons pulse is comparable with the characteristic dimensions of the clouds. The
transverse dimension of such area is RT = rsinf?, where 6 ~ I/7 is the relativistic
beaming angle, so we have Rt ~ r/7.
We have found in many cases that this approximation cannot be valid during the
whole prompt emission. In fact, when the accelerated baryons impact with dense
clouds of CBM, they are decelerated and their gamma factor drops abruptly. In this
1998
situation, after the first peaks (the number of peaks depending from their height,
the higher they are the smaller their number is) the visible area becomes comparable
with the size of the clouds and our approximation is not valid anymore. This is case
for other GRBs we analyzed, as GRB9912166 and GRB0503157 .
Another situation in which our radial approximation fails can occurs. Because
the transverse dimension of the baryonic fireshell's visible outlined above,
depends not only from the Lorentz gamma factor but also from the radius of the
fireshell, it can be that for very large value of this radial coordinate, the size of the
visible area becomes comparable with the CBM clouds, that is, the approximation
of spherical symmetric distribution for the CBM fails.
In all the GRB sources studied up to date, this have never been the case, because
usually the radial coordinate r at which the prompt emission occurs is small.
It is important here to remember the fact that within our fireshell model, the
initial instant of time to (related to the initial value of the radial coordinate, tq =
ct0) is often different from the moment in which the satellite instrument triggers:
in fact in our model the GRB emission starts at the transparency point when the
P-GRB is emitted, but sometimes the P-GRB is under the instrumental threshold
or comparable with it and so is not enough to trigger the instrument. For example
in the case of GRB050315, a possible precursor was observed ~ 50 s before the
trigger,8 that indeed occurred when the main prompt emission started.
In this case instead the BAT instrument triggers on a precursor that we identify
as the P-GRB because of the excellent agreement in terms of the energetics and
of the time delay between it and the main prompt emission; so in this case our £0
coincides with the BAT trigger and the main prompt emission occurs at AT ~ 600
s so at a value r = cAT for the radial coordinate of the fireshell; with this value of
r the transverse dimension of the baryonic fireshell's visible area is such that the
radial approximation is not valid anymore.
In particular, we found that at tda ~ 600 s, that is when the main burst
occurs, the radius of the fireshell is r ~ 1018 cm and the Lorentz gamma factor has
dropped abruptly to a value of ~ 100 from the initial 70 = 430, due to the CBM
cloud assumed to be present at the moment of the prompt emission. The transverse
dimension of the visible area of the incoming baryons pulse indeed results Rt ~ 1016
cm, so even bigger then the characteristic dimensions of the CBM clouds usually
assumed (from6 1014 to 1015 cm), in this case ~ 1015 cm.
A correct treatment of the 3-dimensional structure of the CBM clouds is needed
in this case.
We have already tested this idea in order to explain an apparently physical
different feature of the GRBs: the flares. This phenomenon has been discovered to
occurs in the early part of the X-ray afterglow, that means very late from the satellite
trigger and very far. From our point of view, there are no differences between a flare
and the prompt emission in this case, that has occurred at 600 s.
Many interpretations have been provided in order to explain the flares. The
most common explanation is a central engine activity which results in internal
1999
shocks (or similar energy dissipation events) at later times9 . Another possibility is
emission from reverse shock, but the predicted amplitude is too low to interpret all
the cases9'10 . Alternatively such emission could be produced by a multi-component
jet11-13 : the X-ray flare is caused by the deceleration of the wider cocoon component
with the ambient medium. In this case, however, the decay after the peak should
follow the standard afterglow model, so it cannot interpret the observed rapid fall-
off in the flares9 . The same problem9 affects also the scenario in which the flare is
produced by the energy injection into the decelerating shell by the collision with a
high-7 shell14 .
Within our fireshell model the flares are interpreted as being due to the same
process responsible for the following afterglow emission. So the difficulties to fit
them are due to the radial approximation, not valid anymore at such late time (or
at such big value of the radial coordinate).
We tested our idea of abandoning the radial approximation and introducing
a 3-dimensional structure of the CBM clouds in order to fit the flare (occurred
at ~ 250 s) of GRB01112115-16 , an old burst observed by BeppoSAX which for
the first time showed the feature of an X-ray flare. We obtain good results that
demonstrate at least the validity of such proposal. Anyway the implementation of a
such description of the CBM clouds is not yet finished, but we are currently working
on it.
4. Conclusion
We applied the fireshell model to GRB060124. The work is not finished yet and
we showed only the preliminary results. The main peculiarity of this source is the
biggest ever recorded time delay between the precursor and the prompt emission.
We reproduced correctly the energetics of the precursor, identified with the P-GRB,
and of the prompt emission, identified with the extended afterglow peak emission.
The most important consequence of having such a big time delay between P-
GRB and afterglow peak is that the radial approximation assumed in modeling
the CBM structure is not valid anymore at the time of the prompt emission. For
this reason our model failed in reproducing the narrow two peaks of the prompt
emission. Our peaks, in particular the second, resulted much more spread.
In order to have a good fit of the light curves, we have to change our way of
modeling the CBM structure. We have to take into account the fact that only a part
of the visible area of the fireshell interacts with the CBM cloud. This is only possible
introducing a 3-dimensional structure of the clouds, that will mean to introduce a
new parameter. In this way we will obtain narrow peaks also for big values of the
fireshell radius.
We have already successfully applied this idea in order to fit the flare of
GRB011121, that is a bump of an order of magnitude in luminosity, lasting for 20
s. occurred after 250 s from the trigger. The likeness of this flare with the prompt
emission of GRB060124, a short bump of an order of magnitude in luminosity oc-
2000
curred at very late time as well, is evident: so we expect to obtain also in this case
the same good agreement we had in the case of GRB011121.
References
1. P. Romano, et al., Astronom. and Astrophys. 456, 917 (2006).
2. D. Lazzati, MNRAS, 357, 722 (2005).
3. R. L. Aptekar, D. D. Frederiks, S. V. Golenetskii, et al., Space Sci. Rev., 71, 265
(1995).
4. S. V. Golenetskii, R. L. Aptekar, E. Mazets, et al., GCN Circ, 4599, 1 (2006).
5. R. Ruffini, M. G. Bernardini, C. L. Bianco, L. Caito, P. Chardonnet, M. G. Dain-
otti, F. Fraschetti, R. Guida, M. Rotondo, G. Vereshchagin and S. S. Xue, "The
Blackholic energy and the canonical Gamma-Ray Burst" in XII Brazilian School of
Cosmology and Gravitation-2006, edited by M. Novello and S. E. Perez Bergliaffa,
AIP Conference Proceedings, 910, American Institute of Physics, New York, 2007,
pp.55.
6. R. Ruffini, C. L. Bianco, P. Chardonnet, F. Fraschetti and S. S. Xue, ApJ, 581, L19
(2002).
7. R. Ruffini, M. G. Bernardini, C. L. Bianco, P. Chardonnet, F. Fraschetti and
R. Guida, ApJ, 645, L109 (2006).
8. S. Vaughan, et al., ApJ, 638, 920 (2006).
9. B. Zhang, et al. ApJ, 642, 354 (2006).
10. D. N. Burrows, et al., Science, 309, 1833 (2005).
11. P. Meszaros, & M. J. Rees, ApJ, 556, L37 (2001).
12. E. Ramirez-Ruiz, A. Celotti, & M. J. Rees, MNRAS, 337, 1349 (2002).
13. P. Kumar, & T. Piran, ApJ, 532, 286 (2000).
14. P. Kumar, & T. Piran, ApJ, 535, 152 (2000).
15. C. L. Bianco, L. Caito and R. Ruffini, Nuovo Cimento, 121B 1441 (2006).
16. L. Caito, M. G. Bernardini, C. L. Bianco, P. Chardonnet, F. Fraschetti, R. Ruffini,
S. S. Xue, Proceedings of the XI Marcel Grossmann Meeting on General Relativity,
in press.
GRBs and Host Galaxies
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NUMERICAL COUNTERPARTS OF GRB HOST GALAXIES
S. COURTY*, G. BJORNSSON and E. H. GUDMUNDSSON
Science Institute, University of Iceland,
Reykjavik, Iceland
* courty@raunvis.hi.is
Properties of host galaxies of gamma-ray bursts (GRBs) are investigated, using N-
body/Eulerian hydrodynamic simulations and the stellar population synthesis model,
Starburst99, to infer observable properties. The simulations include gravitation, hydro-
dynamical shocks, and radiative cooling, as well as a phenomenological description of
galaxy formation. We first focus on the overall population at intermediate redshifts and
emphasize the strong relationships between the specific star formation rate (SFR) and
the epoch of formation, color index and mass-to-light ratio, quantities known to reflect
the star formation history of galaxies. The faintest and bluest galaxies are objects with
the highest specific rates. Faint and blue colors are common properties among the
population of GRB host galaxies. We then consider a well-defined sample of observed GRB host
galaxies with optical estimates of SFR and SFR-to-luminosity ratios and look for their
numerical counterparts by selecting objects that have both values nearest to those of the
observed host galaxies. Comparing the numerical counterparts to the overall simulated
galaxy population at different redshifts suggests that GRB host galaxies are a particular
sub-population of galaxies, likely to be drawn from the high specific SFR population,
rather than the high SFR galaxy population. In a separate, preliminary study, we address
the link between the cosmological evolution of galaxy properties and the properties of
the gas surrounding galaxies by tracing the history of galaxies through their main
progenitors. We show that high specific SFRs tend to occur in the early evolutionary stages
of galaxies. GRB host galaxies may thus be a powerful way to select those proto-galaxies
and contribute to our understanding of galaxy evolution.
Keywords: Cosmology: large-scale structure of Universe - galaxies: formation - galaxies:
evolution - gamma rays: bursts
1. Numerical procedure
We consider N-body/Eulerian numerical simulations that include gravitation, hy-
drodynamical shocks, radiative cooling processes (without assuming any collisional
ionization equilibrium) and galaxy formation based on a phenomenological
description. A A—CDM cosmological model scenario is adopted. The comoving size of
the computational volume is 32 ft_1Mpc and the simulation has 2563 dark matter
particles and an equal number of grid cells. Galaxy-like objects are characterized
through their mass, M, epoch of formation (mass-weighted average of the epochs of
formation of all the stellar populations contained in a galaxy), instantaneous star
formation rate, SFR* (amount of stellar material formed in the previous 108 yr),
and specific star formation rate, e = log(10u yr SFR*/M). Observable properties
are inferred using the stellar population synthesis model, Starburst99,l considering
each stellar particle as a homogeneous stellar population (with metalicity Z = 0.004
and a Salpeter IMF).
2003
2004
2. Results
First we consider the strong relationships between the specific star formation rate
(SFR) and the color index and mass-to-light ratio, quantities known to reflect the
star formation history of galaxies. At intermediate redshift, the faintest and bluest
galaxies are also the objects with the highest specific rates. Faint and blue
colors are common properties among the population of GRB host galaxies.
Observational studies of host galaxies of GRBs tend to show that host galaxies have
particular characteristics: they seem to be optically sub-luminous, low-mass, blue,
star-bursting galaxies, with young stellar populations, a modest activity of optical
CO
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0.5
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Fig. 1. The median values (crosses) of the SFR-to-luminosity ratio (top) and specific SFR
(bottom) for the star-forming galaxy populations at the redshifts of the observed GRB hosts. The
range of values for each catalog is shown by the length of the vertical lines. Diamonds refer to the
numerical counterparts of the 10 hosts from Ref. 2 and the two squares are the counterparts of
GRB 000911 and 030329 hosts that are discussed in our paper.3
2005
star formation, and present (although less firmly established), low-metalicity and
modest amount of dust obscuration. Ref. 2 estimates the optical SFR and the ratio
between this SFR and the B-band magnitude for 10 observed GRB host galaxies
whose redshifts range between ~0.4 and ~2. They show that the hosts are
similar to those HDF galaxies that have the highest SFR-to-luininosity ratios. Their
magnitude-limited (R < 25.3) sample includes host galaxies with redshifts in the
range 0.43 < z < 2.03. In simulated catalogs obtained at the same redshifts as
the observed host galaxies in this sample, we then identify objects that have both
SFRuv and SFR-to-luminosity ratio nearest to those of the observed hosts. When
compared to the overall galaxy population, the 10 counterpart hosts are low-mass
galaxies (M < 4-1010 M0), with low mass-to-light ratios, and SFR*/(SFR) around
unity or higher; most of them are blue and young galaxies, with epochs of formation
within 40% of the age of the universe at the different redshifts. Although the SFR*
of the counterparts varies between ~ 0.4 and 8 M© yr-1, the specific SFR is equal
to or higher than the median values estimated for the different catalogs (see Fig. 1).
The comparison of counterparts-versus-observed hosts is limited by the fact that the
magnitude-limited sample of Christensen et al. only includes 10 host galaxies,
spanning a wide range in redshift. The Swift mission will provide a much larger sample,
with more hosts at similar redshifts. Our results3 suggest that GRB host galaxies
are a particular sub-population of galaxies, likely to be drawn from the high specific
SFR population, rather than the high SFR galaxy population. Moreover in an
extended sample, the specific SFRs of the majority of GRB host galaxies are expected
to be even higher than found in the sample studied here. In a separate, preliminary
study, addressing the link between the cosmological evolution of galaxy properties
and the properties of the gas surrounding galaxies, we determine the whole history
of galaxies through their main progenitors at different epochs. We show that high
specific SFRs tend to occur in the early evolutionary stages of galaxies. GRB host
galaxies may thus be a powerful way to select those proto-galaxies and therefore
contribute to our understanding of galaxy evolution.
References
1. Vazquez, G.A. et al. 2005, ApJ, 621, 695
2. Christensen, L. et al. 2004, A&A, 425, 913.
3. Courty, S. et al., 2007, MNRAS, in press
THE HOST GALAXIES OF LONG GAMMA-RAY BURSTS: THE
MID-INFRARED VIEW FROM THE SPITZER SPACE TELESCOPE
EMERIC Le FLOC'H
Spitzer fellow, Institute for Astronomy, University of Hawaii
2680 Woodlawn Drive
Honolulu, HI 96822, USA
elefloch@ifa. Hawaii, edu
We report on an on-going survey of 55 long GRB host galaxies at mid-infrared
wavelengths (i.e., 3 < A < 30 ftm) using the Spitzer Space Telescope. Our very low rate of
detections argues against a population harboring luminous dusty starbursts in already-evolved
and massive galaxies, which contrasts with previous claims based on submillimeter and
radio observations. Given the contribution of such luminous and dusty galaxies to the
cosmic growth of structures, we infer that long GRBs are biased tracers of star formation
possibly favored in low-metallicity environments. In the case of the host of GRB 980425
the MIPS-24 ftm observations reveal a luminous compact source responsible for ~ 80%
of the mid-IR emission of the entire galaxy. It is associated with a bright HII region
located ~900pc away from the position where the GRB occured. The possible connection
between GRB 980425 and this luminous star-forming region is not understood yet but it
could open a new window on the exploration of the origin of long GRBs.
1. Introduction
Long Gamma-Ray Bursts (LGRBs) are known to originate from the core collapse
of very short-lived massive stars. Because of their dust-penetrating power and their
detectability up to very high redshifts, LGRBs thus appear as promising probes
of star formation in the early Universe. In particular, should their occurence rate
in a given comoving volume of universe remain proportional to the star formation
rate density in this volume throughout cosmic history, LGRBs could be used as
a powerful and unbiased quantitative tracer of the growth of structures since the
emergence of the first galaxies.
Before reaching this point however, we need to understand whether LGRBs are
able to pinpoint any "kind" of starburst episodes in the Universe and to equally
sample star-forming activity independently of galaxy types and environments. In
other words we need to test whether their formation is not biased toward any
particular type of starbursting sources. I see two different ways to address this point.
The first one is to reach a very detailed understanding of the physical mechanisms
triggering LGRBs. Considerable progress have been reached on this subject over the
last decade, but the role of lots of free parameters still remains unclear. The second
approach, which I will discuss here, is to characterize the properties of LGRB host
galaxies and to assess whether these sources are representative of the ones that
produced the bulk of star formation as a function of redshift.
Since 1997 the detections of LGRB afterglows in the optical, near-IR and
radio have enabled the accurate localization of several dozens of bursts as well as
the subsequent identification of their hosts. So far, independent studies of these
host galaxies have led to conflicting views on their nature. At optical wavelengths
2006
2007
LGRB hosts appear as faint blue and low mass systems, probably characterized
by young stellar populations, a moderate activity of on-going star formation (i.e.,
SFR<^ 10 M0 yr_1) and a negligible amount of dust extinction.1 From submmil-
limeter and radio observations though, it has been claimed that they should rather
experience intense episodes of massive star formation enshrouded in dust (i.e.,
SFR> 100M0 yr-1).2 The true nature of these hosts is therefore still debated.
2. An on-going survey of LGRB hosts with Spitzer
To further investigate this apparent contradiction and to unveil the potential
presence of luminous phases of star-forming activity in these objects, we initiated a
survey of LGRB host galaxies at mid-infrared wavelengths using the Spitzer Space
Telescope. The first sources that we targeted were biased on purpose toward those
GRB hosts either associated with optically faint counterparts or located at high
redshift. Results were reported by Le Floc'h et al. 2006.3 The second part of the
survey is more centered on host galaxies with bright optical magnitudes and
located at z< 1.2. Here I briefly summarize the results we have obtained so far for
this second program.
In spite of the superb sensitivity of the Spitzer instruments, the rate of
detections in our sample appears to be very low, both with IRAC and MIPS. At the IRAC
3.6 or 4.5 /xm wavelengths, the non-detections of LGRB hosts located at moderately
high redshifts confirm the trend of LGRBs to occur within relatively low-mass
systems (i.e., M < 1010 Mq). We will soon quantify this statement in more detail using
stellar population synthesis models to fit the optical, near-IR and IRAC broad-band
properties of GRB hosts. At 24 /xm, the rate of detections is also very small, which
argues against the presence of strong dust-enshrouded star-forming activity in these
systems. Up to redshift z ~ 1 indeed, our MIPS observations should easily reveal any
starburst with SFR > 10 M0 yr-1. The 24 /im properties of our sample are
therefore consistent with the picture emerging from the optical wavelengths. Given the
major contribution of luminous infrared galaxies to the growth of structures at high
redshifts, they also reveal that LGRBs are biased tracers of star formation. Indeed,
the formation of LGRBs is believed to be favored in star-forming regions with low
chemical abundance, which could explain the blue colors and low luminosities of
their hosts. On the other hand, we also note that this lack of dusty sources probed
with LGRBs could be an observational bias due to our selection mostly based on
GRBs with optical afterglows. To better understand this potential selection effect
we are currently studying another sample of dark GRBs, with positions determined
using the XRT camera on-board Swift (paper in prep.).
3. Dusty star formation and low-metallicity environments
Because of the rather low star formation rate of LGRB hosts, studying with
Spitzer/MlPS the presence of obscured star-forming activity in these systems
requires observations of sources at low redshift. The host of GRB 980425 at z = 0.0085
2008
represents therefore a particularly interesting case. In this galaxy we found that
~ 80% of its 24 fim emission originates from a single point source associated with a
bright HII region producing 0.3 M© per year and located in a rather low metallicity
environment (see, Fig. 1). The MLPS-24 mil observations of this galaxy thus confirm
that a low metal content does not, necessarily implies the absence of dust grains
that, can be responsible for a noii-iiegligeablo reprocessing of the UV photons
emitted by young stars. The connection between this luminous IR point-like region and
the trigger of the GRB in this galaxy still needs to be addressed however. We are
currently getting far-infrared MIPS observations of this object at 70 and 160 mil
to better characterize the bolometric properties of this luminous HII region and its
relation to massive star formation in the host (paper in prep.)
\,
i^Mi-4Siiffi
&'fiPS-2«J»tW
Fig, 1, HST and Spitzer images of the host of GRB 980425 (z = 0.0085), with the position of the
burst shown with a cross in each panel. The IR, data reveal the presence of an extremely bright
point source located 900 pc from the GRB, possibly a super-star cluster enshrouded in dust.
Acknowledgments
It is a great pleasure to thank Gunnlaugur Bjornsson for inviting me to this
conference and for giving me the opporunity to present our preliminary results on this
Spitzer survey.
References
1
V. Sokolov el al., Host galaxies of gamma-ray bursts: Spectral energy distributions and
internal extinction (A&A 372, 438, 2001)
E. Berger et al, A Sub-millimeter and Radio Survey of Gamma-Ray Burst Host
Galaxies: A Glimpse into the Future of Star Formation Studies (ApJ 588, 99, 2003)
E.Le Floc'h et al., Probing Cosmic Star Formation Using Long Gamma-Ray Bursts:
New Constraints from the Spitzer Space Telescope (ApJ 642, 636, 2006)
GAMMA-RAY BURST HOST GALAXY GAS AND DUST*
RHAANA STARLING, RALPH WIJERS AND KLAAS WIERSEMA
University of Amsterdam
Kruislaan 403, 1098 SJ Amsterdam, The Netherlands
rlcsl@star.le.ac.uk ; rwijers@science.uva.nl ; kwrsema@science.uva.nl
We report on the results of a study to obtain limits on the absorbing columns towards
an initial sample of 10 long Gamma-Ray Bursts observed with BeppoSAX, using a new
approach to SED fitting to nIR, optical and X-ray afterglow data, in count space and
including the effects of metallicity. When testing MW, LMC and SMC extinction laws we
find that SMC-like extinction provides the best fit in most cases. A MW-like extinction
curve is not preferred for any of these sources, largely since the 2175A bump, in principle
detectable in all these afterglows, is not present in the data. We rule out an SMC-
like gas-to-dust ratio or lower value for 4 of the hosts analysed here (assuming SMC
metallicity and extinction law) whilst the remainder of the sample have too large an
error to discriminate. We provide an accurate estimate of the line-of-sight extinction,
improving upon the uncertainties for the majority of the extinction measurements made
in previous studies of this sample.
1. Introduction
The accurate localisation of Gamma-Ray Bursts (GRBs) through their optical and
X-ray afterglows has enabled detailed studies of their environments. Selection solely
by the unobscured gamma-ray flash has allowed the discovery of a unique sample
of galaxies spanning a very wide range of redshifts from z ~ 0.009 to 6.3.3 Hence,
detailed and extensive host galaxy observations provide a wealth of information on
the properties of star-forming galaxies throughout cosmological history.
Afterglow spectroscopy and/or photometry can be used to provide an estimate of
the total extinction along the line-of-sight to the GRB. Absorption within our own
Galaxy along a particular line of sight can be estimated and removed, but absorption
which is intrinsic to the GRB host galaxy as a function of wavelength is unknown,
and is especially difficult to determine given its dependence on metallicity and the
need to distinguish it from that of intervening systems. In general, low amounts
of optical extinction are found towards GRBs, unexpected if GRBs are located
in dusty star-forming regions, whilst the X-ray spectra reveal a different picture.
At X-ray wavelengths we often measure high values for the absorbtion columns
absorption, where the absorption is caused by metals in both gas and solid phase.11
The apparent discrepancy between optical and X-ray extinction resulting in high
gas-to-dust ratios in GRB host galaxies (often far higher than for the MW, LMC
or SMC, e.g. GRB0201242) is not satisfactorily explained, though the suggestion
that dust destruction can occur via the high energy radiation of the GRB9 could
possibly account for the discrepancy.
"The authors acknowledge funding from the EU RTN 'Gamma-Ray Bursts: An Enigma and a
Tool', support from PPARC, and RS thanks the conference organisers for financial assistance.
2009
2010
Traditionally the optical and X-ray spectra have been treated seperately in
extinction studies. Since the underlying spectrum is likely a synchrotron spectrum
(power law or broken power law) extending through both wavelength regions, it is
most accurate to perform simultaneous fits. We perform simultaneous broadband
fits of the spectral energy distributions (SEDs) in count space, so we need not first
assume a model for the X-ray spectrum. Inclusion of nIR data and R band optical
data together with the 2-10 keV X-ray data, regimes over which extinction has
the least effect, allows the underlying power law slope to be most accurately
determined. This sample of 10 long GRBs observed with the BeppoSAX Narrow Field
Instruments is chosen for the good availability (3 bands or more) of optical/nIR
photometry.
2. Results and Discussion
Detailed results of fits to the SEDs for all GRBs in the sample, and further
references, can be found in Starling et al. (2007). Figure 1 shows a comparison of the
absorption measurements with Galactic, LMC and SMC gas-to-dust ratios. This
plot has been constructed in a number of previous works1846 and here we show
the observed distribution of E(B — V) and Nh for the first time derived
simultaneously from a fit to X-ray, optical and nIR data. We find a large excess in absorption
above the Galactic values in two sources: GRBs 000926 (E(B-V) only) and 010222,
whilst no significant intrinsic absorption is necessary in GRBs 970228 and 990510.
The cooling break can be located in three of the afterglows: GRBs 990123, 990510
and 010222 and to all other SEDs a single power law is an adequate fit.
We find a wide spread in central values for the gas-to-dust ratios, and for 4 GRBs
the gas-to-dust ratios are formally inconsistent with (several orders of magnitude
higher than) MW, LMC and SMC values at the 90 % confidence limit assuming the
SMC metallicity. This must mean that either gas-to-dust ratios in galaxies can span
a far larger range than thought from the study of local galaxies, or the ratios are
disproportionate in GRB hosts because the dust is destroyed by some mechanisms
(likely the GRB jet), or that the lines of sight we probe through GRBs tend to be
very gas-rich or dust-poor compared with random lines of sight through galaxies. A
dust grain size distribution which is markedly different than considered here may
also affect these ratios.
We have compared the results of this method to those of other methods of
determining E(B — V). In particular we find that with respect to continuum
fitting methods such as this, optical extinction is overestimated with the depletion
pattern method,5 and we have quantified this for a small number of cases.7 We
note, however, that since this is a line-of-sight method, the measured columns may
not be representative of the host galaxy as a whole, therefore comparison with the
integrated host galaxy methods is important.
Swift, robotic telescopes and Rapid Response Mode on large telescopes such as
the William Herschel Telescope and the Very Large Telescopes now allow early, high
2011
quality data to be obtained, which will help immensely in discriminating between
the different extinction laws at work in the host galaxies.
0.0
0.2
0.3
E(B-V)
Fig. 1. Intrinsic absorption in optical/nIR [E{B — V)) and X-rays (log Nn) measured for the GRB
sample with 90 % error bars. We compare these with three different optical extinction laws overlaid
with solid curves: Galactic (top panel), LMC (middle panel) and SMC (lower panel). Appropriate
metallicities are adopted for LMC (1/3 Z©) and SMC (1/8 Z©) calculations (diamonds), and stars
mark the centroids of the Solar metallicity fits. For GRB 000926 the data were too sparse to fit for
Nh, so we plot the E(B — V) range at log Wh = 17.0 for clarity.
References
1. Galama T. J. and Wijers R. A. M. J., 2001, ApJ, 549, L209
2. Hjorth J. et al, 2003, ApJ, 597, 699
3. Jakobsson P. et al, 2006, A&A, 447, 897
4. Kann D. A., Klose S. and Zeh A., 2006, ApJ, 641, 993
5. Savaglio S., Fall S. M. and Fiore F., 2003, ApJ, 585, 638
6. Schady P. et al, MNRAS submitted
7. Starling R. L. C. et al, ApJ in press, astro-ph/0610899
8. Stratta G. et al, 2004, ApJ, 608, 846
9. Waxman E. and Draine B. T., 2000, ApJ, 537, 796
10. Wijers R. A. M. J. and Galama T. J., 1999, ApJ, 523, 177
11. Wilms J., Allen A. and McCray R., 2000, ApJ, 542, 914
LOW REDSHIFT GRBS AND THEIR HOST GALAXIES
NIAL R. TANVIR
Department of Physics and Astronomy,
University of Leicester,
Leicester, LEI 7RH.
United Kingdom
nrt3@star.le. ac. uk
There is growing evidence that a proportion of GRB-like events occur at relatively low
redshifts and have lower luminosities than the cosmological GRBs. Some of these are
long-duration bursts which are associated with type-Ibc supernovae, and presumably
produced by a similar mechanism to their higher redshift counterparts. Others are short-
duration bursts which may well be produced by flares from soft gamma-ray repeaters
(SGRs). I review the evidence for the existence of these various populations, and discuss
the implications for progenitor models and their relative number densities.
1. Introduction
Gamma-ray bursts (GRBs) split into at least two categories, those of shorter-
duration (S-GRBs; typically i90 <2 s) which are usually spectrally hard, and the
long-duration (L-GRBs) which are relatively softer.1 The discovery of afterglows of
both long2 and short3 bursts, along with their host galaxies, have provided critical
clues to the nature of GRB progenitors. The large majority of GRBs studied to date
have been at cosmological redshifts, 0.1 < z < 6.3. However, there is evidence that
there are populations of GRB-like events, of both types, with lower-luminosities and
detectable only at low redshifts, which may dominate the overall number density.
2. Local Short-Duration GRBs
Although much less well-studied than the L-GRBs, the discovery of afterglows of
short bursts produced rapid progress in the field. As a group, the S-GRBs appear to
be associated with a wider range of host galaxies4 including those with only older
stellar populations, and to have no association with optical supernovae.5 This has
bolstered the long-standing idea that they are most likely produced by the merger
of two compact objects (two neutron stars, or a neutron-star black-hole binary).
The mean redshift appears to be lower than the L-GRBs, although increasingly it
appears that some can be comparably luminous and distant.6,7
In a parallel development, a reanalysis of the S-GRBs localised by
CGRO/BATSE has found evidence for a more local population.8 Individually the
error circles for these bursts, at several degrees, are too large to provide an
unambiguous identification of host galaxies. However, cross-correlation of burst positions
with the positions of galaxies in the local universe indicates that between 10 and
25% of BATSE S-GRBs seem to be associated with galaxies at a distance of less
than wlOO Mpc (ie. z « 0.025).
How plausible is this finding? The occurence of an immensely powerful gamma-
2012
2013
ray flare from SGR 1806-20 in December 2004 puts a new complexion on this
question. The most intense spike of this event would have been detected by BATSE as
a short-hard gamma-ray burst had it occured in a galaxy out to about 50 Mpc.9
Most estimates of the volume average star formation rate in the local universe10
put it at about 0.02 MQ yr-1 Mpc-1. So in a sphere of radius 100 Mpc we would
expect to find a total rate of star formation roughly 20000 times the current rate in
the Milky Way. SGRs are thought to be young (and short-lived), highly-magnetised
neutron stars, and so their number within a galaxy should reflect its star-formation
rate (although it has been suggested that magnetars in old stellar populations may
be produced via WD-WD mergers11). Hence, even if an event like SGR 1806-20, or
somewhat brighter, were only to occur in the MW on average once every millenium
or so, ~20 per year should occur within this volume. This number is essentially
equivalent to the upper limit to the rate of local S-GRBs found in the correlation
analysis.
All this does raise another question, however, which is whether any of the small
number of well-localised S-GRBs to-date could plausibly have originated in local
galaxies? The answer to this is unclear, but there are three plausible candidates.
S-GRB 050906 was a weak event detected by Swift and had no clear X-ray or optical
afterglow. However, the BAT error circle also contains a galaxy IC 328 at only 130
Mpc distance. This galaxy is a star-forming spiral likely to host a number of SGRs
and a priori it is unlikely to find such a galaxy by chance within a BAT error
circle (Levan et al. 2007). However, the, galaxy is on the edge of the error circle,
which itself would be rather surprising if the GRB came from the galaxy itself.
Furthermore, the spectrum of GRB 050906 is not as hard as would be expected for
an SGR 1806-20-like event (Hurley et al. in prep.). Clearly this S-GRB could have
originated in one of the many more distant galaxies within the error circle.
Another candidate local S-GRB is the bright GRB 051103, which was discovered
in fact by the Inter-planetary network.13 To date, no afterglow has been reported
for this burst,14 although searches were compromised by the relatively large
positional error box, and the fact that the position only became available 58 hours
after it occurred. However, the relevance of this burst is that its error box overlaps
the outer regions of M81, which is at only ~ 4Mpc, and at that distance the
intrinsic luminosity of this event would have been very comparable to the spike in
the SGR 1806-20 giant flare. Finally, perhaps the most compelling example is the
recent bright burst, GRB 070201, whose IPN error box overlapped a large part of
the northern spiral disk of M31.15
3. Local Long-Duration GRBs
It was realised early that long-duration GRBs (L-GRBs) are preferentially found
in late-type, star-forming galaxies, suggesting a link to massive star core-collapse.16
However, the first direct evidence of such a link was the association of the highly
under-luminous and nearby GRB 980425 (z ~ 0.008) with an extreme type-Ibc
2014
supernova SN1998bw.17 The convincing proof that the same (or very similar)
type of progenitor was responsible for cosinological L-GRBs was the association
of GRB 030329 with the type-Ic supernova 2003dh at z ~ 0.17.18
For a long time GRB 980425 remained essentially in a class of its own.
Recently another burst, GRB 060218, was also found to be associated with a low
redshift supernova, SN2006aj, at z ~ 0.033.l9 Again this was an very low
luminosity burst, and in this case an extremely long-lived prompt phase. The implications
of these two bursts is that the rate density of such low-energy, long-duration events
in the local universe,19,20 although very uncertain, is likely to be between 100 and
700 Gpc~3 yr"1. This conclusion has recently been bolstered by the finding of a
weak correlation, consistent with the above event rate, between the local galaxy
distribution out to about 150 Mpc, and a subset of the long-duration BATSE bursts
chosen to have low fluence and smooth, single-peaked light curves.21
4. Conclusions
Although the bulk of observed GRBs originate at cosinological redshifts, it is clear
that a significant fraction of previously detected long- and short-GRBs are low
luminosity events in the nearby universe. The numbers are such that they likely
dominate the rate density, being intermediate between the cosmological GRBs and
core-collapse supernovae.
References
1. C. Kouveliotou, et al. Astrophysical Journal 413, L101 (1993).
2. J. van Paradijs, et al., Nature 386, 686 (1997).
3. N. Gehrels, et al., Nature 437, 851 (2005).
4. E. Nakar, arXiv:astro-ph/0701748.
5. J. Hjorth, et al., Astrophys. J. 630, L117 (2005).
6. A. J. Levan, et al. Astrophysical Journal 648, L9 (2006).
7. E. Berger, et al. arXiv:astro-ph/0611128.
8. N. R. Tanvir, R. Chapman, A. J. Levan, and R. S. Priddey, Nature 438, 991 (2005).
9. K. Hurley, et al., Nature 434, 1098 (2005).
10. J. Iglesias-Paramo, et al., astro-ph/0601235.
11. A. J. Levan, et al., Monthly Notices Royal Astronomical Society 368, LI (2006).
12. A. J. Levan, et al., astro-ph/0705.1705.
13. D. D. Frederiks, et al., Astronomy Letters 33, 19 (2006).
14. E. O. Ofek, et al., Astrophysical Journal 652, 507 (2007).
15. S. Golenetskii, et al., GCN Circular, 6088 (2007).
16. B. Paczynski, Astrophysical Journal 494, L45 (1998).
17. T. J. Galama, et al., Nature 395, 670 (1998).
18. J. Hjorth, et al., Nature 423, 847 (2003).
19. E. Pian, et al., Nature 442, 1011 (2006).
20. A. M. Soderberg, et al., Nature 442, 1014 (2006).
21. R. Chapman, et al., Monthly Notices Royal Astronomical Society, submitted.
THE ANALYSIS OF GRB REDSHIFT DISTRIBUTION
IRENE V. ARKHANGELSKAJA
Moscow Engineering Physics Institute (State University), Kashirskoe shosse, 31
Moscow, 115409, Russia
At the middle of December 2006 the volume of GRB set with known redshift consisted of
approximately 100 bursts, mostly localized by SWIFT. In this article the GRB redshift distribution is
presented and its shape is discussed. Analysis of single peak approximation of GRB redshift
distribution, have shown that it has very heavy tail which consists of 37% of volume set. As example
of real uniform set the shape of normalized z-distribution for first 604 QSO from 2QZ 6QZ catalog
and for some SNIa are analyzed and it is shown that 2-9% events in dependence of amount of
sampling must be in the tails of distributions above 3oTevels. So, this fact allows to make conclusion
that GRB sources set is not uniform and at least two subgroups could be separated in GRB redshift
distribution at 95% confidence level which limited by volume of GRB set with known redshift. This
conclusion confirmed by analysis of chi-square for two-peaks function approximation, which gives
more significant result than one-peak fit (95% and 70% correspondingly).
1 Introduction
Up to now the redshifts were defined for approximately 115 GRB [1], a half of them
were localized by SWIFT. Such volume of set is sufficient to analysis of GRB source
population uniformity in the first approximation.
But we must take into account some important notices:
1. Most part of GRB source models (see for example [2] and so on) does not give a
limitation for source distances except evidences of their cosmological origin. These
models only take into account difference between GRB types in dependence of their
duration and spectra.
2. Most part of GRB with known z are not a short bursts: only 6 GRB with t90< 2 s had
sufficient localization accuracy for their sources redshift definition ~ 5% from whole set.
3. Unfortunately t90 has dependence from instrument registered this burst - it is
function of detector sensitivity threshold and operation energy band - for example,
tgo_GRB06o4i8_swiFT/BAT ~ 52 s [3] and tgo_c,RBo6o4i8__RHEssi ~ 36 s [4]. This difference
caused by GRB spectral behavior and differences of sensitivity threshold and
operation energy band between RHESSI and SWIFT. Up to now 13 GRB with
defined t90 were observed at the same time by SWIFT and RHESSI: for 6 of these
bursts t90 swift/bat ^ 2 twjwEsst, for 5 GRB these values are comparable and
t9o_GRB051221_RHESSI ~ 1,4 X t90CRB051221_SmFT_BAT, t90GRB061121 RHESSI ~ 1,2 X
19o_grbo6u2i_swift/bat [3-4], So, we must take into account these differences in our
investigation of various distributions in duration, for example for z-t90 distribution.
2 GRB redshift distribution
The GRB sources redshift distribution analysis allows us to investigate the uniformity of
this population. But at first we must decide which sources will give us real uniform
distribution on redshift. There are at least 2 more or less uniform populations of sources
2015
2016
correspond to real uniform distributions on redshift - la type supernova and quasars
located at high redshifts. At first we consider redshift distribution properties for these
populations. High redshift supernova with class la were used for definition of Q and A
for our Metagalaxy [5]. The redshift distributions for 42 supernovae used in this work
and for 52 ones from Daly and Djorgovski catalogue [6] are shown at Figure la. This
distribution is well fitted by one-peak function (see Table 1). Distribution for SN la has
(9±5)% objects in tails - outside 3 a level, but 52 events contain very small set. The QSO
redshift distribution (see Figure lb and Table 1) has (2.3±0.2)% outside 3 a level.
24
22-
?S1
116
114
*12-
110
i 8
5 6
4
2
$2.
2 0
aj : 140
fl-l
-52 SNIa
-42 SNIa
outside
3 cr level
J
.[■:yi..{,.,}.,4:
0.2 0.4 0.6 7 0.8 1.0 1.2 1.4
Figure 1. One-peak functions fits for: a) high redshift SN la (black histogram correspond to 42 objects were
used for definition of i3and A for our Metagalaxy [5], gray one - to 52 SN la from [6]), b) first 604 QSO from
"QSO and AGN"[7] and 2QZ-6QZ catalogues [8].
GRB redshift distribution is fitted by one-peak function only at 70% significance level because
-20%) of events (23 GRB) are in tail of this distribution (see Figure 2a and Table 1) - shape of
this distribution is quite different from one for uniform set, for example for SNIa and QSO. But
significance level of this distribution for two-peaks fit is 95%> (it is limited by only volume of
GRB set with known z) and (4±2)%> GRB are outside 3 cr level for this fit (see Figure 2b and
Table 1). So, at least two subgroups can be separated in GRB redshift distribution and GRB
sources population is not uniform. We suppose that one criterion of separation some subgroups
in GRB with tgtp- 2 s is the presence of high energy emission during GRB - for more than 40
Table 1. The parameters of redshift distribution for SNIa, QSO and GRB.
Fits parameters
Zi
0"l
Z2
a2
Zmux
significance level
amount of sampling
% outside 3 a level
SNIa (one-peak fit)
0.42 ± 0.01
0,19 ±0,02
-
-
1.7(SN1997ff)
95%
52
(9±5)% -> 5 SN
QSO (one-peak fit)
1 J8±0.03
1.25±0.042
-
-
5.85 (SDSS J00058-0006)
97%
604
(2.3±0.2)% -> 5 QSO
GRB (one-peak fit)
1.0±0.1
0.8±0.1
-
-
GRB (two-peaks fit)
0.89±0.07
0.72±0.13
2.8±0.4
2.8±0.6
6.29 (GRB50904)
70%
115
(20t4)% -> 23 GRB
95%
115
(4±2)% -> 4 GRB
GRB ^emission up to 200 MeV and for 6 GRB ^emission up to 2 GeV is observed within
2017
BATSE tgo intervals [9], and for some GRB ^emission up to 140 MeV is detected within
RHESSI tgo intervals [10]. Another criterion is GRB duration - subgroup of intermediate GRB
with duration of 2-10s was found some years ago in BATSE GRB duration distribution [11-13].
Figure 2. GRB redshift distribution for 115 bursts and its fits: a) one-peak function, b) two-peaks one.
3 Conclusions
The analysis of GRB sources redshift distribution allows us to make 3 conclusions:
1. The population of GRB with known redshift is not uniform;
2. At least 2 subgroups exist in population of GRB with /<_> 2s. May be the hardness of
spectra and the presence of high-energy y-emission could be criteria to separation
these subgroups;
3. It is impossible to use whole GRB subset with known redshifts as "standard candles"
for various cosmological tests - at first different GRB subsets must be separated.
References
1. http://www.mpe.mpg.de/~jcg/grbgen.html
2. V. Baran, M. Colonna, M. Di, T. Piran, Rev. of Modern Physics, 76, Issue 4, 1143
(2005).
3. http://gcn.gsfc.nasa.gov/swift_grbs.html
4. http://grb.web.psi.ch/grb_list_2005.html, http://grb.web.psi.ch/grbjist_2006.html
5. S. Perlmutter, G. Aldering, G. Goldhaber et al., Astrophys. J., 517, 565 (1999).
6. R. A. Dali, S. G. Djorgovski // astro-ph/0403664, (2004).
7. http://cdsweb.u-strasbg.fr/viz-bin/VizieR-3
8. http://www.2dfquasar.org/Spec_Cat/cat/2QZ_6QZ_pubcat.txt
9. B.L. Dingus, L., P. Sreekumar, E. J. Schneid et al, AIP Conf. Proc. 307, 22 (1994).
10. I. V. Arkhangelskaja, A.I. Arkhangelskiy, A. S. Glyanenko et al, Proceedings of XI
Marsel Grossman Meeting, in press (2007).
11. Z. Bagoly, A. Meszaros, L. Balazs et al, Astronomy and Astrophysics, 453, 797
(2006)
2018
12. I. V. Belousova, A. Mizaki, T. M. Roganova and I. L. Rosental', Astronomy Reports
43, JV°11,752 (1999).
13. I. Horvath; L.G. Balazs, Z. Bagoly et al. Astronomy and Astrophysics, 2006, 447,
Issue 1, 23 (2006).
FUNDAMENTAL PROPERTIES OF GRB-SELECTED GALAXIES:
A SWIFT/VLT LEGACY SURVEY
PALL JAKOBSSON
Centre for Astrophysics Research, University of Hertfordshire, College Lane,
Hatfield, Herts, ALIO 9AB, UK
JENS HJORTH and JOHAN P. U. FYNBO
Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen,
Juliane Maries Vej 30, 2100 Copenhagen, Denmark
JAVIER GOROSABEL
Instituto de Astrofisica de Andalucia (CSIC), Apartado de Correos 3004, 18080 Granada, Spain
ANDREAS O. JAUNSEN
Institute of Theoretical Astrophysics, PO Box 1029, 0315 Oslo, Norway
We present the motivation, aims and preliminary result from the Swift/VLT legacy
survey on gamma-ray burst host galaxies. This survey will produce a homogeneous and
well-understood host sample covering more than 95% of the lookback time to the Big
Bang, and allow us to characterize their fundamental properties.
1. Introduction
With a very broad redshift distribution and a mean redshift of around z = 2.8, *
gamma-ray bursts (GRBs) are becoming extremely useful tracers of star-forming
galaxies. Long-duration GRBs are known to be associated with the deaths of
shortlived massive stars2 and thus have the essential advantage that their detection
requires only a single stellar progenitor. Therefore, their detection is in principle
independent of host galaxy luminosity.
The Swift satellite and a suite of ground-based observatories are detecting,
localizing and studying a large homogeneous sample of GRBs. To take advantage of
this unique sample, we have launched a dedicated programme aimed at building
up a sample of host galaxies, based on Swift detections and VLT follow-up. This is
a Large Programme to be executed over a period of two years. The resulting host
sample will be largely unaffected by dust extinction and entirely independent of
host galaxy luminosity. A more thorough description of the survey and preliminary
results are presented in Hjorth et al. (in prep).
The details of the sample selection are relatively straight-forward, i.e. the GRBs
have to be well-placed for optical follow-up observations: (i) Detected by Swift after
1 March 2005 when it was fully operational and automatically slewing, (ii) An X-
ray position is available, obtained by the Swift XRT detector, (iii) The Galactic
extinction is less than Ay < 0.5 mag. (iv) Declination favorable for VLT and not at
a polar declination, i.e. —70° < dec < 25°.
2019
2020
1
ORB 050416A (R)
(z = 0.65)
4
ORB050«)15A(R)
S! {/. = 0.94j
GRB 050416A(K>
ORB 050915A (K)
i ■
1
GRB0510I6B.(K|
*
► * 4
Fig. 1. A mosaic of three of the targets; left column displays the fi-band while the /C-band is in
the right column. The host galaxy is detected in both bands for all targets, and is located inside
the revised6 XRT error circle in each case (solid circle). Each host galaxy also coincides with the
corresponding optical afterglow. The GRB 050915A host and all the if-band host detections have
not been reported before. North is up and east left in each panel which is 20" on a side.
2021
2. Aims
The concrete goals of the programme are to: (i) Identify the GRB hosts, reaching a
limit of around R = 27.0 and K = 21.5, which will allow us to detect extremely red
objects. For non-detections of hosts we will spend additional time to reach a limit of
around R = 28.0. While hosts have been detected for nearly all pre-Swift, localized
GRBs, almost none have been detected in the Swift era. (ii) Measure redshifts for
GRBs without absorption redshifts. (iii) Search for the Lya emission line when
possible, i.e. for bursts with a known redshift z > 2. (iv) Study the effects of dust
reddening within hosts, (v) Determine the host luminosity function. Finally, we will
perform detailed studies of particularly interesting targets, e.g. short-duration GRB
hosts and very bright hosts. Specifically, we will carry out emission line diagnostics,
e.g. metallicity estimates via the R23 method.3
3. Results
The final host sample is expected to consist of approximately 70 galaxies of which
a major fraction will have redshifts. The programme so far has consisted mostly of
target build-up, observational preparation, data taking and preliminary analysis. To
date, only six months after the start of the programme, we have completed roughly
half of item (i) above; R- and X-band imaging of three of the hosts is displayed in
Fig. 1 as an example. The current average and median i?-band magnitude of the
sample is fainter than 25.5.
With this programme, we hope to detect a number of faint galaxies (such as
the GRB 030323 host4) that possibly dominate5 the total star-formation density at
z > 2, but are impossible to find and study by other methods than GRB selection.
But most importantly, we will produce a coherent sample of GRB host galaxies for
future follow-up with the HST, Spitzer, VLT, and later with ALMA and JWST.
References
1. P. Jakobsson, et al. A&A, 447, 897 (2006).
2. J. Hjorth, et al. Nature, 423. 847 (2003).
3. J. Gorosabel, et al. A&A, 444, 711 (2005).
4. P. Vreeswijk, et al. A&A. 419, 927 (2004).
5. P. Jakobsson, et al. MNRAS, 362, 245 (2005).
6. N. R. Butler, A.J, submitted, astro-ph/0611031.
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GRB Observations
by SWIFT
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THE SWIFT XRT: EARLY X-RAY AFTERGLOW
GIANPIERO TAGLIAFERRI
INAF - Osservatorio Astronomico di Brera, Via Bianchi ^6, 1-23807 Merate, Italy
gianpiero.tagliaferri@brera.inaf.it
Thanks to the X-ray Telescope (XRT) on board the Swift satellite, we have now the
X-ray light curves of hundreds of bursts on time scales from ~ 1 minute up to weeks and
in some cases months from the burst explosion. This database allow us to investigate the
physics of the highly relativistic fireball outflow and its interaction with the circumburst
environment. Unexpectedly, these X-ray light curves in the early phases are characterised
by different slopes, with a very steep decay in the first few hundred of seconds, followed
by a flatter decay and, a few thousand of seconds later, by a somewhat steeper decay.
Often strong flare activity up to few hours after the burst explosion is also seen. One
possible interpretation is that the central engine activity last much longer than expected,
still dominating the X-ray light curve well after the prompt phase, up to a few thousand
of seconds. The flatter phase is probably the combination of late-prompt emission and
afterglow emission. When the late-prompt emission ends the light curve steepens again.
Also the late evolution of the XRT light curves is puzzling, in particular many of them
do not show a "jet-break". Although there are various possibilities to explain these
observations, a clear understanding of the formation and evolution of the jet and of the
afterglow emission is still lacking.
1. Introduction
The gamma Ray Burst (GRB) studies in the pre-Swift era showed that the
afterglows associated with GRBs are rapidly fading sources, with X-ray and optical light
curves characterised by a power law decay oc t~a with a -=-1 — 1.5. Moreover, while
most of the GRBs, if not all, had an associated X-ray afterglow only about 60% of
them had also an optical afterglow, i.e. a good fraction of them were dark-GRBs
(see1 for a general discussion on GRBs and their afterglows). Therefore, it was clear
that to properly study the GRBs, and in particular the associated afterglows, we
needed a fast-reaction satellite capable of detecting GRBs and of performing
immediate multiwavelength follow-up observations, in particular in the X-ray and optical
bands. Swift is designed specifically to study GRBs and their afterglows in multiple
wavebands. It was successfully launched on 2004 November 20, opening a new era in
the study of GRBs.2 Swift has on board three instruments: a Burst Alert Telescope
(BAT) that detects GRBs and determines their positions in the sky with an
accuracy better than 4 arcmin in the band 15-150 keV;3 a X-Ray Telescope (XRT) that
provides fast X-ray photometry and CCD spectroscopy in the 0.2-10 keV band with
a positional accuracy better than 5 arcsec;4 an UV-Optical Telescope (UVOT)
capable of multifilter photometry with a sensitivity down to 24th magnitude in white
light and a 0.5 arcsec positional accuracy.5
In the first two years of operation Swift has detected about 200 GRBs. Soon after
detection the satellite autonomously determines if it can repoint the narrow field
instruments to the burst location and, if possible, it usually slews to the source in
less than 100-150 seconds. Therefore, we have now X-ray light curves of hundreds of
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2026
bursts that cover a time interval from few tens of seconds up to weeks and months for
some of the bursts. As expected, the most spectacular results have been obtained in
the first few thousand seconds, i.e. in the gap not covered by the previous missions.
In particular, the XRT observations have shown that the burst X-ray light curves
in the early phases are much more complex than a, simple backward extrapolation
of the power law light curves observed few hours after the GRB explosion. Here we
will outline the most relevant results that have been obtained so far thanks to the
XRT observations.
2. The early phases X-ray light curves
The XRT data presented us with expected but also unexpected results. The XRT
confirmed that essentially all long GRBs are accompanied by a X-ray afterglow,
there are only a couple of them that have been fastly repointed by Swift and do not
have an associated X-ray afterglow.6 But, for instance, the XRT data do not show
the presence of spectral lines whatsoever in the X-ray spectra of GRB afterglows,
neither in the first few thousand second, nor at later (hours-days) time scales. They
do show the presence of a bright fading X-ray source. However, the source decay
does not follow a smooth power law, rather it is usually characterised by a very
steep early decay7 followed by a flatter decay and then a somewhat steeper decay8
(see Fig. 1, left panel). Although this is the most common behaviour, in some of
the Swift GRBs, the early X-ray flux follows the expected and more gradual power
law decay.9,10
Time since burst (t-t0) (s]
Fig. 1. Left panel: the X-ray light curve of some Swift GRBs. Note the different decaying
behaviours detected in the early phases and described in the text (figure from8). Right panel: the
X-ray light curve of GRB050713A, showing various strong flares both during the steep and flatter
decay phases. The underline X-ray light curve does not seem to be altered by these flares.
The most likely explanation for the steep early decay is that this is still due
to the prompt emission. Thanks to the fast reaction of the Swift satellite often we
are able to detect the prompt emission also with the XRT telescope and the steep
2027
decay that we are observing is probably due to the "high-latitude emission" effect:
when the prompt emission from the jet stops, we will still observe the emission
coming from the parts of the jet that are off the line of sight. This interpretation is
supported by the fact that the prompt BAT light curve converted in the XRT band
joins smoothly with that one seen by XRT for almost all of the Swift GRBs.n'12 The
origin of the flatter part that follow the early steep decay, that is well represented
by a power law with slope 0.5 ;$ a ;$ 1, is more controversial. The total fluence that
is emitted during this phase is comparable to, but it does not exceed that one of the
prompt phase.12 It is probably a mixture of afterglow emission (the forward shock)
plus a continuous energy injection from the central engine that refreshes the forward
shock. When this energy injection stops, the light curve steepens again to the usual
power law decay already observed in the pre-Swift era.8 Not all bursts show the
steeper+flatter parts, a significant minority of them show a more gradual decay
with a ;$ 1.5. These are more consistent with the classical afterglow interpretation
in which the X-ray emission is simply due to the external shock. The natter part
is not seen either because in these cases the continuous activity from the internal
engine is not present, or because the afterglow component is much brighter and it
dominates over the internal contribution.
2.1. The flares
The first flares detected by XRT were those of GRB050406 and of
GRB050502B.13-15 This results came as a full surprise (although X-ray flares were
already detected by BeppoSAX in a couple of bursts, which were interpreted as due
to the onset of the afterglow16). We now know that X-ray flares are present in a
good fraction of the XRT light curves.17 Flares have been detected in all kinds of
bursts: in X-ray flashes (XRF),14 in long GRBs (e.g.15,18'19), including the most
distant one at redshift z=6.2920 and in short GRBs.21'22 These flares are usually
found in the early phases up to a few thousand of seconds, but in some cases they
are also found at > 10 thousand seconds. The ratio between their duration and peak
time is very small, ~ 0.1, with late flares having longer duration.17 They can be
very energetic and in some cases can exceed the fluence of the prompt emission.15
The fact that in the X-ray light curve of the same GRB there are more than one
flare argues against the interpretation that the flares correspond to the onset of
the afterglow. Moreover, they do not seem to alter the underlying afterglow light
curve that after the flare follows the same power law decay as before the flare (see
Fig. 1, right panel). Therefore, since the beginning it was clear that these flares
were correlated to the central engine activities and not to the process responsible
for the afterglow emission. For a comprehensive analysis of the flare properties see
references.17,23'24
2.2. Any eveidence for a jet break?
If the afterglow emission is collimated in a jet, then we expect to see an achromatic
break in the power law decay at the time when the full jet opening angle becomes
2028
visible to the observer. The detection of this break is important for the evaluation
of the beaming factor, in order to determine the total energy emitted by the burst.
Breaks were detected a few days after the explosion in the optical and radio light
curves of pre-Swift bursts. If interpreted as jet-breaks, then the correct total energy
emitted in the gamma band by the prompt clusters around 1051 ergs.26
If these breaks are really due to a jet, then they should be seen simultaneously
also in the X-ray band. Before the advent of Swift the observations in the X-ray
band were limited and there were only few measurements. Now thanks to XRT we
have many detailed X-ray light curves and the picture is not so clear any more.
First of all as we have seen, in the early phases there can be more than one break,
but none of them seems to be due to a jet-break. Rather they are probably due to
the activity of the internal engine, as we have seen previously. Moreover, for some
of these bursts we have also the early optical data and the breaks are not seen in
the optical, therefore they are not achromatic. This behaviour can be explained
either by assuming an evolution of the microphysical parameters for the electron
and magnetic energies in the forward shock or by assuming that the X-ray and
optical emission arise from different components.27 From a systematic analysis of
the XRT light curves of 107 GRBs, 72 afterglow breaks are found, but of these only
12 are consistent with being jet-breaks and only 4 are not related to the early flat
phase.28 In other words there are only 4 breaks that are good candidates for being
jet breaks. Therefore, contrary to the earlier expectations, jet-breaks seem to be the
exception and not the rule in the X-ray light curves of GRB afterglows.
3. Conclusions
After more than two years of Swift operations, the data provided by the XRT allowed
us to make break-through discoveries in various field of the GRB studies including
the detection of the afterglows of short GRBs. We did not discuss this argument
here, but for the first time we have been able to study in more details the properties
of these elusive sources and to find and study their host galaxies with on ground
follow-up. Thanks to the Swift fast repointing and its instrumentation capabilities,
we have now the fast localisation of GRB with an accuracy of few arcsec, which
allows us to immediately start ground-based observations. Uniform multiwavelength
light curves of the afterglows are available starting from ~ 1 minute after the burst
trigger. In particular, in the X-ray band, thanks to XRT, we have hundreds of light
curves spanning the range from few tens of seconds up to weeks and months after
the explosion. These data allow us to investigate the physics of the highly relativistic
fireball outflow and its interaction with the circumburst environment.
Unexpectedly, these X-ray light curves are characterised by different slopes in
the early phases and often by the presence of strong flare activity up to few hours
after the burst explosion. The picture that is consolidating is that the central engine
activity lasts much longer than expected and it is still dominating the X-ray light
curve well after the prompt phase, up to a few thousand of seconds. The external
2029
shock, the real afterglow, takes over the emission only after the end of the natter
phase, although some flare activity can be still detected during these later phases.
Finally, even the evolution of the XRT light curve at the later phases is providing
more questions than solutions. In particular, the lack of a "jet-break" in many
of these light curves is puzzling. There are various possibilities to explain these
observations (e.g.time evolution of the microphysical parameters, structured jet).
However, a clear understanding of the formation and evolution of the jet and of the
afterglow emission is still lacking.
Acknowledgments
This work was supported by ASI grant I/R/039/04 and MIUR grant 2005025417.
We gratefully acknowledge the contributions of dozens of members of the XRT team
at OAB, PSU, UL, ASDC, IASF-Pa and GSFC.
References
1. Zhang, B., Meszaros, P., Int. Journ. Mod. Phys. A, 19, 2385 (2004)
2. Gehrels N., Chincarini G., Giommi P., et al., ApJ, 611, 1005 (2004)
3. Barthelmy S., Barbier L.M., Cummings J.R., et al., SSRv, 120, 143 (2005)
4. Burrows D.N., Hill J.E., Nousek J.A., et al., SSRv, 120, 165 (2005)
5. Roming P.N., Kennedy T.E., Mason K.O., et al., SSRv, 120, 95 (2005)
6. Page K.L., King A.R., Levan A.J., et al., ApJ, 637, L13 (2006)
7. Tagliaferri G., Goad M., Chincarini G., et al., Nature, 436, 985 (2005)
8. Nousek J.A., Kouveliotou C, Grupe D., et al., ApJ, 642, 389 (2006)
9. Campana S., Antonelli A., Chincarini G., et al., ApJ, 625, L23 (2005)
10. Chincarini G., Moretti A., Romano P., et al., asiro-p/i/0506453 (2005)
11. Barthelmy S., Cannizzo J.K., Gehrels N., et al., ApJ, 635, 1133 (2005)
12. O'Brien P.T., Willingale R., Osborne J., et al., ApJ, 647, 1213 (2006)
13. Burrows D.N., Romano P., Falcone A., et al., Science, 309, 1833 (2005)
14. Romano P., Moretti A., Banat P.L., et al., A&A, 450, 59 (2006)
15. Falcone A., Burrows D.N., Lazzati D., et al., ApJ, 641, 1010 (2006)
16. Piro L., De Pasquale M., Soffitta P., et al., ApJ, 623, 314 (2005)
17. Chincarini G., Moretti A., Romano P., et al., ApJ sub. astro-ph/0702371 (2007)
18. Guetta D., Fiore F., D'Elia V., et al., A&A, 461, 95 (2006)
19. Pagani C, Morris D.C., Kobayashi S., et al., ApJ, 645, 1315 (2006)
20. Cusumano G., Mangano V., Chincarini G., et al., A&A, 462, 73 (2007)
21. Barthelmy S.D., Chincarini G., Burrows D.N., et al., Nature, 438, 994 (2005)
22. Campana S., Tagliaferri G., Lazzati D., et al., A&A, 454, 113 (2006)
23. Falcone A., Morris D., Racusin J., et al., ApJ sub. (2007)
24. Liang E.W., Zhang B., O'Brien P.T., et al., ApJ, 646, 351 (2006)
25. Rhoads J.E., ApJ, 525, 737 (1999)
26. Frail D.A., Kulkarni S.R., Sari R., et al., ApJ, 562, 55 (2001)
27. Panaitescu A., Meszaros P., Burrows D., et al., MNRAS, 369, 2059 (2006)
28. Willingale R., O'Brien P.T., Osborne J.P., et al., ApJ in press astro-ph/0612031 (2007)
INITIAL RESULTS FROM SWIFT/UVOT
F.E. MARSHALL
Astrophysics Science Division, Goddard Space Flight Center, Greenbelt, MD 20771 USA
frank.marshall@gsfc.nasa.gov
The UltraViolet/Optical Telescope on Swift is a 30-cm Richey-Cretien reflector that
provides sub-arcsecond positions, light curves on time scales from minutes to days after
a GRB, and spectra covering 160 - 700 nm. UVOT is detecting optical afterglows from
~67% of rapidly observed long GRBs that have little extinction in the Milky Way. We
consider possible reasons for the non-detections. UVOT is also producing remarkable
results on the evolution of GRB afterglows including GRB060218/SN2006aj as well as
the evolution of young supernovae.
1. Introduction
UVOT is one of three instruments on Swift,1 a NASA-managed mission with
international partners whose primary goal is to study gamma-ray bursts (GRBs). Images
taken with UVOT are used to determine the position of GRBs with an accuracy of
~ 0.5 arc seconds, and there are 6 niters that are used to determine the spectrum of
a GRB afterglow in the wavelength range from 160 to 700 nm. An additional white
filter covers this entire wavelength range. UVOT also has two grisms that provide
better spectral resolution for bright sources, but they have yet to be used for GRBs.
Most of the time Swift is observing afterglows of recent GRBs while the Burst
Alert Telescope (BAT) is simultaneously searching for new GRBs. When a new
burst is detected, BAT performs an on-board determination of its position on the
sky with an accuracy of ~3 arc minutes, the spacecraft autonomously slews the
observatory to point the X-Ray Telescope (XRT) and UVOT at the burst, and the
UVOT begins a pre-determined sequence of exposures. Currently the first exposure
uses the White filter for 100 seconds, and exposures with the each of the other filters
follow. Limited data from the first exposure and up to 3 other exposures are sent
immediately to the ground using NASA's Tracking and Data Relay Satellite System,
and then distributed to the world using the Gamma-ray Coordinates Network.
The average time delay between a GRB trigger and the start of the first UVOT
exposure is ~109 seconds if no observing constraint prevents an immediate
spacecraft maneuver. Typically afterglows as weak as 19.5 magnitude can be detected
with the initial exposure with the White filter. If UVOT detects an afterglow, it
provides the most accurate position of any of the Swift instruments with a typical
uncertainty of ~ 0.5 arc seconds.
2. Detecting GRBs with UVOT
X-ray afterglows are seen for almost all (>90%) of long GRBs, but finding optical
afterglows (OAs) has been more difficult. In the Beppo-Sax era, only ~30% of GRBs
with accurate positions had OAs.2 With accurate, quickly distributed positions from
HETE-2, the fraction increased3 to ~90%. In the early part of the Swift mission
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2031
-25% of Swi/t-detected GRBs were detected with UVOT and an additional -22%
were detected with ground-based observations.4
UVOT greatly improved its sensitivity to bursts on March 15, 2006, by using the
White (160-650 nm) filter for the initial (finding chart) exposure. UVOT's detection
capability is significantly reduced for bursts that cannot be observed quickly or for
which there is significant extinction in the Milky Way. Consequently we define a
sample of "good" bursts that are observed in <30 minutes and that have E(B-V) <
1.0. We also exclude short/hard bursts, which are a small fraction of Swift-detected
GRBs. All these criteria require only information that is immediately available after
a GRB trigger. Eighteen of the 33 Swi/t-detected GRBs between March 15, 2006,
and July 14, 2006, satisfy these criteria, and UVOT has detected 12 of these bursts
in finding chart exposures. As shown in Fig. 1, the magnitude measured in the initial
exposure spans —6 magnitudes with a typical value of —18. Two additional bursts
were detected with UVOT in subsequent, longer exposures. Including ground-based
observations, OAs were eventually detected for 17 of the 18 GRBs.
16 18
White Magnitude
Fig. 1. The distribution of magnitudes for UVOT-detected afterglows.
UVOT is now providing a useful sample of OAs with relatively simple selection
effects. All the observations start —100 s after the trigger, and provide continuous
coverage for -2000 s. All 7 UVOT lenticular filters (White, V, B, U, UVW1, UVM2,
and UVW2) are used, but the longest exposures are with the White and V filters.
With the current, modest sample of f2 early detections, no correlation is found
between the initial optical magnitude and the burst fiuence (15 - 150 keV), peak
burst flux, or simultaneous X-ray flux. More sensitive searches will be possible as
the sample grows during the life of the Swift mission.
There are several plausible reasons for a GRB not to be detected with UVOT.
Large extinction in the host galaxy5 can significantly reduce the sensitivity of UVOT
even while have a small effect on the sensitivity of XRT or BAT. This is especially
true for the White filter with its broad band pass extending into the UV. There are
2032
two reasons to expect that extinction does not seriously affect most GRBs. First,
most non-detections are also dim in the R band in later ground-based observations.
The non-detections also do not show large absorption in the X-ray band, and the
detected afterglows typically have very low dust-to-gas ratios. A second possibility
is that the GRBs are at large distances, and the Lya edge has been redshifted into
the UVOT band. At a redshift of 5, the Lya edge is in the middle of the band of the
reddest UVOT filter. The number of such bursts is of great interest as it provides
insight into star formation in the early universe. Recent studies6 predict that 7% to
10% of GRBs could have redshifts > 5. One of the six UVOT non-detections (GRB
060510B) has a redshift 4.9,7 and the others have unknown redshifts.
The initial results suggest that few long bursts are truly "dark" and that a large
fraction of the non-detections could be due to high redshift bursts.
3. Conclusion
UVOT is a highly capable instrument that is detecting the majority of long GRBs
detected with Swift. Many of the afterglows are sufficiently bright that UVOT
produces detailed light curves and spectra for many days after the trigger. Examples
include the bright, relatively nearby GRB 050525A,8 and the most comprehensive
light curves to date of a short/hard burst.9 UVOT also provided the first optical/UV
view of the breakout of the blast wave from a GRB (GRB060218/SN2006aj).10
Finally, UVOT is providing the first extensive sample of early light curves in the UV
of Type 1A supernovae,11 which should eventually enable the distance scale of the
universe to be extended to higher redshifts.
References
1. N. Gehrels et al., ApJ 611, 1005 (2004).
2. J. U. Fynbo et al., A&A 369, 375 (2001).
3. D. Lamb et al., in Proc. of the 2nd BeppoSAX Conference: The Restless High-Energy
Universe, Nuclear Physics B Proceedings Supplement, Volume 132, p. 279-288 (2004)
4. P. Roming and K. Mason in Gamma-Ray Bursts in the Swift Era (AIP Conf.
Proceedings 836), p. 224 (2006).
5. P. J. Groot et al., ApJ, 491, L27 (1998).
6. V. Bromm and A. Loeb, ApJ, 575, 111 (2005); P. Jakobsson et al. A&A 447, 897
(2006).
7. P. A Price, GCN Circ. 5104 (2006).
8. A. J. Blustin et al., ApJ 637, 901 (2006).
9. P. Roming et al., ApJ 651, 985 (2006).
10. S. Campana et al., Nature 442, 1008 (2006).
11. S. Immler et al., ApJ 648, L119 (2006).
INVESTIGATION OF JET BREAK FEATURES IN SWIFT
GAMMA-RAY BURSTS *
G. SATO1-2, R. YAMAZAKI3, K. IOKA4, T. SAKAMOTO1'5, T. TAKAHASHI2.6,
K. NAKAZAWA2, T. NAKAMURA4, K. TOMA4, D. HULLINGER1-7-8, M. TASHIRO9,
A. M. PARSONS1, H. A. KRIMM1'10, S. D. BARTHELMY1, N. GEHRELS1,
D. N. BURROWS11, P. T. O'BRIEN12, J. P. OSBORNE12, G. CHINCARINI13-14 and
D. Q. LAMB15
1NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
2Institute of Space and Astronautical Science JAXA, Kanagawa 229-8510, Japan
3 Department of Physics, Hiroshima University, Higashi-Hiroshima 739-8526, Japan
4 Department of Physics, Kyoto University, Kyoto 606-8502, Japan 5 Oak Ridge Associated
Universities, P. O. Box 117, Oak Ridge, TN 37831, USA 6 Department of Physics, University of
Tokyo, Bunkyo, Tokyo 113-0033, Japan 7Department of Physics, Brigham Young University,
Rexburg, ID 83440, USA 8Department of Physics, University of Maryland, College Park, MD
20742, USA 9 Department of Physics, Saitama University, Saitama 338-8570, Japan
10 Universities Space Research Association, Columbia, MD 20744, USA 11 Department of
Physics, Pennsylvania State University, University Park, PA 16802, USA 12 Department of
Physics and Astronomy, University of Leicester, Leicester, LE 1 7RH, UK 13 Unwersitd degli
studi di Milano-Bicocca, Dipartimento di Fisica, Italy I4INAF-Osservatorio Astronomico di
Brera, Via E. Bianchi 46, 23807 Merate(LC), Italy 15Department of Astronomy and
Astrophysics, University of Chicago, Chicago, IL 60637, USA
We analyze Swift gamma-ray bursts (GRBs) and X-ray afterglows for three GRBs with
spectroscopic redshift determinations — GRB 050401, XRF 050416a, and GRB 050525a.
We find that the relation between spectral peak energy and isotropic energy of prompt
emissions (the Amati relation) is consistent with that for the bursts observed in pre-
Swift era. However, we find that the X-ray afterglow lightcurves, which extend up to
10-70 days, show no sign of the jet break that is expected in the standard framework of
collimated outflows. We do so by showing that none of the X-ray afterglow lightcurves
in our sample satisfies the relation between the spectral and temporal indices that is
predicted for the phase after jet break. The jet break time can be predicted by inverting
the tight empirical relation between the peak energy of the spectrum and the collirnation-
corrected energy of the prompt emission (the Ghirlanda relation). We find that there are
no temporal breaks within the predicted time intervals in X-ray band. This requires
either that the Ghirlanda relation has a larger scatter than previously thought, that the
temporal break in X-rays is masked by some additional source of X-ray emission, or that
it does not happen because of some unknown reason.
1. Introduction
It is widely believed that the GRBs arise from collimated outflows (i.e., jets). This
picture is supported by the break from a shallower to a steeper slope that is observed
in many afterglow light curves at around a day after the burst.1 These breaks are
interpreted as being due to the geometrical effect caused by the inverse of the bulk
Lorentz factor of the jet becoming larger than the physical opening angle of the
jet, and to a hydrodynamical transition of the jet (i.e., a broadening of the jet),
* This research has been partially supported by the Postdoctoral Fellowships for Research Abroad
(2006-) of the Japan Society for the Promotion of Science.
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2034
which is expected to occur shortly afterward. The break is therefore expected to be
independent of wavelength (i.e., achromatic).
2. Investigation of Jet Break Features
We investigate the presence or absence of a jet break in the X-ray afterglows of
recent Swift GRBs. According to Frail et al.(2001)2 and Bloom et al.(2003),3 given
the observed jet break time, we can calculate the jet opening angle and thereby the
collimation-corrected gamma-ray energy (E~,). After correcting for the jet collima-
tion, E1 shows a tight correlation with the peak energy E^ak of the vFv spectrum
in the source-frame: £"recak = AE7.520-706 (the Ghirlanda relation4). For Swift GRBs
in which one can obtain both E^ak and the isotropic-equivalent gamma-ray energy
Eiso = £7/(1 — cos#j), where #j is the opening-half angle of the jet, the Ghirlanda
relation can be inverted to predict the value of 0j, and hence the jet break time as
follows:
'» = OT<1+*Kdhr''(&r'3j^(^r),J" d^ (1)
The efficiency ??7 of the shock, and especially, the number density n of the ambient
medium, are poorly known for most bursts. In particular, n could easily lie anywhere
in a fairly wide range, where the majority are within 1 < n < 30 cm-3.5,6
Following the assumption made by Ghirlanda et al.(2004)4 for most of their
samples, we initially assume n = 3 cm-3 and ??7 = 0.2. Allowing A to vary from
1950 keV to 4380 keV in Eq. (1) then gives the time interval in which the jet
break is expected to occur if the Ghirlanda relation is satisfied, assuming these
values of n and rq1 (or equivalently, that nr/j = 0.6). Allowing n to vary between
1 — 30 cm-3 (or equivalently, 0.2 < nq1 < 6), in Eq. (1) gives the time interval
in which the jet break is expected to occur if the Ghirlanda relation is satisfied
without assuming a particular value of nrj-y. The intervals thus obtained are also
plotted in Fig. 1-2. The dash-dotted, dashed, and solid lines show the allowed time
intervals, without assuming a particular value of nrj-y and taking into account the
errors in Eiso and E^ak; assuming a particular value of n?77 and taking into account
the errors in E[so and E^ak; and assuming a particular value of nr/j without taking
into account the errors in E-sso and E^ak. The time interval in which the jet break
is expected to occur was completely observed for XRF 050416a and GRB 050525a,
but no temporal break is seen within the interval. The break at about 11000 sec
for GRB 050525a, which is close to the edge of the expected time interval, was
suggested to be a possible jet break because of its achromatic feature between X-
ray and optical bands.7 However, if we consider the discrepancy in the spectral and
temporal relations with the theoretical predictions as well, it is suggested that the
break is not a jet break. For GRB 050401, time intervals on both sides of the time
interval were observed and can be joined with a single power-law decay. Thus, none
of the three bursts exhibit a jet break within the time period required if they are
to satisfy the Ghirlanda relation.
2035
10°
109
} 1ff,°
e 10"
sL 10'12
X
-3 13
u- 10
10""
io-15
Time since trigger [day]
10"3 102 101 1 10
102 103 104 105
Time since trigger [s]
10°
107
Time since trigger [day]
10"3 10~2 10"1 1
102 103 104 10°
Time since trigger [s]
10°
102
107
Fig. 1. X-ray afterglow light curves of GRB 050401 (left) and XRF 050416a (right) in the 2-10
keV energy band. Sec text for more explanations.
Time since trigger [day]
10"3 102 101 1 10
102 103 10" 10° 10°
Time since trigger [s]
107
Fig. 2. The same as Fig. 1 but for GRB 050525a.
This requires either that the Ghirlanda relation has a larger scatter than
previously thought, that the temporal break in X-rays is masked by some additional
source of X-ray emission, or that it does not happen because of some unknown
reason.
References
1. Sari, R., Piran, T., & Halpern, J. P. 1999, ApJ, 519, L17
2. Frail, D. A., et al. 2001, ApJL, 562, L55
3. Bloom, J. S., Frail, D. A., Kulkarni, S. R. 2003, ApJ, 594, 674
4. Ghirlanda, G., Ghisellini, G., & Lazzati, D. 2004, ApJ, 616, 331
5. Panaitescu, A. & Kumer, P. 2001 ApJL, 560, L49
6. Panaitescu, A. & Kumer, P. 2002 ApJ, 571, 779
7. Blustin, A. J., et al. 2006, ApJ, 637, 901
RECENT RESULTS FROM THE SWIFT BURST ALERT
TELESCOPE *
HANS A. KRIMM for the SWIFT/BAT TEAM
Universities Space Research Association,
10211 Wincopin Circle, Suite 500, Columbia, Maryland 21044-3432, USA and
Center for Research and Exploration in Space Science and Technology
NASA Goddard Space Flight Center, Greenbelt, Maryland, 20111, USA
krimm@milkyway.gsfc.nasa.gov
The Burst Alert Telescope (BAT) on the Swift Gamma-Ray Burst MIDEX mission
has detected more than 200 gamma-ray bursts (GRBs), nearly all of which have been
followed up by the narrow-field instruments on Swift through automatic repointing, and
by ground and other satellite telescopes after rapid notification. Within seconds of a
trigger the BAT produces and relays to the ground a position good to three arc minutes
and a four channel light curve. An overview of the properties of BAT bursts and BAT's
performance as a burst monitor will be presented in this talk. BAT is a coded aperature
imaging system with a wide (~ 2 sr) field of view consisting of a large coded mask
located 1 m above a 5200 cm2 array of 32.768 CdZnTe detectors. All electronics and
other hardware systems on the BAT have been operating well since commissioning and
there is no sign of any degradation on orbit. The flight and ground software have proven
similarly robust and allow the real time localization of all bursts and the rapid derivation
of burst light curves, spectra and spectral fits on the ground.
1. Introduction
The Swift satellite1 carries three astronomical instruments that work together to
study all aspects of gamma-ray bursts (GRBs) over a wide range of energies. The
instruments are the Burst Alert Telescope2 (BAT), a coded aperture hard X-ray
telescope that serves as the GRB trigger for Swift, and the two Narrow-Field
Instruments (NFIs), the X-Ray Telescope3 (XRT), a grazing incidence X-ray telescope,
and the UltraViolet Optical Telescope4 (UVOT), a Ritchcy-Chretien telescope that
provides coverage into the optical band. A significant feature of Swift is the ability
to swiftly and autonomously slew to a newly detected GRB within ~70 s to allow
detailed multi-wavelength observations to be carried out with all three instruments.
The typical BAT trigger is a two-step process. First there is a rate trigger on one
of more than 400 criteria based on energy, time scale (4 ms to 16 s), and detector
quadrant. There are hundreds of rate triggers per day, most of which are rejected by
the image confirmation in which a background subtracted sky image is produced on
board and checked for the presence of an unknown source above a certain threshold.
If the image confirmation passes, then the normal burst procedure is initiated
including ground notification and automatic spacecraft repointing. Bursts longer than
16 s are found through the image trigger, in which sky images are produced and
scanned on board on time intervals ranging from 64 s to a full spacecraft pointing
(~ 20 minutes). The normal burst rate is 2-3 bursts per week.
*This research has been partially supported by the Swift project funded by NASA.
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2037
2. BAT observations of Gamma-Ray Bursts
Swift has detected 196 GRBs between Dec. 17, 2005 and Dec. 20, 2006. Nearly all
have been followed up with the Swift NFIs. GRB afterglows are observed starting
seconds after the burst and lasting for days to weeks in most cases. A large fraction of
these bursts have also been observed by ground-based telescopes and other satellites.
2.1. Short GRBs
Among the most significant findings from Swift is the localization of 15 short
gamma-ray bursts and the detailed studies of their afterglows. It has been seen
that short bursts are a nearer population than long bursts, with an average redshift
one-sixth as large as long bursts and isotropic energies a factor of 100 smaller. Unlike
long bursts, the host galaxies of short bursts have low star formation rates and are
located in both elliptical and dwarf host galaxies. These findings combine to point
to an old stellar population as progenitors and a likely origin as the coalescence
of degenerate binaries (either a pair of neutron stars or a neutron star-black hole
binary).
However, there are still unexplained mysteries about the short bursts studied
with Swift. GRB 0507245 is a good example of a short burst with unusual 7-ray and
X-ray properties. The X-ray afterglow lies off the centre of an elliptical galaxy at a
redshift of z = 0.258. The low level of star formation typical for elliptical galaxies
makes it unlikely that the burst originated in a supernova explosion. The afterglow
light curve showed evidence (such as late time X-ray flares) for continued energy
injection for at least ~ 200 seconds after the burst, a finding inconsistent with most
current neutron star- neutron star merger models, but a possibility for a neutron
star-black hole merger.6
Gravity waves and neutrinos are the only way to probe the actual merger since
the fireball itself is too dense for electromagnetic radiation to escape. Krimm et
al.7 have derived an expected detection rate of GRBs with the Advanced Laser
Interferometer Gravitational Wave Observatory (ALIGO) to be ~ 10 to > 100 yr~l.
If NS/BH mergers arc responsible, there will be a coincidence rate of ~ 2 yr^1
between Swift and ALIGO.
2.2. Burst Redshifts
As the brightest explosions in the universe, GRBs allow us to probe the early
universe back in time from the epoch of reionization at z > 6 to the end of the cosmic
"dark ages" when the universe first became transparent to light. Rapid
localization by Swift allows early absorption spectroscopy to determine the redshifts of
more bursts than ever before. Jakobsson et al.8 have shown that the mean redshift
of Swift bursts is zmean = 2.8, while the mean redshift of pre-Swift bursts was
zmean = 1-4- This is because Swift is able to locate and follow-up a fainter burst
population than ever before. There are pubished models9 in which the GRB rate
2038
tracks either star formation or decreasing metallicity. Jakobsson et al.s show that
the cumulative fraction of GRBs as a function of redshift closely matches several
of the models and as more burst redshifts are measured, the Swift sample will soon
be able to distinguish between models.
3. Correlative Observations
The response with the Swift observatory is enhanced by observations of the prompt
emission with other instruments including rapid response ground-based optical
telescopes and other satellites.
The burst GRB 041219A was observed by the RAPTOR instrument10 during
the prompt emission. In this case the optical light closely tracked the high energy
emission, showing that in this case the optical emission likely arose from internal
shocks. However another burst, GRB 050401, showed a different pattern. In this
case, observations with ROTSE-III11 showed the optical light following a smooth
afterglow light curve which was not correlated with the prompt emission. Here, the
prompt optical emission is likely due to an external shock, implying a very rapid
rise in forward-shock emission.
It has been observed that many bursts show a strong correlation between the
peak of the v—Fv spectrum, Epeak and the intrinsic burst luminosity.12 14 However,
given the narrow range (15-150 keV) over which BAT can do spectroscopy, it is
often difficult to determine Epeak from BAT data alone. Fortunately, there have
been more than 20 bursts which have been observed simultaneously with Swift-
BAT and either Konus-WIND, Suzaku-WAM, or HETE-II. These other satellites
have much greater energy range (though poorer angular resolution), so can provide
an accurate measurement of Epeak- Preliminary work on simultaneously detected
bursts15 shows that Swift bursts are consistent with previously published relations.
References
1. Gehrels N. et al ApJ 611 1005 (2004).
2. Barthelmy S. et al Space Sci. Rev. 120 143 (2005).
3. Burrows D. N. et al Space Sci. Rev. 120 165 (2005).
4. Roming P. W. A. et al Space Sci. Rev. 120 195 (2005).
5. Barthelmy S. et al Nature 438 994 (2005).
6. Davies M. B., A. Levan and A. King MNRAS 356 54 (2005).
7. Krimm, H. A. "Gamma-ray observations with Swift and their impact," in Proceedings
of TeV Particle Astrophysics II Workshop (Journal of Physics: Conference Series 2007)
8. Jakobsson, P. et al A & A 447 897 (2006).
9. Natarajan, P. et al MNRAS 364 L8 (2005).
10. Vestrand, T. et al. Nature 435 178 (2005).
11. Rykoff, E. et al ApJ 631 L121 (2005).
12. Yonetoku, D., et al, ApJ 609 935 (2004).
13. Amati, L., et al., A & A 390 81 (2002).
14. Ghirlanda, G, G. Ghisellini, and D. Lazzati, ApJ 616 331 (2004).
15. Krimm, H.A. et al, in Gamma-Ray Bursts in the Swift Era, eds S.S. Holt, N. Gehrels,
and J.A. Nousek, 145 (AIP Conference Proceedings 836 2006)
OPTICAL OBSERVATIONS OF GAMMA-RAY BURSTS AT THE
FIRST RUSSIAN ROBOTIC TELESCOPE MASTER
N.V. TYURINA, V.M. LIPUNOV, V.G. KORNILOV, E.S. GORBOVSKOY
and D.A. KUVSHINOV
Sternberg Astronomical Institute,
Moscow, 119992, Russia
tiurina @sai. msu. ru
The results of optical observations of gamma-ray bursts and supernovae at the first
russian robotic telescope MASTER in 2005-2006 are presented. The world's first
observations of optical emission of gamma ray bursts GRB050824 and GRB060926 are shown.
Our data combined with later observations give the law of brightness ~ (-"■55±0'5 for
GRB050824. We discovered optical flare for GRB060926 about 500-700 sec. The power
law spectral index (F ~ E~P) is equal to /5 = 1.0 ± 0.2. A new method of the OT search
after IPN-triangulation of the gamma ray observation is proposed and tested.
Keywords: Gamma-ray bursts; telescope-robots; supernovae search.
1. Introduction
Telescope robots are telescopes, which automatically observe the sky, process
images, and choose subsequent strategy of observations. The MASTER,12 is the first
and unique robotic telescope in Russia. It was designed at Sternberg Astronomical
Institute and Moscow Union "Optics" in 2002. Modern version of the MASTER
system consists of the four parallel telescopes on the automatic parallax mount,
which points at the source with a speed up to , and the 2 wide field cameras on
their own mounts with their own covers. One of the wide field cameras is located
in the Mountain Astronomical Station of the Pulkovo Observatory (Kislovodsk).
Both systems are connected through the Internet and can respond to new
transient objects (not included into the catalogues) during several tens of seconds.
The MASTER, works in fully automatic regime. The most similar in characters
to the MASTER telescope {http : //observ.pereplet.ru) is ROTSE-III system,3
(http : //www.rotse.net). There are some differences between them: the field of
view of MASTER is larger, it has several tubes mounted on the same axis (this
design enables us to observe the source in different wave lengths at the same time).
2. Gamma-ray bursts observations and supernovae search
During the period from the beginning of 2005 and October 2006 our system
MASTER (Domodedovo) had observed 31 gamma ray bursts. Sixteen of them are the
world's first observations within the GRB optical emission limits. We should note
that only several GRB were detected by SWIFT in 2006 during night time in
Moscow. In 2005 it gave us 90% of all bursts. In spite of this fact, we made the
world's first observations of optical emission of 2 gamma ray bursts. We make
photometry in automatic mode, using USNO-A2.0 for all stars in image (up to 10000
stars) with combined stellar magnitude m = 0.89R + 0.1113., which is optimum for
2039
2040
our instrumental magnitude. The image reduction takes less than 1 min. As the
result the robot finds the objects, which are not included in the catalogues within
the error box of the GRB, writes a telegram to GCN? using the magnitude of the
suspected new source and limiting magnitude of the image. At that time the full
image with marked error box and DSS-II-Red image and our old image of this field
appear in our data-base (the base is accessible through the Internet). If the object
could not be found in separate images, the limit can be raised up to 20m in sum of
10-15 images in clear moonless night. The results of our observations are in press.
GRB050824. The first image was obtained5 764s after SWIFT (trigger 151905)
GR.B050824 detection. We detected the optical transient (OT) candidate, proposed
by OSN (J.Gorosabel et al.). We've analyzed all photometric points obtained during
the first 2 hours from ROTSE, MASTER, OSN and Swift UVOT in similar colors.
The upper limits of ROTSE and MASTER for 500-750 sec (GRB time) are in
agreement. Both instrumental systems are more or less similar. If we include only
2 MASTER points and Swift UVOT V-point we can obtain the power law , here
m = (2.1 ± 0.2)logt+ 19.5, t is the time in hours. The images are available at
http : // observ .pereplet.ru / images / GRB050&2A/1, jpg.
GRB060926. The MASTER robotic system responded to GRB060926.6 The
first image was started 76s after the GRB time. We find a faint OT on the
first and on the co-added images at the position: alpha = 17 35 43.66 dec
= 13 02 18.3 err = ± 0.7". We discovered optical flare around 500-700s
after the GRB time (note that the optical flares are very rare phenomenon which
is sharp rise of the luminosity during GRB fading The light curve is available
at http : //observ.pereplet.ru/images/GRBQ6Q926/light-curvejnew.jpg. Between
91s and 255s a power law decline with a temporal index is estimated to be equal to
— 1.4 ± 0.24. Between 707s and 1200s a power law decline with a temporal index is
estimated to be equal to —3.3±0.7. After 1000s a power law decline with a temporal
index 0.73 ± 0.1 was obtained. We remember that X-ray flare in GRB060926 was
discovered by XRT team. The X-ray spectrum covering the time period from T+67s
to T+878s is well fit by an absorbed power law with a photon index of 2.1 ± 0.3
and a column density of 2.2 ± 0.9 • 1021cm-2, see Figure 1. They note the Galactic
column density in the direction of the source is 7.3 • 1020cm~2. This means that
absorption is about 1 magnitude in our band. The optical and X-ray data is well fit
by power law with a photon index of 1.7 ± 0.2 during all our time observation.
Supernovae search scheme at the MASTER is following: 1) the robot marks
the signal above the galaxy phone, 2) coordinates and stellar magnitudes of found
stars are compared with objects of this field from the catalogues and so we find new
objects, 3) if this field was observed by the MASTER previously, new objects are
compared with marked objects during previous observations, if there wasn't one, this
object can be considered as a supernova. This process is fully automatic. The last
fact makes our software to be unique in the world. So we discovered SN2005bv (the
first supernova discovered in Russia, la), SN2005ee(II-type), SN2006ak (la). Also
we imaged one of the brightest (and nearest) supernova 2006X in M100 galaxy. Its
2041
g 2.5E-15
^ 2E-15
1.5E-15 -|
1E-15
-i 1 1 1—i—i—i
• MASTER
^7 MASTER (optical limit*)
♦ XRT (Swift)
fp=4+HXt?H
H#H
^^H
* $
*-
'"I
100
' ' I—
1000
Log (*),[•]
Fig. 1. Optical light curves of GRB060926 made by the MASTER system and the OPTIMA-
Burst and X-ray light curve obtained by XRT Swift.
stellar magnitude was (2006/02/06.06162). Our point was the second in the world
at ascending part of the light curve of SN2006X (SN la).
We proposed and tested new method7 of optical observations of GRBs by wide
field robotic telescopes: survey search of the optical transient like supurnovae in large
error box GRBs or dark not X-ray GRB. Especially this method is very important
for bright short GRB, that frequently detected by all sky gamma-ray detectors
(Konus-Wind, Ulysses, Odyssey, etc) and for which connection with supernovae is
not clear. The difference between "OT" and "SN" is the following: OT - is the new
optical object without known galaxy. The SN pipe line reduction is based on the
searching of the uncatalogized candidate near known galaxies. MASTER observed
GRB060425 error box (IPN triangulation8) in survey mode at considerable zenit
distances. We have 4 nights (1-2 hours per night) observations. The robot not find
OT brighter then 17.0 and SN brighter then 17.5 in IPN error box.
References
1. Lipunov V.M. et al., 2005, in Astrophysics 48, 389
2. Lipunov V.M. et al., 2004, in AN 325, 580
3. Yost S.A. et al., 2006, AN, 327, 803
4. Barthelmy S.D. et al., 1995, Ap&SS, 231, 235
5. V. Lipunov et al., 2005, GCN3883
6. V. Lipunov et al., 2006, GCN5632
7. V. Lipunov et al., 2006, GCN5080
8. J. Cummings et al., 2006, GCN5005
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Cosmological Singularities
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FLAT, RADIATION UNIVERSES WITH QUADRATIC
CORRECTIONS AND ASYMPTOTIC ANALYSIS
SPIROS COTSAKISf and ANTONIOS TSOKAROSJ
University of the Aegean, Karlovassi, Samos 83200, Greece
fskot@aegean.gr, Xatsok@aegean.gr
It was shown long ago by T. V. Ruzmaikina and A. A. Ruzmaikin in1 that
within the framework of a homogeneous and isotropic cosmological model quadratic
corrections of the gravitational field cannot provide solutions that are both regular
initially and go over to Friedmann type at later times. We find here, by applying
a dynamical systems approach,3 the general form of the solution to this class of
models in the neighborhood of the initial singularity under the above conditions.
Our starting point in this brief paper is the Lagrangian of the general quadratic
gravity theory given in the forma
L(R) = R + BR2 + CRijRij + DRijkmRljkm. (1)
Since for a general spacetime we have the following identity,
5 J(R2 - AI&Rij + RlikmRzjkm)^dn = 0, (2)
in the derivation of the field equations through variation of the action associated
with (1), only terms up to Rl°Rij will matter. If we further restrict ourselves to
isotropic spacetimes we have a second identity of the form
S f(R2 - 3RijRij)y/=gdn = 0 , (3)
which enables us to keep terms only up to R2 in (1). In the specific model treated
herein, we consider a spatially flat universe with metric given by
ds2 =dt2-b(t)2(dx2+ dy2+dz2), (4)
assumed to be radiation dominated, i.e. P = p/3. Under these assumptions, the
variation of the action functional constructed using (1) gives the following higher
order field equations:
c4 2y 6
2RR* - l-R2g^ - 2(<7*Vm - 9ij9km)VkVmR
(5)
where n = 6B + 2C. Using (3), (4) the tt-component of (5) can be written in the
form
b2
V + b3 V V
¥' (6)
athe conventions for the metric and the Riemann tensor are those of.2
2045
2046
where &i is a constant defined from §0£ = jji (VjTi0 = 0). Note that the
Friedmann solution \/2b\t satisfies the above equation.
Assuming that Eq. (6) has a solution with a regular minimum at t = t0, (b0 =
b(t = t0) = 0 and &o = b(t = t0) ^ 0) we can expand this solution as a Taylor series
m = bo + bf(t-t0f + ^(t~t0)3 + --- . (7)
Direct substitution to Eq. (6) restricts the value of the constant k to k =
(&i/(&oM)2 > 0b- For this value of n, Ruzmaikina-Ruzmaikin conclude that the
asymptotic form of the solution to (6) is b(t) « exp((t - t0)2/12n) which is
obviously not approaching the corresponding Friedmann solution as t tends to infinity.
We now move on to perform a local dynamical systems analysis in order to find
the general behaviour of the solutions to Eq. (6) near the initial singularity. This
analysis is based on the use of the method of asymptotic splittings expounded in
Ref.3 As a first step, setting b = x, b = y and b = z, Eq. (6) can be written as a
dynamical system of the form x = f (x):
V bj yz z2 3y3
x = y,y = z,z=- 2 + ^~ + TT- (8)
Ik lK,yxz x ly Zxz
If a = (a, /?, 7), and p = (p, g, r) we denote by x(r) the solution
x(r) = arp = (arp,/3r9,7rr) (9)
and by direct substitution to our system (8) we look for the possible scale invariant
solutions0. From all possible combinations, the most interesting is the one with
dominant part given by
2y x
and subdorninant part
•sub _ I n n &i + V_
2nyx2 2k
0,0,-^ + ^1 (ii)
where f = f(°) + fsub. The dominant balance (of order 3) turns out to be
a a\ (\ 1 3N
(«)= (-.§.-?). -f- («)
where a is an arbitrary constant.
The Kowalevskaya exponents for this decomposition, eigenvalues of the
matrix K — -Df(a) — diag(p), are { — 1,0,3/2} with corresponding eigenvectors
{(4,—2. 3), (4, 2,-1), (1,2, 2)}. The arbitrariness coming from the coefficient a in
the dominant balance reflects the fact that one of the dominant exponents is zero
bDue to a sign mistake, Ruzmaikina-Ruzmaikin in1 conclude the opposite.
CA vector field f is called scale invariant if f (arp) = Tp_1f (a)
2047
with multiplicity one. According to the method of asymptotic splittings,3 we proceed
to construct series expansions which are local solutions around movable
singularities. In our particular problem the expansion around the singularity turns out to
be a Puiseux series of the form
oo oo oo
*(*) = 5ZcH(t-t0)l+i, y(t) = ^c^i-io)*-*, z(t) = J^c3i(t-t0)*-*, (13)
i=0 i=0 i-0
where to is arbitrary and cjo = en, cio = ot/2, C30 = —a/4. For these series expansions
to be valid the compatibility condition
(-2ci3 + c23 \
-c23 + c33 1=0 (14)
-|c13 + |c23 - C33/
must be satisfied. Substitution of Eq. (13) into Eq. (8) leads to recursion relations
that determine the unknowns c\i,C2i,czi. After verifying that Eq. (14) is indeed
true, we write the final series expansion corresponding to the balance (12). It is:
„,4 Ai.2
x(t) =a(t-to)i+ Cl3 (t - t0)2 + 24wa31 (t - t0)i + ■ ■ ■ . (15)
The series expansions for y(t) and z(t) are given by the first and second time
derivatives of the above expressions.
Our series (15) has three arbitrary constants and is therefore a local expansion of
the general solution around the movable singularity to- Also since the leading order
coefficients can be taken to be real, by a theorem of Goriely-Hyde,4 we conclude
that there is an open set of real initial conditions for which the general solution
blows up at the (finite time) initial singularity at to- Finally, we observe that near
the initial singularity, the flat, radiation solutions of the higher order gravity theory
considered here are Friedmann-like regardless of the sign of the R2 coefficient, while
away from the singularity they strongly diverge from such forms.
This work was co-funded by 75% from the EU and 25% from the Greek Government,
under the framework of the "EPEAEK: Education and initial vocational training
program - Pythagoras".
References
1. T. V. Ruzmaikina and A. A. Ruzmaikin, Zh. Eksp. Tear. Fiz. 57, 680 (1969).
2. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, (Pergamon Press,
1975).
3. S. Cotsakis, J. D. Barrow, The Dominant Balance at Cosmological Singularities,
arXiv:gr-qc/0608137; to appear in the Proceedings of the Greek Relativity Meeting
NEB 12, June 29-July 2, 2006, Nauplia, Greece.
4. A. Goriely and C. Hyde, J. Diff. Eq., 161, 422 (2000).
THE RECOLLAPSE PROBLEM OF CLOSED ISOTROPIC MODELS
IN SECOND ORDER GRAVITY THEORY
JOHN MIRITZIS
Department of Marine Sciences, University of the Aegean,
University Hill, Mytilene 81100, Greece
imyr@aegean.gr
We study the closed universe recollapse conjecture for positively curved Friedmann-
Robertson-Walker (FRW) models in the Jordan frame of the second order gravity theory.
We analyse the late time evolution of the model with the methods of the dynamical
systems. We find that an initially expanding closed FRW universe, starting close to the
Minkowski spacetime, may exhibit oscillatory behaviour.
Keywords: Recollapse conjecture; Higher order gravity theories.
It is well known that higher order gravity (HOG) theories in vacuum derived
from Lagrangians of the form L = f (R) s/—g, are described by fourth-order field
equations. It is also well known,1 that under a suitable conformal transformation
the field equations reduce to the Einstein field equations with a scalar field as a
matter source. The two frames8, are mathematically equivalent, but physically they
provide different theories.2 Therefore, if a problem can been solved in the Jordan
frame, it should be interesting to compare this result with the solution of the same
problem obtained in the Einstein frame.
One such problem is the closed-universe recollapse conjecture for the / (R) =
R + (3R? theory. This problem was partially solved for homogeneous and isotropic
spacetimes in the Einstein frame.3 In order to investigate the same problem in the
Jordan frame we need the field equations for the FRW metric.4 We denote by x the
inverse of the scale factor and by H the Hubble function, so that our dynamical
system in vacuum is
R = v, i = -3Hv-—R, x = -xH, H = -R-2H2 - kx2. (1)
6(3 6
Since we are interested only for the closed, k = +1, models, from now on we omit
k from the formulas.
The only equilibrium point is the origin (0, 0, 0, 0). It corresponds to the
asymptotic state of very large, slowly expanding closed universe. The eigenvalues of the
Jacobian matrix at the origin have zero real parts and therefore the usual stability
analysis fails. We define new variables, (u,w,y,x), by the equations
R = \ —u, V = ~~E-, H = w + y, x = x, (2)
aThe term frame denotes the set of dynamical variables used. In the literature, the original set
of variables is called the Jordan frame and the conformally transformed set is called the Einstein
frame.
2048
2049
c = o
Fig. 1. Phase portrait of (5).
and we find the normal form of the system (see5 for technical
details). It turns out that the normal form in cylindrical coordinates
(u = v cos 6, w = r sin 8, y = y, x = x) , is
3-1.1
:W,
y
2 ■"' " v/g^' " 2
From the first and fourth of (3) we obtain
r = Ax3/2, A>0
2yz~x\
-yx.
(3)
(4)
We substitute (4) into the third equation of (3) and we obtain the projection of the
fourth-dimensional system on the x — y plane, namely
-yx, y = bx3 — 2y2 — x2,
b >0.
(5)
System (5) has a first integral, viz.
i , s 2b 1 y2
<P(x,y) = h -o + — ■
x x^ x4
The level curves of 0 are the trajectories of the system (see again5 for details).
Theorem. For the system (5) (i) there are no solution curves asymptotically
approaching the origin (ii) there exist periodic solutions and (Hi) the basin of attraction
of every periodic trajectory is the set y2 + x2 — 2bx3 < 0.
Proof. The function 0 has a local isolated minimum at (1/6,0) and therefore its
level curves near this point are closed. For 0 (x, y) = C we have
y2 =x2 (~l + 2bx + Cx2) ,
2050
which implies that — 1 + 2bx + Cx2 must be non-negative. It follows that for C > 0
any orbit starting in the first quadrant satisfies
x>^(-b+y/b2 + c) >0,
i.e., there are no solutions approaching (0,0). For C S (—&2,0) an orbit of (5)
starting in the first quadrant crosses the or—axis at (—b — \Jb2 + C) jC and
reenters in the first quadrant crossing the x—axis at (—6 + \Jb2 + C) /C, i.e. it is a
closed curve and represents a periodic solution. The curve corresponding to C = 0
separates the phase space into two disjoint regions I and II (see Figure 1). In region
I every initially expanding universe eventually recollapses. In region II, (C < 0),
every trajectory corresponds to a periodic solution and we conclude that the basin
of attraction of every periodic trajectory is the set y2 + x2 — 2bx3 < 0. □
The periodic solutions of (5) induce periodicity to the full four-dimensional
system (3). Obviously one cannot assign a physical meaning to the new variables
(u,w.x,y) since the repeated transformations have "mixed" the original variables
of (1) in a nontrivial way. However, the periodic character of the solutions of (3)
whatever the physical meaning of the variables be, has the following interpretation.
Close to the equilibrium of the original system (1), there exist periodic solutions
for all variables. This implies that an initially expanding closed universe can avoid
recollapse through an infinite sequence of successive expansions and contractions.
This interesting result was not revealed in the Einstein frame.3 Since the basin of
attraction of all periodic trajectories of (5) is an open subset of the phase space,
there is enough room in the set of initial data of (1) which lead to an oscillating
scale factor.
Acknowledgements
I thank Spiros Cotsakis and Alan Rendall for useful comments. This work was
co-funded by 75% from the EU and 25% from the Greek Government, under the
framework of the "EPEAEK; Education and initial vocational training program -
Pythagoras".
References
1. J.D. Barrow and S. Cotsakis, Phys. Lett. B214, 515 (1988); K. Maeda, Phys. Rev.
D37, 858 (1988); S. Gottlober, V. Miiller, H. Schmidt and A. Starobinsky, Int. J. Mod.
Phys. D2, 257 (1992).
2. G. Magnano and L.M. Sokolowski, Phys. Rev. D50, 5039 (1994); V. Faraoni, E. Gun-
zig and P. Nardone, Fund. Cosmic Phys. 20, 121 (1999); S. Cotsakis, Preprint gr-
qc/0408095 (2004).
3. J. Miritzis, J. Math. Phys. 44, 3900 (2003); J. Miritzis, J. Math. Phys. 46, 082502
(2005).
4. J.D. Barrow and A. Ottewill, J. Phys. A 16, 35 (1983).
5. J. Miritzis, Preprint gr-qc/0609025 (2006).
BIG-RIP, SUDDEN FUTURE, AND OTHER EXOTIC
SINGULARITIES IN THE UNIVERSE
MARIUSZ P. DABROWSKI* and ADAM BALCERZAK
Institute of Physics, University of Szczecin,
Wielkopolska 15, 70-451 Szczecin, Poland
mpdabfz@sus.univ.szczecin.pl
We discuss exotic singularities in the evolution of the universe motivated by the progress
of observations in cosmology. Among them there are: Big-Rip (BR), Sudden Future
Singularities (SPS), Generalized Sudden Future Singularities (GSFS), Finite Density
Singularities (FD), type III, and type IV singularities. We relate some of these singularities
with higher-order characteristics of expansion such as jerk and snap. We also discuss the
behaviour of pointlike objects and classical strings on the approach to these singularities.
1. Introduction
Through many years in the past only the two basic cosmological type of singularities
were known among the isotropic models of the universe. These were Big-Bang and
Big-Crunch appended by a future asymptotic (and non-singular) state of a de-Sitter
type. The appearance of Big-Bang and Big-Crunch was in no way related to any
of the energy conditions violation. The progress in cosmological observations at
the turn of the 21st century1 did not add anything new to the picture apart from
the fact that then it was realized that these singularities could emerge also in the
strong-energy-condition-violation cases of g + 3p < 0. However, a deeper analysis
of the data from supernovae, cosmic microwave background (WMAP) and large-
scale structure (SDSS)2 shows that there exists other possibilities of the universe
evolution which admit new type of singularities and the problem of the link between
energy conditions violation and the singularity appearance becomes unclear. We will
discuss these new singularities and the problems to relate them with the possible
generalized energy conditions as well as some new observational characteristics of
the expansion of the universe.
2. Phantom-driven Big-Rip. Phantom duality
The main motivation to exotic singularities comes from phantom.3 Apparently, it
emerged that the observational data does not make any "borderline" at p = — g in
cosmology and that the smaller pressure is allowed to dominate current evolution.
Phantom may easily be simulated by a scalar field 0 of negative kinetic energy which
gives the energy-momentum tensor for a perfect fluid with the energy density g =
-(l/2)02 + V{4>) , and the pressure p = -(l/2)<j>2 - V(<f>) , so that it surely violates
the null energy condition since g+p = — 02 < 0. Phantom is allowed in Brans-Dicke
theory in the Einstein frame (for Brans-Dicke parameter lj < —3/2), in superstring
*Presenting author.
2051
2052
cosmology, in brane cosmology, in viscous cosmology and many others. The most
striking consequence of phantom is that its energy density g grows proportionally
to the scale factor a(t). Then, unlike in a more intuitive standard matter case,
where the growth of the energy density corresponds to the decrease of the scale
factor, here, the growth of the energy density accompanies the expansion of the
Universe. This allows a new type of singularity in the universe which is called a
Big-Rip. This singularity appears despite all the energy conditions are violated. It
is a true singularity in the sense of geodesic incompletness apart from some range of
the possible equations of state for isotropic geodesies which are complete.4 A very
peculiar feature of phantom models against standard models is phantom duality.5
It is a new symmetry of the field equations which allows to map a large scale factor
onto a small one and vice versa due to a change
a(t) <-> —— or w + 1 <-> — (w + 1) , (1)
a(tj
with a consequence of replacing energy conditions violating matter onto a non-
violating one.
3. Sudden (and Generalized) Future Singularities, Finite Density
singularities, type III and IV singularities
Big-Rip leads to violation of all the energy conditions. It appears that one is able
to get some other exotic singularities which violate the dominant energy condition
(p < I Q I) only or even do not violate any energy condition. The former are SFS and
the latter are GSFS. The idea to get them is not to constrain the set of cosmological
field equations by any equation of state,6 which allows an independent evolution
of the energy density and the pressure. Actually, the energy density depends on at
most first derivative of the scale factor, while the pressure depends on the second
derivative, too. Then, it may happen that at a certain moment of the evolution only
the second derivative of the scale factor is divergent - this is a Sudden Future
Singularity - the energy density remains finite, while the pressure blows-up to infinity.
It was shown that it is a weak singularity4 in the sense of the formal definitions of
singularities known in general relativity. The main point is that there is no geodesic
incompletness at this singularity and the evolution of an individual pointlike object
can be extended through it. Same refers to Generalized Sudden Future Singularities.
These singularities are temporal (appear at some fixed time on a hypersurface t =
const.), but there exist also a spatial pressure singularities (may exist somewhere
in the universe nowadays) in cosmology, though in inhomogeneous models.7 It is
possible to have inhomogeneous models of the universe which exhibit both types of
singularities. Finally, other exotic types of singularities are also possible.8 These are
type III (with finite scale factor and blowing-up the energy density and pressure)
and type IV (with finite scale factor, vanishing the energy density and pressure,
blowing-up the pressure derivative). It is interesting to know the difference between
the evolution of pointlike objects and extended objects such as fundamental strings
2053
through these various exotic singularities.9 As it was mentioned already, the
pointlike objects are really destroyed in a Big-Rip singularity only. However, at SFS the
infinite tidal forces appear, and one may worry about the fate of strings approaching
these singularities. It was shown9 that this is subtle in the sense that strings are not
infinitely stretched (remain finite invariant size) at any of these singularities apart
from a Big-Rip. In other words, extended objects like strings, despite infinite tidal
forces, may cross through SFS, GSFS, type III, and type IV singularities.
4. Generalized energy conditions and exotic singularities
From the above considerations it is clear that the application of the standard
energy conditions to exotic singularities is not very useful. Then, one should try to
formulate some different energy conditions which may be helpful in classifying exotic
singularities.10 This may be put in the context of the higher-order characteristics
of the expansion (statefinders) which involve higher-order derivatives of the scale
factor such as jerk, snap etc.10'11 For example, one could think of a hybrid energy
condition like ag > p with a = const., to prevent an emergence of SFS, or a higher-
order dominant energy condition in the form g > | p |, whose violation can be a good
signal of GSFS.
5. Conclusion
Universe acceleration gave some motivation to study non-standard cosmological
singularities such as Big-Rip, Sudden Future Singularity, Finite Density singularity
and type III, IV singularities. However, most of these singularities (apart from Big-
Rip) are weak singularities which do not exhibit geodesic inconipletness and allow
the evolution of both pointlike objects and strings through them.
Acknowledgments
This work has partially been supported by the Polish Ministry of Science and
Education grant No 1P03B 043 29 (years 2005-07).
References
1. S. Perlmutter et al., Astroph. J. 517, 565 (1999).
2. M. Tegmark et al, Phys. Rev. D 69, 103501 (2004).
3. R.R. Caldwell, Phys. Lett. B 545, 23 (2002).
4. L. Fernandez-Jambrina and R. Lazkoz, Phys. Rev. D 70, 121503(R) (2004).
5. M.P. Dabrowski, T. Stachowiak and M. Szydlowski, Phys. Rev. D 68.
6. J.D. Barrow, Class. Quantum Grav. 21, L79 (2004); ibid. 21, 5619 (2004).
7. M.P. Dabrowski, Phys. Rev. D 71, 103505 (2005).
8. S. Nojiri, S.D. Odintsov and S. Tsujikawa, Phys. Rev. D 71,063004 (2005).
9. A. Balcerzak and M.P. Dabrowski, Phys. Rev. D73, 101301(R) (2006).
10. M.P. Dabrowski, Phys. Lett. B625, 184 (2005).
11. U. Alain, V. Sahni, T.D. Saini, and A.A. Starobinsky, Mon. Not. R. Astron. Soc. 344,
1057 (2003).
BRANEWORLD COSMOLOGICAL SINGULARITIES
IGNATIOS ANTONIADIS1-**, SPIROS COTSAKIS2^ and IFIGENEIA KLAOUDATOU3-*
1 Department of Physics, CERN - Theory Division, CH-1211, Geneva 23, Switzerland,
2,3 University of the Aegean, Karlovassi, 83 200 Samos, Greece
ignatios.antoniadis@cern.ch*, skot@aegean.gr*, iklaoud@aegean.gr *
The purpose of this brief report is to present some results of our on-going project
on the asymptotic behaviour of braneworld-type solutions on approach to their
possible finite 'time' singularities. Cosmological singularities in such frameworks have
served as means to attack the cosmological constant problem (see1 and references
therein). The main mathematical tool of our analysis is the method of asymptotic
splittings introduced in Ref.2
Below we study a model consisting of a 3—brane configuration embedded in a
five dimensional bulk space with a scalar field being minimally coupled to the bulk
and conformally coupled to the fields restricted on the brane. The total action is
taken to be Stotai = Sbuik + Sbrane, where
2k2 2
Sbuik = d xdY^fgi —^ - -(V0) , Sbra„e = - d xv/^I/(0), at Y = Y„
with Y denoting the fifth bulk dimension, n\ = M~3, M* being the five dimensional
Planck mass and /(0) is the tension of the brane depending on the scalar field 0.
We assume a bulk metric of the form ds2 = a2(Y)ds2 + dY2, where ds2 is the four
dimensional flat, de Sitter or anti-de Sitter metric. Then varying the above action
we obtain the field equations:
„'2 rt*.2^2 J.H-2
(1)
a"
a
a'2 /3kW2 kH2
a2 ~ 12 ' a2
(3k14>12 ., a'
0' = 0, (2)
where k = 0,1 or —1, and H~l is the de Sitter curvature radius. Assuming further
that the unknowns are invariant under a7^ —Y symmetry and solving the field
equations on the brane we may express the solution in the form
«'(n) = -f/(0(n))a(n), 0,(K) = //(02(J*)). (3)
We now apply the method of asymptotic splittings to look for the possible
asymptotic behaviours of the general solution. Setting x = a, y = a', z = 0', where the
differentiation is considered with respect to T = Y — Ys (Ys being the position of
the singularity), the field equations (2), become the following system of ordinary
differential equations:
x' = y, y' = -pAz2x, z' = -Ayz/x, (4)
*On leave from CPHT (UMR CNRS 7644) Ecole Polytechnique, 91128 Palaiseau Cedex, France.
2054
2055
where A = k|/4. Hence, we have the vector field f = (y, —/3Az2x, — 4yz/x)T.
Equation (1) does not include any terms containing derivatives with respect to T; it
is the constraint equation of the above system. In terms of the new variables, the
constraint has the form
y2/x2 = Ap/3z2 + kH2/x2. (5)
Substituting the forms (x,y,z) = (aTp,"/Tq,STr), with (p,q,r) e Q3 and
(a, 7,6) e C3 — {0}, in the dynamical system (4), we seek to determine the possible
dominant balances in the neighborhood of the singularity, that is pairs of the form
B = {a, p}, where a = (a, 7, S) and p = (p, q, r). For our system we find:
B, = {(a, a/4, y/3/Ay/Ap), (1/4, -3/4, -1)} (6)
82 = {(a,a,0), (1,0,-1)} (7)
i33 = {(a,0,0),(0,-1,-1)}. (8)
Since (4) is a weight-homogeneous system, the scale invariant solutions given by
the above balances are exact solutions of the system. The balance B\ satisfies the
constraint equation (5) only for k = 0, corresponding thus to a general solution for
a flat brane, whereas B2 corresponds to a particular solution for a curved brane
since it satisfies Eq. (5) for k ^ 0 and a2 = kH2. Finally the balance B3 represents
a static universe conformal to Minkowski space and will not be analyzed further.
Next we calculate the Kowalevskaya exponents, i.e., the eigenvalues of the matrix
given by /C = Di(a) — diag(p); for B\ we find that spec(/C) = { — 1, 0, 3/2}, whereas
for B2, spec(/C) = { — 1,0,-3}. These exponents correspond to the indices of the
series coefficients where arbitrary constants first appear. The —1 exponent signals
the arbitrary position of the singularity, Y8. Since we have two non-negative integer
eigenvalues the solution we are constructing will be a general solution (full number
of arbitrary constants).
Let us now focus on each of the two possible balances separately and build
series expansions in the neighborhood of the singularity. For the first balance, we
substitute in the system (4) the series expansions x = Tp(a + T,JL1CjT^s), where
x = (x, y, z), Cj = (cj\, Cj2, c/3), s is the least common multiple of the denominators
of positive eigenvalues (here s = 2), and we arrive at the asymptotic solution
x = aT1'* + U2r'* + ..., y = x', z = ^=T^--^=c32T^2 + ---.(9)
7 4y/A la^JAp
The last step is to check if, for each j satisfying j/s = p with p a positive eigenvalue
corresponding to an eigenvector v of the /C matrix, the compatibility conditions
hold, i.e. vT • Pj = 0, where Pj are polynomials in c,,... ,cy_i given by K-Cj —
(j/s)cj = Pj. Here the corresponding relation j/2 = 3/2 is valid only for j = 3
and the compatibility condition indeed holds. We therefore conclude that near the
singularity at finite distance Ys from the brane, the asymptotic forms of the variables
are a —> 0, a' —> 00, cjj —> 00. This is exactly the asymptotic behaviour of the solution
found previously by Arkani-Hammed et al in Ref.1
2056
However, the previous behaviour is not the only possible one. The second balance
has two distinct negative Kowalevskaya exponents and we therefore expect to find
an infinite expansion of a particular solution around the presumed singularity at
Ys. Expanding the variables in series with descending powers of T, in order to meet
the two arbitrary constants occurring j = — 1 and j = —3, and substituting back in
the system (4) we find the forms
x = aT + C-n~\ , y = a~\ , z = c_33T~4 H (10)
Therefore as T —► 0, or equivalently as S = 1/T —> oo, we have that a —> oo,
a' —► oo and <fi' —> oo.
We thus conclude that there exist two possible outcomes for these braneworld
models, the dynamical behaviours of which strongly depend on the spatial geometry
of the brane. For a flat brane the model experiences a finite distance singularity
through which all the vacuum energy decays, whereas for a de Sitter or anti-de
Sitter brane the singularity is now located at an infinite distance. We can choose
the coupling such that to allow only for that behaviour met in flat solutions and,
in fact, the particular form of the coupling used by Arkani-Hammed et al in1 is the
only choice to make this possible. This easily follows by using equations (3) and
solving the Friedmann equation (1) on the brane for kH2, i.e.
4 V 9 J ^ 4^2
Clearly then k is identically zero if and only if /'(</>)/f(<t>) = (2/3/3)^5, or
equivalently, if and only if /(</>) oc e(2/3/3^5* (Arkani-Hammed et al in1 have (3 = 3).
By working with other couplings we can allow for non-flat, maximally symmetric
solutions to exist and avoid in this way having the singularity at a finite distance
away from the position of the brane.
LA. was supported in part by the European Commission under the RTN contract
MRTN-CT-2004-503369, while S.C. and I.K. were supported by the joint E.U. and
Greek Ministry of Education grants 'Pythagoras' and 'Herakleitos' respectively. S.C.
and I.K. are very grateful to CERN-Theory Division, where part of their work was
done, for making their visits there possible and for allowing them to use its excellent
facilities. This work of I.K. represents a partial fulfilment of the PhD requirements,
University of the Aegean.
References
1. N. Arkani-Hammed, S. Dimopoulos, N. Kaloper, R. Sundrum, Phys. Lett. B480
(2000) 193-199, arXiv:hep-th/0001197v2; S. Kachru, M. Schulz, E. Silverstein, Phys.
Rev. D62 (2000) 085003, arXiv:hep-th/0002121.
2. S. Cotsakis, J. D. Barrow, The Dominant Balance at Cosmological Singularities,
arXiv:gr-qc/0608137; to appear in the Proceedings of the Greek Relativity Meeting
NEB12, June 29-July 2, 2006, Nauplia, Greece.
GENERALIZED PUISEUX SERIES EXPANSION FOR
COSMOLOGICAL MILESTONES
CELINE CATTOEN and MATT VISSER
School of Mathematics, Statistics, and Computer Science,
Victoria University of Wellington,
P.O.Box 600, Wellington, New Zealand
celine. cattoen@mcs. vuw. ac.nz, matt.visser@mcs. vuw. ac.nz
We use generalized Puiseux series expansions to determine the behaviour of the scale
factor in the vicinity of typical cosmological milestones occurring in a FRW universe. We
describe some of the consequences of this generalized Puiseux series expansion on other
physical observables.
1. Introduction
Over the last few years, the zoo of cosmological singularities considered in the
literature has been considerably expanded, with "big rips" and "sudden singularities"
added to the "big bang" and "big crunch", as well as renewed interest in non-singular
cosmological events such as "bounces" and "turnarounds".1_5
We consider a cosmological spacetime of the FRW form and assume applicability
of the Einstein equations of general relativity. We will provide a generic definition
of all the physically relevant singularities considered above (which we shall refer to
as cosmological milestones), using generalized Puiseux series for the scale factor of
the universe a(t). We will show that, most importantly, all physical observables (H,
g, the Riemann tensor, etc..) will likewise be described by a generalized Puiseux
series.
2. Generalized Puiseux series expansion of the scale factor a(t)
Solutions of differential equations can often be expanded in Taylor series or Laurent
series around their singular points. We shall extend this idea by expanding the scale
factor a{t) in generalized power series, similar to a Puiseux series, in the vicinity of
the cosmological milestones.
Generic cosmological milestone: Suppose we have some unspecified generic
cosmological milestone, that is defined in terms of the behaviour of the scale factor
a(t), and which occurs at some finite time t&. We will assume that in the vicinity
of the milestone the scale factor has a (possibly one-sided) generalized power series
expansion of the form
a(t) = co|t - *Q|"° +ci\t- *©r +c2\t~ *0p + c3\t - *0r» + ... (1)
where the indicial exponents rn are generically real (and are often non-integer) and
without loss of generality are ordered in such a way that they satisfy
Vo < Vi < m < Vs ■ ■ ■ (2)
2057
2058
Finally we can also without loss of generality set cq > 0. There are no a priori
constraints on the signs of the other Cj, though by definition Cj 7^ 0.
The first term of the right hand side of equation (1) is the dominant term,
and is therefore responsible for the convergence or divergence of the scale factor at
the time iQ. The indices r\i are used to classify the cosinological milestones and the
absolute value symbols are used to distinguish a past event from a future event. This
generalized power series expansion of the scale factor is sufficient to represent almost
all the physical models that we are aware of in the literature. Table 1 represents
this cosmological milestone classification depending on the value of the scale factor.
Note that sudden singularities are of order n where the nth derivative of the scale
Cosmological
milestones
Big Bang/
Big Crunch
Sudden
Singularity
Extrernality
events
Big rip
Scale factor
value
a(tQ) = 0
a(tQ) = c0
a(")(i0) =00
a(tQ) = c0
a (to) = 00
Indices
Vi
Vo >0
%=0
T]i non-integer
m e z+
Vo <0
factor is the first one that is infinite:
a(n\t — *0) ~ Co 771(771 - l)(J7i - 2)... (771 - n + 1) \t
t(
1771-n
(3)
and therefore rj\ has to be a non-integer.3,4 Note that for most calculations it is
sufficient to use the first three (or fewer) terms of the power series expansion.
3. Power series expansion of all physical observables
We have exhibited a generic expansion of the scale factor a(t) based on generalized
power series for all the physically relevant cosmological milestones found in the
literature to date (big bang, big crunch, sudden singularity, extrernality events and
big rip). We can now use the parameters of this series to explore the kinematical and
dynamical properties of the cosmological milestones, for example, to see whether
they are true curvature singularities or whether the energy conditions hold in the
vicinity of the time of the event iQ.
For instance, on a kinematical level, we can analyze the Hubble parameter for
finiteness in the vicinity of the cosmological milestones. Keeping the most dominant
terms, we have for j)0^0:
H
a^ ami* - to)"0'1
a
m
c0(t - tQ)i°
t-tn
(Vo^O).
(4)
2059
That is, for bangs, crunches, and rips the Hubble parameter exhibits a generic
l/(i — t@) blow up.
In a similar fashion, we can also determine whether a cosmological milestone is
a true curvature singularity by testing R^ and G^ in orthonormal components for
fmiteness:
% = -3^; Gff = 3(^ + ^J- (5)
On a dynamical level, we can quantify how "strange" physics gets in the vicinity
of a cosmological milestone by introducing the Friedmann equations and the
standard energy conditions in general relativity — which are the null, weak, strong, and
dominant energy conditions.5~7 The density and pressure are given as a function of
the scale factor a(t) and can therefore likewise be power series. Whether or not a
specific energy condition is satisfied is simply a matter of calculating the dominant
indicial exponents of the series expansion (full details provided in8).
To conclude, if in the vicinity of any cosmological milestone, the input scale
factor a(t) is a generalized power series, then all physical observables (e.g. H, q, the
Riemann tensor, etc.) will likewise be a generalized Puiseux series. By checking the
related indicial exponents, which can be calculated from the indicial exponents of
the scale factor, one can determine whether or not the particular physical observable
then diverges at the cosmological milestone.
References
1. R. R. Caldwell, "A Phantom Menace?," Phys. Lett. B 545 (2002) 23 [arXiv:astro-
ph/9908168].
2. R. R. Caldwell, M. Kamionkowski and N. N. Weinberg, "Phantom Energy and Cosmic
Doomsday," Phys. Rev. Lett. 91 (2003) 071301 [arXiv:astro-ph/0302506].
3. J. D. Barrow, "More general sudden singularities," Class. Quant. Grav. 21 (2004)
5619 [arXiv:gr-qc/0409062].
4. J. D. Barrow and C. G. Tsagas, "New Isotropic and Anisotropic Sudden Singularities,"
Class. Quant. Grav. 22 (2005) 1563 [arXiv:gr-qc/0411045].
5. C. Molina-Paris and M. Visser, "Minimal conditions for the creation of a Friedman-
Robertson-Walker universe from a 'bounce'," Phys. Lett. B 455 (1999) 90 [arXiv:gr-
qc/9810023].
6. D. Hochberg, C. Molina-Paris and M. Visser, "Tolman wormholes violate the strong
energy condition," Phys. Rev. D 59 (1999) 044011 [arXiv:gr-qc/9810029].
7. M. Visser and C. Barcelo, "Energy conditions and their cosmological implications,"
arXiv:gr-qc/0001099.
8. C. Cattoen and M. Visser, "Necessary and sufficient conditions for big bangs, bounces,
crunches, rips, sudden singularities, and extremality events," Class. Quant. Grav. 22
(2005) 4913 [arXiv:gr-qc/0508045].
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Chaos in General Relativity
and Cosmology
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CHAOS IN THE YANG-MILLS THEORY AND COSMOLOGY:
QUANTUM ASPECTS
SERGEI MATINYAN
Yerevan Physics Institute, Alikhanian Brs.St. 2, Yerevan 375036, Armenia
ICRANet, Piazzale della Repubblica 10, 65100 Pescara, Italy
* smatinian@nc.rr.com
I describe the footprints of the classical chaos of the Yang-Mills fields in the quantum
description. I also review the behavior of the BKL chaotic approach to the classical
singularity on the basis of the Loop Quantum Gravity.
Keywords: Chaos, Yang-Mills theory, Loop Quantum Gravity, Cosmology, General
Relativity.
1. Introduction: Classical chaos in the Yang-Mills field theory and
General Relativity
At the MG IX I had a talk on the chaos of the non-abelian gauge theory and
the gravity (mainly cosmology). This talk was devoted to the classical fields. They
are the gauge theories and are non-linear. Thus, chaoticity is not surprising gen-
erally,although we know examples of the stable solutions (solitons and others) to
the non-linear equations. Concerning the Yang-Mills (YM) theory it is known that
this theory is not integrable. All attempts to prove its integrability, not given up
until now, are not conclusive and if one analysis them he will be convinced that
the proof contains some conjecture (mostly of the mathematical nature).Of course
some approximation scheme (e.g. large ./V-expansion) leads to the integrability.
Therefore, it is not surprising that in 1981 it was found that the classical source-
less YM equations exhibit the strong chaotic phenomena. This fact was proved
initially for the simplest form of the spatially homogeneous YM equations for n = 2, 3
numbers of degrees of freedom [1,2] and then was extended to the spherically
symmetric field theory [3]. The spatial-temporal chaos of YM fields was established by
the lattice calculations (see book [4] for review of this activity). This gave rise to a
program to describe the YM dynamics not in terms of the potential and fields but
rather in terms of the loops (strings) variables [5].
These results demonstrate that the classical YM fields lack any special stable
configuration, all states are chaotic and no particular configuration dominates in
the Minkowski space-time in contrast to the Euclidean case, where the instantons
give a dominant contribution to the functional integral.
Turning now to the problem of the chaos in the General Relativity (GR) we have
to remark that despite an even longer history of the study the chaos in GR, the
chaoticity is observed mostly in the problem of the approaching to the space like
singularity t — 0. These studies arc initiated and developing now due to the famous
BKL chapter of GR [6]. The problem of the chaos in GR closely related to the
problem of the singularity in the classical GR, where it is unavoidable according to
the fundamental singularity theorems [7]. The backward evolution of an expanding
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universe leads to a singular state where the classical theory fails to be applied.
If, as remarked by Landau (see [8]) one considers the metric g as a function of
the synchronous time t only, at some finite time interval g = det^g^) tends to zero
as t —> 0, independent of the equation of state or the character of the gravitational
field, and this results the singularity.
The corresponding metric ds2 = dt2 — dl2 (a,(3= 1,2,3), dl2 = -yapdxadxf3 near
the singularity t = 0 is
dl2 = t2pi dx\ + t2p2 dx\ + t2ps dx\ (1)
with
3 3
This is so-called Kasner solution [9] corresponding to the Bianchi I spatial
geometry. Thus one arrives to the regular flat, homogeneous, anisotropic space with the
total volume homogeneously approaching the singularity. BKL consider the Bianchi
IX geometry which generalize (1) and corresponds to the diagonal homogeneous
anisotropic spatial metric with ■yjj = a2
dl2 = a2InIanIpdxadxP (2)
with the unit vectors along the axes n^, (I = 1,2,3). a,i positive scale factors are
functions of t only.
The diagonal form of the matrix is a result of the vacuum Einstein equation
Roa = 0 . However, even in the presence of matter, as argued by BKL, the
possible non-diagonal terms do not affect the character of the Kasner epochs and the
character of the "replacement" of the Kasner exponents. As a result, the
evolution towards singularity proceeds via a series of successive oscillations during which
the distances along two of the principal axes oscillate while they shrink monotoni-
cally along the third axis (Kasner epochs). The volume V = J vdet^dxxdx^dx^, =
I6ir2aiaiiaiij oc t as t -^ 0.
A new " era" begins when the monotonically falling metric components begin to
oscillate while one of the previously oscillating directions begin to contract. This
approach to the singularity reveals itself as an infinite succession of alternating
Kasner "epochs" and has the character of a random process [10]. Thus infinite
number of the oscillations are confined between any finite time t and moment of
t = 0.
The central role in the BKL approach to the singularity plays the justification
that the time derivatives dominate over space ones at the approach to the
singularity. This fact allows to think that the inhomogeneous model can be well described
by Bianchi IX model where the spatial geometry can be viewed as an assembly
of the small patches each of which evolves almost independently. In other words,
the dynamical decoupling of the different spatial points on the space-like slices has
place. Sizes of the patches are defined by the scale of the space derivatives during
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the evolution while the curvatures grow. To sustain the homogeneity of the
evolution, parches have to be subdivide more and more at the vicinity of the singularity.
If the geometry by some reason is discrete, such fragmentation must stop [11].
In the recent paper [12] this view was justified by the generalization the Misner's
Mixmaster model to the generic inhomogeneous case. It is shown that neglect of
spatial gradients possible in the asymptotic regime. In other words, authors of [12]
claim that the generic cosmological solution near the singularity is isomorphic, point
by point in space, to the one of the Bianchi VIII and IX models because the spatial
coordinates in the Mixmaster model enter as parameters, and one can apply the
long wavelength approximation.
The global chaoticity of the classical YM fields and the chaotic behavior in the
approach to the singularity in the classical GR, of course, have to be changed when
the quantum effects enter the game.
For the YM fields we have to take into account the quantum effects since QCD
describes the quantum world of the hadrons and their interactions. For distances
close to the Planck scale the gravitational field acquires large curvature and the
evolution to the t = 0 singularity has to be replaced by the quantum dynamics.
In this situation, we expect that the chaos of the classical fields should be
diminished if not eliminated completely due to the quantum fluctuations of the gauge
and quark fields. Below we show how it is happened and to which extent. We will
see in general that the chaotic phenomena of the YM fields do not disappear in full,
exhibiting the explicit footprints of the classical chaos in the quantum world.
2. Quantum chaos of the YM fields
The quantum insight into the YM dynamics clearly is achieved if we consider the
spatially homogeneous potentials A^(t) (a = 1, 2, 3; /x = 0,1, 2, 3) (YM classical and
quantum mechanics).
It is obvious that if this type of fields exhibits classical chaos, as we know, the
wider class of the non-homogeneous YM fields A^(x,t) also will have this property.
We mentioned that in the Introduction.
The YM sourceless equation for potential Aa(t) in the gauge A^(t) = 0 is reduced
to the discrete Hamiltonian system (see [4] for review)
d2AaJdt2 + g2{AajAbjAbl - A)A)A$) (3)
With the conserved "external" and "internal" angular momenta
Mi = eijkA<*Aak, (4)
which are vanished for the sourceless fields. Thus there exist seven integrals of
motion and system is not integrable.
Homogeneous limit of YM equations, or their long wavelength regime
corresponds to the gluon high density ng and/or strong coupling regime g2ng\ » 1.
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In this sense it is stated that the homogeneous fields are the relevant degrees of
freedom for infrared regime [13]. Very recently, the equations (3) were obtained as
a strong coupling limit of YM fields [14]. It was shown that in the leading order of
1/g YM equations are reduced to (3). The resulting theory is stable and leads to
the mass gap with the confinement. Author even calculated in this approach the
glueball spectrum which is in good agreement with the lattice QCD computations.
Consider now the simplest case of two and three degrees of freedom n = 2, 3 with
x = A\,y = A\,z = A3. We obtain the systems of two (three) coupled oscillators
with the potential energies
V(x,y) = (g2/2)x2y2, (6)
V[x, y, z) = (g2/2) (x2y2 + y2z2 + x2z2). (7)
Classically these systems, despite their extremely simple forms, exhibit strong
chaotic behavior. Potential (6) (a:2y2-model) has been used in various fields of
science, including chemistry, astronomy, astrophysics and cosmology (chaotic
inflation). We mainly describe here the case n = 2.
Quantum mechanical system with the potential (6) has only discrete spectrum
[15] despite its open hyperbolic channels along the axes and its infinite phase space.
Physically it is clear why it is so: quantum fluctuations, e.g. zero modes forbid the
trajectory to escape along the axis where the potential energy vanishes. The system
is thus confined to a finite volume and this implies the discreteness of the energy
levels. Classically of course the "particle" always can escape along one of the axes
without increasing its energy.
Despite so drastic influence of the quantization on the behavior of the system
(6) chaos left its footprints: periodic (unstable) orbits of the classical potential (6)
after quantization have so called scars [16] (See also [17,18]). Energy level spacing
distribution for the system (6) has the Wigner-Dyson type distribution in contrast
to the Poissonian one for the systems whose classical counterparts are regular. They
are in accordance with the Random Matrix Theory for GOE.
One can go further and propose that the traces of the classical chaos, in principle,
should show in the real spectra of hadrons. For instance, if we would collect the
rich enough glueball spectra then their mass spacing distribution has to reflect
the chaoticity of the classical gluon field (gluon statistics has to be deal with the
assembly of the particles with the same quantum numbers) [19]. Not having today
(when?) such a rich collection of glueballs as a cleanest sample, author of [20] used
the relatively rich baryon and meson spectra for the examine the nearest-neighbor
level spacing distributions for mass m < 2.5 GeV. It is seen that these distributions
are well described by the Wigner surmise corresponding to the statistics of the GOE.
Of course, one should consider this result as very preliminary since barions and
mesons with their quark content are not the best case to check this idea (see [4] to
inquire why the glueballs are necessary to solve this problem).
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We would like to stress the important role which play here the so called billiards
(classical and quantum). If we write the potential V(x2y2) in the form (x2y2)l/a
where 0 < a < 1, then limit a = 0 corresponds to the so-called hyperbolic billiards,
where the classical trajectory undergoes elastic collisions on an infinite barrier
(hyperbolic cylinder x2y2 = 1). Trajectories lie in the x — y plane and consist of
rectilinear segments constructed by the rules of geometrical optics. The notion of
billiards plays an important role also in the GR [21].
Consider now the quantum mechanical adventures of the coupled YM oscillators
in the study of the partition function for our non-integrable system (6)
oo
Z(t) = Tr [exp (-tH)] = J2 e~tE" (8)
n=0
with the quantum Hamiltonian H
h2 ( d2 d2
.2 „.2
H = Y{d72 + d?)+2xy (9)
using the quite effective method of the adiabatic separation of the motion in the
hyperbola channels of the equipotential curves xy = const [22].
The partition function defines the integrated density states N(E) by the inverse
Laplace transform of Z(t)
N(E) = jE dE p(E') = L-1 (^f) (10)
o v t
and for the large enough energy levels E is given by the Thomas-Fermi term - the
zero order term of the Wigner-Kirkwood expansion [23]
Z0(t) = ^=^ (in —^ + 9/n2 - C
(11)
with C the Euler constant.
From (11) and (8) one obtains N(E) - E3/2logE. For the hyperbola
billiard (a = 0) the computations give [24], with the logarithmic precision, N(E) =
^ElogE.
This result some time ago was encouraging from the point of view of the famous
Hilbert-Polya-Berry program to look for the quantum (classically chaotic)
Hamiltonian whose eigenvalues reproduce the Riemann zeta-function's zeros. However,
from Random Matrix Theory we know that such Hamiltonian (or some operator )
must be non-invariant with respect to time inversion (GUE not GOE!).
Returning to the calculation of Z(t), we apply the method of [22] (see also [25]):
The range of the integration over x and y variables and momenta px and py in
the calculation of Z(t) in the Winger representation is divided into two regions:
the central region (/x/, jyj < Q ,where Q is arbitrary, not specified scale) and
the channels region (Q < /x/, Q < /y/).In the central region it is natural to apply
Wigner-Kirkwood method; in the channels with the "slow" motion in the x variables
(along the channel) again will be used the Wigner-Kirkwood expansion, but the
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"fast" motion, transverse to the channel, must be treated quantum mechanically.
Remarkably, the dependence on Q from both regions is cancelled in the leading
terms {tQA)~x « 1 up to eighth order of h. No doubts are left that the higher
order of h terms behave similarly (see [25]). Thus, only Q-independent, non-leading
terms are contributing to Z(t).
We encounter here the phenomenon of the "transmutation" of small parameters:
Classical parameter \/t Q which rules the adiabatic separation of the variables in
the channels transmutes into the small quantum parameter of the final asymptotic
series for Z(t).
It is interesting to follow the motion in the channel (along the x axis) a little in
detail. Motion in the channel can be describe by the Hamiltonian
Hv=\pl + \ultf (12)
where u = gx is x dependent frequency and eigenvalues of (12) are (n+ 1/2) hgx.
In the channel {/x/ » /y/) where the derivatives w .r. t. x are small relative
to the derivatives w. r .t. y, we may first average the motion over the quantum
fluctuations of y [25,26] described by (12) and by the corresponding wave
function involving Hermite polynomials with the frequency u = g x. The corresponding
average
< n/Hyjn >= (n + -) hgx
then appears as an effective potential for the motion in the "slow" variable x
( h2 d2 \
\ ~2 Ox2 +(n+1/2)ff^j ^n(x) = E^n{x). (13)
This is the well known Schrodinger equation for a linear potential having the
solutions in terms of Airy functions which shows the linear confinement and discrete
spectrum for eigenvalues.
We would like to emphasize that this confinement is not like standard
phenomenon commonly refered to as quark confinement. Here the potential described by
the gauge field amplitude x(t). One may call this phenomenon as "self-confinement":
the fields themselves " prepare" the effective potential barrier, prohibiting escape to
the infinity.
As we remarked above, just due to this consistent treatment of the motion
in the channels when each n-th quantum evolves along the x axis according the
Hamiltonian
H^ = l-P2 + {n+1/2) hg\x\ (14)
there are the precise cancellation of all leading quantum corrections (in the regime
-j-hi << 1) and only non-leading but Q-independent corrections survive and lead
to the final answer for Z(t) in the form of asymptotic series with the expansion
parameter g2hAt2.
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One remark is worthy on the discreteness of the spectrum. In the corresponding
supersymnietric quantum mechanics, due to the cancellation between the bosonic
and fermionic modes, there appears the continuous spectrum which coexists with
the discrete one [27] and the confinement generally has not place.
There is another approach to calculate Z(t): add to the potential (6) the extra
term V2(x2 + y2) - Higgs vacuum term, compute Z and then put V = 0 [28]. This
limit opens the hyperbola channels classically. In quantum mechanics, this limit
leads to the singularities: Logarithmic for the Thomas-Fermi term and power like
for the higher quantum corrections V for the fc-th order of h. However, quantum
mechanics cures that: it introduce the Higgs-like term [29] to the potential and due
to this the limit V = 0 in Z(T) has no singularity at all.
In conclusion of this Section we state that the lessons derived from this quantum
mechanical study of the higher order corrections to the homogeneous limit of YM
equations would be useful for the better understanding the internal dynamics of the
YM quantum field theory.
3. BKL chaos in the Loop Quantum Gravity (Loop Quantum
Cosmology)
In this Section we consider which kind modifications on should expect for the
oscillating chaotic approach to the classical singularity t = 0 (BKL scenario) if one
includes the quantum corrections. It is natural that at each novel scheme of the
quantum effects to the gravitational field, BKL chapter is tested with the various
conclusion and verdicts about this scenario. I described this efforts briefly in the
Talk to MG IX mentioned in the Introduction [30]. Now the number of these
considerations of the BKL behavior is increased significantly and includes Matrix models,
String theory, its brane aspects, higher derivative corrections, matter content
modifications (dilaton, p-form fields) etc. (see e.g., [31]). In the most of these approaches
classical singularity, as a rule, are not avoided and this has a strong influence on
the BKL scheme.
Here we describe only one approach based on the Loop Quantum Gravity (LQG)
[32] and its sibling, Loop Quantum Cosmology (LQC) which is based not only on
the LQC, but includes several additional assumptions and simplifications [33].
Here we describe very briefly the basic notions of LQG and LQC.
As it is known, this approach to the canonical Hamiltonian G R is based on the
use not the ADM variables (spatial metric and extrinsic curvature) but the Ashtekar
variables [34,35] based on the inclusion the spin connection variables what allows
the formulation closely to the YM-like gauge field theory.
Briefly, the following has a place. In the expression of the contra variant spatial
metric gab = ef e\ in terms of triad (orthogonal and normalized at each point) there
is redundancy due to an arbitrary 3-dimensional rotation of the triad which does
not change metric.
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Densitized triad
E? = g1/2el (g = detgab) (15)
together with the SU(2) connection A\ (Ashtekar connection), j = 1, 2, 3; a, b =
1, 2, 3) form the pair (A\, Ef) canonically connected to the metric conjugated pair
(dab, pab)- As in the gauge theories, "momentum" Ef of the spin connection Ala
is an analog of the electric field of SU(2) YM theory (index labels a basic element
of the SU(2) Lie algebra). Connections Ala involve the curvature of space and spin
connection Yla
A\ = ra+(3Kl (16)
where Kla ("torsion") defines the extrinsic curvature Kai = Kai,eb. Positive
parameter (5 was introduced by Barbero [36] as substitute of the former imaginary unit,
to have a certain reality conditions for Ashtekar variables.
This /3-ambiguity leads to the Poisson bracket dependent on j3:
{Aia(x),Ebj{y)} = K{35bJ)5(x,y) (k = 8ttG) (17)
and after the Dirac first class constraint quantization, leads to the /3-dependent
Hamiltonian constraint which rules the evolution of the system ADM-like way.
The rest two constraint: Gauss constraint (generating triad rotations) and dif-
feomorphism constraint (generating spatial diffeomorphisms) are independent on
the Barbero-Immirzi ambiguity.
We would like to remark that parameter /3 can be considered as the rescaling of
the triad
E? = ±yfte? (18)
and this allows the possibility to introduce the conformal symmetry [37].
However, in the Dirac quantization this will lead to the new first class constraint
corresponding to the conformal symmetry.
The main advantage of the new variables is that they allow a natural smearing
of the basis fields (A, E) to the linear objects without introducing a background and
retain the well-defined algebra. The connections integrated along a curve,
exponentiated with a path-ordered way. Thus we arrive to the holonomies (this is analogous
of the quantum mechanics where the Heisenberg operator, e.g., x is represented by
Weyl operator elx):
he[A]=Pexp j TtA\eadt (• = ±-)
Je at ^
Ti = -- in,
<Ji are the Pauli matrices.
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Trace of he [A] corresponds to the Wilson loop for the closed curve in the YM
theory. Similarly, we arrive to the fluxes by integrating densitized triad over two-
surfaces S:
Fs [E] = J t1 El na d2x (20)
where na is the normal to the surface S.
The above introduced smearing, without introducing background, eliminate all
delta functions in the Poisson relations giving the well-defined algebra to construct
the Hilbert space. We do not dwell on the fundamental problem of LQG of this
construction in terms of the holonomies and fluxes in the conditions of the diffeo-
morphism invariance. There are wide spectrum of the conceptual and the technical
problems for LQG (see e.g. [38]).
We only make some remarks:
* Barbieri- Immirzi ambiguity can be resolved " experimentally" comparing
the LQG results with the Bekenstein-Hawking entropy for the Black Hole
[39]. Taking the formula of the area eigenvalues A = 8n/3lp^/j(j + 1) one
obtains 13 = ln2/iry/?>. The new data yield 0.27398.
* It was remarked [40] that the role of parameter (5 in the canonical quantum
gravity is analogous in various senses to that of the parameter describing
the different sectors associated with the topological structure of the finite
gauge transformations in the YM theory. In contrast to the LQG, the 0-term
enters as the Pontryagin topological term which is a total derivative.
* After quantization, homologies and fluxes act as well defined operators,
fluxes have the discrete spectra. Since the spatial geometry is determined by
the densitized triads, spatial geometry is discrete as well,with the discrete
area and volume operators. In this sense, it is sometime declared that the
Quantum Gravity is a Natural Regulator of matter [41]. By this reason,
differential equation for the Wheeler-De Witt constraint is replaced by the
difference equations.
* Although in the LQG there is some parallel with the Wilson loops of
YM theory (hence the term Loop Quantum Gravity), there is essential
difference. In the YM theory different sizes of loops are inequivalent in the
light of the interpretation relying to the quark confinement (e.g., inside the
large loops fields are chaotic, inside the small loops are regular, small-large
w.r.t . the confinement radius). The value of the Wilson loop is invariant
under continuum deformations only for the vanishing field strength. In the
LQG, with its diffeomorphism invariance, there is no physical information
in the shape and size since two networks of the different shape but the same
topology can always be related by a suitable diffeomorphism, independently
of the "value" of the Ashtekar field strength.
After these remarks we turn to the BKL problem for Bianchi IX space geometry.
We will base here on the mini-superspace spanned by the scale factors ai of (2),
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diagonal anisotropic model, gjj = a2 (£).
Hamiltonian constraint is written in terms of the spin connections Yj (below we
follow [42]).
2 1
H = - [(TjTK - TI)aI - - aiajiiK + {ViYj - YK) aK
k 4
--aKajdj + (YKYj - Yj) aj - - ajciKa,!}
1 /a J clk aj \ 1 (VK PJ PJ PK
2 \aK aj ajaKJ 2 \pJ pK (p1)
(21)
(22)
(I, J, K) an even permutation of 1, 2, 3; p1 — eIKL aK cll ■
Derive now the classical equations transforming to a new canonical variables 717
and q and to new time coordinate:
717 = -(loga/) , q1 = - logp1 , dt = a1a2a3dT
2 (23)
W, nj} = kSj,
Separating terms with momenta like variables 717, we obtain the potential term
W(a1, a2, a3) = -
E°/
2 2 3 2 2 2 /o/l\
i! a2 -a2a3- ax a3 . (24)
From (22) and (23) it is seen that at the small aj (or p1) due to the divergencies
of the spin connection components there is singularity.
Classical equation of motion are
~(\ogai)" = (a22-al)2-4 (25)
and two e.o.m. by cyclic.
Right hand sides in (25) are vanished for Bianchi I geometry, giving aj ~ tai
with, Y2i = 1 = Y2i aj> i-e- the Kasner solution.
For Bianchi IX geometry one may write the evolution potential in terms of p1:
w(P\ p2, p3) = 2 [(pip2 (r! r2 - r3) +PV (r3r! - r2)
(26)
+P2p3(r2r3-r!)]
which has an infinite walls at p1 ~ 0 due to the divergence of the spin
connection components. Evolution consists of the succession of the Kasner epochs with
reflections on the walls and this process never stops what leads to the BKL chaos.
To be closer to the common picture one can introduce the Misner variables
il = -- logy= -- log(aia2a3)
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and anisotropies /3j:
a, = e-"+/3++^/3- ) a2 = e-n+/j++V3/j_ (
a3=e-n-2(3+
Then the potential (26) takes the form
W(Q, 0+, /?_) = \e-An[e-*f3+ -4e"2/3+ cosh(2v/3/?_)
2 (27)
2e4/3+(cosh(4v/3>_)-l)].
Volume dependence factorizes and the rest anisotropy potential shows
exponential walls for the large anisotropies. For instance, at typical wall has a form if one
takes /3_ = 0 and /3+ < 0:
w~ie-4n-80+ (28)
To prevent the "eternal" reflections at the walls where the expansion/contraction
behavior of the different directions change, it is necessary to stop the unconstrained
rise of the heights of the walls. Just the quantum effects are called up for this
prevention.
In other words, they should lead to the upper limit on the curvature.
What is the concrete scenario to achieve this aim in the LQC?
First of all, one has to have in mind that in the LQG and in the LQC there exist
effective minimal length (or area, A-^/2 = 8ir(3li v3/2) or the maximal curvature.
Quantization according to the rules of the game in LQG results in the
replacement of the spin connections components Tj by the effective coefficients, which leads
to the effective potential instead of (26).
Central moment here is the special rules of the quantization of the inverse den-
sitized triad variables (p1)^1 or the inverse volume, giving that in the LQG they
are not singular at p1 = 0 despite the classical curvature divergence.
One should remember that the LQC which actively considers various important
phenomenological effects is not in the strong sense the direct limiting case of the full
LQG where the desired boundedness of the inverse scale factors or the inverse
volume are ensured. LQC is the usual cosmological mini superspace with its symmetry
reduction and oversimplifications, quantized by the LQG methods and techniques.
For this reason, for instance isotropic model in the LQC, in contrast to some claims,
has no bounded from above inverse scale factor whereas in the full scale LQG it is
proved that such a inverse scale and inverse volume operators have bounded from
above eigenvalues [43]. The physical explanation of this not common situation may
be that the isotropic homogeneous quantum fluctuations for isotropic mini super-
space model are not enough to eliminate the classical singularity. For anisotropic
Bianchi IX model it is not excluded that the fluctuations of the same symmetry
may ensure this elimination although it is not based on the firm grounds as it has
place for the LQG [43].
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Anyway, taking the assumption that for Bianchi IX geometry this conjecture
is realized, let us continue to follow what is happened with the chaos near the
singularity. We again follow [42]. For convenience, we take ^(312 = 1 making p1
dimensionless.
Quantization replaces (pl)~l in the spin connection components by the function
F(p/2j) where the parameter j appears explicitly and controls the peak of the
function F. The same parameter j enters the expression of the area in the LQG:
A(j) = 8ir f3l2^/j (j + 1). Further, follow the recipe of the LQG to obtain the
effective description, one needs to replace all negative powers of triad p1 with the
appropriate factors of the spectrum of the inverse volume operator.
For instance,
p-3/2 -> d = D(p/p*)/p3/2
with p* = a* = I6nj /io /3. Function D ~ 1 for p/p* » 1, recovering the classical
behavior. For the small p (or a), D(^-) ~ pll2 or d ~ p6 thus giving the smooth
behavior at the singularity.
For the anisotropic homogeneous model the components of (p1)-1 (I = 1,2,3)
are replaced in the spin connection components by a function F(p! /2 j) giving the
effective spin connection. Parameter j belongs to the set of the ambiguities of this
framework and controls the peak of the function F. The resulting effective potential
as a function of p1 at fixed volume V has a form
2"
Wj
1
- 1 v
Kp1
V F2 (q)
= Wj(p1,p1,2jq)a
(29)
\3-2qF(q)]
32 jV
where q= ± (j^.
At the peak and beyond it F(q) ~ \jq and we have the classical wall \ e-An-%P+.
The peak of the finite walls is reached for a constant q which in the usual variables
gives that e~2n+2/3+ = const. Maxima of the wall lie on the line [3+ = fl + const in
the classical phase space and the height of the wall drops off as e~l2n ~ VA with
the decreasing V —► 0.
At very small volume the walls collapse more rapidly and the effective potential
becomes negative everywhere at the volumes close of in the Planck units. For the
smallest value of j = 1/2 it is about Planck volume.
Thus, with the decreasing walls during the evolution towards the singularity the
classical reflections will stop at a finite time interval and the chaos should disappear.
Universe -at some time- can "jump over the wall" [42], Kasner regime becomes
stable. If we think about non homogeneities, the patches of the corresponding space
become of the order of a Planck volume, i.e. the scale of the discreteness. Below
that scale further fragmentation does not happen and the discreteness is preserved.
In the large volumes when the evolution is chaotic, two nearby points - patches
will diverge away with no correlations between the points ("non interacting two
2075
dimensional gas"). In the vicinity of singularity, the chaotic motion is replaced by the
Kasner evolution, points-patches begin to correlate ("interacting two dimensional
gas").
The evolution to the singularity on the basis of LQC with its final non-chaotic
scenario near the Planck scale and beyond rises the important question of the
increasing the role of non-homogeneities at that scale. To sustain the homogeneous
regime one needs the further and further fragmentation of patches of the spatial
regions. But at the conditions of the discrete spatial geometry, the fragmentation
must be stopped and the homogeneity should be replaced by non - homogeneity.
Thus, at the approaching to the singularity role of the inhomogeneous quantum
fluctuations may be essential. This brings us to the relatively old notion of the
"turbulent" universe [50], [51].
4. In lieu of conclusion
In lieu of conclusion, we enumerate here several important problems considered by
the LQC and not concerning the chaos in the cosmology. LQC, using the similar
approach (based on the minisuperspace and the recipes of the quantization from the
full LQG) has a several important contribution to the "explanation" of the inflation
[44], to the quantum nature of the Big Bang [45], possible observational signatures
in the CMBR [46], avoidance of the future singularity [47] where the interesting
effect of the negative quadratic density correction inspired by LQC is observed in
the FRW equation, and the quantum evaporation of the naked singularity [48]. Last
paper gives an interesting view on the problem of the gravitational collapse of the
matter (scalar field as an example) near the classical singularity. The authors of [48]
observed the rise of the strong outward energy flux which dissolves the collapsing
cloud before the formation of the singularity. This effect based on the LQC may be
considered as a mechanism of the censorship of the naked singularity [49].
Authors of [48] think about the observational signature of this effect in the
astrophysical bursts. Time will show how reliable are these interesting investigations.
Acknowledgments
It is my great pleasure to thank Remo Ruffini and Vahagn Gurzadyan for invitation
to the superbly organized, in spite of hot weather, MG11 meeting. I am grateful to
Remo Ruffini for the support which made my participation possible.
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7. S.W. Hawking and G.F.R. Ellis The Large Scale Structure of the Space-Time,
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8. L.D. Landau and E.M.Lifshitz, The Classical Theory of Fields, Pergamon, Oxford
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9. E. Kasner, American J. Math. 43, 217 (1921)
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11. M.Bojowald, G.Date and G.M. Hossain, gr-qc/0404039
12. G. Imponente and G. Montani, gr-qc/ 0607009
13. M. Lusher, Nucl.Phys.B 219, 233 (1983)
14. M. Frasca, Phys.Rev. D 73, 027701 (2006)
15. B. Simon, Ann. Phys. NY, 146, 209 (1983); J. Funct.Anal. 53,84 (1983)
16. E. Heller, Phys.Rev. Lett. 53, 1515 (1984)
17. B. Eckhardt, G. Hose and E. Pollak, Phys. Rev. 39 A , 3776 (1989)
18. J. Zakrzewski and R. Marchinek, Phys. Rev. 42 A,7172 (1990)
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20. V. Pascalutza, Eur.Phys.J. A16, 226 (2003)
21. T. Damour, M.Henneaux and H.Nicolai, Clas. Quant. Grav.20,R 145 (2003)
22. S. Tomsovich, J. Phys. A: Math. Gen. 24, L733 (1991)
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27. B. DeWitt, M. Lusher and H. Nicolai, Nucl.Phys. B320 (1989)
28. S. Matinyan and J. Ng, J.Phys. A: Math. Gen. 36, L417 (2003)
29. S.G. Matinyan and B. Muller J. Phys.A: Math. Gen. 39, 61 (2006)
30. S.G. Matinyan, "Chaos in the Non-Abelian Gauge Fields, Gravity and Cosmology",
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31. T. Damour and M. Henneaux, Phys. Rev. Lett. 85, 920 (2002); A. Coley,
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46. S.Tsujikawa, P.Singh and R. Maartens,Clas. Quant. Grav. 21, 5767 (2004)
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51. V.A. Belinskii, JETP Lett. 56, 422 (1992)
CHAOS, GRAVITY AND WAVE MAPS WITH TARGET SU{2)
S. J. SZYBKA
Astronomical Observatory, Jagellonian University,
ul. Orla 171, 30-244 Krakow, Poland
szybka@if.uj.edu.pl
We present the numerical evidence for the chaotic solutions and the fractal threshold
behavior in the Einstein equations coupled to a wave map (with target SU(2)). This
phenomenon is explained in terms of heteroclinic intersections.
Keywords: chaos; critical phenomena; wave maps.
1. Introduction and setup
Let us consider the action
o l [ ( R U\abdXAdXB \
S=2jM[lteG~~2~g ~^r^rGAB)dVM' ()
where X : M —► N is a map from a spacetime (M, ga(,) into a Riemannian manifold
(N, Gab), G is Newton's constant and /„- is the wave map coupling constant. The
equations resulting from the variational principle 6S = 0 are the Einstein equations
and the wave map equation. We will refer to any map X satisfying these equations as
to a wave map coupled to gravity. The strength of the coupling can be parameterized
by a dimensionless coupling constant a = AirGfn .
One of the motivations to study this model conies from the fact that wave maps
on fixed background share some properties with Einstein equations but are simple
enough to be considered rigorously. Hence, it is interesting to study singularity
formation and critical phenomena for wave maps on fixed background (a = 0) and
later to ask how does the coupling to gravity change the dynamical evolution.
Hereafter, we take the target manifold N to be a three sphere S3 (it is diffeo-
morphic to SU(2)) and we assume that the domain manifold M is spherically
symmetric. Moreover, we assume that the wave maps are corotational.l This particular
setting was investigated in a series of articles1_8 in the context of the singularity
formation and critical phenomena. The study of wave maps on Minkowski
background9 (a = 0) revealed that the dynamics of the flat model is ruled by the family
of continuously self-similar solutions10 (CSS). Next, it was shown that ,,turning
on gravity" does not break the structure of CSS solutions.1 The investigation of
Cauchy problem2 suggested that analysis of the CSS class can be helpful also in
understanding critical phenomena in the model coupled to gravity.
2. Chaos in the model
We briefly report the results7 concerning the structure of CSS solutions for the
strong coupling. Therefore, wc restrict our analysis to the CSS class. Since we
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2079
study the self-similar problem in spherical symmetry the equations reduce to an
autonomous first order system7
W' = -l + a(l-W2)D2, (2)
D> = 2aWD3 . Sin(2f) „ L& + T_l
-l + 2asin(JF)2V l~W2
F' = D,
where the functions W, D parameterize the metric ga\, and the function F
parameterizes the wave map X. The coordinate system chosen here1 covers spacetime up
to the Cauchy horizon.
The consequence of self-similarity is the existence of a strong curvature
singularity for non-trivial solutions.11 The general solution to the system (2) does not
satisfy regularity conditions at the center and at the past light cone of the
singularity. However, it is more convenient to study general solutions and later to identify
these that satisfy regularity conditions.
The system (2) exhibits bistable behavior with a basin boundary separating
both types of solutions. For one type of solutions the curvature singularity is naked,
for second type it is hidden under an apparent horizon. All solutions lying at the
boundary tend asymptotically to the intermediate attractor that is periodic6 for the
weak coupling. The numerical analysis shows that for larger values of the coupling
constant the intermediate attractor becomes chaotic. This phenomena can be
understood with the help of a two dimensional Poincare map which reveals that the
heteroclinic intersection arises at the bifurcation point (a ~ 0.426).
Figure 1 presents the Poincare sections of the phase space for different
values of a. For small a periodic attractor correspond to two saddles J\, P^ in the
Poincare section. The stable manifolds Eis, E29 of these points form the basin
boundary. At the bifurcation point the unstable E\u and stable E^s manifolds cross
and the transversal heteroclinic intersection arises. The basin boundary becomes
fractal and the intermediate attractor becomes chaotic. The capacity dimension of
the one dimensional intersection of the boundary was estimated numerically to be
d = 0.337 ± 0.003 for a — 0.4264. The fractal dimension cannot be changed by
any continuous coordinate transformation, hence the description of chaos presented
above is diffeomorphism invariant. The presence of the transversal heteroclinic
intersection implies the appearance of the horseshoelike dynamics.7
It follows from the analysis of the regularity conditions1 that the regular
solutions asymptoting to the chaotic intermediate attractor necessarily have more
then one unstable mode. Therefore, they cannot drive the dynamics of the critical
phenomenon in the full Cauchy problem. However, chaotic solutions separate two
general types within the CSS class and in this sense they are critical. The behavior
of the CSS class studied in this context resembles many aspects of type II critical
phenomena.1
2080
-0.8 -0.4 0 0.4 0.8 -0.8 -0.4 0 0.4 0.8
W W
Fig. 1. Two saddles Pi, P2 and the creation of the transversal heteroclinic intersections of
unstable Ein and stable E2e manifolds; F = 1 + kir where kgZ.
3. Summary
We have presented the diffeomorphism invariant argument for the existence of
chaotic solutions to Einstein equations coupled to a wave map with target SU(2).
These solutions provide an example of fractal critical behavior within CSS class of
solutions.
This work was supported by the MNII grant no. 1 P03B 012 29.
References
1. P. Bizori and A. Wasserman, Phys. Rev. D62, p. 084031 (2000).
2. S. Husa, C. Lechner, M. Piirrer, J. Thornburg and P. C. Aichelburg, Phys. Rev. D62,
p. 104007 (2000).
3. C. Lechner, PhD thesis, University of Vienna, (2001).
4. C. Lechner, J. Thornburg, S. Husa and P. C. Aichelburg, Phys. Rev. D65, p. 081501
(2002).
5. P. Bizori and A. Wasserman, Class. Quant. Grav. 19, 3309 (2002).
6. P. Bizori, S. Szybka and A. Wasserman, Phys. Rev. D69, p. 064014 (2004).
7. S. J. Szybka, Phys. Rev. D69, p. 084014 (2004).
8. P. C. Aichelburg, P. Bizori and Z. Tabor, Class. Quant. Grav. 23, S299 (2006).
9. P. Bizori, T. Chmaj and Z. Tabor, Nonlinearity 13, 1411 (2000).
10. P. Bizori, Commun. Math. Phys. 215, 45 (2000).
11. C. Gundlach and J. M. Martin Garcia, Phys. Rev. D68, p. 064019 (2003).
CHAOS IN CORE-HALO GRAVITATING SYSTEMS
T. GHAHRAMANYAN1, V.G. GURZADYAN]-2>*
1 Yerevan Physics Institute, Yerevan, Armenia; 2ICRANet, Dipartimento di Fisica, Universita
La Sapienza, Roma, Italy
* E-mail: gurzadya@icra.it
Chaotic dynamics essentially defines the global properties of gravitating systems,
including, probably, the basics of morphology of galaxies. We use the Ricci curvature criterion
to study the degree of relative chaos (exponential instability) in core-halo gravitating
configurations. We show the existence of a critical core radius when the system is least
chaotic, while systems with both smaller and larger core radius will typically possess
stronger chaotic properties.
Keywords: Chaos; gravitating systems; galaxies.
1. Introduction
The importance of chaotic effects in the dynamics of N-body gravitating systems
has been attracting attention during the last decades1-5 . The difficulty of rigorous
study of chaos in 3D gravitating N-body systems is partly determined by the limited
content of such descriptors as Lyapunov characteristic exponents, otherwise
applicable for low degree of freedom systems. Numerous numerical studies estimating
not clearly defined Lyapunov-like exponents remain not helpful in deciphering the
complex nature and far going consequences of chaos in many dimensional nonlinear
systems (see the critics in6).
Below we represent a brief summary of the study of statistical properties of core-
halo type gravitating systems,7 using the Ricci curvature criterion of relative
instability. That criterion was introduced in10 upon discussions with Vladimir Arnold,
and later was applied in extensive numerical studies of N-body systems, see e.g.11
We have investigated spherical stellar systems with a core of various radii, following
the behavior of the Ricci curvature depending on a ratio of core rc and system's R
radii, k = rc/R. That dependence is also observed while varying the total energy of
the system.
2. The Ricci Criterion
Well known geometric methods of theory of dynamical systems enable one to study
the properties of a Hamiltonian system reducing the equations of motion to a
geodesic flow in the configuration space Af.3,9 For further development of these
methods see.8 The Ricci curvature ru(s) in M in the direction of the velocity vector
u of the geodesies is defined as
ru(s)=RljUluj/\\u\\2. (1)
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2082
Averaged with respect to the set of perturbed N-body systems, within an interval
[0, s*] of the affine parameter of the geodesies, it will yield
-4 inf r„(.).
At smaller negative values of ru a system is unstable with higher probability, as the
deviation vector z(s) of the close geodesies increases faster
z(s) > e^rs, (2)
in that interval. For collisionless N-body systems rv(s) is10
ru(s) =
3N -2Wi kul
\*»-*^
3N-4J \jW\
(3)
2 W ' 4V~" ' W2 4 W3
where W = E — V(q), V(q) is the Newtonian potential, and E is the total energy
of the system, W{ are the derivatives of W.
3. Numerical Analysis
Spherical 3D systems have been created with randomly generated velocities and
coordinates of N point particles of equal mass. Each system consisted of two concentric
spheres, initially both spheres having the same radius k = 1, then with appearance
of a core by means of the decrease of k. To enable the comparison of the sequence
of the created systems, the total energy parameter has been fixed via the
multiplication of all velocities of the system by certain constants. The estimation of the
Ricci for such static configurations describes the role of the core in the instability
properties of the system immediately moving away from the initial time moment.
Typical behavior of the Ricci curvature is exhibited in Figure 1 vs k, and the total
energy as a parameter.
0.00-,
-0 25
-0 50
-0 75
-1 00
0 0 0 1 0 2 0.3 0 4 0 5 0.6 0.7 0 8 0 9 10
k
Fig. 1. The dependence of the Ricci curvature on the ratio of the core and system's radii, k, for
two values of total energy of the system for N=1000.
One can see that, the Ricci curvature has a maximum at some value of k = kcr.
The latter corresponds to the most stable system among those with different core
2083
radii, so that for both k —> 0 and k —> 1, the system becomes more unstable;
we know that spherical N-body systems are exponentially instable as Kolmogorov
systems.1 The value of kcr has been investigated for different systems, varying the
total energy, the number of stars and the radius of the system. The behavior of kcr
vs the total (negative) energy E is shown in Figure 2.
-4000 -3500 -3000 -2500
Fig. 2. The variation of kcr vs the total energy of the system E for N=1000.
Core-halo configurations are typical for the observed stellar systems, globular
clusters and elliptical galaxies, and were an object of numerous theoretical studies,
including the pioneering one by Lynden-Bell.7 Note, we do not discuss core collapse
type evolutionary effects, as they have much larger characteristic time scales than
those of the reaching the quasi-stationary states discussed here. The existence of a
critical core radius as revealed above, can bring closer the link with thermodynamic
and other approaches to the stability of spherical gravitating systems.
References
1. V.G. Gurzadyan, G.K. Savvidy, Dokl. AN SSSR, 277, 69, f984; A&A, 160, 203, 1986.
2. D. Pfenniger, A&A, 165, 74, 1986.
3. V.G. Gurzadyan, D. Pfenniger, (Eds.) Ergodic Concepts in Stellar Dynamics, Springer,
Berlin, 1994.
4. V.G. Gurzadyan, Highlights of Astronomy, vol.13, p.366, 2004.
5. D. Benest, C. Froeschle, E. Lega, Hamiltonian Systems and Fourier Analysis. New
Prospects for Gravitational Dynamics, Cambridge Sci. Publ., 2005.
6. D. Ruelle, Chance and Chaos, Princeton Univ. Press, 2004.
7. D. Lynden-Bell, MNRAS, 136, 101, 1967.
8. A.A.Kocharyan, in: Proc. IV Monash Gen. Relat. Workshop, (Eds A.Lun, L.Brewin,
E.Chow), p.38, Melbourne, 1993; astro-ph/0411595.
9. V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer, Berlin, 1989.
10. V.G. Gurzadyan, A.A. Kocharyan, Ap&SS, 135, 307, 1987; Dokl. AN SSSR, 301,
323, 1988.
11. A. El-Zant, A&A, 326, 113, 1997; A.A. El-Zant, V.G. Gurzadyan, Physica, D, 122,
241, 1998.
TRANSIENT CHAOS IN SCALAR FIELD COSMOLOGY
ON A BRANE
A. TOPORENSKY
Sternberg Astronomical Institute, Universitetsky prospekt, 13, Moscow 119992, Russia
lesha@sai.msu.ru
We study cosmological dynamics of a flat Randall-Sundrum brane with a scalar field
and a negative "dark radiation" term. It is shown that in some situations the "dark
radiation" can mimic spatial curvature and cause a chaotic behavior which is similar to
chaotic dynamics in a closed Universe with a scalar field.
The phenomenon of transient chaos in homogeneous cosmological models had been
described by D. Page1 (he studied a closed isotropic Universe with a massive
minimally coupled scalar field) even earlier than this concept was formulated and
investigated systematically.2'3 The key feature of this type of chaos is that the dymamical
system (in comparison with the well-known case of strange attractors) has a regular
regime as its future attractor while particular trajectories can experience a chaotic
behavior before reaching this stable regime. Apart from attractors, the final
outcome can also be represented by some another situation which can be treated as a
"final state" (as in the case of a cosmological singularity where the entire dynamics
brakes down).
In the described dynamics of the Universe a cosmological singularity is the
ultimate fate of any (except for a set of zero measure) trajectory, though the Universe
can go through an arbitrary number of "bounces" (i.e. transitions from
contraction to expansion) before final contraction stage ends in a singularity. The set of
initial conditions leading to bounces has a rather regular structure,4 which allows
calculation of topological entropy.5'6
This type of dynamics is different from the Mixmaster chaos where shear
variables experience chaotic oscillations while volume of the Universe decreases mono-
tonically. Similar picture exists for chaos in two-field system and for non-abelian
field dynamics - both these cases do not require volume oscillations, which are
crucial for describing type of transient chaos. It differs also from the chaos in a closed
Universe with a conformal massive scalar field. The main feature of the latter
system is that the dynamics can be prolonged through a cosmological singularity to
the range of negative scale factors. As a result, we have chaotic oscillations of scale
factor (it changes its sign twice during one oscillation) without any future stable
regime, and this chaos can not be treated as a transient one.
In all our previous studies6-8 we were interested only in steepness of the scalar
field potential V(ip) for large ip and its influence on the possibility of bounces. On
the other hand, any positive potential with V(0) = 0 in a close Universe leads to a
recollaps ultimately, while open and flat Universe will expand forever. This is the
reason why the transient chaos exists only for closed Universe in the standard
cosmology. However, violation of positive energy condition can change this situation.
There are several possible sources of an effective negative energy in modern cosmo-
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2085
logical scenarios. One of the most popular possibilities is so called "dark radiation"
which appears in braneworld scenarios. The sign of dark radiation is not fixed in the
theory, and in the case of a negative sign the dark radiation can cause the recollaps
of a flat brane Universe. The goal of the present communication is to study the
possibility of a transient chaos in a flat brane Universe, where recollaps is achieved
solely by a negative dark radiation.
From now on we study a flat Randall-Sundrum brane with a minimally coupled
scalar field. The equations of motions are9
a a
2 1,4 J.2
k4 , s k2
a • ^ = -KPb{Pb+Pb)-YA (1)
a2 k2 K kA 2 C
az 6 36 a1
Here k2 = 8n/M?5y where M(5) is a fundamental 5-dimensional Planck mass, C is
the "dark radiation". The matter density on a brane is
pb = ip2/2 + V(<p) + A,
where A is the brane tension, the effective pressure is
Pb = <p2/2-V(<p),
the Klein-Gordon equation for the scalar field remains the same as in the standard
cosmology.
In the eq.(l)-(2) A is the cosmological constant in a bulk, and we assume that
A = —(fc2/6)A (the Randall-Sundrum constraint) in order to get the effective
cosmological constant on a brane vanishing.
The cosmological dynamics on a brane depends on the ratio p/\, where p =
tp2/2 + V(ip) is the energy density of a scalar field (so as pb = p + A). We will study
two limiting cases p/\ ^ 1 and p/A> 1 separately.
In the former case (a low-energy regime) expanding (p + A)2 and neglecting
p2 term in comparison with pA, we get the standard linear dependence between
Hubble parameter square and the matter density.10'11 Introducing an effective 4-
dimensional Planck mass m2P = 487r/(fc4A), the equation (2) can be rewritten in a
form analogous to the case of a standard cosmology with the 4-dimensional Planck
mass and rescaled C = 18/(fc4A)C:
8n az a4
It is clear that the second term in the LHS resembles the spatial curvature in
the case of C < 0, however, with different power-law dependence on a. The question
we should answer is whether this difference is crucial for existence of the transient
chaos in this system.
It is rather easy to show that the possibility of a bounce does not depend
significantly on the particular form of a "curvature-like" term Cjav in the LHS of
eq. (3) for an arbitrary positive p.12 However, the second condition for the chaotic
2086
dynamics - transitions from expansion to contraction - appears to be sensitive to
the value of p. It is clear from (3) that a transition to contraction never happens if
the matter density p decreases less rapidly than a~p at the expansion stage. It is
well-known that a late-time regime for the scalar field with the potential V ~ tpn is
damping oscillations with the effective equation of state in the form p = ^r^P-13 It
means, in particular, that a massive scalar field (V = m2ip2/2) behaves like dust at
the oscillatory stage (p ~ a~3), while a self-interacting scalar field (V" = A^4) has
the equation of state of an ultra-relativistic fluid (p ~ a-4). As the dark radiation
in the RS brane cosmology decreases as a~4, we immediately see that oscillations of
a massive scalar field can not be followed by the contraction epoch, and this brane
Universe will expand forever.
In the case of a self-interacting scalar field its energy remains proportional to the
"dark radiation", so a late-time recollaps of the brane Universe remains impossible.
Only for potential V ~ ipn with n > 6 (the potential V ~ ip6 corresponds to
asymptotic equation of state in the form p = p/2. and leads to the energy density
proportional to a-4'5) a recollaps of a flat brane Universe becomes inevitable, and
we get the same picture as for a closed Universe without " dark energy".
In the high-energy regime the matter part of the equations of motion depends
quadraticaly on the matter density, while the "dark radiation" term C/a4 remains
unchanged. This leads to a situation, qualitatively different from the regime
described above. Now even in the case of massive scalar field the matter term in the
RHS of (2) falls more rapidly than the "dark radiation", providing an ultimate
recollaps. Our numerical results for the potential V = m2ip2/2 confirm existence
of a transient chaos. For steeper potential the energy density during scalar field
oscillations fall even more rapidly, and the conditions for a chaos are satisfied as
well.
We see that in some situations a negative " dark radiation" term on a flat brane
can cause the same type of chaotic behavior, which is known in the standard
cosmology for a closed Universe.
References
1. Page D.N., Class. Quant. Grav. 1, 417 (1984).
2. Kantz H. and Grassberger P., Physica D 17, 75 (1985).
3. Gaspard P. and Rice S.A., J. Chem. Phys. 90, 2225 (1989).
4. Starobinsky A., Sov. Astron. Lett. 4, 82 (1978).
5. Cornish N.J. and Shellard E.P.S., Phys. Rev. Lett. 81, 3571 (1998).
6. Kamenshchik A., Khalatnikov I., Savchenko S. and Toporensky A., Phys. Rev. D59,
123516 (1999).
7. Toporensky A., Int. J. Mod. Phys. D8, 739 (1999).
8. Toporensky A., SIGMA 2, 037 (2006).
9. Binetruy P., Deffayet C, Ellwanger U. and Langlois D., Phys. Lett. B477, 285 (2000).
10. Csaki C, Graesser M., Kolda C. and Terning J., Phys. Lett. B462, 34 (1999).
11. Cline J., Grossjean C. and Servant G., Phys. Rev. Lett. 83, 4245 (1999).
12. Toporensky A., gr-qc/0609048.
13. Turner M., Phys. Rev. D28, 1243 (1983).
TOWARD A HOLOGRAPHIC ORIGIN OF COSMOLOGICAL
LARGE SCALE STRUCTURE
J. R. MUREIKA
Department of Physics, Loyola Marymount University, Los Angeles, CA, USA
jmureika@lmu. edu
The fractal dimension of large-scale galaxy clustering has been demonstrated to be
roughly Dp- ~ 2 from a wide range of redshift surveys. This statistic is of interest
for two main reasons: fractal scaling is an implicit representation of information content,
and also the value itself is a geometric signature of area. It is proposed that the fractal
distribution of galaxies may thus be interpreted as a signature of holography ("fractal
holography"), providing more support for current theories of holographic cosmologies.
The general fractal scaling relationship assumes a power-law form N(r) ~ rDp,
where Dp is the fractal dimension and r is the scale measure. The quantity N(r)
represents the characteristic of the distribution that exhibits the fractal behavior.
In many cases, the fractal dimension is treated as a statistical quantity, but it is
important to remember that it also has a geometric significance. When a fractal
dimension coincides with an integer dimension, it is possible to make an inference
between the structure under consideration and the geometry associated with the
dimension.
The advent of deep sky redshift surveys has brought with it a surge interest
surrounding the exact nature of large-scale galaxy distributions in the observable
universe. An overwhelming number of independent estimates of the galaxy
clustering fractal dimension obtained from a variety of sources seem to unanimously
suggest that this statistic has a value of or around Dp = 2, up to depths of at least
10 hTl Mpc or more.1 The newest SDSS redshift data confirms the Dp ~ 2 to a
high precision up to 20 h~l Mpc,2 but with the correlation weakening to
homogeneity at distances of 70 h~l Mpc.3 Alternate analyses suggest that the transition
to homogeneity occurs at much larger scales of 200 h~l Mpc.4
The exact origins of large scale structure in the universe are unknown, although
it is commonly believed that it has arisen from anisotropically-distributed quantum
fluctuations in the pre-infiation epoch. A number of theoretical solutions have been
offered for such inhomogeneous structure, based on CDM N-body gravitational
collapse scenarios.5~8 These are of particular interest due to their natural connection
to hierarchical clustering growth from small initial mass/density perturbations in
the early universe.
Fractality is generally not associated with equilibrium growth, and thus most
models of large scale structure evolution do not predict its existence. However, as the
aforementioned evidence undeniably suggests, there is a definite fractal distribution
of matter in the universe. The use of entropy to represent fractal structure stems
from the implicit relation between entropy and information (this is discussed in the
concluding section of this paper). Fractal - and moreover multifractal - statistics
quantify the nature in which information is encoded or distributed in a system. It is
the intention of this presentation to highlight this connection between information,
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2088
entropy, and fractality.
The holographic principle9 (HP) suggests that there exists a deeper geometric
origin for the total number of possible quantum states which can occupy a spatial
region. In its most general formulation, the HP states that S(B) < dB/4, where B
is some region, dB its boundary, and S(B) the entropy contained in B. Assuming
a homogeneous and isotropic universe with constant mean density,10 it is possible
to define a (co-moving) volumetric entropy density a such that the total entropy
with a volume V is S = aV. The holographic condition is thus aV < —4 , where
A(V) = 47rr2 is the bounding area of the volume V (in flat space). Including the
r-dependence, the spacelike entropy bound is r3a < ^-, which is violated9 for
sufficiently large values of r > 3/4cr. Extensions of the HP to cosmology have been
proposed,11'12 in an attempt to incorporate inflationary scenarios and explain the
flatness and horizon problems in various FRW-type cosmologies. So, using the HP
to explain cosmic inhomogeneities seems a reasonable next step.
The observed Df = 2 scaling makes it more appropriate to describe the entropic
content by a mean "surface" entropy density £. Fractal large scale structure thus
states that within a sphere of radius r, the number of cosmological objects is a
function of area (r2). Thus, within a spherical volume of radius r, the number of
galaxies N(r) must be proportional to the surface area of the region's boundary,
N(r) oc dV(r) = A(r), so that the entropy contained with a region V is S(V) =
a£A, where a > 0 is the proportionality constant. The above relation suggests
that the distribution of matter in the universe has perhaps a more fundamental
and geometric origin. Specifically, the violation of the space-like entropy bound
is eliminated, and instead is replaced by rigid constraints on the surface entropy
density, and hence the geometry of the matter distribution: a£' < \.
The fractal distribution of galaxies extends to at least 10 Mpc, or 1058 Planck
length units. The area of the bounding sphere is thus on the order of 10116 area
units. The entropy content of the entire visible universe13 is on the order of 1090, so
even if a sizable fraction is represented in this fractal distribution, this implies the
"surface" density is no greater than £ ~ 10~24 or so. The value of the proportionality
constant a thus is the key to the inequality. Unless a is of exceedingly high order
of magnitude, though, it is unlikely that this bound will ever be violated.
This model can be used to assess the nature of possible transitions to
homogeneity, by matching the "surface" entropy distribution to a volumetric one:
Sf(R) = Sh(R), where SF(R) ~ £'R2 and Su ~ crR3. The requirement of
statistical continuity in the description of the entropy places strong constraints on the
nature of the matter distributions at scales r < R, namely £' = aR. Therefore,
the combination of the total entropy in the universe and the homogeneity scale R
explicitly determines the "density" of matter on smaller scales. This gives new
interpretation of R as a type of critical parameter that marks some variety of "phase
transition" in the distribution of matter.
No cosmological model would be complete without paying due attention to the
existence of dark matter (DM). Unfortunately, not much is known about how DM
2089
might be spread through the universe. For density profiles prjM ~ ^ 7, the
associated fractal scaling dimension is Ddm = 3 — 7, and so the presence of dark
matter complements the entropy constraint for luminous matter (LM) to give the
bound 5*lm + >5dm < A/4. Hence, precise knowledge of the form of the dark matter
density profile will determine whether or not the modified bound is violated. The
best models currently available for density distributions are those of "small scale"
dark matter halo structures derived from galaxy rotation curves, which suggest a
physical density profile of the form Pdm ~ r~2, and thus -Ddm = 1- More elaborate
forms have also been proposed, such as the NFW profile.14 Numerous other refer-
ences15~17 peg the possible distribution profile anywhere between Ddm ~ 1.5— 2.5.
If Ddm < 2, the holographic constraint behaves as in the case of luminous matter.
The connection between information theory, gravitation, and geometry is a
common "theme" for fractal large scale structure, and this proposal ties these three
concepts together. At the very least, the observed fractal distribution behavior of
galaxies could be understood to be a large scale bookend principle to holography.
Redshift survey results provide strong evidence that the number counts scale as an
area, but in order to verify a deeper connection future analyses should also focus on
the pre-factor of the fractal relationship. Fractal clustering of large-scale structure
may well represent either a manifestation of holographic entropy bounds, or the end
result of a cosmological holography model, and future studies should adopt such a
re-interpretation to explore new implications of the data.
References
1. Sylos Labini, F., Montuori, M., and Pietronero, L., Phys. Rep. 293, 61-226 (1998)
2. Hogg, D. et al., Astrophys. J. 624, 54-58 (2005)
3. Joyce, M., Sylos Labini, F., Gabrielli, A., Montuori, M., and Pietronero, L., Astron.
Astrophys. 443, 11-16 (2005)
4. Baryshev, Y. V. and Bukhmastova, Y. L., Astron. Lett. 30, 444 (2004)
5. Valdarnini, R., Borgani, S., and Provenzale, A., Astrophys. J. 394, 422 (1992)
6. Borgani, S. et al, Phys. Rev. E 47, 3879-3888 (1993)
7. Dubrelle, B., and Lachieze-Rey, M., Astron. Astrophys. 289, 667 (1994)
8. Colombi, S., Bouchet, F. R., and Schaeffer, R., Astron. Astrophys. 263, 1 (1992)
9. Bousso, R., Rev. Mod. Phys. 74, 825-874 (2002)
10. Fischler, W. and Susskind, L., hep-th/9806039 [SU-ITP-98-39,UTTG-06-98]
11. Fischler, W. and Banks, T, hep-th/0111142; hep-th/0405200; Phys. Scripta T117,
56-63 (2005)
12. Bak, D. and Rey, S.-J., Class. Quant. Grav. 17 (15), L83-L89 (2000)
13. Kaloper, N. and Linde, A, Phys. Rev. bf D60, 103509 (1999)
14. Navarro, J. F., Frenk, C. S., White, S. D. M., Astrophys. J. 462, 563 (1996)
15. Navarro, J. F., Frenk, C. S., White, S. D. M., Astrophys. J. 490, 493-508 (1997)
16. Sumner, T. J., Living Rev. Relativity 5 (2002), http://www.livingreviews.org/lrr-
2002-4. Viewed 01 July 2006.
17. Kirillov, Astron. Astrophys., Phys. Lett. B535, 22-24 (2002)
18. Fukushige, T. and Makino, J., Astrophys. J. 557, 553-434 (2001)
VECTOR FIELD INDUCED CHAOS IN MULTI-DIMENSIONAL
HOMOGENEOUS COSMOLOGIES
R. BENINI12t, A. A. KIRILLOV3* and G. MONTANI24*
1 Dipartimento di Fisica - Universita di Bologna and INFN
Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy
2ICRA—International Center for Relativistic Astrophysics c/o Dipartimento di Fisica (G9)
Universita di Roma "La Sapienza", Piazza A.Moro 5 00185 Roma, Italy
3 Institute for Applied Mathematics and Cybernetics
10 Ulyanova str., Nizhny Novgorod, 603005, Russia
4ENEA C.R. Frascati (U.T.S. Fusione), Via Enrico Fermi 45, 00044 Frascati, Roma, Italy
t riccardo.benini@icra.it
t kirillov@unn. ac.ru
0 montani@icra. it
We show that in multidimensional gravity vector fields completely determine the
structure and properties of singularity. It turns out that in the presence of a vector field
the oscillatory regime exists for any number of spatial dimensions and for all
homogeneous models. We derive the Poincare return map associated to the Kasner indexes
and fix the rules according to which the Kasner vectors rotate. In correspondence to
a 4-dimcnsional space time, the oscillatory regime here constructed overlap the usual
Belinski-Khalatnikov-Liftshitz one.
1. Introduction
The wide interest attracted by the homogeneous cosmological models of the Bianchi
classification relies over all in the allowance for their anisotropic dynamics; among
them the types VIII and IX stand because of their chaotic evolution toward the
initial singularity1 that correspond to the maximum degree of generality allowed
by the homogeneity constraint; as a consequence it was shown2-4 that the generic
cosmological solution can be described properly, near the Big-Bang, in terms of the
homogeneous chaotic dynamics as referred to each cosmological horizon. However
the correspondence existing between the homogeneous dynamics and the generic
inhomogeneous one holds only in four space-time dimensions. In fact a generic
cosmological inhomogeneous model remains characterized by chaos near the Big-
Bang up to a ten dimensional space-time5-7 while the homogeneous models show a
regular (chaos free) dynamics beyond four dimensions.8,9
Here we address an Hamiltonian point of view showing how the homogeneous
models (of each type) perform, near the singularity, an oscillatory regime in
correspondence to any number of dimensions, as soon as an electromagnetic field is
included in the dynamics.
2. The Standard Kasner Dynamics
Let us consider the standard n + 1-dimensional vector-tensor theory in the ADM
representation:
/ = Jdnxdt lua^tgaP + tt" JU« + <pDaita - NH0 - NaHa\ , (1)
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2091
Ho = ~ Wf - -L_ (H«)2 + l-gaP^ + g (\FapF«P - r\\ , (2)
Ha = -VpU^+^Fa(3. (3)
Here Hq and Ha denote respectively the super-Hamiltonian and super-momentum,
Fap = dpAa — daAp is the electromagnetic tensor, g = det(gap) is the determinant
of the n-metric, R is the n-scalar of curvature and Da = da + Aa.
Since the sources are absent, it is enough to consider only the transverse
components for Aa and ira; therefore, we take the gauge conditions ip = 0 and Daira = 0.
When going over the homogeneous case, we choose the gauge N = 1 and TV" = 0.
Let's adopt the Kasner parameterization,that is based on the metric and
conjugate momentum decomposition along spatial n-bein:
9a(3 = dablalpi ^-a(3 = Pabi^p- (4)
We also define a dual basis Laa = ga/3lap, such that L%lba = Sba and I£Z°j = 6%.
We want to put in evidence the oscillatory regime that the bein vectors possess
and so we distinguish scale functions and the parallel from the transverse component
(Aa = (**£%))
la = exp (g°/2) £a, La= exp(-qa/2) Ca. (5)
La ^a ~ ^aA-i
^TT, (7?C±)=0. (6)
The standard Kasner solution is obtained as soon as the limit in which all the terms
exp(qa) become of higher order is taken
pa = const, Xa = const, £a± = const,
9 „ — 2W /„ 1
$t<La = fg [Pa-^T.^J, (7)
gap = E t2Sa£aJap - So = 1 - (n - 1) =^- . (8)
Z^bPb
The Kasner indexes sa satisfy the identities X) sa = X) s^ = 1.
3. Billiard representation: the return map and the rotation of
Kasner vectors
If we order the s0's, the largest increasing term (as t —> 0 tSl —> oo) among the
neglected ones comes from si and it is to be taken into account to construct the
2092
oscillatory regime toward the cosmological singularity.
fcPi = -1T^\ exP W) > T^Pa = °> (9)
d_ _
dtqa~ V9
The first of equations (9) gives Ai = const, while the second admits the solution
Aa {Pa ~Pi) = const. (10)
The remaining part of the dynamical system allows us to determine the return map
governing the replacements of Kasner epochs and the rotation of Kasner vectors £a
through these epochs
"Si , Sa + ^2«1
2
i + ^V
(n-l)si \
(n- 2)sa +ns1J
o I™-1)*!
5
Aa
(11)
^ = Ai, K = Kii-2,„K"y:' , (12)
£ = ?a + *«/l, *a = ^-^ = -2, V" ^ ^- (13)
Ai (n-2)sa+ns! Ai
Thus the homogeneous Universes here discussed approaches the initial
singularity being described by a metric tensor with oscillating scale factors and rotating
Kasner vectors. The presence of a vector field is crucial because, independently on
the considered model, it induces a closed domain on the configuration space.
References
1. V.A. Belinski, I.M. Khalatnikov and E.M. Lifshitz, Adv. Phys., 19, 525, (1970).
2. V A Belinskii, I M Khalatnikov and E M Lifshitz, Adv. Phys. 31 (1982) 639.
3. A.A. Kirillov, Zh. Eksp. Teor. Fiz. 103, 721 (1993). [Sov. Phys. JETP 76, 355 (1993)].
4. G. Montani, Class. Quantum Grav. 12, 2505 (1995).
5. J. Demaret, M. Henneaux and P. Spindel, Phys. Lett, 164B, 27, (1985).
6. J. Demaret et al., BPhys. Lett, 175B, 129 (1986).
7. Y.Elskens, M.Henneaux, Nucl. P/i2/s.290B(1987) 111
8. P.Halpern, Phys. Rev.HQQ (2002) 027503
9. P.Halpern, Gen. Rel. Grav. 35 (2003) 251-261
Einstein—Maxwell Systems
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DYNAMO ACTION ON RELATIVISTIC SPHERICAL STARS
N. MONTELONGO GARCIA
Ins. de Fisica y Matemdticas, Universidad Michoacana de San Nicolas de Hidalgo,
A.P. 2-82, 58040 Morelia, Mich, MEXICO
nadiezhda@ifm.umich.mx
T. ZANNIAS
Ins. de Fisica y Matemdticas, Universidad Michoacana de San Nicolas de Hidalgo,
A.P. 2-82, 58040 Morelia, Mich, MEXICO
zannias@ginette.ifm.umich.mx
Within the framework of relativistic gravity, an axisymmetric magnetic field B cannot
maintained by dynamo action if:
l)The fluid flow is axially symmetric and divergence free.
2)The background geometry corresponds to a static spherical star of constant density
with compactness ratio e = 2 vf , in the range s £ [0, |].
Let a static non singular spherical stellar model of areal radious R that joins
smoothly to a part of Schwarzschild spacetime. For this model the interior
metric takes the form:
dr2
g = -V2(r)dt2 + ^y + r2(d02 + sin<92dip2), r<R (1)
r
while in the exterior r >R region, g reduces to the the familiar Schwarzschild form
with V2(r) = 1 — "y and m(i?) > 0. On this background Maxwell's equations
take the form:1
V ■ E = 4yrp, V ■ B = 0, (2)
**™-> + \%- **™~\% <3»
where the divergence V- and curl operator Vx formed using the spatial metric:
/ 2m(r)\~1/2
7 = h2rdr2 + h2ed02 + h2vdLp2, hr=il ^J , he=r, hv=rsm8, (4)
and B(i, r, 6,ip) y E(t,r,6,<p) the magnetic and electric fields as measure by a
Killing observer relative to their orthonormal basis. We assume in the stars interior
a test conducting fluid flow is defined with velocity field:
1 v' ! d v% d m fv2\ ■ n ,rs
For such flow the conduction current J takes the form:
■ff(E+^^),v = v*ei> (6)
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2096
and a combination of (6) with (2,3) yields the following induction equation:
dB
~dt
V x (vx VB)- Vxt|(Vx VB), V • B = 0, |x| < R,
(7)
where r\ = -f^ denotes the magnetic diffusivity. In the stars exterior region and in
the absence of any current distribution, B obeys:
V x (VB) = 0, V ■ B = 0, |x| > R.
(8)
As long rj ^ 0, the field B will be continuous across the star's surface and would
decay to zero at spacelike infinity at least as a dipole field, i.e, limix^^ B = O (^).
Within the context of relativistic dynamo of particular interest are steady state
solutions of (7,8) i.e. solutions where the effects of advection are balanced by ohmic
dissipation. In complete analogy to the non relativistic Cowling's theorem, we shall
show that any state solution of (7,8) developing from some regular initial field
distribution, under the assumption that B and v are axisymmetric about the same
axis and V ■ v = 0 is necessary trivial i.e B(x,£) = 0 for any t S> To with To the
corresponding ohmic time scale. In order to show that at first we split the field
B into toroidal BT and poloidal parts Bp via B = BT + Bp where BT = Bvev,
Bp = Brer + B0ev with a similar decomposition for the velocity field v. Combining
that splitting with eq (7) we obtain for the interior fields:
= Vx(vpxFBT|vTxFBp)-Vx(tiVxFBT), V • BT = 0, (9)
dt
dW_
~&7
V x (vp x VBP) - V x (nV x VBP), V-B1
o,
(10)
while in the exterior region BT and Bp obey V x (VBP) = V • (Bp) = 0, B1
The representation Bp = V x ( \'r' ev), for some X(t,r,9) combined with (10)
yields:
2VX-Vloghv~
1
(v^ • V)X + v
K
M<R, (11)
vdt vv/ ,v
where V2 stands for the Laplacian operator of the spatial metric and in above he
have set: v = vp — 77^7-. The exterior poloidal field B = V x ( -^ ] satisfies:
1
V2X
hv ' h.
Similarly the representation B
dB , P _ (VB
-VX ■ Vlogh^
ldB=Vx
0, |x| > R.
(12)
B(t,r,8)ev, combined with (9) yields:
dt
hvvp ■ V
V K
+ (V-vp)VB + v
■V-hiV
fVB
+
vv(Bp ■ VV) + hv VBP ■ V
xl <R.
2BVV2loghv
(13)
2097
Multiplying (11,12) by X, taking into account that V ■ v
the star's interior we eventually obtain (for details see2):
id_ r x^
2dtJ V
where K and A are defined by
W2V 1W-W
2"
0, and integrating in
K = -'-
dfl = ri KX2dtt - 77 / VX • VXdfl,
V2{loghv), A = r)XVX-r)X2Vloghlf
(14)
X2.
while the
2 V 2 V2
and the integral in the left hand side is evaluated on the star's interior,
integrals in the right hand side are computed over the entire t=constant spacelike
hypersurface. The identify (14) shows that if K < 0 then any steady state solutions
of (11,12) on the background on (1) would be trivial i.e X = 0. However a
straightforward evaluation of K for the case where (1) correspond to constant density star
with compactness ration e = ^jf obeying e £ [0, 8/9] shows that K < 0. Thus
(14) implies X = 0 and thus B =0. The vanishing of B has as a consequence
the decay of the toroidal field. Setting Bp = 0 in (13) and upon multiplying both
sides by Xp- we obtain:2
\_d_
2dt
V
VB"
dfl
VB
(
(vbV
~\K)
V
fVB\
1
Vloghv - -
\k)
(VB
V2logh„
■ ds,
dfl-
(15)
and the surface integral is evaluated on the star's surface. Since however B\r = 0, it
follows that the right hand side of (15) is negative definite provided \72(loghv) < 0
in the star's interior region. Again for the background of a constant density star
always \72loghv < 0 and thus the above relation shows that a steady state has
B = 0 in the star's interior and hence BT = 0. In summary any fluid flow satisfying
the conditions described in the text will fail to act as a dynamo for an axisymmetric
B field on the background geometry of a constant density star.
Acknowledgments
This work supported by a grant of Coordinacion Cientifica- UMSNH and
CONACYT-Mexico.
References
1. N. Montelongo Garcia and T. Zannias, Relativistic dynamo theory,XXIX Spanish
Relativity Meeting E.R.E 2006.
2. N. Montelongo Garcia and T. Zannias: Report, unpublished (2007).
EXTERNAL ELECTROMAGNETIC FIELDS OF A SLOWLY
ROTATING MAGNETIZED STAR WITH NONVANISHING
GRAVITOMAGNETIC CHARGE
B.J. AHMEDOV, A.V. KHUGAEV and N.I. RAKHMATOV
Institute of Nuclear Physics and Ulugh Beg Astronomical Institute
Astronomicheskaya 33, Tashkent 700052, Uzbekistan
We write Maxwell equations in the external background spacetime of a slowly rotating
magnetized NUT star and find analytical solutions after separating them into angular and
radial parts. The star is considered isolated and in vacuum, with monopolar configuration
model for the stellar magnetic field. The contribution to the external field from the NUT
charge and frame-dragging effect are considered in detail.
1. Introduction
At present there is no any observational evidence for the existence of
gravitomagnetic monopole though there are attempts to detect it through astronomical
observations as gravitational lensing or to explain anomalous acceleration of Pioneer
satellites through the gravitational field of magnetic mass. However it is interesting
to study the electromagnetic fields in NUT space with the aim to get new tool
for studying new important general relativistic effects which are associated with
nondiagonal components of the metric tensor and have no Newtonian analogues.
2. Solutions to Maxwell Equations In a Space of Slowly Rotating
NUT Star
Our approach is based on the reasonable assumption that the metric of spacetime is
known i.e. neglecting the influence of the electromagnetic field on the gravitational
one and finding analytical solutions of Maxwell equations on a given, fixed
background. The next our main approximation is in the specific form of the background
metric which we choose to be that of a stationary, axially symmetric system
truncated at the first order in the angular velocity Q and in gravitomagnetic monopole
moment /. In a coordinate system (ct, r, 9, </>), the "slow rotation metric" for exterior
space-time of a rotating relativistic star with nonvanishing gravitomagnetic charge
is (see, for example,,21)
ds2 = -N2dt2 + N'2dr2 + r2d62 + r2 sin2 6d(j)2
- 2 [co{r)r2 sin2 6 + 21N2 cos(9] dtd<j> , (1)
that is, the Schwarzschild metric plus the Lense-Thirring and Taub-NUT terms.
Here parameter N = (l — ~-) , w(r) = M; can be interpreted as the angular
velocity of a free falling (inertial) frame and is also known as the Lense-Thirring
angular velocity, J = I(M, R)£l is the total angular momentum of metric source with
total mass M as measured from infinity and I(M, R) its momentum of inertia. The
nondiagonal component of the metric tensor is finite at the infinity: lmv^oo <7o3 =
—2lcos9 which is meant that the metric (1) is not asymptotically flat.
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2099
We will look for stationary solutions of the Maxwell equation, i.e. for solutions
in which we assume that the magnetic and monopole moments of the star do not
vary in time as a result of the infinite conductivity of the stellar interior. Below
we suggest that external electric field is generated by the magnetic field, taking as
a special monopolar configuration. For this case we can obtain and investigate an
analytical solution with detail consideration of the contributions from the dragging
effects and nonvanishing NUT charge in the magnitude of the external electric field
of the slowly rotating magnetized star.
As a toy model we could consider the following magnetic field configuration5
Bf = Bf(r)^0 , B§=Q. (2)
Although this form of magnetic field can not be considered as a realistic, we will
show that this toy model can be used to obtain first estimates of the influence of
gravitational field of the NUT charge on the external electromagnetic field of the
star. For this case, Maxwell equations reduce to
(r2Bf)r=0. (3)
The solution admitted by this equation is
Br = £ • (4)
where /i is some integration constant being responsible for source of magnetic field.
Electric field created by monopolar magnetic field is defined by the following
Maxwell equations
/ -x o..i„n„ a / at2\
0, (5)
(6)
= 0 • (7)
The analytical solutions of equations (5), (6) and (7) are responsible for the
electric field of NUT star with the monopolar magnetic field (4).
The analytical solution for radial and tangential components of electric can be
found in the form
E?(r,0) = ^F1(r,0), (8)
E§ (r, 6) = AC(r) „ - —?— f Fi (r, 6) r sin Bd0 . (9)
v ' rNsmO rsm6 J K h
Ef
sin
sin
e - [rNE
HW),r
6 (ur2£f)
')
+ fismO(uj)^ +
+ N~1r(sm8Ed)
r +
21 cose 2e,
sin 9 K '
N \ J ,6
2fil cos
sin 9
= 0,
+
r
21N
+
r
e-
m
(cot 6E§
where function i<\ is
3/iwr3
Fi(r,6>) = flRfi-
2M2
In N2 + — + 1 -i I
2Af2 \M 3r I
cosO-^, (10)
r
2100
and R is radius of star. Detailed derivation of analytical solutions and integration
constant C(r) will be given in separate paper.
3. Conclusion
We have presented analytic general relativistic expressions for the electromagnetic
fields external to a slowly-rotating magnetized neutron star with nonvanishing
gravitomagnetic charge I. The star is considered isolated and in vacuum, and for
simplicity with the monopolar magnetic field directed along the radial coordinate.
We have shown that the general relativistic corrections due to the dragging of
reference frames and gravitomagnetic charge are not present in the form of the
magnetic fields similar to dipolar case3'4 but emerge only in the form of the electric
fields. In particular, we have shown that the frame-dragging and gravitomagnetic
charge provide an additional induced electric field which is analogous to the one
introduced by the rotation of the star in the flat spacetime limit.6
Acknowledgments
This research is supported in part by the UzFFR (project 01-06) and projects
F.2.1.09, F2.2.06 and A13-226 of the UzCST. BJA acknowledges the partial financial
support from NATO through the reintegration grant EAP.RIG.981259.
References
1. D. Bini, C. Cherubini, R.T. Janzen and B. Mashhoon, Class. Quantum Grav. 2, 457
(2003).
2. N. Dadhich and Z.Ya. Turakulov, Class. Quantum Grav. 19, 2765 (2002).
3. V.L. Ginzburg and L.M. Ozernoy, Zh. Eksp. Teor. Fiz. 47, 1030 (1964).
4. J.L. Anderson and J.M. Cohen, Astrophys. Space Science 9, 146 (1964).
5. N. Messios, D.B. Papadoupolos and N. Stergioulas, Mon. Not. R. Astron. Soc. 328,
1161 (2001).
6. L. Rezzolla, B.J. Ahmedov and J.C. Miller, Mon. Not. R. Astron. Soc. 322, 723 (2001);
Erratum 338, 816 (2003).
ALIGNED ELECTROMAGNETIC EXCITATIONS OF THE
KERR-SCHILD SOLUTIONS*
ALEXANDER BURINSKII
Gravity Research Group, NSI Russian Academy of Sciences,
B. Tulskaya 52, Moscow 115191, Russia, bur@ibrae.ac.ru
Aligned to the Kerr-Schild geometry electromagnetic excitations are investigated, and
asymptotically exact solutions are obtained for the low-frequency limit.
1. In this paper we consider the aligned electromagnetic excitations of the Kerr-
Schild geometry, taking into account the back reaction of the excitations on metric.
To our knowledge, it is the first attempt to get in the Kerr-Schild formalism a self-
consistent solution for the case 7^0. Electromagnetic field of the exact Kerr-Schild
solutions1 has to be aligned to the Kerr null congruence2'3 which is generated by
tangent vector fcM(x). Aligned e.m. excitations on the Kerr background were
investigated in.2-4 Contrary to the usual 'quasi-normal' modes, the aligned excitations
are compatible with the Kerr congruence and type D of the metric. On the other
hand they have very specific exhibition in the form of semi-infinite 'axial' singular
lines producing narrow beams which can lead to some new astrophysical effects like
the holes in the horizons and jet formation.4 'Axial' singularities appear also in the
particle aspect of the Kerr-Schild solutions.5
2. The vector field k^ is determined by the Kerr Theorem via a complex function
Y(x), k^dx^ = P^1(du + Yd( + YdC, — YYdv), where P is a normalizing factor,
providing ko = 1. For the geodesic and shear-free congruences, satisfying to Y,2 =
Y,4 = 0, the Einstein-Maxwell field equations were integrated out in1 in a general
form and reduced to the system of equations for electromagnetic field
A,2-2Z-lZY,3A = 0, ,4,4=0,
VA+Z-^,2 -Z-XY,3 7 = 0, 7,4 = 0, where V = 33 - Z~YY,3 dx - Z~XY,3 <%
and for gravitational field, which will be discussed bellow.
Electromagnetic Sector, was discussed in.2-4 The first equation has the
general solution A = ip/' P2, where tjj,2 = ip,i = 0. Therefore tp has to be a holomorphic
function of variable Y, since Y,2 = Y,& = 0. Function Y is a projective (complex)
angular coordinate Y G CP1 = S2,Y = e^ tan f. A holomorphic function may be
represented as an infinite Laurent series ip(Y) = J^^L-oo ^"- ^ the function Y e S2
is not constant, it has to contain at least one pole which may also be at Y = oo (or
0 = 7r). So, for exclusion of the Kerr-Newman solution having Y = e = const., we
has to consider solutions ip(Y) = J2i y^y which are singular at angular directions
Yi = e^* tan y, and represent a narrow beams in there angular directions. Note,
that for qi = const, these solutions are exact self-consistent solutions of the full
system of Kerr-Schild equations.
A wave excitation propagating in the direction Yi will be described by the func-
*Talk at theGT3 session of the MG11 Meeting, this work is performed in the frame of collaboration
with E.Elizalde, S.R.Hildebrandt and G.Magli.
2101
2102
tion xj)i(Y, t) = q(r) exp{iu>T} Y^Y . where r is a retarded time. For the rotating
Kerr source the retarded time is complexa In the nonstationary case, solution for A
has the only difference that the function tp acquires extra dependence from the 'left'
retarded time tj,. In the rest frame the function P has the form P = 2_1//2(1 + YY).
The real operator V acts on the real slice as follows VY = T>Y = 0, and UP = 0.
The explicit form of the retarded time is tx = t — r + ia cos6. Since cos6 = \ZYy'
we have T>cos8 = 0, , and T>t = T>p = -p.
The second e.m. equation takes the form A = ~(-yP),Y ■ Integration yields
-y=^+<t>(Y,r)/P, (1)
where we neglected recoil and </> is an arbitrary analytic function of Y and r.
3. Gravitational sector is:
M,2 -ZZ-XZY,3 M = AyZ, (2)
VM = l-~n, M,4 = 0. (3)
Solutions of this system were given in1 only for stationary case, corresponding to
7 = 0. We assume that the energy of electromagnetic wave excitation is much
lower then the mass of rotating object m, and does not affect on the motion of the
center of mass of the solution. However, influence ofthe electromagnetic field on
the metric occurs also via the function H = r^T2^{2 g, in the K-S metric form
9iiv = f]\iv + 2_fffcAjfc„, where r]^ is the metric of auxiliary Minkowski space-time.
This is a more thin effect, leading to a deformation of the metric tensor around
rotating black hole by electromagnetic excitations. The poles in function ip which
cause the :axial' singular electromagnetic beams deform strongly the function H.
The equation (2) acquires the form (MP3))2 — AjZ. The equation (3) takes the
form m = |P477- It is known,6,7 that it determines the loss of mass by radiation.
The right sides of (2) and (3) will be small for the small (low-frequency) aligned
wave excitations, since the functions ij) and 7 will be of order ~ iujtjj. In this sense the
aligned excitations will be asymptotically exact solutions in the low-frequency limit.
However, since ip contains the singular poles in Y, the limit 7 —> 0 is not uniform
one, and an extra trick is necessary - a regularization. Such a regularization may be
performed by the free function </>(Y, r) in (1). The function 7 is represented as a sum
of simple poles J^i "p^ry-ly-)' '' w^lere tne coefficients ai are determined by
function -0, and coefficients bi are chosen from free function </> to provide cancelling
of the poles. It allows us to perform regularization of the most of poles in 7. If
all the poles in the function 7 will be cancelled, the result of integration will be a
stochastic radiation which will reduce to zero for weak excitations, and solutions of
aThe Kerr solution is described by a complex 'point-like' source propagating along a complex
world line.6 There are different 'left' and 'right' complex conjugate world lines and corresponding
'left' and 'right' retarded times tj, and tr.
tdL I Uw
(2) and (3) will be asymptotically exact. However, the pole at Y = oo can not be
regularized by this method and demands especial treatment.
4. Structure of the solutions near the beams (pp-waves) is discussed in.2,4 It was
shown that such beams pierce the horizons forming the tube-like holes connecting
internal and external regions. So the classical structure of black hole turns ont to
be destroyed.
Our solution turns out to be exact in the asymptotic limit 7 —> 0, which
corresponds to the weak and slowly changed electromagnetic field. In particular, it
shall tend to exact one for a black hole immersed into the zero point field of
virtual photons. In this case we have a sum of excitations in diverse directions
?/;(Y, r) = ]T\ F^y: which leads to a flow and migration of many singular beams
leading to an instantaneous appearance and disappearance of the holes- in
horizon, as it is shown on fig.l. One can assume that it may be a mechanism of BH
evaporation.
Fig. 1. The vacuum flow of virtual photons pierces the black hole horizon.
Note, that this picture is reminiscent of the haired black hole which was
suggested by the approach from the loop quantum gravity, where singular hairs were
formed from the horizon contrary to the appearance of the holes in horizon.9
References
1. G.C. Debncy, R.P. Kerr, A.Schild, J. Math. Phys. 10(1969) 1842.
2. A. Burinskii, Grav.&Cosmol.lO, (2004) 50; hep-th/0403212.
3. A. Burinskii Phys.Rev. D 70, 086006 (2004); hep-th/0406069.
4. A. Burinskii, E. Elizakie, S.R. Hildebrandt and G. Magii, Phys. Rev. D74 (2006)
021502(E); gr-qc/0511131.
5. W.Israel, Phys. Rev. D2 (1970) 641: A.Burinskii, Sov. Phys. JETP, 39(1974)193.,
A.Burinskii, Grav.&Casmol.ll, (2005) 301; hep-th/0506006.
6. A. Burinskii,Phys. Rev. D 87 (2003) 124024; gr-qc/0212048.
7. D.Kramer, H.Stcphani, E. Herlt, M.MacCalhnn, "Exact Solutions of Einstein's Field
Equations", Cambridge Univ. Press, 1980.
8. A. Burinskii and G. Magli, Phys. Rev. D 81(2000)044017; gr-qe/9904012.
9. A.Ashtekar, J.Baez, K.Krasnov, Adv.Theor.Math.Phys.4(2000)1; gr-qc/0005126.
STATIC PERTURBATIONS OF A REISSNER-NORDSTROM
BLACK HOLE BY A CHARGED MASSIVE PARTICLE
DONATO BINI
Istituto per le Applicazioni del Calcolo "M. Picone," CNR 1-00161 Rome, Italy and
ICRA, University of Rome "La Sapienza," 1-00185 Rome, Italy and
INFN - Sezione di Firenze, Polo Scientifico,
Via Sansone 1, 1-50019, Sesto Fiorentino (FI), Italy
binid@icra.it
ANDREA GERALICO
Physics Department
and ICRA, University of Rome "La Sapienza", 1-00185 Rome, Italy
geralico @icra. it
REMO RUFFINI
Physics Department and ICRA, University of Rome "La Sapienza", 1-00185 Rome, Italy
and ICRA Net, 1-65100 Pescara, Italy
ruffini@icra.it
The interaction of a Reissner-Nordstrom black hole and a charged massive particle at
rest is studied in the framework of first order perturbation theory following the approach
of Zerilli. The solutions of the combined Einstein-Maxwell equations for both perturbed
gravitational and electromagnetic fields are exactly reconstructed (to first order) by
summing all multipoles leading to closed form expressions.
Up to now the study of the interaction of a charged particle with a static black
hole has been done only within the test field approximation [1-6]. The Einstein-
Maxwell equations reduce to Maxwell equations in a fixed background when the
effect of the mass and the electromagnetic field of the test charge on the geometry
can be neglected. However, when this approximation is not valid one must take into
account the backreaction both of the mass and of the charge of the particle on the
background electromagnetic and gravitational fields. This issue has been recently
addressed in [7,8].
In the study of an uncharged black hole, since the electromagnetic stress-energy
tensor is second order in the electromagnetic field, one can treat the electromagnetic
perturbations separately, keeping the background metric unchanged to first order
of the perturbations. However, for a charged black hole the change in the stress-
energy tensor is first order, and thus any electromagnetic perturbation causes a
gravitational perturbation and vice versa [9], leading to the necessity of studying
the whole set of combined Einstein-Maxwell equations.
Consider thus the problem of a massive charged particle of mass m and charge q
in the field of a Reissner-Nordstrom geometry describing a static charged black hole,
2104
2105
with mass M and charge Q. The Reissner-Nordstrom black hole metric is given by
ds2 = -f(r)dt2 + /(r)_1dr2 + r2(d82 + sin2 6d<p2) ,
with associated electromagnetic field
Q
F = ~^dt/\dr . (2)
Let the point particle of mass m and charge q be at rest at the point r = b on
the polar axis (9 = 0. The only nonvanishing components of the stress-energy tensor
and of the current density are then
^cT = ~f (b)3/2S(r - b) 5 (cos 0- 1) ,
Jpart = ^(^H(cos0-l) . (3)
The system of combined Einstein-Maxwell equations is given by
G^=87r(r^rt + 2;T
i>% = 4ttJ£art , *F^,/3 = Q, (4)
where the quantities denoted by a tilde refer to the total electromagnetic and
gravitational fields, to first order of the perturbation
rpem
~9puFpilFav - hj^F^F^
G^ — R^ — -g^R ■ (5)
The perturbation equations are then obtained from the system (4), keeping terms
to first order in the mass m of the particle and its charge q which are assumed
sufficiently small with respect to the black hole mass and charge.
Following Zerilli's procedure [10] we expand the fields h^ and f^ as well as
the source terms of Eq. (3) in tensor harmonics, imposing then the Regge-Wheeler
gauge [11]. Such a standard approach leads to a set of radial coupled differential
equations for the gravitational as well as electromagnetic perturbation functions.
The compatibility of the system provides the following stability condition
bfjbf'2
Mb-Q2
involving the black hole and particle parameters Q,M.)q)m as well as their
separation distance b. If the black hole is extreme (i.e. Q/M = 1), then the particle
must also have the same ratio q/m = 1, and equilibrium exists independent of the
separation. In the general non-extreme case Q/M < 1 there is instead only one
position of the particle which corresponds to equilibrium, for given values of the
charge-to-mass ratios of the bodies. In this case the particle charge-to-mass ratio
must satisfy the condition q/m > 1. It is remarkable that quite surprisingly Eq. (6)
= «Q-£^o2. (6)
2106
coincides with the equilibrium condition for a charged test particle in the field of
a Reissner-Nordstrom black hole which has been discussed by Bonnor [12] in the
simplified approach of test field approximation, neglecting all the feedback terms.
We then succeed in the exact reconstruction of both the perturbed gravitational
and electromagnetic fields by summing all multipoles [8]. The perturbed metric is
given by
ds2 = -[1- H}f(r)dt2 + [1 +H][f(r)-1dr2+ r2{d92 + sin2 6d<p2)] , (7)
where
nrn (r-M)(b-M)-T2coSe
n = 2-m 3 ,
V = [(r - M)2 + (b- M)2 - 2(r - M)(b - M) cosO - T2 sin2 0]1/2 . (8)
In the extreme case Q/M. = q/rn = 1 this solution reduces to the linearized form of
the well known exact solution by Majumdar and Papapetrou [13,14] for two extreme
Reissner-Nordstrom black holes. The total electromagnetic field to first order of the
perturbation turns out to be
F
Q
with
Er =
q Mr - Q2 1
M(b-M) + r2cos6
dthdr- Egdthde , (9)
Q2[(r -M){b-M)-T2cos(
Mr-Q2
[(r-M)-(b-M) cos 0]
r3 Mb-Q2T>
r[(r - M)(b -M)-T2 cos 6>]
V2
„Mr-Q2b2f(b)f(r) .
Ee=qMb-Q2 W Smd- (10)
References
1. R. Hanni, Junior Paper submitted to the Physics Department of Princeton University,
1970 (unpublished).
2. J. Cohen and R. Wald, J. Math. Phys. 12, 1845 (1971).
3. R. Hanni and R. Ruffini, Phys. Rev. D 8, 3259 (1973).
4. J. Bicak and L. Dvorak, Gen. Relativ. Grav. 7, 959 (1976).
5. B. Linet, J. Phys. A: Math. Gen. 9, 1081 (1976).
6. B. Leaute and B. Linet, Phys. Lett. A58, 5 (1976).
7. D. Bini, A. Geralico and R. Ruffini, Phys. Lett. A360, 515 (2007).
8. D. Bini, A. Geralico and R. Ruffini, in preparation.
9. M. Johnston, R. Ruffini and F. J. Zerilli, Phys. Rev. Lett. 31, 1317 (1973); Phys. Lett.
B49, 185 (1974).
10. F. J. Zerilli, Phys. Rev. D 9, 860 (1974).
11. T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063 (1957).
12. W. B. Bonnor, Class. Quant. Grav. 10, 2077 (1993).
13. S. M. Majumdar, Phys. Rev. 72, 390 (1947).
14. A. Papapetrou, Proc. R. Irish Acad. 51, 191 (1947).
CHARGED STRING SOLUTIONS OF THE EINSTEIN-MAXWELL
EQUATIONS IN HIGHER DIMENSIONS
CHUL H. LEE
Department of Physics, and BK21 Division of Advanced Research and Education in Physics,
Hanyang University, Seoul 133-791, Korea
chulhoon@hanyang. ac. kr
We consider solutions of the Einstein-Maxwell equations with a charged string source
in 1 + 4 dimensions. The uniform extension of the Reissncr-Nordstrom metric to the
extra dimension is shown to be one of such solutions. The Maxwell fields associating this
metric turn out to contain magnetic as well as electric monopole fields as seen in the 1+3
dimensions. The magnetic charge per unit length of the source string is the same as that
of the electric charge per unit length in this case. We try to find solutions generalizing
the above solution to the arbitrary electric and magnetic charge case. So far we have
only been able to find the asymptotic forms of such solutions in the large r region and
they are presented here.
1. Introduction
Spacetimes of dimensions higher than 1+3 have become objects for serious
consideration in physics as some physical theories such as the string theory and brane
cosmology are necessarily formulated in those higher dimensional spacetimes. In
exploring the existence of extra dimensions, the higher dimensional black hole
solution can be a useful tool. Along with the higher dimensional black hole, another
interesting object is the black string which is obtained by extending the 1 + 3
dimensional black hole to the extra dimensions. The simplest case, first studied by
Gregory and Laflamme,1 is the black string obtained by extending uniformly the
Schwarzschild black hole to the fifth dimension. Its metric is given by
ds2 = g^dx^dx"
= -(1 - -)dt2 + -^ + rW + r2 sin2 Od<p2 + dz2. (1)
r
It is a vacuum solution of the 1+4 dimensional Einstein equations and represents the
geometry of the space outside a string source lying along the z-direction. It is shown,
in Ref. [2], that this string is characterized by the tension(r) whose magnitude is
one half of the mass per unit length(A).
A general class of solutions containing two arbitrary parameters, the tension and
the mass per unit length, is also presented in Ref. [2]. In order to find the solutions
of the vacuum Einstein field equations which reduce to the appropriate asymptotic
form at large distance characterized by the arbitrary tension and mass per unit
length of the souce string, the following the ansatz is used;
ds2 = -F{p)dt2 + G{p){dp2 + p2d62 + p2 sin2 9d<p2) + H{p)dz2, (2)
This form of metric is substituted into the vacuum Einstein field equations to derive
differential equations for the three functions F(p), G(p) and H(p). The solutions
2107
2108
turn out to be
K K
ir = (i _ fl£)»(i + fl£)-»
P P
H=(1.I^r^sil+Ka)^s (3)
where
2(2 a) Ka = Sr "'" G5X. (4)
V3(l-a + a2)'
In the above a = J and G5 the five dimensional gravitational constant. For a = ^,
these solutions in Eq's (3) and (4) can be seen to give Eq. (1), the Gregory-Laflamme
metric, with a = 4K"i/2 = 2G5A = 2G4M. The same solution as Eq. (3) was also
discussed by other authors in the different context of Kaluza-Klein gauge theories.
In the next section we consider the solutions of the Einstein-Maxwell equations
with a charged string source in 1+4 dimensions.
2. Spacetime geometry produced by a charged string source
We now consider the case where the source string is charged. We start with the
action given by
-1 ,„ 1
^J**™^*-1^^ (5)
from which the Einstein-Maxwell equations
Riiv ~ 29^R = ~87rG5T^, (6)
F"".„ = 0 (7)
with
J-liv = i' )ipi'„ — -^Q^ivr par (8)
are derived. Just as the uniform extension of the Schwarzschild solution to the fifth
dimension is a solution of the Einstein equations in 1 + 4 dimensions, it can be
easily seen that the uniform extension of the Reissner-Nordstrom solution to the
fifth dimension,
ds2 = -(!-- + ^)dt2 + ^-=- + r2d62 + r2 sin2 Qdtf + dz2, (9)
r rz 1 — £■ + -%
together with
Ft,
^b/8irG5\, Fe4, = y/b/8wG5smO (10)
2109
and all other independent components zero, is a solution of the Einstein-Maxwell
equations in 1 + 4 dimensions. This solution represents the radial electric(Ftr) and
magnetic (Fqj,) monopole fields as seen in the 1 + 3 dimensions. And the magnitude
of the magnetic charge per unit length of the source string is the same as the
magnitude of the electric charge per unit length, ^b/SirGs. We now try to find
solutions generalizing the above solution(Eq. (9) and (10)) to the arbitrary electric
and magnetic charge case. So far we have only been able to find the asymptotic
forms of such solutions in the large r region. We start with the ansatz
ds2 = -F{r)dt2 + G{r)dr2 + r2{d02 + sin2 8d<j)2) + H{r)dz2 (11)
Then we find that
f + m
u 6
2m
3
together with
F(r) « l-^ + JL_Z; (12)
G(r)«l + ^+- ;a a, (13)
H(r) « 1 - ^-r-2- (14)
Ftr&y/b/8irG5^, Fe4>^y/m/8nG5siae (15)
and all other independent component zero, satisfy the Einstein-Maxwell equations
in the asymptotic region of large r. The electric and magnetic charges per unit
length of the source string are \Jb/&itG§ and yJm/8irG^ respectively.
Acknowledgments
This research has been supported by the Korea Science and Engineering Foundation
grant funded by the Korea Government(No. R01-2006-000-10651-0)
References
1. R. Gregory and R. Laflamme, Phys. Rev. D 37, 305 (1988)
2. C. H. Lee, Phys. Rev. D 74, 104016 (2006)
3. J. Gross and M. Perry, Nuc. Phys. B 226, 29 (1983)
4. A. Davisson and D. Owen, Phys. Lett. B 155, 247 (1985)
ON THE HYPOTHESIS OF GRAVIMAGNETISM
M.M. ABDIL'DIN, M.E. ABISHEV
Al-Farabi Kazakh National University,
Kazakstan, Almaty, Tole be 96 a
abdnur@kazsu. kz
In the work is considered hypothesis of gravimagnetism, which represents that gravitation
could be a source of magnetism.
Some time ago, to explain the magnetism of the celestial bodies, a number of
hypotheses were put forward, leading to correct quantitative results. Moreover, quite
remarkable is the unusual nature of these hypotheses from the viewpoint of the
existing physical outlook. Thus, according to Wilson's hypothesis [1], the magnetic
fields of the Earth and the Sun are such as if they possessed a negative volume
charge density a = — ^fyp , where 7 is the gravitational constant and p is the mass
density. An unusual feature is that this "charge" does not create an electric field
but, rotating, creates a magnetic field. Another hypothesis, also leading to correct
quantitative results, is Blackett's hypothesis [2]. According to Blackett, any rotating
body, irrespective of the existence of any charge in it, should possess a magnetic
moment proportional to its mechanical angular momentum: M = —t-Zp-S.
Einstein's remark [3] is in full agreement with these hypotheses: "The Earth and the
Sun possess magnetic fields whose orientation and polarity are approximately
determined by the directions of these bodies' rotation... It rather seems as though
magnetic fields emerge from rotary motion of neutral masses... Here, Nature
apparently points at a fundamental law so far unexplained by theory". Recently [4], the
interest in discussing the physical roots of "Blackett's rule" increased again.
Some time ago [5], in search for a foundation of these hypotheses, we put forward
a more general hypothesis that gravity may be a source of magnetism. It has been
shown [6,7] that:
1. The relation A = —-^=tj is valid, where A- is the vector potential of the
magnetic field of a rotating body and U is the vector potential of the gravitational
field. For instance, for a rotating homogeneous fluid ball, the vector potential of the
gravitational field is U = — 7^3 [r So] • To calculate the potential A in a more general
case, one can use the equation A A = -^-Airpv, where v is the velocity inside the
body.
2. The off-diagonal component of the metric tensor goi = —gAi is connected
with the magnetic field.
3. The approximate results for the magnetic fields of the moon (10~5 Oersted)
and a pulsar (1010 Oersted) are obtained.
4. The traditional interpretation of GR, as a theory of the gravitational field
only [8], also changes to a certain extent. Now GR, or, more precisely, its
mathematical framework (the Einstein equations!) correspond to a gravimagnetic field
theory. Gravitational wavws, as they are now understood, should in fact exist as
2110
2111
gravimagnetic waves.
5. The gravimagnetism hypothesis being discussed leads to one more, though
indirect, conclusion. Indeed, in modern electrodynamics there is an asymmetry
between electricity and magnetism, which manifests itself physically in the existence of
electric charges and the absence of magnetic charges; mathematically, it is reflected
in the lack of symmetry in the right-hand sides of the Maxwell-Lorentz equations
with respect to the electric and magnetic field sources. This fact is probably not
accidental but rather bears a deeper meaning, allowing one to think of a distinguished
role of magnetism. Indeed, let us present the Maxwell equations:
^ IdH ,. T-±
rotE = —, divH = 0, 1)
c
dt
47r - 1 dE
rotH =—?'H —, divE = An:<j. (2)
c c at
where E is the electric field strength, H is the magnetic field strength, a and j are
the electric charge density and the electric current density, respectively. It follows
that the magnetic field emerges as a by-product of the electric field that has a source
of its own, the electric charge. Long ago, Dirac [9] tried to remove this asymmetry
and arrived at the hypothesis on the existence of a magnetic charge (a solitary
magnetic pole, or monopole). However, a magnetic monopole has so far not been
found. This negative result is also a result which can lead to an extreme idea that
a magnetic monopole does not exist at all. The asymmetry in electrodynamics is
thus a feature of principle: the electric and magnetic fields are not equal in rights,
the magnetic field is rather a by product of the electric field.
Let us now address to another branch of physics, nuclear physics. Here we
consider the situation with the neutron. The electrically neutral neutron has a magnetic
field. To explain this, one could also suggest that the magnetic field is here a
byproduct of the neutron's nuclear field. The neutron has a nuclear charge which is a
source of a nuclear field, and, in turn, rotating (the current of the nuclear charge!),
creates a magnetic field.
The celestial bodies show a similar situation They have a gravitational mass,
i.e., a gravitational charge. The latter creates a gravitational field. When a celestial
body has a rotation of its own (a mass current, or a current of gravitational charge)
then, as a by-product of gravity there emerges a magnetic field. This is what we
call the gravimagnetism hypothesis. Gravitation is also a source of magnetism.
Thus, summing up the situation in electrodynamics, nuclear physics and
gravitational physics, we can assert that the magnetic field is a by-product of all physical
fields having their own sources (the electric, nuclear and gravitational charges).
Now let us mention a certain discrepancy between the theoretical results and the
actual data on the magnetic fields of the Earth, the Sun, neutron stars and other
celestial bodies.
It has been found that this situation is explained by our considering the simplest
model of celestial bodies: we described them as rotating homogeneous fluid balls.
2112
One should take into account the inhomogeneous distribution of matter inside all
the bodies.
Indeed, the seismic data indicate that the Earth's core occupies about one eighth
of its volume. The matter in it must be in a liquid state and possess large density
[10]. It is believed that the core may rotate with a velocity slightly different from
that of the Earth's crust.
A similar situation, i.e., inhomogeneity of density and rotation velocities, may
take place for the Sun and the neutron stars (pulsars).
References
1. H.A. Willson. Prog. Roy. Soc. A, 104 (1923),
2. P.M. Blackett, Uspekhi Fiz. Nauk 38, 1 (1947).
3. A. Einstein, Collected works, v. 2, Nauka, M., 1966.
4. V.I. Grigoryev and E.V. Grigoryeva, On gravitational relations of celestial bodies.
Vestn, Mosk. Univ., ser. 3, Phys. Astron., No. 3, page 75 (1996).
5. M.M. Abdil'din. On the interpretation of general relativity. Izv. AN Kaz. SSR, ser.
Fiz. Math., No. 4, 76 (1968).
6. M.M. Abdil'din. On Interpretation of the Einstein Equations in General Relativity.
Gravitation & Cosmology, 5, 3(19), 219-221 (1999).
7. M.M. Abdil'din. Gravimagnetism and the interpretation of Einstein's equations.
Gravitation, Cosmology and Relativistic Astrophysics, Kharkov National University, 2001.
8. L.D. Landau and E.M. Lifshits, Classical Field Theory. Moscow, 1973, 502 pp.
9. P.A.M. Dirac, Proc. Roy. Soc, A 133, 60 (1931).
10. N.V. Pushkov, Magnetism in Space. Znanie, ser. IX: Fiz, Khim., M., 1961.
STATIC PERTURBATIONS OF A REISSNER-NORDSTROM
BLACK HOLE BY A CHARGED MASSIVE PARTICLE
DONATO BINI
Istituto per le Applicazioni del Calcolo "M. Picone," CNR 1-00161 Rome, Italy and
ICRA, University of Rome "La Sapienza," 1-00185 Rome, Italy and
INFN - Sezione di Firenze, Polo Scientifico,
Via Sansone 1, 1-50019, Sesto Fiorentino (FI), Italy
binid@icra.it
ANDREA GERALICO
Physics Department
and ICRA, University of Rome "La Sapienza", 1-00185 Rome, Italy
geralico @icra. it
REMO RUFFINI
Physics Department and ICRA, University of Rome "La Sapienza", 1-00185 Rome, Italy
and ICRANet, 1-65100 Pescara, Italy
rufp,ni@icra. it
The interaction of a Reissner-Nordstrom black hole and a charged massive particle at
rest is studied in the framework of first order perturbation theory following the approach
of Zerilli. The solutions of the combined Einstein-Maxwell equations for both perturbed
gravitational and electromagnetic fields are exactly reconstructed (to first order) by
summing all multipoles leading to closed form expressions.
Up to now the study of the interaction of a charged particle with a static black
hole has been done only within the test field approximation [1-6]. The Einstein-
Maxwell equations reduce to Maxwell equations in a fixed background when the
effect of the mass and the electromagnetic field of the test charge on the geometry
can be neglected. However, when this approximation is not valid one must take into
account the backreaction both of the mass and of the charge of the particle on the
background electromagnetic and gravitational fields. This issue has been recently
addressed in [7,8].
In the study of an uncharged black hole, since the electromagnetic stress-energy
tensor is second order in the electromagnetic field, one can treat the electromagnetic
perturbations separately, keeping the background metric unchanged to first order
of the perturbations. However, for a charged black hole the change in the stress-
energy tensor is first order, and thus any electromagnetic perturbation causes a
gravitational perturbation and vice versa [9], leading to the necessity of studying
the whole set of combined Einstein-Maxwell equations.
Consider thus the problem of a massive charged particle of mass m and charge q
in the field of a Reissner-Nordstrom geometry describing a static charged black hole,
2113
2114
with mass M and charge Q. The Reissner-Nordstrom black hole metric is given by
ds2 = -f{r)dt2 + /(r)"1^2 + r2{d02 + sin2 6d<p2) ,
with associated electromagnetic field
F=-Qdt.Adr. (2)
Let the point particle of mass m and charge q be at rest at the point r = b on
the polar axis (9 = 0. The only nonvanishing components of the stress-energy tensor
and of the current density are then
m
27T&2'
J°part = ~^S(r-b) 5 (cos 0 - 1) . (3)
The system of combined Einstein-Maxwell equations is given by
F^.^AirJ^, *i^% = 0, (4)
where the quantities denoted by a tilde refer to the total electromagnetic and
gravitational fields, to first order of the perturbation
TZt = ^J(b?/25(r-b)5(cose-l)
mem
hPaF F n F Fpa
G\u, — Rfn, — —g^vR . (5)
The perturbation equations are then obtained from the system (4), keeping terms
to first order in the mass m of the particle and its charge q which are assumed
sufficiently small with respect to the black hole mass and charge.
Following Zerilli's procedure [10] we expand the fields h^ and f^u as well as
the source terms of Eq. (3) in tensor harmonics, imposing then the Regge-Wheeler
gauge [11]. Such a standard approach leads to a set of radial coupled differential
equations for the gravitational as well as electromagnetic perturbation functions.
The compatibility of the system provides the following stability condition
bfibf'2
Mb-Q2
involving the black hole and particle parameters Q,A4,q,m as well as their
separation distance 6. If the black hole is extreme (i.e. QjM. = 1), then the particle
must also have the same ratio q/m = 1, and equilibrium exists independent of the
separation. In the general non-extreme case Q/Ai < 1 there is instead only one
position of the particle which corresponds to equilibrium, for given values of the
charge-to-mass ratios of the bodies. In this case the particle charge-to-mass ratio
must satisfy the condition q/m > 1. It is remarkable that quite surprisingly Eq. (6)
™ = *QJZFLa2, (6)
2115
coincides with the equilibrium condition for a charged test particle in the field of
a Reissner-Nordstrom black hole which has been discussed by Bonnor [12] in the
simplified approach of test field approximation, neglecting all the feedback terms.
We then succeed in the exact reconstruction of both the perturbed gravitational
and electromagnetic fields by summing all multipoles [8]. The perturbed metric is
given by
ds2 = -[1 -H]f{r)dt2 + [l +H}[f{r)-1dr2 +r2{d62 +sin26d<p2)] , (7)
where
U = 2™ f(b)-i/2(r-M)(b-M)-r2cos6
brJ ' V
f>= \{r - M)2 + {b - M)2 ~ 2{r - M){b - M) cos9 -T2 sm2 9}1'2 . (8)
In the extreme case Q/M = q/m = 1 this solution reduces to the linearized form of
the well known exact solution by Majumdar and Papapetrou [13,14] for two extreme
Reissner-Nordstrom black holes. The total electromagnetic field to first order of the
perturbation turns out to be
Q
F
with
q Mr - Q2 1
£L/r =
■ET
M(b-M)+T2 cos 9 +
dt A dr - Eedt A dO , (9)
Q2[{r -M){b-M)- V2 cos8}
Mr - Q2
[(r-M)-(b-M) cos9}
r3 Mb-Q2V
r[(r - M){b-M)-T2 cos9}
W2
References
1. R. Hanni, Junior Paper submitted to the Physics Department of Princeton University,
1970 (unpublished).
2. J. Cohen and R. Wald, J. Math. Phys. 12, 1845 (1971).
3. R. Hanni and R. Ruffini, Phys. Rev. D 8, 3259 (1973).
4. J. Bicak and L. Dvorak, Gen. Relativ. Grav. 7, 959 (1976).
5. B. Linet, J. Phys. A: Math. Gen. 9, 1081 (1976).
6. B. Leaute and B. Linet, Phys. Lett. A58, 5 (1976).
7. D. Bini, A. Geralico and R. Ruffini, Phys. Lett. A360, 515 (2007).
8. D. Bini, A. Geralico and R. Ruffini, in preparation.
9. M. Johnston, R. Ruffini and F. J. Zerilli, Phys. Rev. Lett. 31, 1317 (1973); Phys. Lett.
B49, 185 (1974).
10. F. J. Zerilli, Phys. Rev. D 9, 860 (1974).
11. T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063 (1957).
12. W. B. Bonnor, Class. Quant. Grav. 10, 2077 (1993).
13. S. M. Majumdar, Phys. Rev. 72, 390 (1947).
14. A. Papapetrou, Proc. R. Irish Acad. 51, 191 (1947).
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Theoretical Issues in GR
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A FRAMEWORK FOR THE DISCUSSION OF SINGULARITIES IN
GENERAL RELATIVITY
BENJAMIN E. WHALE and SUSAN M. SCOTT
Centre for Gravitational Physics, Department of Physics, The Australian National University,
Canberra ACT 0200, Australia
ben.whale@anu.edu.au, susan.scott@anu.edu.au
The occurrence and nature of singularities in General Relativity remains arguably
the most important outstanding problem in the field. This is in direct contrast to most
other areas of physics where singularities are merely mathematical peculiarities. We
examine why singularities in General Relativity are different—they have a complex and
subtle nature and are notoriously difficult to investigate due to the lack of a suitable
framework. We consider the effectiveness of various attempts since the '60s to devise
such a framework or "boundary construction" for the study of space-time singularities.
Finally, we discuss the most recent such boundary construction, the a-boundary; the
most objective, flexible and ultimately the most practical of all these constructions. The
a-boundary has recently been recast in terms of "distances" rather than its original
topological description, rendering it accessible to a much broader range of researchers.
If we put the general question to physicists, "How should singularities be discussed
in physics ?", most physicists would reply, uWhy should we discuss singularities at
all ?" . Indeed, in almost all areas of physics singularities are just mathematical
peculiarities; simply the results of incomplete theories. The situation is somewhat
more complicated in General Relativity, however. The importance of singularities in
General Relativity is demonstrated by two of the most important outstanding
problems in the field: the properties of Penrose-Hawking singularities* and the Cosmic
Censorship Conjecture. It is well known that the implications of the Cosmic
Censorship Conjecture extend beyond General Relativity; it is less well known that the
existence of Penrose-Hawking singularities in the quantum realm is still debated.
This establishes the need to ponder and discuss singularities, but gives no
indication of a possible framework for our thoughts and discussions on this matter.
Whilst in most areas of physics singularities are rather simple, this is certainly not
the case in General Relativity. The reason for this goes to the very heart of both
the beauty and complexity of General Relativity, and illustrates one of the reasons
why General Relativity does not quite fit with the rest of physics.
In most areas of physics the theory is constructed in two parts: a space-time^,
and an associated mathematical object or objects (e.g. a field, scalar, twistor, etc.).
In theories such as these there are only two places that a "singularity" can occur:
either in the metric of the space-time or in the associated object(s). Outside of
General Relativity the singularity almost always occurs in the associated object (s),
due to the assumptions of the theory. In General Relativity, however, the singularity
*That is, those singularities whose existence can be predicted from a singularity theorem.
tin this context, by "space-time" we mean the manifold and metric structure most appropriate
for the area of physics being considered.
2119
2120
theorems demonstrate that, under very general physical circumstances, singularities
will always occur in the metric. These singularities are fundamentally different from
those which, in other areas of physics, only occur in the associated object(s).
This difference can be illustrated by the two most fundamental questions
relating to the study of singularities in General Relativity, "Where is it ?" (location) and
" What is it ?" (definition). For singularities only occurring in an associated object,
both these questions can readily be answered by simply examining the associated
object with reference to its place in the underlying space-time. For metric
singularities occurring in General Relativity, however, there is no pre-determined "place"
at which to examine them since they do not lie in the given space-time. We must
somehow analyse the singularity from within the existing manifold structure. These
two problems of location and definition encapsulate the main issues needing to be
dealt with when considering singularities in General Relativity. The fact that these
problems were discussed at length in the '60s1 and that papers on the subject are
still being produced today2 demonstrate that these two problems are indeed of a
deep, highly complex and subtle nature.
The area of physics described as "Boundary Constructions" is dedicated to the
study and potential resolution of these two problems. The idea is that one devises a
boundary construction by determining a method of adding a boundary to the space-
time under consideration. With the additional presence of the boundary, one then
attempts, firstly, to answer the question of the location of the metric singularities,
and then to answer the question of their definition.
There are a variety of boundary constructions in existence: the ^-boundary,3
6-boundary,4 and c-boundary,5 for example. Each boundary construction attempts
to "fix" the location of the singularities, so that in different coordinate patches the
singularity always "looks" the same. For example, if in one set of coordinates the
singularity is a point, then in all other sets of coordinates the singularity is also a
point.
It is interesting to note, however, that during the years since their first
introduction, it has been demonstrated that all of these constructions have their failings
and that all of their failings can be traced back to one problem; a rigidity of
approach that forces these boundaries to contain an element of subjectivity when
being applied. This subjectivity means that these boundary constructions fail to give
consistent definitions of singularities across multiple space-times and that,
potentially, when different relativists apply the same boundary construction to the same
space-time they can get different answers. So far, it has been relativists' common
understanding of the properties of the particular space-times being considered, and
the fact that most relativists have attempted to apply the boundary uniformly to
different space-times, which have prevented this subjectivity from coming to light.
Nonetheless, why use a faulty tool when there is an alternative? And there is
an alternative, namely the a-boundary6 or abstract boundary construction. The a-
boundary avoids this subjectivity issue with a trade-off. The cost of an objective
boundary is that the representation of singularities in different coordinate patches
2121
may be different. At first glance this state of affairs may seem to cause more
problems than the previous approaches. Indeed, it is remarkable that Scott and Szekeres
managed to salvage anything at all, let alone a comprehensive and accessible
mathematical structure from such a situation.
The a-boundary is framed in topological language. Since topology is an area
of mathematics that not too many physicists know in detail, the a-boundary can
appear complex and daunting when first encountered. In turn, this has meant that
its many benefits have often been ignored, an example of which is an important
theorem due to Ashley and Scott that links the Penrose-Hawking singularity
theorems to the existence of essential singularities*. This is the first theorem which
has demonstrated that the Penrose-Hawking singularity theorems actually produce
singularities which are "real".
Recently the current authors have found that it is possible to frame the a-
boundary in terms of distances rather than topology §. This recasting of the structure
into the tangible concept of "distance" brings the a-boundary firmly back into the
realm of physicists and physics. We hope that, as a result, many more physicists
will take an interest in the a-boundary and the appropriate framework it provides
for the examination of singularities, in a way that yields consistent definitions and
intuitive application.
References
1. R. Geroch, What is a singularity in general relativity?, Annals of Physics 48, 526-540,
1968
2. S. G. Harris, Discrete group actions on spacetimes: causality conditions and the causal
boundary, Classical and Quantum Gravity 21, 1209-1236, 2004
3. R. Geroch, Local characterization of singularities in general relativity, Journal of
Mathematical Physics 9, 450-465, 1968
4. B. G. Schmidt, A new definition of singular points in general relativity, General
Relativity and Gravitation 1, 269-280, 1971
5. R. Geroch, E. Kronheimer and R. Penrose, Ideal points in space-time, Proceedings
of the Royal Society of London Series A - Mathematical and Physical Sciences 327,
545-567, 1972
6. S. M. Scott and P. Szekeres, The abstract boundary—a new approach to singularities
of manifolds, Journal of Geometry and Physics 13, 223-253, 1994
*An essential singularity is a singularity that cannot be removed by an extension of the manifold
or by a change of coordinates.6
§This work is not yet published.
AXIAL SYMMETRIC GRAVITOMAGNETIC MONOPOLE IN
CYLINDRICAL COORDINATES
V.G. KAGRAMANOVA* and B.J. AHMEDOVt
Institute of Nuclear Physics and Ulugh Beg Astronomical Institute, Astronomicheskaya 33,
Tashkent 700052, Uzbekistan
* Carl von Ossitzky Universitat Oldenburg, Institut fur Physik, D-26111, Oldenburg, Germany
^International Centre for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy
General relativistic effects associated with the gravitomagnetic monopole moment of
gravitational source through the analysis of the motion of test particles and
electromagnetic fields distribution in the spacetime around nonrotating cylindrical NUT source have
been studied. We consider the circular motion of test particles in NUT spacetime, their
characteristics and the dependence of effective potential on the radial coordinate for the
different values of NUT parameter and orbital momentum of test particles. It is shown
that the bounds of stability for circular orbits are displaced toward the event horizon
with the growth of monopole moment of the NUT object. In addition, we obtain exact
analytical solutions of Maxwell equations for magnetized and charged cylindrical NUT
stars.
Keywords: Relativistic stars; gravitomagnetic charge; particle motion; electromagnetic
fields.
The general stationary axially symmetric solution1,3 to the vacuum Einstein
equations in cylindrical coordinates {t, p, ip, z] is given by
ds2 = f-1 [e2"<{dp2 + dz2) + p2dV2} -f(dt- Lodip)2 , (1)
where the metric coefficients /, j,uj are functions of p and z only. The explicit
expressions for /, 7, and u can be found in1. The expression for z is
_ 21E /L2(A2 + p2)-4PA2E2
Z ~ L V L*-4l2E2 ' (2)
where L, E are conserved quantities representing, respectively, the total energy
and orbital angular momentum of the particle, and A = \/M2 + I2 for a source
Fig. 1. The radial dependence of the effective potential for particles with nonzero rest mass for
different values of angular momentum L. The left hand side figure is responsible for the case
when the NUT parameter I = 0. For the right hand side figure the NUT parameter I = 0.5.
Maxima in the effective potential indicate unstable circular orbits and minima stable circular
orbits. Curves for particles with equal angular momentum and different gravitomagnetic charges
have more monotonous behavior with the increase of the value of NUT parameter.
2122
2123
endowed with mass M and gravitomagnetic mass /. Fig. 1 illustrates the radial
p/M dependence of the effective potential
Veff = 72 _ f2,-12 ' (3)
p2 — f2ui2
where L and E are normalised to the unit of mass of the particle m and 'tilded'
quantities are normalised to the total mass of the source.
The influence of the NUT parameter on the motion of a test particle can be
seen from the Table 1 where ]?circular defines the radius of last circular orbit,
Table 1. Gravitomagnetic influence on
motion
I
0
0.05
0.1
0.5
0.7
o1
'circular
3
2.99
2.98
2.49
1.97
7?
"max.bound
8
7.99
7.98
7.5
6.94
~2
Pstable
24
24.02
24.08
27.21
34.41
Pmax.bound defines the radius of the last bound orbit and p^tab[e defines the radius
of first stable orbit.
We look for stationary and axially symmetric solutions of the Maxwell equations,
taking into account that in the vacuum region around the source all components
of electric current are equal to zero. Due to the possible smallness of the
gravitomagnetic mass of the star estimated from some astrophysical observations we may
perform calculations to the first order in NUT parameter. The exact solutions for
the nonvanishing components of magnetic field in our case are
B" = -m^W^MfNAN + m
+ lz(M + N)[(M + N)2\nf + 3M2-N2]j , (4)
Bz
3/xe
8M3/(M + Nf{2MfpN'p{N + 3M) + (M + N) [2M{N + 2M^pf'0 ~ 2/)
+ (M + N)2 [p/>p(ln / - 1) - 2/ In /] ] } , (5)
where p, is the dipolar magnetic moment and the expression for N is defined in1.
The dependence of relation BQR/BpNewt on parameter p/M is shown on Fig. 2.
If Q is the electric charge per unit length of the line tube then the solution
for radial electric field admitted by Maxwell equations is E? = Q~^~ with a =
p2 — f2u)2. Fig. 3 shows that for small values of z and p (near to the source) the
influence of NUT parameter is noticeable.
2124
z M=3.2
z M=2 .2
z/M=I .2
z/M=0.2
Fig. 2. Dependence of general relativistic modification factor B^,R/BpNewt of magnetic field on
the radial coordinate p/M normalized in units of stellar mass. Near to the NUT source the
magnetic field will be amplified, then in some intermediate region it will be weakened and in the
asymptotically far zone the influence of NUT parameter is negligible, the behavior of field is
Newtonian and relation tends to unity. The influence of the NUT parameter is more strong near to
the source of the z axis.
Values of liM
-- 1/M=0. 5
- -1/M = 0. 01
1/M=0
2/M=0
"a.
Values Of l/M
- - 1/M=0.5
1/M = 0. 01
—- 1/M=0
z/M=3
?/M
■}/M
Fig. 3. The radial dependence of electric field E? for different values of the gravitomagnetic
monopole I. The effect of the NUT parameter on the electric field is becoming important near to
the source of the z axis.
This research is supported by the NATO Reintegration Grant EAP.RIG.981259,
by Uz FFR (project 1-06) and projects F2.1.09 and F2.2.06 of the Uz CST.
References
1. R. Gautreau and R. Hoffman, Phys. Lett. A 39, 75 (1972).
2. A. Khugaev and B. Ahmedov, Int. J. Mod. Phys. D 13, 1823 (2004).
3. V. Manko and E. Ruiz, Class. Quantum Grav. 22, 3555 (2005).
4. D. Bini, C. Cherubini, R. Jantzen, B. Mashhoon, Class. Quantum Grav. 20, 457 (2003).
5. L. Rezzolla and B. Ahmedov, Mon. Not. R. Astron. Soc. 352, 1161 (2004).
OPTICAL REFERENCE GEOMETRY AND INERTIAL FORCES IN
KERR-DE SITTER SPACETIMES*
JIRI KOVAR+ and ZDENEK STUCHLIKt
Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava,
Bezrucovo nam. 13, Opava, 746 01,Czech Republic
t Jiri.Kovar@fpf.slu. cz
t Zdenek. Stuchlik@fpf. slu. cz
Results of investigation of the behaviour of inertial forces related to the optical reference
geometry in the Kerr-de Sitter spacetimes and the features of the embedding diagrams
of the geometry are summarized.
Keywords: Optical reference geometry; Inertial forces; Embedding diagrams; Kerr-de
Sitter spacetimes; Circular motion.
In 1988, by an appropriate conformal (3+1) splitting of the Schwarzschild space-
time, the optical reference geometry (ORG) was defined.1 It was shown that the
geometry enables to introduce the concept of inertial forces providing an alternative
description of relativistic dynamics in accord with 'Newtonian' intuition, whereas
properties of the ORG and inertial forces reflects some hidden features of space-
times. Later, the definition of the ORG was generalized,2,3 thoroughly studied in
particular types of spacetimes,4~10 visualized by embedding diagrams11-13 and the
inertial forces formalism was applied for solving specific problems.14
We summarize results of our investigation15'16 extending the previous studies5'10
reflecting thus some basic properties of the Kerr-de Sitter (KdS) spacetimes
incorporating a combined influence of the rotation of source (a) and the cosmic repulsion
(y). Along with the KdS black-hole (BH) solution, containing two stationary regions
(three event horizons), the naked-singularity (NS) solution with the only stationary
region (one horizon) appears. It is worth to stress that the KdS BH spacetimes
could be important in understanding astrophysical phenomena exposed around su-
permassive black holes in giant active galactic nuclei as demonstrated in.17-19
In general stationary spacetimes with a metric gik, the ORG is defined as an
properly adjusted conformal rescaling of the directly projected geometry by
hik = e~2<i(gik + riink), (1)
where nl is identified with a unit and hypersurface orthogonal 4-velocity field of
special observers, whereas ri'Vjrife = Vfe$ and $ is a scalar function. The 4-acceleration
a,k of a particle can be projected into the hypersurface and decomposed into parts,
which after multiplying by the rest mast and changing signs are considered to be
definitions of the gravitational, centrifugal, Coriolis and Euler inertial forces
-maj: = Gk + Zk + Ck + Ek. (2)
*This research has been supported by the Czech grant MSM 4781305903.
2125
2126
In stationary and axially symmetric KdS spacetimes with the Killing vector
fields rf = 5\ and £l
5\, the special observers are chosen with the 4-velocity field
"V+iWitfO, (3)
n
where $ = \ In [-{r\% + ^IlnrfC^Vi + ^lnrf^i)}, which corresponds to the
4-velocity field of locally non-rotating observers with d<j)/dt = Qlnrf- In the case of
the uniform circular motion, Ek = 0 and the only non-vanishing r and 8 components
of the forces are given by
Gfe = -mVfe$, Zk = m^vf R~lVkR, Ck = ~mi2v RV^lnrf, (4)
where R = (r&)1/2e"*. 7 = 1/(1 -«2)1/2 and v is the orbital velocity with respect
to the LNRF. In the equatorial plane, 8 components of the forces vanish. Moreover
Zr = 0 and Cr = 0 for v = 0. Due to the behaviour of the w-independent part of
Zr, the force can vanish and change its sign at some radii independently of v. The
same can happen for Gr, which is w-independent by definition, while Cr = 0 only
in the case of v = 0 (see Fig. 1).
Embeddings of the equatorial plane of the ORG into the 3D Euclidean space
are governed by the embedding formula
dz/dr = yhrr - (dp/dr)2
(5)
where z, p are cylindrical coordinates and p = h^. The formula suggests that the
equatorial plane of the ORG cannot be entirely embeddable and the embedding
diagrams then consist from several separated parts. Shapes of the diagrams are
characterized by the number of their turning points, coalescing with the radii of
circular orbits where Zr = 0 independently of v. Therefore, Zr is closely related to
the diagrams, and some properties of the relativistic dynamics can be effectively
illustrated. Because radii of photon circular orbits in the equatorial are not located
at the radii where Zr = 0, as common in static spherically symmetric spacetimes,
we have discussed their embeddability as well (see Fig. 2).
0.1
0.08
0.06
0.04
0.02
ns;
A^
■' i::F}
sf2
r
BH|:a2=0.9,y=0.02
Fig. 1. Left: Classification of the KdS spacetimes according to the number of circular orbits
in the equatorial plane where Gr = 0 (subscript) and orbits where Zr = 0 independently of v
(superscript). Right: Example of behaviour of Gr (solid) and n-independent parts of Zr (dashed)
and Cr (dotted) in the outer BH stationary region. Vertical lines denote radii of horizons.
2127
1,25 1,5 1.75
1.15 1.175
275 1,3 1.32
Fig. 2. Classification of the KdS spacetimes according to the number of embeddable regions (first
digit), turning points of the embedding diagrams (second digit) and the number of embeddable
photon circular orbits (digit following the dash), For an example see Fig. 3.
.nr
'■•■^:;-:vff>K
NSlS-X'.a?' =1.4,y^G.G 3
3.5 9 9.5 10 10.5 11
Fig. 3. Example of embedding diagram of the class NS13-1 and its profile.
References
o
6
7
8
9
10
11.
J 2
13.
14
15.
16
17
18
19.
M. A. Abramowicz, B. Carter and J. Lasota, Gen. Rel. Gram. 20, 1173 (1988).
M. A. Abramowicz and J. Miller, Royal Astron. Soc. Monthly Notices 245, 729 (1990).
M. A. Abramowicz, P. Nurowski P and N. Wex, Class. Quantum Grav. 12, 1467
(1995).
Z. Stuchli'k, Bull. Asironom. Inst. Czech. 41, 341 (1990).
Z. Stuchlik and S. Hledik, Phys. Rev. D 60, 044006 (1999).
S. Kristiansson, S. Sonego and M. A. Abramowicz, Gen. Rel. Grav. 30, 275 (1998).
Z. Stuchlik and S. Hledik, Ada Phys. Slovaca 52, 363 (2002).
S. Iyer and A. R. Prasanna, Class. Quantum Grav. 10, L13 (1993).
Z. Stuchlik and S. Hledik, Acta Phys. Slovaca 49, 795 (1999).
Z. Stuchlik, S. Hledik and J, Jurarl, Class. Quantum Grav. 17, 2691 (2000).
Z. Stuchlik and S. Hledik, Class. Quantum Grav. 16, 1377 (1999).
S. Hleclik, Gravitation: Following the Prague Inspiration (A Volume in Celebration oj
the 60th Birthday of Jiri Bicdk), p. 161-192 (World Scientific, 2002).
S. Hledik, Proc. of RAGtime 2/3: Workshops on black holes and neutron stars, Opaua,
ll-tS/8-10 October 2000/01, p. 25-52, (Silesian University in Opava, Czech Rep.,
2001).
K. Nayak Rajesh and C. V. Vishveshwara, Class. Quantum Grav. 13, 1783 (1996).
J. Kovaf and Z. Stuchli'k, Int. Journal of Modem Phys. A 21, 4869 (2006).
J. Kovaf and Z. Stuchlik, Class. Quantum, Grav. 24, (2007) (in print).
P. Slany and Z. Stuchlik, Class. Quantum Grav. 22, 3623 (2005).
Z. Stuchlik, Modern Physics Lett. A 20, 561 (2005).
Z. Stuchli'k and P. Slany, Phys. Rev. D 69, 064001 (2004).
ON THE CONSTRUCTION OF SYZYGIES OF THE
POLYNOMIAL INVARIANTS OF THE RIEMANN TENSOR
ALLAN E. K. LIM and JOHN CARMINATI
Mathematics and Computational Theory Group,
School of Engineering and Information Technology,
Deakin University, Geelong, VIC 3217, Australia.
allan.lim@deakin.edu.au, jcarm@deakin.edu.au
This paper outlines our full solution to the classic problem of determining a complete set
for the polynomial invariants of the Riemann tensor in a 4-D Lorentzian space. In
addition to establishing a basis, we provide a constructive two-stage algorithm for expressing
any invariant as a polynomial function of the basis invariants. In the first stage, a formal
correspondence between the SL(2, C) form of these invariants and generalized directed
multigraphs is established. A novel combination of spinor algebra and elementary graph
theory is used to derive an "arc-pairing" algorithm which reexpresses any invariant as
a polynomial function of invariants containing maximal numbers of paired contractions.
The problem is thus reduced to finding a basis for traces of products of complex 3x3
matrices which transform under the SO(3, C) group. Techniques from matrix polynomial
algebra and rotor calculus are subsequently applied to solve the reduced problem and
provide the second stage of the algorithm.
1. Introduction
This paper summarizes our recent work on the polynomial invariants of the Riemann
tensor in a 4-D Lorentzian space.1_4 Recall that a set of invariants, I = {Ii, I2, .../„},
is said to be a complete set in the classical sense if any polynomial invariant can
be expressed as a 'polynomial in I\,l2, ■■■In, ancl no invariant in the set can be so
expressed in terms of the remaining Ii.
We consider the following two problems:
(i) How does one find a complete set of invariants for the Riemann tensor,
and prove that this set is both complete and minimal?
(ii) How does one construct the polynomial syzygies5 relating any other
invariant to the members of this set?
The first problem has received a significant amount of attention recently.
However, none of the existing work fully solves the problem without introducing
additional restrictions. The second problem has hardly been addressed at all in the
literature. Most existing results established completeness using non-constructive
methods and provide no insight on how to relate invariants outside the complete
sets to those within these sets. Detailed surveys of the literature concerning these
problems can be found in the introduction sections of Refs. 1-3.
2. Graphical Notation of Invariants
The SL(2. C) form of an invariant N may be uniquely associated with a directed
multigraph Gjy such that the vertices of Gjy correspond to the spinors contracted to
form N, and the arcs of Gn represent contractions between pairs of spinor indices.
2128
2129
The direction of an arc indicates contraction between a lower index associated with
the origin vertex and an upper index associated with the destination vertex. Vertices
corresponding to the Weyl, conjugate Weyl and Ricci spinors are depicted as \I/, \I/
and $, respectively. Contractions between undotted indices are represented by solid
arcs, whereas contractions between dotted indices are represented by dashed arcs.
For example, the invariant I\ = \I/
B
^
DE B
*
C
,$
CDE
$
A A
is associated
[1 — * A DE^ a *B DE~*B ~C C
with the graph G]1 in Fig. 1. Formal definitions and further examples are provided
in Refs. 1 and 2.
Fig. 1. Directed multigraph Gil associated with the invariant I\.
3. 'Arc-Pairing' Algorithm
This stage of the solution uses the following identity to transform a pair of
contractions spanning four distinct spinors:
3 £a[b£cd] = £ab£cd + £ac£db + £ad£bc = 0.
The graphical form of this identity is shown in Fig. 2.
1
2
3
4
e
1 x 3
2 4
Fig. 2. Graphical form of a key spinor identity.
The unpaired arcs in Gm can always be reoriented to form Eulerian circuits, and
the preceding identity can be used in a systematic manner to explicitly construct an
expression relating N in terms of invariants containing maximal numbers of paired
contractions. 1~4 A simple case is depicted below; full proofs for the general case are
provided in Refs. 1-3.
2130
1 \ 2
4 3
Fig. 3. Decomposition of a 4-circuit into graphs consisting solely of arcs of even multiplicity.
4. Expression in terms of basis invariants
The arc-pairing algorithm reduces the basis determination problem to one for traces
of products of complex 3x3 matrices which transform under the 5*0(3, C) group.1_4
The solution of this reduced problem relies heavily on techniques from matrix
algebra6 and rotor calculus.7'8 We derive a complete set consisting of 38 real invariants,
including the Ricci scalar R}~^ While this complete set is equivalent to the set
obtained by Sneddon,8 we now have the unprecedented ability to construct polynomial
syzygies relating any other invariant to the members of the complete set.
References
1. A. E. K. Lim and J. Carminati, J. Math. Phys. 45, 1673-1698 (2004).
2. J. Carminati and A. E. K. Lim, J. Math. Phys. 47, Art. No. 052504 (2006).
3. A. E. K. Lim and J. Carminati, "The determination of all syzygies for the dependent
polynomial invariants of the Riemann tensor. III. Mixed invariants of arbitrary degree
in the Ricci spinor," manuscript in preparation.
4. A. E. K. Lim, "Syzygies of the Polynomial Invariants of the Riemann Tensor" Ph.D.
Thesis, Deakin University (2007).
5. The term "syzygy" is commonly used to describe polynomial relationships between
invariants within a complete set. We use it, in a broader sense, to refer to any polynomial
relationship between invariants.
6. A. J. M. Spencer and R. S. Rivlin, Arch. Rational Mech. Anal. 2, 309-336 (1958); G.
E. Sneddon, J. Math. Phys. 39, 1659-1679 (1998).
7. H. A. Buchdahl, J. Aust. Math. Soc. B, Appl. Math. 6, 402-423 (1966); ibid. 6, 424-448
(1966).
8. G. E. Sneddon, J. Math. Phys. 40, 5905-5920 (1999).
A GENERAL COVARIANT STABILITY THEORY
M.I. WANAS* and M.A. BAKRYt
* Astronomy Department, Faculty of Science, Cairo University, Giza, Egypt
t Mathematics Department, Faculty of Education, Ain Shams University, Cairo, Egypt
* wanas@frcu. eun. eg
In the present work we suggest a general covariant theory which can be used to study the
stability of any physical system treated geometrically. Stability conditions are connected
to the magnitude of the deviation vector. This theory is a modification of an earlier joint
work, by the same authors, concerning stability. A comparison between the present work
and the earlier one is given. The suggested theory can be used to study the stability of
planetary orbits, astrophysical configurations and cosmological models.
1. Introduction
In a previous paper [1] the authors have suggested the use of geodesic deviation
equations to study stability of gravitating systems. In that paper, they have
generalized the classical perturbation scheme, usually used to deal with such problem.
We have suggested the use of components of the deviation vector, representing the
solution of the equation of geodesic deviation,
where £a is the deviation vector, U13 is the unit tangent to the geodesic, {? \ is
the Christoffel symbol of the second kind and (s) is an invariant parameter. Now,
£a(s) is the solution of equation (1) in the interval [a, 6] in which the functions
£a(s) behave monotonically. This vector reflects the reaction of the system under
perturbation. The quantities, that have been suggested in [1], to be used as sensors
for stability of the system, are
qa =f lim£a(f). (2)
s—>h
The criterion suggested is that, if qa —> oo, the system would be unstable, otherwise
it would be stable. This criteria has been used to study stability of a number of
cosmological models. Applications in cosmological models, using this criterion, is
somewhat easy since most of these models depend on one function, the scale
factor. Further applications show the non-covariance, of the scheme, under coordinate
transformations.
It appears that if stability conditions are obtained depending on the quantities
(2), these conditions would not be, in general, covariant. This is because the
components of the deviation vector depend on the coordinates system used. In other words,
the stability conditions obtained would be coordinate dependent. We are going to
call the scheme suggested in [1] the "Coordinate Dependent Scheme", (CDS).
The aim of the present note is to modify the quantity (2) in order to get covariant
stability conditions.
2131
2132
2. Covariant Stability Conditions
To get covariant results, independent of the coordinate system used, one has to
replace the contravariant components of the deviation vector used in (2) by its
magnitude, then we examine the limit
q^1 lirn^a)*. (3)
Now, if q —> oo, then the system is unstable. Otherwise, it would be stable.
To summarize how to apply the covariant scheme suggested, one has to follow
the following steps:
1. Having a well defined problem, we solve the field equations controlling this
problem to know the type of geometry associated with the system under consideration
(the metric).
2. Knowing the metric of space time, we solve the geodesic equation to get the unit
tangent vector Ua.
3. Using the information, obtained in the above two steps, substituting in the
geodesic deviation equation (1) and solving it, we get the deviation vector £Q.
4. Evaluating the scalar £M£M and examining its limit as given by (3).
If q —> oo, the system will be unstable. Otherwise, it will be stable.
5. A strong stability condition can be achieved if,
lim (£<*£«)* =0. (4)
t-~>oo
We are going to call this scheme "The Coordinate Independent Scheme", (CIS).
3. Discussion
If we use the scheme suggested in the present work CIS and apply it to some
of the world models examined in the previous work [1] we get the results that
are summarized and compared, to those obtained using the CDS, in Table 1. In
the table, the cosmological models treated are classified as follows. The first set of
models represents world models constructed using "General Relativity" (GR). In the
second set, we examine a world model depending on "Miln Kinematical Relativity"
(KR) and another one constructed using " Brans-Dicke Theory" (BD). The third set
contains models resulting from "Miller's Tetrad Theory of Gravitation" (MTT).
The last set contains models obtained using the "Generalized Field Theory" (GFT)
[3]. The sample, in Table 1, is chosen in such a way that it represents models
depending on different geometric field theories. It is clear from the following table
that the use of the covariant scheme, suggested in the present work, gives results
different from those obtained in the previous work.
2133
Table 1: Stability of Some World Models Using CDS and CIS.
Theory
GR
KR
BD
MTT
GFT
Model
Einstein [4]
De Sitter [4]
Einstein-De Sitter [4]
Radiation [5]
Miln [4]
Brans-Dick [6]
D < 0 [7]
D > 0 [7]
fc=-l [8]
k = 0 [8]
CDS
Unstable
Stable
Stable
Unstable
Stable
Stable
Stable
Conditional
Stable
Unstable
CIS
Unstable
Unstable
Unstable
Unstable
Stable
Unstable
Unstable
Unstable
Stable
Unstable
The similar results obtained, using the suggested scheme and the previous one [1],
are just coincidence. It is obvious that changing the coordinate system used to
construct, a world model will not affect the results of the last column of Table 1,
while it may change those given in the third column.
The scheme suggested in the present work has been successfully used to study
stability of non-singular black holes [9]. Further details will be published elsewhere.
References
[1] Wanas, M.I. and Bakry, M.A. (1995) Astrophys. Space Sci. 228, 239.
[2] M0ller, C.(1978) Mat, Fys. Skr. Dan. Vid. selk. 39,13, 1.
[3] Mikhail, F.I. and Wanas, M.I. (1977) Proc. Roy. Soc. Lond. A 356, 471.
[4] McVittie, G.C. (1961) "Facts and Theory of Cosmology", Eyre & spittswoode,
London.
[5] Sciama, D.W. (1971) "Modern Cosmology", Cambridge, London.
[6] Wienberg, S. (1972)" Gravitation and Cosmology" John Wily & Sons.
[7] Saez, D. and de-Juan , T. (1984) Gen.Rel Grav. 16, 5.
[8] Wanas, M.I. (1989) Astrophys. Space Sci. 154, 165.
[9] Nashed, G.G.L. (2003) Chaos, Solitons and Fractals, 15, 841.
RELATIVISTIC GENERALIZATION OF THE INERTIAL AND
GRAVITATIONAL MASSES EQUIVALENCE PRINCIPLE
NIKOLAI V. MITSKIEVICH
Department of Physics, CUCEI, Universidad de Guadalajara
Guadalajara, Jalisco, Mexico,
Apartado Postal 1-2011, C.P. 44^00, Guadalajara, Jalisco, Mexico
mitskievich03@yahoo. com. mx
The Newtonian approximation in the gravitational field description not necessarily
involves admission of non-relativistic properties of the source terms in Einstein's equations:
it is sufficient to merely consider the weak-field condition for gravitational field. When,
e.g., a source has electromagnetic nature, one simply cannot ignore its intrinsically rel-
ativistic properties, since there cannot be invented any non-relativistic approximation
which would adequately describe electromagnetic stress-energy tensor even at large
distances where the fields become naturally weak. But the test particle on which
gravitational field is acting, should be treated as non-relativistic (this premise is required for
introduction of the Newtonian potential <3?n from the geodesic equation).
We use here (in parentheses if in a tetrad basis) Greek indices as 4-dimensional
and Latin as 3-dimensional, >c = 8irG (G is the Newtinian gravitational constant),
Rfiiy = Ranisa, and spacetime signature as +,—,—,—. Einstein's equations then
read as R{{^ - ±i?<5£ = -xT^, thus R = xT, and R(^ = -x fo$ - ±T6g
We shall need only 00-component of Einstein's equations,
P(0) _ K (T(0) T(i)\ m
(°) ~ 2" V (°) W/' [)
We call a source with T)\l = 0 intrinsically relativistic since the spatial part
of its stress-energy tensor is of the same order of magnitude as the temporal
component (cf. the concept of a zero rest mass particle). An example is the Maxwell
electromagnetic field which has this property even of its static solutions when any
kind of motion is excluded. Similarly, a perfect fluid with its energy-momentum
tensor
Tpf =(n+p)u®u-pg (2)
possesses this property in the particular case of incoherent radiation (/i = 3p), and
the tensor (2) is written in the rest reference frame of the fluid. There is also the
case of stiff matter (p = (i) in which sound propagates with the velocity of light;
we say that such objects are hyper-relativistic. Thus in the non-relativistic case
rp(i)
(0)
•C T(J ) the 00-component of Einstein's equations reads
(0) 2 non_ (°)' ^ '
then in the intrinsically relativistic case,
7?(0) - kT- ,(0) (4.\
U(0) ~ -Ximtr.rel(0), (,4J
and finally in the hyper-relativistic case
?(°) - _
(0) — ^^-"-hyper-reljo)
■*£(n\ — Z>il hyper—relfgy \0)
2134
2135
The Newtonian approximation is found from the geodesic motion of a non-
relativistic test particle. Thus let us consider a static spacetime with goo = 1 + 2$n,
|$n| -C 1 and choose a 1-form basis as
0(°)=eadt, 0M = gW jdx*. (6)
Taking the inverse triad, so that dx^ = g(k)J0^k\ dt = e~a6^Q\ we find the necessary
components of 1-form connections u/°)(n = lo^\o) = a,j9{if^°\ and finally from
Cartan's second structural equations,
<{ = 9{l){k)R(Q)mkW) « e- (e")^- g" (7)
where g%i = —6j + higher-order terms (to be neglected). Since ca « 1 + $n,
R(o\ ~ — A$n (A is the usual Laplacian). Thus the Newton-Poisson equations
corresponding to (3), (4), and (5), are
non-relativistic A$n = 4-7rG/i, (8)
intrinsically relativistic A$n = 87rG/i and (9)
hyper-relativistic A$n = 167rG/i, (10)
respectively (we wrote here the inertial mass density /i of the source instead ofTAJ).
For any perfect fluid the Newton-Poisson equation takes the form
A$N=47rG(/i + 3j>), (11)
so that for incoherent dust the old traditional equation follows, but if the fluid
represents an incoherent radiation (p = /x/3), the source term doubles (as this is
the case for electromagnetic source), and for the stiff matter (p = /i), it quadruples.
Since the equations (4) and (5) are exact ones, they strictly express the
equivalence principle already generalized (to use an expression similar to "already unified"
of J.A. Wheeler) in standard general relativity. The conclusions we came upon in
this talk automatically add on relativistic features to the principle traditionally
formulated in standard textbooks on general relativity as a completely non-relativistic
approximation (for both test particle and sources of Einstein's equations) just as it
was used by Einstein in his first attempts to generalize the special relativity. But
the Newtonian-type potential is generated by a wide class of distributions of matter,
including intrinsically relativistic and hyper-relativistic cases: the only restriction
here consists of weakness of the field and not the "state of motion" of the sources
in Einstein's equations (especially such an intrinsic property as to be relativistic
which is so often realized by static configurations when the very idea of motion is
out of question). Clearly, here we haven't used any hypotheses at all.
As to the applications of this generalized principle of equivalence, it is worth
pointing out the (post-) post-Newtonian approximations. Since some conclusions
about validity of the principle of equivalence come from observations of stellar
systems, a mere presence in them of intrinsically relativistic distributed or localized
objects (say, high density of any kind of radiation, strong or widely distributed
magnetic fields, existence of stiff matter in cores of exotic stars, jets of ultrarclativistic
2136
particles) would radically change interpretation of the observational data if their
proper understanding depends on adequate description of the sources of
gravitational field, without any disregard for the pressure and stresses. These conclusions
should definitively lead to a revision of the old problem of stability of young globular
star clusters via the virial theorem (when the electromagnetic radiation between the
stars is very intense) which seems to be done through approximated methods only.
This is also the central point of evolution of the gravitation theory from Soldner9
and Einstein-19111 to Einstein-1915,2 resulted in doubling [cf. (8) and (9)] of the
light beams bending in the final self-consistent version of the theory. This doubling
has two sides: one is mentioned just above, and another pertains to light beams and
jets of ultra-relativistic particles via the 3rd Newtonian law, see comments on both
in Refs. 4, 5 and 8. Another problem is connected to the interesting and stimulating
question by D. Brill, the Chairman of the parallel Session GT4 at which this talk
was delivered: How to relate Einstein's first tentative considerations of photons'
absorbtion by a material sample, leading to its temperature rise, and the
corresponding increase of its masses, both inertial and gravitating ones? My answer was
that the gravitational mass does not satisfy a conservation law, at least that which
follows from the Noether theorem3,6 under the general relativistic invariance of the
action integral, in a contrast to the inertial mass, therefore it is clear that both
masses cannot simultaneously be conserved, e.g. in the process of light absorbtion.
Thus the gravitational mass in general shouldn't be additive when the relativistic
properties (similar to the equation of state for a fluid, but not necessarily reducible
to this equation) suffer changes in physical processes. For example, when we
electrically charged a perfect fluid, starting with an incoherent radiation, its equation of
state (inhomogeneously) changed too, thus the initial combination of energy density
and pressure which determined the gravitational mass density also suffered changes;
this was a side effect in a generation of new solutions of Einstein's equations.7
Finally, it should be emphasized once more that in this talk we made a revision
of a too long persistent old viewpoint, but not of the sane and mature theory itself.
References
1. A. Einstein, Ann. Phys. (Leipzig) 35, 898 (1911).
2. A. Einstein, Sitzungsber. Preufi. Akad. Wiss. 831 & 844 (1915).
3. N.V. Mitskievich (Mizkjewitsch), Ann. Phys. (Leipzig), 1, 319 (1958). In German.
4. N.V. Mitskievich, Newton's third law and self-consistency of interactions in physics. In:
Newton and Philosophical Problems of the Twentieth-Century Physics (Nauka, 1991)
pp. 116-124. In Russian.
5. N.V. Mitskievich, Claro — Obscuro, Serial Cuadernos de Metodologia sobre
Investigation y Desarrollo Tecnologico (IPN Mexico) No. 3, p. 1 (1993). In Spanish.
6. N.V. Mitskievich, Relativistic Physics in Arbitrary Reference Frames (Nova Science
Publishers, 2006).
7. N.V. Mitskievich and M. Cataldo, Class, and Quantum Gravity, 9, 545 (1992).
8. N.V. Mitskievich and L.I. Lopez Bem'tez, Gravitation & Cosmology 10, 127 (2004).
9. J.G. von Soldner, Berl. Astronom. Jahrb. 1804, 161 (1802).
STATIC PERTURBATIONS BY A POINT MASS ON A
SCHWARZSCHILD BLACK HOLE
DONATO BINI
Istituto per le Applicazioni del Calcolo "M. Picone," CNR 1-00161 Rome, Italy and
ICRA, University of Rome "La Sapienza," 1-00185 Rome, Italy and
INFN - Sezione di Firenze, Polo Scientifico,
Via Sansone 1, 1-50019, Sesto Fiorentino (FI), Italy
binid@icra.it
ANDREA GERALICO
Physics Department
and ICRA, University of Rome "La Sapienza", 1-00185 Rome, Italy
geralico @icra.it
REMO RUFFINI
Physics Department and ICRA, University of Rome "La Sapienza", 1-00185 Rome, Italy
and ICRA Net, 1-65100 Pescara, Italy
rufjini@icra. it
The static perturbations by a point mass on a Schwarzschild black hole background are
studied in the framework of first order perturbation theory. It is shown that a solution
free of singularities cannot exist using the standard approach by Regge and Wheeler.
Adopting a different gauge allows to find the explicit form of the perturbation
corresponding to a stable configuration, characterized by the presence of a "strut" between
the particle and the black hole. The resulting perturbed metric with a conical singularity
is shown to be the linearized form of the exact solution for two collinear uncharged black
holes in static configuration belonging to the Weyl class.
The perturbations due to a point mass in a Schwarzschild black hole background
has been studied by Zerilli [1] in the dynamical case following the Regge-Wheeler [2]
treatment. We are interested here in the simplest case of a neutral particle of mass
m at rest near a Schwarzschild black hole of mass A4, whose metric in standard
coordinates is given by
ds2 = -fs{r)dt2 + Mry'dr2 + r2{d82 + sin2 8d<p2) ,
/.w -1 - ^. m
Let the point particle be at rest at the point r = b on the polar axis 9 = 0. The
presence of the massive particle causes a change in the background gravitational
field which can be determined by solving the whole set of Einstein equations
GM„ = 8nT^ , (2)
where the perturbed Einstein tensor denoted by a tilde refer to the total
gravitational field, to first order of the perturbation
9>iv = 9nv + hfn, , (o)
2137
2138
and the stress-energy tensor describing the particle has the only nonvanishing
component
J00
2tt62
fa{bfl28 (r-b) 5 (cos(9 - 1)
(4)
The perturbation equations are then obtained from the system (2), keeping terms
to first order in the mass m of the particle which is assumed sufficiently small with
respect to the black hole mass.
First of all, following Zerilli's procedure [1] we expand the perturbing
gravitational field h^v as well as the source term (4) in tensor harmonics. The next step
consists in suitably fixing a gauge in order to simplify the description of the
perturbation. Adopting the Regge-Wheeler [2] gauge (as customary studying perturbations
of spherically symmetric bodies) leads to a solution which exhibits a singular
behaviour of the perturbed Riemann tensor at the particle position, as shown in detail
in [3]. Indeed, a singularity-free solution for this problem is obviously impossible,
since there is no external force to oppose the infall of the particle towards the black
hole, so that equilibrium cannot be reached in any way. Furthermore, the solution
expressed in the Regge-Wheeler gauge is not suitable for its reconstruction in closed
analytic form summing over all multipoles.
We find in [3] it very helpful to use a new gauge condition particularly adapted
to this problem which differs from the Regge-Wheeler one. This new gauge (which
has been referred to as BGR gauge) gives rise to a more convenient form of the
gravitational perturbation functions, yielding a closed form expression for the
perturbed metric by summing over all multipoles. In addition, the singular character of
the solution turns out to be manifest in this gauge. In fact, the resulting perturbed
metric we obtain is shown to be the linearized form of the exact solution
representing two collinear Schwarzschild black holes in a static configuration belonging to the
Weyl class [4], characterized by the presence of a conical singularity (or a "strut")
between thern.
The perturbed metric in the BGR gauge summed over all multipoles turns out
to be
ds-
-Mr)[l
+r2 sin2 8
/(BGR)!
H^^dt2
l + H
(BGR)
0
fs(r)
Aj3
'[l + H,
(BGR)
dr2
l + H
(BGR)
de2
(5)
where
H
(BGR)
0
2£/^)1/2
H.
(BGR) _
= 2-
m
fs(b)1/2
AMm
b{b-2M)
Mb)
1/2
M-{b-M)cosd
Id's
Ds = [{r - M)2 + {b - Mf -2(r-M)(b -M)cos8-M2 sin2 8]1/2 . (6)
Once the solution is known in a given gauge, one can then express the same
solution in a different form passing to another gauge, whose relations to the previous
2139
one are also known. The relations between the Regge-Wheeler gauge and the BGR
gauge are given explicitly in [3]. The perturbed metric written in the Regge-Wheeler
gauge thus turns out to be
d~s2 = -/s(r)[l - W^]dt2 + /sir)"1!! + W^dr2
+r2
l + i?(RW)
(d0z + rz sin'Odtf) , (7)
where
w(RW) = 5(bgr) _ 4A42m 1/2r-Ds + McosO
0 b{b~2M)M) r{r-2M)
4:Mm ,,,1/2 f, As
M 1. f
c+ -In£
r
2'
with c an arbitrary integration constant and
4 6r 6-2.M ( , 2n t on (r-A4)coS0-(6-A4)-£>g|
c = F^F^^^r + s ^~2M) j '(9)
while Hq ' is given by Eq. (6).
References
1. F. J. Zerilli, Phys. Rev. D 2, 2141 (1970).
2. T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063 (1957).
3. D. Bini, A. Geralico and R. RufRni, in preparation.
4. D. Kramer, H. Stephani, E. Hertl, and M. McCallum, Exact solutions of Einstein's
field equations (Cambridge University Press, Cambridge, 1979).
SPATIAL NONCOMMUTATIVITY IN A ROTATING FRAME
M. BECIU
Department of Physics, Technical University,
Bucharest, B-d Lacul Tei 124, Romania
* E-mail: beciu@utcb.ro
We develop an analog model for the Landau problem and its ensuing noncanonical
brackets but in a relativistic context. The chosen model is the Minkowski spacetime in a
rotating coordinate system where the motion turns out to be quite similar to the motion in
a constant magnetic field. For high angular velocity the Hamiltonian analysis reveals a
problem with constraints of second class. In this case the Dirac bracket between spatial
coordinates is nonvanishing and inversely proportional to the angular velocity. Finally,
the issue of apparent causality violation due to spatial noncommutativity is briefly
discussed.
Keywords: noncommutativity, Dirac bracket, nonlocality
1. Introduction
The very old idea of space coordinates noncommutativity1 has been revived quite
recently in connection to the possibility of fuzzy spacetime at very small scales and
also in string theory with D-branes.2 A more down to earth realization of coordinate
noncommutativity is given by the motion of charged particles in a constant magnetic
field B, perpendicular to the plane of motion, the so-called Landau problem.3 One
arrives at spatial noncommutativity through mainly two routes i) by using the
Hamiltonian analysis of constraint systems4 ii) by solving the problem of a quantum
particle in a constant B and projecting to the lowest Landau level or, anyway, to a
finite number of levels.5 Our purpose here is to present a model, as close as possible
to the simple Landau problem, but one manifestly relativistic.
2. The model
Let us consider the following spacetime whose metric is given by
ds2 = (l - ft2z|) dt2 - 2n£ijxidxjdt - dx2 (1)
where i,j,k = 1,2 summation over repeated indices is implied, ft is a constant
angular velocity, and eki is the antisymmetric tensor in two dimensions. It is easy
to find that this is Minkowski spacetime in disguise (the Riemann and Ricci tensors
are zero) but flat space adapted to a coordinate system F rotating with respect to
the inertial one F0. Let us consider also the Lagrangian of a particle of mass m in
spacetime (1), with
/ 2X1/2
L = -m[l- (xk - fleHxl) J = -mr. (2)
Then the canonical momenta and the Hamiltonian are, respectively
pk = m(xk - Qekixl)/r, H = R + ttefpkX1. (3)
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2141
The equations of motion turn out to be
x* = {xk, H} = pk/R + flekxl-pk = {pk, H} = QelkPl;R = ((p2k + m2)1'2. (4)
Comparing with the Landau problem we find the role of the magnetic field is played
by 2n = —eB/m We are interested in the limit of large angular velocity Q. ,large
compared to the mass m, but with rim, finite. This situation is analogous to strong
magnetic field in Landau problem, B/m-large, but with eB, finite.
m (xk - nekixl) -> ~mVl£klxl;r = M - (xk - VLekixl)2\ -► (l - ti2x2)1/2 .
(5)
We get therefore two primary constraints
Xk=Pk + mflekixL jr « 0, (6)
where ~ denotes equality in the weak sense. The secondary constraints arise from
the requirement of conservation in time of the primary ones
Xi = {Xi,H} + u1{Xi;X2} « 0; x2 = {X2, H} + u2{Xi, X2} « 0. (7)
Here the Dirac algorithm closes, no further constraints occur, the above equations
determine uniquely the Lagrange multipliers w1, u2. The set of constraints, according
to the Hamiltonian analysis of constrained systems,6 are termed as second class
constraints (non-gauge). We are less interested here in the value of the Lagrange
multipliers. The important thing is that having
cal3 = mn(2~ n2xj) (1 - n2x2) "3/2, (8)
with Cap = {xa, Xp}-! it is possible to define a Dirac bracket for the coordinate
{Xi,Xj}D.B. = {Xi,Xj} -{Xi^ajC^lxp^Xj}, (9)
where {xi,Xj} is the usual Poisson bracket, equal to zero in the standard algebra
and Ca(3 is the inverse matrix of Cap. Keeping in mind that in the Dirac bracket the
constraints must be taken as strong equalities, a straightforward calculation leads
to
{xi, Xj}D.B. = (1 - nifty1 £ijQ{x), (10)
with G(x-) = (1 - tt2x2)3/2 / (2 - n2x2) .
3. Conclusions
A few comments are in order:
1) In the limit of large angular velocity n, but with m£l, finite we neglected terms like
mv so we may ask if the system is still relativistic. The system is indeed relativistic
(see for instance (2)) and the velocity (small) is with respect to the noninertial
frame F and not with respect to an inertial frame Fq.
2) The maximum value of the Dirac bracket is reached at x = 0 and is
{xi,Xj}D.B. = {2m£l)~ £ij. (11)
2142
This represents what we would naively expect if the analogy 2fi = —eB/m were
pursued
3) It is instructive to compare the result (10) with the result obtained in,78 The
starting point of these authors is the Hamiltonian
H = neklpkxi. (12)
't Hooft7 remarks that a Hamiltonian of type (12) is unsatisfactory from the
quantum mechanical point of view because it is unbounded from below. A number of
considerations led him to advocate the replacement of the original Hamiltonian with
a new one p = mVPxj, (here the specific parameters are adapted and the
appropriate measurement units, restored), obviously bounded from below, also free of
ordering ambiguities and such that {p, H} = 0. This procedure would be consistent
if it was to generate the same equations of motion
Xi = {xi,p} = {xi, H] = QeikXk (13)
but, in that case, the symplectic structure must be modified to exactly (10).
Comparing now the Hamiltonian (12) with our Hamiltonian (3), we conclude that the
first one would correspond to the approximation where both the mass and the
momenta are vanishingly small, with m£l and p£l, finite.
4) Let us suppose now that we replace the coordinates with the Hermitian operators.
We can write
[Xi,Xj]=i(mn)-1eije(x). (14)
The remark is that in (14) 1/m is just the Compton wavelength of the particle and
we worked under the assumption of high angular velocity f2 » m. It follows that
an uncertainty relation presumably derived from (14) implies
Any problem of nonlocality that might arise from (14) is hidden inside the Compton
radius.
References
Snyder H. S., Phys. Rev. 71, (1947), 38
Szabo R. J., Phys.Rep. 378 (2003) 631
Landau L.D. et Lifshitz E., Mcanique Quantique, (dition MIR, Moscou), 1967, pp
496-499
Jackiw R., Nucl. Phys. Poc. Suppl. 108, (2002), 30
Margo G., quant-ph/0302001 preprint 2003
Heneaux M. and Teitelboim C, Quantization of Gauge Systems- Princeton University,
Princeton N.Y. (1992)
't Hooft G., hep-th/0003005 preprint 2000; hep-th/0105105 preprint 2001
Banerjee R., Mod. Phys. Lett. A17 (2002) 631
ON ENERGY AND MOMENTUM OF THE FRIEDMAN AND
SOME MORE GENERAL UNIVERSES
JANUSZ GARECKI
Institute of Physics, University of Szczecin,
Wielkopolska 15, 70-451 Szczecin, Poland
garecki@sus.univ.szczecin.pl
Recently some authors concluded that the energy and momentum of the Fiedman
universes, flat and closed, are equal to zero locally and globally (flat universes) or only
globally (closed universes). The similar conclusion was also done for more general only
homogeneous universes (Kasner and Bianchi type I). Such conclusions originated from
coordinate dependent calculations performed only in comoving Cartesian coordinates by
using the so-called energy-momentum complexes. By using new coordinate independent
expressions on energy and momentum one can show that the Friedman and more general
universes needn't be energetic nonentity.
In the last years many authors have calculated the energy and momentum of the
Friedman universes and also more general, only spatially homogeneous universes,
like Kasner, Bianchi type I and Bianchi type II universes.1
The above mentioned authors performed their calculations in special comoving
coordinates called "Cartesian coordinates" despite that they used coordinate
dependent double index energy-momentum complexes, matter and gravitation. The
all energy-momentum complexes are neither geometrical objects nor coordinate
independent objects, e.g., they can vanish in some coordinates locally or globally and
in other coordinates they can be different from zero. It results that the double
index energy-momentum complexes and the gravitational energy-momentum pseu-
dotensors determined by them have no physical meaning to a local analysis of a
gravitational field, e.g., to study gravitational energy distribution. In fact, up to
now, complexes and pseudotensors were reasonably used only to calculate the global
quantities for the very precisely defined asymptotically flat spacetimes (in spatial or
in null direction). The best one of the all possible double index energy-momentum
complexes from physical and geometrical points of view is the Einstein canonical
double index energy momentum complex eK^ (See, e.g.,2,3).
The conclusion of the authors which calculated the energy and momentum of
the Friedman and more general universes by using double index energy-momentum
complexes is the following: the energy and momentum of the closed Friedman
universes are equal to zero globally, and in the case of the flat Friedman universes
and their generalizations (Kasner, Bianchi type I, Bianchi type II universes) these
quantities are equal to zero locally and globally.
One can have at least the following objections against the calculations of such
a kind and against the above conclusion:
(1) The authors despite that they used coordinate dependent expressions had
performed their calculations only in Cartesian comoving coordinates.
The results obtained in other comoving coordinates, e.g., in coordinates
2143
2144
(t, x, $, <p) or in coordinates (t, r, ■&, ip) are dramatically different.
(2) The local "energy-momentum distribution" as given by any energy-momentum
complex has no physical sense but the authors try to give a physical sense of
this distribution, e.g., they assert that the total energy density for flat Friedman
universes, for Kasner and Bianchi type I universes, is null.
(3) The conclusion leads us to Big-Bang which has no singularity in total energy
density.
(4) The global energy and momentum have physical meaning only when spacetime
is asymptotically flat either in spatial or null direction and when these quantities
can be measured. But this is not a case of the cosmological models.
So, the problem of the global energy and global linear (or angular) momentum
for Friedman, and for more general universes also, is not well-posed from the
physical point of view because these universes are not asymptotically flat space-
times, and, in consequence, their global quantities cannot be measurable. This
problem can only have a mathematical sense.
Thus, one can doubt in physical validity of the conclusion that the energy and
momentum of the Friedman, Kasner, Bianchi type I and Bianchi type II universes
are equal to zero; especially that all these universes are energy-free.
By using double index energy-momentum complexes one should rather
conclude that the energy and momentum of the Friedman, Kasner, Bianchi type I, and
Bianchi type II universes explicite depend on the used coordinates and, therefore,
they are undetermined not only locally but also globally. The last conclusion is very
sensible because, as we mentioned beforehand, one cannot measure the global energy
and global linear (or angular) momentum of the Friedman and any more general
universe. One can do this only in the case of an isolated system when spacetime is
asymptotically flat.
One cannot use the coordinate independent Pirani and Komar2'3 expressions
in order to correctly prove (at least from the mathematical point of view) the
statement that the energy of the Friedman, Kasner, Bianchi type I and Bianchi
type II universes disappears, i.e., that these universes have zero net energy. It is
because we have no translational timelike Killing vector field (descriptor of energy
in Komar expression) in these universes, and the privileged normal congruence
of the fundamental observers which exists in these universes is geodesic (Pirani
expression on energy only can be applied in a spacetime having a privileged normal
and timelike congruence. But for a geodesic congruence Pirani expression fails giving
trivially zero).
One also cannot use for this purpose the coordinate independent Katz-Bicak-
Lynden (BKL) bimetric approach4 because the results obtained in this approach
depend on the used background and on mapping of the spacetime under study onto
this background.
Thus, the "academic'' statement that the Friedman, Kasner, Bianchi type I and
Bianchi type II universes have no energetic content is still not satisfactory proved.
2145
But by using Komar expression, one can correctly (at least from mathematical
point of view) prove that the linear momentum for these universes disappears in a
comoving coordinates.
Recently we have introduced the new, coordinate independent expressions on
the averaged relative energy-momentum and angular momentum in general
relativity (See5). We have called these new tensorial expressions the averaged tensors of
the relative energy-momentum and angular momentum. The averaged tensors are
very closey related to the canonical superenergy and angular supermomentum
tensors which were introduced in our previous papers.6 When applied, the averaged
relative energy-momentum tensors give the positive-definite energy densities for the
Friedman, Kasner and Bianchi type I universes.5 The result of such a kind is very
satisfactory from the physical point of view. The more general universes were not
analyzed yet.
References
1. N.Rosen, Gen.Rel. Gravil, 26, 319 (1994); V.B. Johri et al, Gen. Rel. Gravit.,27,
313 (1995; N. Banerjee and S. Sen, Pramana J. Phys., 49, 609 (1997); S.S.Xulu,
"The energy-momentum problem in general relativity", hep-th/0308070; M. Salti and
A. Havare, Int. J. Mod. Phys., A 20, 2169 (2005) (gr-qc/0502060); M. Salti et al.,
Astrophys. Space Sci, 299, 227 (2005) (gr-qc/0505079); M. Salti, Mod. Phys. Lett,
A 20, 2175 (2005) (gr-qc/0505078); M. Salti, Czech. J. Physis., 56, 177 (2006)
(gr-qc/0511095); O. Aydogdu, "Gravitational energy-momentum density in Bianchi
type II spacetimes", gr-qc/0509047; O. Aydogdu, Fortsch. Phys., 54, 246 (2006) (gr-
qc/0602070); J. Katz et al., Phys. Rev., D 55, 5957 (1997) (gr-qc/ 0509047); M. Salti et
al., "Energy and momentum of the Bianchi type I universes in teleparallel gravity", gr-
qc/0502042; I. Radinschi, Fizika B (Zagreb) 9, 203 (2000); O. Aydogdu et al., "Energy
density associated with the Bianchi type II spacetimes", gr-qc/0601133; R Halpern,
"Energy of the Taub cosmological solution", gr-qc/0609095; M.S. Berman, "On the
energy of the universe", gr-qc/0605063.
2. A. Trautman,"Conservation laws in general relativity", an article in Gravitation: an
introduction to current problems, L. Witten, ed. (Academic Press, New York 1962).
3. J. Goldberg, "Invariant Transformations, Conservation Laws and Energy-Momentum",
an article in General Relativity and Gravitation, A. Held, ed. (Plenum Press, New York
1980).
4. J.Katz, J. Bicak and D. Lynden-Bell, Phys. Rev., D 55, 5957 (1997) (gr-qc/0504041).
5. J. Garecki, "The averaged tensors of the relative energy-momentum and angular
momentum in general relativity and some their applications", gr-qc/0510114. An amended
version will appear in Found, of Physics; Class. Quantum Grav., 22, 4051 (2005);
"Energy and momentum of the Friedman and more general universes", gr-qc/0611056.
6. J.Garecki, Rep. Math. Phys., 33, 57 (1993); Int. J. Theor. Phys., 35, 2195 (1996); Rep.
Math. Phys., 40, 485 (1997); J. Math. Phys., 40, 4035 (1999); Rep. Math. Phys.,43,
397 (1999); Rep. Math. Phys., 44, 95 (1999); Ann. Phys. (Leipzig) 11, 441 (2002); M.P.
Dabrowski, J. Garecki, Class. Quantum Grav., 16, 1 (2002).
QUASI-LOCAL ENERGY FOR AN UNUSUAL SLICING OF STATIC
SPHERICALLY SYMMETRIC METRICS
CHIANG-MEI CHEN
Department of Physics, National Central University,
Chungli. Taiwan 32054, R-O.C
E-mail: cmchen@phy.ncu.edu.tw
JAMES M. NESTER
Department of Physics and Institute of Astronomy, National Central University,
Chungli, Taiwan 32054, R-O.C
E-mail: nester@phy.ncu.edu.tw
We consider an unusual time slicing for the static spherically symmetric metrics. For the
vacuum case this is the Schwarzschild metric in the Painleve-Gullstrand form. For this
slicing the spatial metric is flat, and the lapse is just unity; all the dynamic geometry is
encoded in what is supposed to be a gauge parameter: the shift vector. One consequence is
that the standard ADM energy expression vanishes (contrary to the idea that vanishing
energy should be Minkowski space). On the other hand, for an appropriate choice of
reference and time displacement vector, our preferred quasilocal Hamiltonian boundary
term expression gives a finite energy, namely 2M.
Keywords: Quasi-local energy; Hamiltonian.
1. Quasi-local energy
The identification of gravitational energy is still an outstanding problem. It is not
a local quantity and is not uniquely defined. Various requirements for a "physical"
quasi-local energy have been proposed such as1 zero for flat space, for spherical
symmetric ~ standard value, ADM mass for spatial infinity, Bondi mass for null
infinity, for apparent horizon ~ standard value, and positivity.
Our covariant Hamiltonian formalism gives a certain preferred Hamiltonian
boundary term for quasi-local quantities which depends on the boundary
conditions, plus a reference and displacement vector choice.2~7 The Hamiltonian 3-form
has the fonnH(N) = N'in^ + dB{N) in which the "density" part, N^H^, generates
the dynamical equations yet vanishes on shell, while the boundary part, B(N),
determines the boundary conditions and gives the quasi-local values. For the Einstein
(vacuum) gravity theory, we found a distinguished quasi-local expression
B(N) := ~ [Ar% A iNr,J + (DpN)aAriJ] , (1)
where T01 p is the connection one-form and r/aP := *($a A i?'3). The bar denotes the
reference variables and A means the difference of physical and reference values. N,
the displacement vector, is chosen to be time-like for defining an energy.
2146
2147
2. Flat Slice: Spherical Symmetric
The general spherical symmetric metric can be expressed in the ADM form
ds2 = -(N'fdr2 + L2 [dr + Nrdr}2 + R2di12, (2)
where Nl is the lapse function and Nr is the shift vector. For a special case, the
flat slicing (L = 1,R = r), each constant time slice (dr = 0) is a flat 3-space. The
static spherical static metric
ds2 = -f(r)dt2 + h{r)dr2 + r2dVt\, (3)
can be re-reexpressed in this form by a suitable coordinate transformation dt =
dr + F{r)dr, with F2 = (h - 1)//, consequently TV* = fh and Nr = fF. For the
Schwarzschild geometry / = l/h = 1 - 2M/r, F2 = (2M/r)/(l - 2M/r)2. and
TV* = i,AT = <,y2M/r; nerc «- = ±L This gives the Painleve-Gullstand form of
the metric:
ds2 = -dr2 +(dr + <;J—dr J + r2dfi2. (4)
The two times are related by t = t + 2y/2Mr + 2Mln VL"^W[. The radial light
paths dr/dr = ±\-q-sj2M/r. show that the choice of <j = 1 (dr/dt\r=2M = {0, -2})
describes a black hole, whereas <;■ = — 1 (dr/dt\r=2M = {2,0}) is a white hole.
3. Quasilocal energy for the Schwarzschild geometry
We are investigating the representation dependence of our quasi-local
gravitational energy. Here we compare the results for the Schwarzschild geometry in the
usual Schwarzschild representation with that of the Painleve-Gullstand form. In the
Schwarzschild coordinates, the natural co-frames are -d0 = &dt. i?1 = §~1dr, t32 =
rd9, $3 = rsinOdif. where $ = Jl — —. Our quasi-local energy, in this case
(using N = eo = $-1<9t unit time-like) coincides with the Brown-York result:
E(r) = r(l - $).3 Particular values are E(rH = 2M) = 2Af, E(oo) = M.
The Painleve-Gullstand coframe is tf° = dr, d1 = dr + cJ^dr, d2 = rdO,
i?3 = rsmddif. Some of the connection one-form components, namely T2i = d9,
r3 i = sin6 dip, r32 = cos9dp, have the same values as in Minkowski space (so
the corresponding Ar vanish), whereas the others are T1o = f J^ffl1, T2o =
~^\jt^2 ■' r3o = — <r\/ 2^f'&3, (the corresponding Ar have the same values).
Consequently the components of the quasi-local expression p^ = Ar"-8 A r/a/3^
vanish (when restricted to constant r, t) except for
P-.
G
-2AT2o A Jfeoi - 2Ar30 A r/30i = ^V2MrdQ. (5)
Naively it seems that our '"energy" vanishes (as does the ADM energy), and,
worse, the "radial" momentum diverges. However if we examine the value on the
2148
time-like Killing vector of the reference, dT = eo + ^^/2Mjr e\, we find
pG(dT)=4.-2M-d£l, (6)
when integrated over the sphere (and re-scaled by 167r) we find a finite value for
the "energy" EG = HG(dT) = 2M, the same as our "standard" horizon value.
We note that in the Painleve'e-Gullstrand frame the time-like vector
corresponding to the Schwarzschild choice eg is
e0s = (1 - 2M/r)-1/2 (e0G + ^^2MfreG) , (7)
which, using the Painleve-Gulstrand reference, yields the quasi-local energy
pG(e§) = (l-2M/r)_1/24-2M-dn, (8)
diverging at the horizon.
4. Discussion
For any gravitating system — and hence for all physical systems — the localization
of energy-momentum is an outstanding problem. For gravitating systems, using our
covariant Hamiltonian formalism, we have obtained quasi-local energy-momentum
expressions; and each is associated with a physically distinct, and geometrically
clear, boundary condition.
However, an appropriate choice of N and reference is essential to get a
physically reasonable results. The Schwarzschild geometry with a flat slice gives a simple
example to address this issue. An unusual slicing of Minkowski, as discussed in a
related talk, is another example. We have hopes that these particular simple examples
will help us to the understanding needed for more general quasi-local calculations.
Acknowledgements
This work was supported by the National Science Council of the R.O.C. under the
grants NSC 95-2119-M008-027 (JMN) and NSC 95-2112-M-008-003 (CMC). JMN
and CMC were supported in part by National Center of Theoretical Sciences and
the (NCU) Center for Mathematics and Theoretical Physics.
References
1. C. C. Liu and S. T. Yau, arXiv:math.dg/0412292.
2. C.-M. Chen, J. M. Nester and R.-S. Tung, Phys. Lett. A 203, 5-11 (1995).
3. C.-M. Chen and J. M. Nester, Class. Quantum Grav. 16 1279-1304 (1999).
4. C.-C. Chang, J. M. Nester and C.-M. Chen, in Gravitation and Astrophysics ed Liao
Liu, Jun Luo, X.-Z. Li, J.P. Hsu (World Scientific, Singapore, 2000) pp 163-73.
5. C.-M. Chen and J. M. Nester, Gravitation & Cosmology 6, 257-70 (2000).
6. J. M. Nester, Class. Quantum Grav. 21 , S261-280 (2004).
7. C.-M. Chen, J. M. Nester and R.-S. Tung, Phys. Rev. D72, 104020 (2005).
8. L. B. Szabados, "Quasi-local energy-momentum and angular momentum in GR: A
review article", Living Rev. Relativity 7, 4 (2004), www.livingreviews.org/lrr-2004-4.
QUASI-LOCAL ENERGY FOR COSMOLOGICAL MODELS
JAMES M. NESTER
Department of Physics and Institute of Astronomy, National Central University,
Chungli, Taiwan 32054, R-O.C.
E-mail: nester@phy.ncu.edu.tw
CHIANG-MEI CHEN* and JIAN-LIANG LIU
Department of Physics, National Central University,
Chungli, Taiwan 32054, R.O.C.
* E-mail: cmchen@phy.ncu.edu.tw
Our covariant Hamiltonian formalism gives a certain preferred Hamiltonian boundary
term for quasi-local quantities which depends on the boundary conditions, plus a
reference and displacement vector choice. With appropriate choices we found the quasi-local
energy for the cosmological models. Homogeneous choices give vanishing energy for all
regions of Bianchi class A models and positive energy for class B. Isotropic choices give
energies proportional to the curvature parameter k: ie, vanishing for the flat case, positive
for the closed model and negative(l) for the open model. Our values are consistent with
the requirement that the energy vanishes for closed models. We have some conclusions
regarding the best reference choice and two quasi-local desiderata: positivity and zero
energy iff Minkowski space.
Keywords: Quasi-local energy; Hamiltonian; Cosmology.
1. Quasi-local energy and the Hamiltonian boundary term
Our quasi-local energy is given by the value of the Hamiltonian associated with a
time-like displacement vector field N. The Hamiltonian H{N) is given by aii integral
of a suitable density of the form H(N) = N^H^ + dB(N). The density H^ must
vanish "on-shell", the quasi-local energy is determined by the boundary integral;
E(N) = H(N) = f H{N) = f [N^Hft + dB{N)] = I B(N). (1)
The two parts of the Hamiltonian have distinct roles. The 3-form "H^ generates the
equations of motion. The Hamiltonian boundary term B(N) plays two key roles: it
determines the quasi-local values and the boundary conditions (via the requirement
that the boundary term in the variation of the Hamiltonian vanish). Thus, just as
in thermodynamics, in gravity there are various "energies" which are related to how
the system interacts with the outside through its boundary.x~6
It is necessary (to guarantee functional differentiability on the phase space with
the desired asymptotic boundary conditions) to include suitable reference values,
(which determine the ground state). For GR we have several expressions associated
with various types of boundary conditions. One choice is favored; it has the form
B(N):=ArapAiNV^ + DpNaAr]J, (2)
corresponding to a Dirichlet type condition on the orthonormal frame field.6
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2. Homogeneous cosmologies
For the homogeneous cosmological models the orthonormal (co) frame has the form
$° = dt, $a = hak(t)(Tk, where the spatially homogeneous frames satisfy
d<Tk = ^Ckij(Ti /\<rj. (3)
The associated spacetime metric is ds2 = —dt2 + gij{t)a% [x)a^(x) where gij :=
Sabhaihbj (which need not be diagonal). There are 9 Bianchi types in two classes
distinguished by the particular form of the structure constants Ckij-.7
Class A (types I, II, VI0, VII0, VIII, IX) have Ak := C\i = 0,
Class B (types III, IV, V, VI/j, VII/j) are characterized by Ak ^ 0.
For TV = dt, with a Dirichlet type boundary condition and the Bianchi
homogenous frame as the boundary value, along with static homogenous cartesian frame
reference values, our favored quasi-local expression gives
E{V) = n-lA0Akgik{t)V(t) > 0 (4)
for all types of sources (including dark energy a/o cosmological constant) for all
regions. Our quasi-local energy vanishes for all class A models and is positive for all
class B models.8 Note: this is consistent with the requirement that E = 0 for closed
universes, since all class A models can be compactified and class B cannot.9
3. FRW cosmologies
The FRW (homogeneous and isotropic) metrics have the form ds2 = —dt2+a2(t)dl2.
The spatial metric dl2 has constant curvature. The spatial metric has several forms:
dl2 = dp2 + EW = -^ + rW = ? (dR2 + R2dfl2) . (5)
l-kr2 (l + (fc/4)i?2)2 V ' ' V '
where £ = (sin p, p, sinhp) for k = ( — 1, 0, +1) respectively. We take TV = dt,
Dirichlet boundary conditions, the FRW frame as boundary values, and the flat cartesian
frame as reference. Our FRW quasi-local energy within a constant radius is
Ek = fl£(l - £') = ar [l - (1 - kr2)"2} = ., °f^ D2,2 . (6)
More specifically, Eq = 0 and
(l + (fc/4)i?2)2'
aR3
£Li = asinhp(l - coshp) = ar [1 - \/l + r2 = - _ R2 uy> (7)
E+1 =asmp(l-cosp) = ar ]A - \/l-- r2J = 2 fl2 2 . (8)
4. Discussion
The Bianchi perspective favors homogeneous boundary conditions and reference.
Then our quasi-local energy vanishes for all regions in all class A models, which
2151
includes isotropic type I and IX, which are equivalent to FRW k = 0 and k = +1,
respectively. Class B has positive energy; it includes isotropic Types V and VII/j
which are equivalent to the FRW k = — 1.
According to the FRW isotropic- about- a-point boundary conditions and
reference, we find that the sign of the quasi-local energy is proportional to k, negative
for the open universe, vanishing for the flat case and positive for the closed case
(but vanishing as it should when the whole universe is considered). It is noteworthy
that in the case k = — 1 with vanishing matter, we get a(t) = t. It can be directly
verified that the geometry is really Minkowski, yet our quasi-local expression gives
a non-vanishing energy, which, moreover is negative!
Our analysis suggests that the homogeneous choice is more suitable. To
understand the physical and geometric meaning of the difference in detail we need to do
extensive calculations using the rather complicated relation between the FRW and
Bianchi coordinates.
Our cosinological energies challenge two quasi-local desiderata:10 for the
expressions considered positivity need not hold, and zero energy iff flat Minkowski space
need not hold in either direction.
Acknowledgements
This work was supported by the National Science Council of the R.O.C. under the
grants NSC 95-21 f9-M008-027 (JMN) and NSC 95-2112-M-008-003 (CMC). JMN
and CMC were supported in part by National Center of Theoretical Sciences and
the (NCU) Center for Mathematics and Theoretical Physics.
References
1. C.-M. Chen, J. M. Nester and R.-S. Tung, Phys. Lett. A 203, 5-11 (1995).
2. C.-M. Chen and J. M. Nester, Class. Quantum Grav. 16 1279-1304 (1999).
3. C.-C. Chang, J. M. Nester and C.-M. Chen, in Gravitation and Astrophysics ed Liao
Liu, Jun Luo, X.-Z. Li, J.P. Hsu (World Scientific, Singapore, 2000) pp 163-73.
4. C.-M. Chen and J. M. Nester, Gravitation & Cosmology 6, 257-70 (2000).
5. J. M. Nester, Class. Quantum Grav. 21 , S261-280 (2004).
6. C.-M. Chen, J. M. Nester and R.-S. Tung, Phys. Rev. D72, 104020 (2005).
7. G. F. R. Ellis and M. A. H. MacCallum, Comm. Math. Phys. 12 108 (1969).
8. L. L. So, J. M. Nester and T. Vargas, "On the energy of homogeneous cosmologies",
in preparation.
9. A. Ashetkar and J. Samuel, Class. Quantum Grav. 8, 2191-2215. (1991).
10. L. B. Szabados, "Quasi-local energy-momentum and angular momentum in GR: A
review article", Living Rev. Relativity 7, 4 (2004), www.livingreviews.org/lrr-2004-4.
RELATIVE STRAINS IN GENERAL RELATIVITY
DONATO BINI
Istituto per le Applicazioni del Calcolo "M. Picone," CNR 1-00161 Rome, Italy and
ICRA, University of Rome "La Sapienza," 1-00185 Rome, Italy and
INFN - Sezione di Firenze, Polo Scientifico,
Via Sansone 1, 1-50019, Sesto Fiorentino (FI), Italy
binid@icra.it
FERNANDO DE FELICE
Dipartimento di Fisica, Universita di Padova
and INFN, Sezione di Padova, Via Marzolo 8, 1-35131 Padova, Italy
fernando.defelice @pd. infn. it
ANDREA GERALICO
Physics Department
and ICRA, University of Rome "La Sapienza," 1-00185 Rome, Italy
geralico @icra. it
The analysis of relative accelerations and strains among a set of comoving particles is
presented. The frame-dependent character of the definition of strains and applications
to special congruences of test particles in flat spacetime are briefly discussed.
Long ago Szekeres [1] introduced the concept of "gravitational compass,"
consisting in an arrangement of three test particles joined by springs to a central observer.
At the instant of measurement the reference particle drops the apparatus observing
the strains on the springs, then mapping out the strength of the local gravitational
field. However Szekeres' analysis was limited to the case of relative acceleration
between two nearby geodesies. Later on de Felice and coworkers [2,3] studied the
relative strains among a set of comoving particles in black hole spacetimes with
orbits accelerated (in general) and confined to a normal neighborhood of the
observer's world line. Starting from that analysis, we have considered in [4] how the
definition of relative accelerations and strains among the particles of the congruence
is affected by the geometric properties of the frame adapted to the fiducial observer
(e.g. transport law of the reference spatial triad along the observer's congruence).
In this paper we limit our analysis to the relative strains of a bunch of uniformly
rotating particles in the flat Minkowski spacetime. In fact, consider a bunch of test
particles, i.e. a congruence Cu of timelike world lines, with unit tangent vector U
(U ■ U = —1) parametrized by the proper time iy- Let C* be the reference world
line of the congruence, which we consider as that of the "fiducial observer." The
separation between the line C* and a generic curve of the congruence is represented
by a connecting vector Y, i.e. a vector undergoing Lie transport along U:
£uY = 0 -» \7uY = VYU. (1)
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2153
Taking the covariant derivative along U of both sides of the previous equation gives
rise to the "relative acceleration equation"
T^- = -R(U,Y)U + VYa(U), (2)
aTu
where R(U,Y)U = Rap1sU^3Y1Us represents the tidal force contribution to the
relative acceleration, whereas Vya(J7) is the "inertial" contribution due to the
observer's acceleration a{U) = Vi/U. Next set up an orthonormal frame {Eq = U, Ea}
adapted to the congruence U and write the relative acceleration equation (2)
with respect to this frame. After introducing the frame components of Y, i.e.
Y = Y° U + Ya Ea, Eq. (2) gives
Ya + lC{u,E)abYb = 0, (3)
where )C(utE)ab = [T(fw,£/,£) ~ S(U) + £(U)}ab are the components of the "deviation"
matrix K.^e), the overdot meaning differentiation with respect to the proper time.
Here £{U)a1 = R0lp7sUl3Us is the electric part of the Riemann tensor (as measured
by the observer U). The strain tensor S is defined by
S(U) = V(U)a(U) + a(U) ® a(U) , S(U)ab = V(U)ba(U)a + a(U)aa(U)b , (4)
where the spatial covariant derivative [5] V(U) = P(U)S7 is obtained by projecting
V onto the local rest space of U using the spatial projector P(U)% = Sg + UaUp.
Note that S(U) depends only on the congruence U and not on the chosen spatial
triad Ea, differently from the frame-dependent tensor T, which turns out to be given
by
T({w,u,E)ab = Sbu>ffv!^E) - w(°fW)£/iS)u;(fw,c/,s)6 - e°h/^(fw,c/,£;)
-2eafcu;((v/^E)K(Uyb , (5)
where K(U)f}a = -P(U)£P(U)PVfjU" is the kinematical tensor and W(iy,tu,E)
represents the angular velocity with which the spatial triad Ea rotates with respect to
a Fermi-Walker transported triad along U: P(U)VijEa = W(fw,c/,E) x Ea. We dicuss
now the case of the Minkowski metric written in standard cylindrical coordinates
{t,r,<j),z}
ds2 = -At2 + dr2 + r2d</>2 + dz2 , (6)
with et- = dt , ef = dr , e^ = dz , ei = ^d^ a fixed orthonormal frame.
Consider a family of uniformly rotating particles with angular velocity (; the four
velocity U of the generic particle of the congruence is then given by
U = T{dt+Cd*)=l{ei + vei), 7 = (1 - J)'1'2 , (7)
1 In
where T = (l — r2£2) and ( = v/r. A frame adapted to U can be fixed as
£(£0!=^, E(U)2 = j(vet + e.) 7 E(U)3 = es . (8)
2154
The orbits are accelerated, with a(U) = —~f2C,2r E(U)\. The deviation equations
(3) reduce to Ya = 0, since K,{jj,E) = 0 resulting from the balancing between the
strain tensor and the Fermi-Walker tensor, namely S(U) = T^wUE\ with only
nonvanishing components S(U)u = S(U)22 = ~74C2- Taking into account the
Lie transport equation (1) implies that in addition Ya = 0, so that the spatial
components of the deviation vector remain all constant along the path with respect
to the frame (8). Rotating the spatial triad in the 2-plane E(U)\ — E(U)2 by an
angle a = ~^Qt = ~~f2CTu one obtains a Fermi-Walker triad
E'(U)l = cosaE(U)l+sinaE(U)2 ,
E'(U)2 = -sinaE(U)i +cosaE(U)2 ,
E'(U)3 = E(U)3 . (9)
With respect to this new triad T^wUE^ = 0, so that IC(UE,\ = —S(U) and the
only nonvanishing components are
fc(U,E')U = K-(U,E')22 = 1 C , (10)
implying harmonic oscillations for the deviation vector components Y1 and Y2 with
frequency Hw^jy^H = 72|C[ = 72MA- The corresponding solution is
straightforward:
Y'1 = y'oCos(||w(fw,£7iS)||r£/) -Y'lsmiWu^jj^WTu) ,
Y'2 = Y'lcosiWw^u^W-nj) +y'osin(l|w(fw,t/,B)||7l/) ,
Y/3 =Y'l , (11)
where Y'% are the components of the deviation vector at the starting point and
the Lie transport equation (1) has been taken into account. This implies that an
initially circular bunch of particles on the Yn-Y'2 plane remains always circular for
increasing values of the proper time.
In spite of its simplicity this case exhibits all features of the strains; the
generalization to the more reach and interesting case of either geometrically or
physically motivated timelike congruences in vacuum stationary axisymmetric space-
times (static observers, Zero Angular Momentum Observers (ZAMOs), Painleve-
Gullstrand observers in Schwarzschild and Kerr black hole spacetimes) is contained
in [4].
References
1. P. Szekeres, J. Math. Phys. 6, 1387 (1965).
2. F. de Felice and S. Usseglio-Tomasset, Gen. Rel. Grav. 24, 1091 (1992); Class.
Quantum Grav. 10 353 (1993); Gen. Rel. Grav. 28, 179 (1996).
3. O. Semerak and F. de Felice, Class. Quantum Grav. 14, 2381 (1997).
4. D. Bini, F. de Felice and A. Geralico, Class. Quantum Grav. 23 7603 (2006).
5. R. T. Jantzen, P. Carini and D. Bini, Ann. Phys. (N.Y.) 215, 1 (1992).
DYONIC KERR-NEWMAN BLACK HOLES, COMPLEX SCALAR
FIELD AND COSMIC CENSORSHIP*
IBRAHIM SEMIZ
Department of Physics
Bogazici University
Bebek, Istanbul, Turkey
ibrahim.semiz@boun.edu.tr
We construct a gedanken experiment, in which a weak wave packet of the complex
massive scalar field interacts with a four-parameter (mass, angular momentum, electric
and magnetic charges) extreme Kerr-Newman black hole. We show that the resulting
black hole does not violate the cosmic censorship conjecture for any black hole parameters
and wave packet configuration.
1. Introduction
A "naked singularity" is one that is not hidden behind an event horizon. The "cosmic
censorship" conjecture of Penrose1 forbids them; more precisely, it is conjectured
that they cannot be produced from regular initial conditions with matter satisfying
reasonable properties.
In the absence of a general proof, gedanken- and numerical experiments have
been devised to check the validity of the cosmic censorship conjecture (CCC) under
different limited circumstances, by studying the evolution of various initially regular
systems to see if a naked singularity develops.2 For example, the Kerr-Newman
metric has a horizon —therefore describes a black hole— if and only if
M2>Q2+a2. (1)
Otherwise, it describes a naked singularity, and one can ask if the Kerr-Newman
metric can somehow be made to evolve from a form satisfying (1) to one that does
not, i.e from a black hole into a naked singularity. Wald3 has asked if one can do this
by throwing spinless test particles into an extreme black hole (i.e. one saturating
(1)) and answered in the negative. His argument generalized to the case of the
dyonicallya charged black hole by Hiscock4 and independently, by Semiz.5
In the present work, we ask the same question for a complex massive scalar wave
packet, treated as a perturbation, impinging onto a dyonic black hole. More details
are available.6
*This research has been partially supported by grant 06B303 by Bogazigi U Research fund.
aA dyon is a particle with both electric and magnetic charge.
2155
2156
2. Changes in the mass, electric charge and angular momentum of
the black hole
As is well known, the energy and angular momentum conservation laws are
consequences of spacetime symmetries and locally take the form of continuity equations
(T^XV),^=Q (2)
where T^v is the energy-momentum tensor for test particles or fields and Xv is the
Killing vector generating the symmetry. The Killing vectors of the Kerr-Newman
metric are gfu and gfg-- We also have the current continuity equation, giving
(^r0^ = o, (v^VX^o, (v^'"),, = o (3)
From the first continuity equation, we get the rate of change of the mass of the
black hole
where S^ is the spherical surface at infinity. We choose this surface, since the black
hole mass is defined in asymptotically flat space. Similarly, dL/dt and dQe/dt for
the black hole can be calculated. The energy-momentum tensor T^v and the current
density j^ are calculated from the Lagrangian by the standard prescription.
When a perturbation expansion is done around the Kerr-Newman background,
it is found that the changes dM/dt, dL/dt and dQe/dt are all second order, if no
photons are assumed to be present. To evaluate these changes, we use the
separability7 of the Klein-Gordon equation on the background:
</, = R(r)0(6)e-tu>tel(mTeQ^'t'. (5)
The solutions of the angular equation form a complete and orthonormal set, and
the radial problem can be converted into a one-dimensional scattering problem.
3. Testing the Cosmic Censorship Conjecture
Since we want to compare changes of the left-hand-side vs. right-hand-side of
inequality (1), we are interested in 8{CCC) = S(M2) — 8(Q2 + a2). Since this quantity
changes over time,
2 f/»,r2 , 2^dM n*^ dQe dL
m\}m +a^-MQ^-aTt)dt <6>
where we used Q2 = Q2, + Q2n, dQm/dt = 0 and a = L/M.
Since e~lu>t and the angular functions ("monopole spheroidal harmonics") form
complete sets, the changes dM/dt, dL/dt and dQe/dt can each be written as a sum-
integral over the eigenmodes. Using the orthonormality of the modes and doing the
time integral, 5(CCC) eventually reduces to
2157
M +a fj V^ , i m-* , u eQeM - am.
5{CCC) = -^- J du^ fimMfLMI" +
M2 + a2
, eQer+ - am,
XiU' + 2 i 2 l^^lrn-D^lm
r\ + az
where f[m(u>) and B^im are arbitrary coefficients and r+ is the horizon radius. For
the extreme black hole, r+ —>■ M, therefore
M2 4- n2 f
5{ccc) ~" -^ikr J ^Y,fi™Mfi™w*lB<>>irnB:lm. (?)
and 5{CCC) is strictly positive, i.e. M2 increases at least as fast as (Q2 + a2), which
means that the Cosmic Censorship Conjecture can not be violated by adding charge
and/or angular momentum to a extreme black hole via a Klein-Gordon field.
References
1. R. Penrose, Riv. Nuovo Cimento 1, special number, pp.252-276 (1969).
2. R. M. Wald, gr-qc/9710068; R.Penrose, in Black Holes and Relativistic Stars, ed. R.
M. Wald, The University of Chicago Press, Chicago (1998), pp.103-122; P. S. Joshi,
Modern Physics Letters A 17, pp.1067-1079, (2002).
3. R.M. Wald, Ann. Phys. 82, pp.548-556 (1974).
4. W.A. Hiscock, Ann. Phys. 131, pp.245-268 (1981).
5. I. Semiz, Class. Quantum Grav. 7, pp.353-359 (1990).
6. I. Semiz, gr-qc/0508011.
7. I. Semiz, Phys. Rev. D 45, pp.532-533 (1992). Erratum: Phys. Rev. D 47, p. 5615
(1993).
THE IDEAS OF GR, QUANTIZATION, NON-EQUILIBRIUM
THERMODYNAMICS AND GRAVIMAGNETISM
IN PLANETARY COSMOGONY
M.M. ABDIL'DIN, M.E. ABISHEV and N.A. BEISSEN
Al-Farabi Kazakh National University,
Kazakstan, Almaty, Tole be 96 a
abdnur@kazsu. kz
In the work the existence of relativism, quantization and non-equilibrium
thermodynamics ideas in cosmogony are discussed.
Planetary cosmogony is still remaining to be outside the application of the ideas of
the mechanics of general relativity (GR) as well as the ideas of quantization, non-
equilibrium thermodynamics and gravimagnetism. It is classical. One may think,
however, that such a situation is not forever. Indeed, there are apparently certain
indications of that.
The first indication is the existence of a class of circular orbits of a test body
which lie in the equatorial plane of a rotating central body and are stable with
respect to the vector orbital elements M (angular momentum) and A (the Laplace
vector).1 Indeed, let us address to the well-known problem of GR mechanics, the
Lense-Thirring problem, i.e., the problem of finite motion of a test body of mass
m in the field of a rotating body of mass rriQ. We perform our consideration on the
basis of the Fock's refined first approximation metric due to a rotating fluid ball
m0c2 c2 7m0c2 r c2
U = ^> U = -^[fSol eo = ^o + |T0, (2)
ds2 = [c2^2U(l+^^)+~ + -^^(S0V)(S0y^)]dt2-(l+ — )d^ + -(Udr}dt,
(1)
where
here So is the ball's angular momentum; T0 is the kinetic energy of its rotation, and
£o is the energy of mutual attraction of the particles ion the body taken with an
opposite sign. Recall that
(^V)(S0vi) = -^ + ^)!. (3)
Unlike other similar metrics of the first approximation, the metric (1) correctly
describes the Schwarzschild problem2 and also takes in to account the term nonlinear
in So which is important for the Lense-Thirring problem. Now, the Hamiltonian of
the Lense-Thirring problem will be written as1
2m c2 8?rr 2m mo
-I"*2) ~ |f(&V]I + ^([^V][S0vi]), (4)
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2159
where p = ^4 is the particle momentum and L is the Lagrange function. The
equations of motion have the form
c2rA 7mQcz
M = ^\^-^-r-r2(s<,r)\s«A, (5)
A = (lE + 9mU + !2*)!?™1 + JW] + -^(^)lrM]
mo mcA cArJ mc*r°
67 {S2[fMj - ^{SQf)2[rM} + 2(S0r)[S0M] + 2(S0r)ip[rS0}}}, (6)
7moc2r5 r2
where M and A are vector elements of the orbit:
M=[rpl, A= [—Ml - 7mm°f] ,4 = 7mm0e = ae. (7)
m r
Here E is the non-relativistic energy and e is the orbit eccentricity. Eqs. (5) and (6)
show that the vectors M and A slowly change with time and take part in two
motions, the evolutionary one and the periodical one. Consider the evolutionary motion
of a material particle of mass m in the gravitational field of a massive rotating fluid
ball of mass rriQ. To do so, let us apply to Eqs. (5) and (6) an asymptotic method
of nonlinear mechanics, the averaging method (over the Newtonian period ). Then
the differential equations of the first approximation of the asymptotic method (the
equations of evolutionary motion) acquire the form
dM ,,=♦-•., dA , => -,
- = [OM], - = [IM], (8)
where
o= ^E. = ^mai m m2a4 f2 9 - 3m(M^o) q
dM M3M§c2 +m0M3M03c2t ° 7m0M2 °
6m(MS0)2 ? 3m2a4M -.-. m 2 3m(MS0)2
+ 7m0M4 M} ~ m0M^M^{2{MSo) + 7nV0S° " 7m0M2 }' (9)
Here M0 = Mj J\ — ^ is an invariant of the system. The mean Hamiltonian is
- 2 rna'2 1 r,15ma2 ma2 3ma4
H = mc --—2 +^{(-777^-— &)-
2M2"rc2U8M02 ma^'M2 M$M
m"a4 -[2(S0M) + ^--^(S0M)(S0M)}}. (10)
m0M$M31 y ' 7m0 7m0M2
Let us now consider the stability with respect to absolute values of the vector
elements M and A. As is easily seen, the equations of evolutionary motion (8) and
(9) imply conservation of the absolute values of the vector M and A,
M = const, A = const. (11)
Hence it is clear that the evolutionary motion of a material particle is stable with
respect to the absolute values of the vector elements M and A. On the other hand,
2160
(11) implies orbital stability of the motion of a material particle in the field of
a rotating body. Indeed, orbital stability of the motion of a material particle is
understood as the property of the osculating ellipse to preserve its shape and size,
at ant time instant, close to the shape and size of the unperturbed Keplerian ellipse
defined for the initial time instant. The ellipse shape and size are characterized by
the eccentricity and the length of the focal axis 2a. If the relations that determine e
and a, do not contain secular terms, then, by definition, the elliptic motion possesses
orbital stability.
Eqs. (11) just have as their consequences
a = const, e = const, (12)
i.e., the orbital stability of material particle motion in the field of a rotating fluid
ball.
Let us now introduce into our consideration a new type of stability in GR
mechanics, namely, stability with respect to the vector elements M and A themselves,
i.e., we will require the fulfillment of the following stability conditions in the material
particle motion:
M = const, A = const, (13)
i.e., the general equations (8) for such a motion should take the form
dM dA
—- = 0. — = 0, 14
dt ' dt v '
or
[&M] = 0, [ClA} = 0. (15)
This implies that the orbits stable with respect to the vector elements M and A in
the Lense-Thirring problem are those belonging to the class of circular orbits lying
in the equatorial plane of the rotating body.
Another indication is O.Yu. Schmidt's law of planetary distances in cosmogony.3
According to O.Yu. Schmidt, the difference of square roots of the distances of two
adjacent planets from the Sun is a constant quantity:
V Rn+l — yRn = yRn — y/Rn-1, (16)
or
^%l = R0 + bn, n = 0,1,2,... (17)
where 6 - is the constant difference between two adjacent square roots.
Assuming that all orbits are circular in the Solar system and that all planets
of the Earth's group have equal masses, we can rewrite Schmidt's law, i.e., the
equalities (16), (17), in terms of the angular momenta, using the well-known relation
Mn
Rn = , a = 7mmo. (18)
ma
2161
Then Schmidt's law of planetary distances acquires the form
Mn+1-Mn = Mn-Mn-1, (19)
Mn = ^3>{R0 + bn). (20)
Thus Schmidt, in his well-known cosmogonical theory, actually uses the angular
momentum quantization law.
Here we would like to add that N.G. Chetaev (1902-1959), outstanding Soviet
mechanist and mathematician, author of fundamental works and ideas in stability
theory and analytical mechanics, has expressed an idea of utmost interest:4,5
"Stability, as a phenomenon general in principle, should apparently somehow manifest
itself in the general laws of Nature" Consecutively developing this idea, Chetaev
arrived, in particular, at the hypothesis on quantization of stable orbits of dynamics.
According to Chetaev, only some particular, exclusive trajectories can be stable,
similarly to the stability of only exclusive electronic orbits in quantum mechanics.4
Note that in cosmogony much has been said about the role of rotation (of the Sun
and the planets), both their own rotation and that of orbits, in the evolution of the
Solar system. However, only the framework of GR makes this problem determined
since it relates rotation to a certain vector field: the gravitational vector field with
the vector potential U.
The third indication is a relation between the relativistic spin-spin and magnetic-
magnetic interactions in planetary cosmogony. For the Solar system, the spin-spin
interaction between the Sun and a planet is of the same order as the magnetic-
magnetic interaction of the same bodies. Indeed, according to GR, there is an
addition to the Hamiltonian which takes into account the interaction of two angular
momenta (the Sun and the planet's own rotation) having the form:1
8H = _-£([£v][£0V-]) - ^±^([SV][§0V-]), (21)
where V is the operator JL
The magnetic-magnetic interaction in the Sun-planet system gives the following
additional term in the Hamiltonian:6
SH, = (MMo)r2-3(Mr)(Mor)
It is easily shown that the interactions (21) and (22) are of the same order in the
Solar system; to do so, one can use Blackett's relation [7]
M = -^-S, (23)
2c
where /? is a numerical factor of order unity. It is important that the spin-spin and
magnetic-magnetic interactions are of the same order in the planetary system. By
Alfven [8], the magnetic-magnetic interaction plays an important role in the Solar
system evolution. It is now clear that the spin-spin interaction should also be taken
into account.
2162
The fourth indication is the existence of situations in planetary cosmogony which
may be described using the ideas of non-equilibrium thermodynamics, although
the thermodynamics of processes in space is itself in an embryonic state [9, p.
21]. The first idea which we should catch here is Curie's symmetry principle. In
Weyl's formulation, Curie's principle claims: "If the conditions that unambiguously
determine an effect possess a certain symmetry, then the result of their action will
possess the same symmetry" [9, p. 27]. Therefore, as it seems to us, the planetary
system formation occurring under a permanent influence of the Sun's scalar and
vector gravitational fields
V=~-9, <?=-£KS.l, W
where U possesses spherical symmetry and U axial symmetry, should eventually
establish the same symmetries in the resulting system.
Another idea from non-equilibrium thermodynamics that we also can use is the
existence of the so-called stationary states. These states should not, however, be
confused with equilibrium which is characterized by maximum entropy and zero
entropy production. Stationary states play a prominent role in physics since the
physical systems, being subject to constant (or nearly constant) influences, spend
an overwhelming part of their time in a stationary state [9, p. 32].
Stationary states are stages in the system evolution towards equilibrium. The
transition of a system to equilibrium usually splits into two more or less clearly
distinguished stages [9, p. 32]: formation of quasi-stationary non-equilibrium states
and the evolution of quasi-stationary states to a complete statistical equilibrium.
The word "quasi-stationary" is used here to emphasize that stationary states exist
for a finite time interval. As the system exceeds this interval, the stationary states
slowly evolve to other stationary states or to equilibrium. The same may happen as
well to a planetary system with orbits.
The fifth indication is the gravimagnetism hypothesis [14]. Some time ago, to
explain the magnetism of the celestial bodies, a number of hypotheses were put
forward, leading to correct quantitative results. Moreover, quite remarkable is the
unusual nature of these hypotheses from the viewpoint of the existing physical
outlook. Thus, according to Wilson's hypothesis [10], the magnetic fields of the
Earth and the Sun are such as if they possessed a negative volume charge density
a = — 1/7P , where 7 is the gravitational constant and p is the mass density. An
unusual feature is that this "charge" does not create an electric field but, rotating,
creates a magnetic field. Another hypothesis, also leading to correct quantitative
results, is Blackett's hypothesis [11]. According to Blackett, any rotating body,
irrespective of the existence of any charge in it, should possess a magnetic moment
proportional to its mechanical angular momentum: M = — ^rS. Einstein's remark
[12] is in full agreement with these hypotheses: "The Earth and the Sun possess
magnetic fields whose orientation and polarity are approximately determined by
the directions of these bodies' rotation... It rather seems as though magnetic fields
emerge from rotary motion of neutral masses... Here, Nature apparently points at
2163
a fundamental law so far unexplained by theory". Recently [13], the interest in
discussing the physical roots of "Blackett's rule" increased again.
Some time ago [14], in search for a foundation of these hypotheses, we put
forward a more general hypothesis that gravity may be a source of magnetism. It
has been shown [7,15] that:
1. The relation A = ~-^=U is valid, where A- is the vector potential of the
magnetic field of a rotating body and U is the vector potential of the gravitational
field. For instance, for a rotating homogeneous fluid ball, the vector potential of the
gravitational field is U = —-^[riSo]. To calculate the potential A in a more general
case, one can use the equation AA = t-^-Anpv, where v is the velocity inside the
body.
2. The off-diagonal component of the metric tensor goi = —^ At is connected
with the magnetic field.
3. The approximate results for the magnetic fields of the moon (10~5 Oersted)
and a pulsar (1010 Oersted) are obtained.
4. The traditional interpretation of GR, as a theory of the gravitational field
only [2], also changes to a certain extent. Now GR, or, more precisely, its
mathematical framework (the Einstein equations!) correspond to a gravimagiietic field
theory. Gravitational wavws, as they are now understood, should in fact exist as
gravimagiietic waves.
5. The gravimagnetism hypothesis being discussed leads to one more, though
indirect, conclusion. Indeed, in modern electrodynamics there is an asymmetry
between electricity and magnetism, which manifests itself physically in the existence of
electric charges and the absence of magnetic charges; mathematically, it is reflected
in the lack of symmetry in the right-hand sides of the Maxwell-Lorentz equations
with respect to the electric and magnetic field sources. This fact is probably not
accidental but rather bears a deeper meaning, allowing one to think of a distinguished
role of magnetism. Indeed, let us present the Maxwell equations:
1 f)ff
rotE = — divH = 0, (25)
c
dt
jt> 4:7r^ IdE ,. -± . ,„„.
rotH = —7 + - —. divE = 4ttct. (26)
c c dt
where E is the electric field strength, H is the magnetic field strength, a and j are
the electric charge density and the electric current density, respectively. It follows
that the magnetic field emerges as a by-product of the electric field that has a source
of its own, the electric charge. Long ago, Dirac [16] tried to remove this asymmetry
and arrived at the hypothesis on the existence of a magnetic charge (a solitary
magnetic pole, or monopole). However, a magnetic monopole has so far not been
found. This negative result is also a result which can lead to an extreme idea that
a magnetic monopole does not exist at all. The as}Tnmetry in electrodynamics is
2164
thus a feature of principle: the electric and magnetic fields are not equal in rights,
the magnetic field is rather a by product of the electric field.
Let us now address to another branch of physics, nuclear physics. Here we
consider the situation with the neutron. The electrically neutral neutron has a magnetic
field. To explain this, one could also suggest that the magnetic field is here a
byproduct of the neutron's nuclear field. The neutron has a nuclear charge which is a
source of a nuclear field, and, in turn, rotating (the current of the nuclear charge!),
creates a magnetic field.
The celestial bodies show a similar situation They have a gravitational mass,
i.e., a gravitational charge. The latter creates a gravitational field. When a celestial
body has a rotation of its own (a mass current, or a current of gravitational charge)
then, as a by-product of gravity there emerges a magnetic field. This is what we
call the gravimagnetism hypothesis. Gravitation is also a source of magnetism.
Thus, summing up the situation in electrodynamics, nuclear physics and
gravitational physics, we can assert that the magnetic field is a by-product of all physical
fields having their own sources (the electric, nuclear and gravitational charges).
Now let us mention a certain discrepancy between the theoretical results and the
actual data on the magnetic fields of the Earth, the Sun, neutron stars and other
celestial bodies.
It has been found that this situation is explained by our considering the simplest
model of celestial bodies: we described them as rotating homogeneous fluid balls.
One should take into account the inhomogeneous distribution of matter inside all
the bodies.
Indeed, the seismic data indicate that the Earth's core occupies about one eighth
of its volume. The matter in it must be in a liquid state and possess large density
[17]. It is believed that the core may rotate with a velocity slightly different from
that of the Earth's crust.
A similar situation, i.e., inhomogeneity of density and rotation velocities, may
take place for the Sun and the neutron stars (pulsars).
References
1. M.M. Abdil'din, Mechanics of Einstein's Theory of Gravity. Alma-Ata, 1988, 198 pp.
2. L.D. Landau and E.M. Lifshits, Classical Field Theory. Moscow, 1973, 502 pp.
3. O.Yu. Schmidt, Four lectures on the theory of the Earth's origin. Selected works.
Geofizika I Kosmogoniya, USSR Acad. Sci. Publ., 1960, p. 102.
4. V.V. Beletsky, Essays on the Motion of Celestial Bodies. Nauka, , 1977.
5. N.G. Chetaev, Stability of Motion. Works on Analytical Mechanics. USSR Acad. Sci.
Publ., M, 1962.
6. V.V. Batygin and I.N. Toptygin, Electrodynamics Problem Book, M.,1970, 503 pp.
7. M.M. Abdil'din. On Interpretation of the Einstein Equations in General Relativity.
Gravitation & Cosmology, 5, 3(19), 219-221 (1999).
8. H. Alfven and G. Arrhenius. Evolution of the Solar System. Mir, M.,1979.
9. A.L Ossipov, Self-Organization and Chaos. Znanie, M., ser. Physics, 1986/7,
10. H.A. Willson. Prog. Roy. Soc. A, 104 (1923),
11. P.M. Blackett, Uspekhi Fiz. Nauk 38, 1 (1947).
2165
12. A. Einstein, Collected works, v. 2, Nauka, M., 1966.
13. V.I. Grigoryev and E.V. Grigoryeva, On gravitational relations of celestial bodies.
Vestn, Mosk. Univ., ser. 3, Phys. Astron., No. 3, page 75 (1996).
14. M.M. Abdil'din. On the interpretation of general relativity. Izv. AN Kaz. SSR, ser.
Fiz. Math., No. 4, 76 (1968).
15. M.M. Abdil'din. Gravimagnetism and the interpretation of Einstein's equations.
Gravitation, Cosmology and Relativistic Astrophysics, Kharkov National University, 2001.
16. P.A.M. Dirac, Proc. Roy. Soc, A 133, 60 (1931).
17. N.V. Pushkov, Magnetism in Space. Znanie, ser. IX: Fiz, Khim., M., 1961.
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Wormholes, Energy
Conditions and Time
Machines
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N-SPHERES: REGULAR BLACK HOLES, STATIC WORMHOLES
AND GRAVASTARS WITH A TUBE-LIKE CORE
O. B. ZASLAVSKII
Department of Mechanics and Mathematics, Kharkov V-N.Karazin National University, Svoboda
Square 4-, Kharkov 61077, Ukraine
E-mail: ozaslav@kharkov.ua
We consider a way to avoid black hole singularities by gluing a black hole exterior
to an interior with a tube-like geometry. The inner region is everywhere regular and
supported by matter with the vacuum-like equation of state. Such composite spacetimes
accumulate an infinitely large amount of matter inside the horizon but reveal themselves
for an external observer as a sphere of a finite ADM mass. In this way we obtain also
wormholes and gravastars.
The nature of inner structure of black holes and the problem of their singularity is
one of central issues in black hole physics. Different attempts were undertaken to
remove a singularity by making composite spacetimes that reveal themselves as a
black hole for an external observer but contain a regular inner region. In doing so,
the special role is played by the de Sitter (dS) metric which is supposed to mimic
vacuum-like media,1,23 The aforementioned approaches assume that the central
singularity is replaced by some regular interior in which this singularity is smoothed
out in the centre. In the present work we suggest a quite different way - to get rid off
the singularity in the centre by simply getting rid of the centre by itself. As far as
the spacetime structure of the inner region is concerned, the aforementioned options
(1) and (2) correspond to T-regions in the sense that (Vr) < 0 where r is the areal
radius. In case (3) the interior spacetime represents R region for which (Vr) > 0.
In this sense, our case occupies the intermediate position since (Vr) = 0 inside
just because of constancy of r. For brevity, we will call it N-region. Thus, the whole
spacetime consists of gluing one R and one N region.
Consider the static metric
ds2 = -dt2b2 + dP + r2(l){d92 + d4? sin2 6). (1)
Let the stress-energy tensor be represented in the form T^ = diag(—p, pr, p±,
p±). We choose the metric of interior to obey the Einstein equations with r =
r0 = const. Then it follows from 00 and 11 equations that p~ = —p~ = ^j and
22 equation gives us ^- = 87rp2, where (...)± = lim(...)r._+ro±o,signs "+" and "-"
correspond to the outer and inner regions, respectively. Thus, the interior should
be vacuum-like in the sense that p + pr = 0, and there are three different cases
depending on the sign of p±. If (1) p± > 0, then (a) b = asinh/d, where a is a
constant, k2 = 8np±, (b) b = aexp(/d) or (c) b = a cosh/d. If (2) p± < 0, by a
suitable linear transformation of I we can achieve b = a sin nl with k2 = —87rp±, if
(3) p± = 0, we have (a) b = al or (b) b = a. Particular examples of corresponding
physical sources are electromagnetic field (case 1 with p± = p - BR solution4),
cosmological constant (case 2 with p-± = -p - Nariai solution5), string dust (case
2169
2170
3).
If we glue the inner region to a black hole region outside, one can show that the
surface stresses on the boundary vanish in the horizon limit. The resulting composite
spacetimes reveal the essential gravitational mass defect. The ADM mass measured
in the outer region is finite since matter outside the shell is supposed to be bounded
within some compact region or the density p decreases rapidly enough. However,
the total proper mass mp = 4n f dip2 measured on the hypersurface T = const in
the tube under the shell at ro, obviously, diverges.
Up to now we discussed gluing between two regions only. One can proceed further
and glue in the same manner another Schwarzschild (or extremal black hole) region
from the left, but again with the shell in the R-region. In a similar way, one may glue
the tube and the outer wormhole region. Actually, we have some generalization of
notion of wormholes,67 - with a throat of an arbitrary length lying in the N-region
and connecting two R-regions. Inside the throat the equation of state is exactly
vacuum-like pr + p = 0, the proper mass bounded inside the throat can be made as
large as one wishes.
Thus, we constructed composite objects that interpolate between black holes and
gravastars in that there is no horizon in the particular solution obtained by gluing
different regions of spacetime but the horizon appears as a result of the limiting
procedure when the object turns into what we called a N-sphere. In doing so, we
obtained event horizons without apparent ones. Alternatively, we also obtained
a gravastar with an infinite tube as a core (N-gravastar). Generalization of the
procedure under consideration gave rise to objects interpolating between black holes
and wormholes (not traversable N-wormholes) or connecting two external regions
without horizons (traversable N-wormholes).8 In the case of the electromagnetic
field we return to the gluing between the Reissner-Nordstrom and Bertotti-Robinson
spacetimes considered in.9
After finishing this paper, I became aware of the recent work10 in which
minimally coupled scalar fields with negative kinetic energy (phantom fields) were
considered. It was shown that regular black hole solutions do exist for such a system,
including those with asymptotically constant areal radius r, and this does not need
any "surgery" for matching two regions at all. In our case, some "surgery" is needed
but it is mild in the sense that the corresponding surface stresses vanish in the limit
under discussion. We do not specify the nature of matter that supports our
configuration but only require it to obey the phantom equation of state for radial pressure.
I thank K.A. Bronnikov for drawing my attention to the aforementioned article. At
the conference itself, there has been the talk by N. V. Mitskievich, M. G. Medina
Guevara and H. Vargas Rodriguez "Nariai-Bertotti-Robinson spacetimes as a
building material for one-way wormholes with horizons, but without singularity" on the
closed subject.
I thank Org. Committee and especially H. Kleinert for the excellent conference
and support that made it possible for me to attend it.
2171
References
1. E. B. Gliner, Sov. Phys. JETP 22, 378 (1966).
2. I. Dymnikova, Gen. Rel. Grav. 24, 235 (1992); Phys. Lett. B 472, 33 (2000);
Int. J.Mod.Phys. D 12, 1015 (2003).
3. P. O. Mazur and E. Mottola. "Gravitational condensate stars: An alternative to black
holes", gr-qc/0109035; "Dark Energy and Condensate Stars: Casimir Energy in the
Large", gr-qc/0405111; Proc. Nat. Acad. Sci. Ill 9545 (2004).
4. T. Levi-Civita, Rend. Atti Acad. Naz. Lincei, 2 (1917) 529; I. Robinson, Bull. Acad.
Pol. Sci. 7 (1959) 351; B. Bertotti, Phys. Rev. 116 (1959) 1331.
5. H. Nariai, Sci. Rep. Tohoku Univ., Ser. 1 34, 160 (1950); 35, 62 (1951).
6. M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395 (1988).
7. M. S. Visser, Lorentzian wormholes: From Einstein to Hawking (AIP Press, New York,
1995).
8. O. B. Zaslavskii, Phys.Lett. B634, 111 (2006).
9. O. B. Zaslavskii, Phys. Rev. D70, 104017 (2004).
10. K.A. Bronnikov and J.C. Fabris, Phys.Rev.Lett. 96, 251101 (2006).
AVERAGED ENERGY INEQUALITIES FOR NON-MINIMALLY
COUPLED CLASSICAL SCALAR FIELDS
LUTZ W. OSTERBRINK
Department of Mathematics,
University of York,
Heslington,
York YO10 5DD,
United Kingdom, lwo500@york.ac.uk
The stress-energy tensor for the non-minimally coupled scalar field is known not to
satisfy the pointwise energy conditions, even on the classical level. We show, however,
that local averages of the classical stress-energy tensor satisfy certain inequalities and
give bounds for averages along causal geodesies. It is shown that in vacuum background
spacetimes, ANEC and AWEC are satisfied. Furthermore we use our result to show
that in the classical situation we have an analogue to the so called quantum interest
conjecture. These results lay the foundations for averaged energy inequalities for the
quantised non-minimally coupled fields.
1. Introduction
It is generally believed that the energy density should be positive for all physically
reasonable classical matter. However, it is well known that this is not true for
quantised fields. Wightman fields, for example, do not satisfy pointwise positivity
of the renormalised energy density,2 which resulted in a lot of research on this
peculiarity. In particular the work of L.H. Ford3 was seminal and resulted in what
is usually referred to as the quantum inequalities^. They state that, even though
the energy density (for instance) can be made arbitrarily negative at a point by
varying the quantum states, the weighted time-like average is bounded from below.
This bound is in particular state-independent.
Additionally to the violation on the quantum level, it is well known that the
pointwise energy conditions can even be violated on the classical level. One of the
theories allowing such violations is the classical scalar field, non-minimally coupled
to the Ricci-scalar of the spacetime manifold. Such a coupling changes the form of
the energy density, even in the limit of a flat spacetime, such that the pointwise
energy conditions can be violated. This can actually be so severe that it is possible
to find wormhole spacetimes,5'6 supported by the non-minimally coupled scalar
field. On the other hand, there are various reasons to believe that such effects
should be limited by certain bounds to the energy density, at least its weighted
averages. One of those reasons, and probably the most obvious, is that there must
be restrictions such that the second law of thermodynamics is not violated, at least
on a macroscopicb scale. In particular, this means that there must be limitations
(of some kind) to the duration and amplitude of the negative energy density. These
aFor a good overview see, e.g., the work by C.J. Fewster and references therein.
hThe parameter denning macroscopic in the classical field theory is the maximal field amplitude.
2172
2173
should then rule out any possibility to use negative energy density to cool down a
hot body without (macroscopically) changing its entropy.3
Below, we give an overview of the work done so far, to find such restrictions for
the classical scalar field with non-minimal coupling, based on the results obtained
by the author together with C.J. Fewster.7,8
2. Bounds for the Classical Non-Minimally Coupled Scalar Field
The stress-energy tensor for the non-minimally coupled scalar field can be derived
from its Lagrangian, L = | {(V</>)2 — (m2 + £i?)</>2}, by variation of the action with
respect to the co-metric g^v. A straightforward calculation yields the expressionc
V = (VM<£) (Vv<t>) + \g^v {m24? - (V</>)2) + £ {g^ag - V^V, - GM„} 4>2, (1)
where G^ is the Einstein tensor and Og is the d'Alembertian with respect to the
metric g. Furthermore, the equation of motion is (□ g + rn2 + £R)<fi = 0. Even though
the Lagrangean and the equation of motion in flat spacetime reduce to the one for
minimal coupling, i.e., for £ = 0, the stress-energy tensor (1) does not. This feature
makes it possible to have negative energy density for the non-minimally coupled
scalar field, even in flat spacetimes. A simple example is given by L.H. Ford and
T.A. Roman in [9].
The averaged stress-energy tensor, however, obeys the following result:7
Theorem 2.1. Let j be a causal geodesic with affine parameter X in a spacetime
(M,g). Furthermore, let T^ be the stress-energy tensor of the non-minimally
coupled classical scalar field with coupling constant £ € [0,1/4]. For every real-valued
function f e Cq(R) the inequality
J dx T^rrf > -2£ J d\ |(aA/)2 + hi^YYf - {\ - i)Ri2f] 4?
is satisfied on-shell.
Here, "on-shell" means, that the field is required to satisfy the field equation, as
given above. This result can be used in various ways to analyse averaged energy
densities and can be generalised to spacetime-volume averages.7 Interesting results
for Ricci-flat spacetimes can be derived by scaling arguments. Without going into
too much detail, we can summarise the results by: Long-lasting negative energy
densities of large magnitude must be associated with large magnitudes of the field
or with large curvatures. As a consequence, one finds conditions that ensure ANEC
and AWEC.
A further interesting aspect of our work concerns energy interest. Originally
analysed in quantum field theory, this phenomenon was first described by Ford and
Roman.10 It states that negative energy density is always associated with positive
cSee [7] for conventions.
2174
energy density, which actually overcompensates the former one, ensuring an overall
positive energy density. This overcompensation can then be understood
metaphorically as the repayment with interest of a negative energy density debt. The same
phenomenon can be found for the classical non-minimally coupled scalar field.7 In
detail, one finds that the maximal time-separation of such pulses is proportional
to the coupling constant, the maximal field amplitude and furthermore inversely
proportional to the magnitude of the negative energy density.
Since the non-minimally coupled scalar field allows these strange phenomena
already on the classical level, it is very important to study them for the quantised
field as well. To get a lower bound for the latter situation one has to mix two
different methods. One of these is analogous to the classical manipulation described
above and the other is in line with the methods used by Fewster and Eveson11 to
derive a class of quantum inequalities. As expected, their result is recovered in the
case of minimal coupling. The more general result that we found8 is a lower bound
for the time-like averaged energy density p/ with coupling constants £ S [0,1/4]. It
is given by
P/>-(l-4O0S^(/)l-2£S(/), (2)
in terms of quadratic forms. The non-linear functional Q.f~E (/) is the one that
was obtained as the state independent lower bound for the minimal coupling, as
remarked above. The additional term Q5(/) is a non-negative quadratic form, whose
expectation values are state-dependent. Even though one can show that the right
hand side in (2) is unbounded from below, there is a sense in which the bound is
nontrivial, in that 2$(/) is of "lower order" than the energy density. Our hope is
that by understanding this case, we will be better placed to understand quantum
energy inequalities for general interacting quantum fields.
References
1. See for example: S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-
Time (Cambridge University Press, 1973).
2. H. Epstein, V. Glaser and A. Jaffe, Nuovo Cim. 36, 1016 (1965).
3. L.H. Ford, Proc. R. Soc. Lond. A364, 227 (1978).
4. C.J. Fewster, 'Energy Inequalities in Quantum Field Theory', in XlVth
International Congress on Mathematical Physics, (World Scientific, Singapore, 2005). See
math-ph/0501073 for an expanded and updated version.
5. C. Barcelo and M. Visser, Phys. Lett. B466, 127 (1999).
6. C. Barcelo and M. Visser, Class. Quant. Grav. 17, 3843 (2000).
7. C.J. Fewster and L.W. Osterbrink, Phys. Rev. D74, 044021 (2006).
8. C.J. Fewster and L.W. Osterbrink, (in preparation).
9. L.H. Ford and T.A. Roman, Phys. Rev. D64, 024023 (2001).
10. L.H. Ford and T.A. Roman, Phys. Rev. D60, 104018 (1999).
11. C.J. Fewster and S.P. Eveson, Phys. Rev. D58, 084010 (1998).
SELF SUSTAINED TRAVERSABLE WORMHOLES AND THE
EQUATION OF STATE
REMO GARATTINI
Universita degli Studi di Bergamo, Facoltd di Ingegneria,
Viale Marconi 5, 24044 Dalmine (Bergamo) ITALY.
INFN - sezione di Milano, Via Celoria 16, Milan, Italy
revao.garattini@unibg.it
We compute the graviton one loop contribution to a classical energy in a traversable
wormhole background. The form of the shape function considered is obtained by the
equation of state p = cop. We investigate the size of the wormhole as a function of the
parameter u>.
The discovery that our universe is undergoing an accelerated expansion1 leads to
reexamine the Friedmann-Robertson-Walker equation to explain why the scale
factor obeys a > 0. One way to explain the sign of the acceleration can be done by
introducing an equation of state p = up causing a negative pressure. A value of
u < —1/3 is required for the accelerated expansion, while u = — 1 corresponds to
a cosmological constant. A specific form of dark energy, denoted phantom energy
has also been proposed with the property of having u < — 1. It is interesting to
note that the phantom energy violates the null energy condition, p + p < 0,
necessary ingredient to sustain the traversability of wormholes. A wormhole can be
represented by two asymptotically flat regions joined by a bridge: one example is
represented by the Schwarzschild solution. One of the prerogatives of a wormhole is
its ability to connect two distant points in space-time. In this amazing perspective,
it is immediate to recognize the possibility of traveling crossing wormholes as a
short-cut in space and time. Unfortunately, although there is no direct evidence, a
Schwarzschild wormhole does not possess this property. It is for this reason that in a
pioneering work Morris and Thorne and subsequently Morris, Thorne and Yurtsever
studied a class of wormholes termed "traversable". Unfortunately, the
traversability is accompanied by unavoidable violations of null energy conditions, namely, the
matter threading the wormhole's throat has to be "exotic". Lobo,3 Kuhfittig4 and
Sushkov5 have considered the possibility of sustaining the wormhole traversability
with the help of phantom energy. On the other hand, we explored the possibility
that a wormhole can be sustained by its own quantum fluctuations.6 In practice, it
is the graviton propagating on the wormhole background that plays the role of the
"exotic" matter. This has not to appear as a surprise, because the computation
involved, namely the one loop contribution of the graviton to the total energy, is quite
similar to compute the Casimir energy on a fixed background. It is known that, for
different physical systems, Casimir energy is negative and this is exactly one of the
features that the exotic matter should possess. In particular, we conjectured that
quantum fluctuations can support the traversability as effective source of the semi-
classical Einstein's equations. The classical Einstein equations G^u = kT^v, where
T^v is the stress-energy tensor, G ^ is the Einstein tensor and k = 8irG can be
rearranged to give interesting results in the semi-classical context. By introducing
2175
2176
a time-like unit vector u^ such that u ■ u = — 1. we can write
G^ (gaP) u»u» = k (T^u»unren = - <AGM„ (ga/3, haP) u»uv)ren , (1)
where, in principle AGM„ {gap,hap) is a perturbation series in terms of /iM„ with
g = g^v + /i The chosen background will be that of a traversable wormhole.
In Schwarzschild-like coordinates, the traversable wormhole metric can be cast into
the form2
dr2
ds2 = - exp (-2</> (r)) dt2 + ^1 + r2 [d62 + sin2 Odtp2] . (2)
r
where </> (r) is called the redshift function, while b (r) is called the shape function.
If we impose that </>' = 0, we can get a relevant expression for the shape function
6 (V) = rt ( —) " , where the equation of state has been used together with the
following Einstein equation 8irGp (r) r2 = b' (r). From this point of view, the equation
governing quantum fluctuations behaves as a backreaction equation and to one loop
we get for the graviton
"Mi-*" ^>-n£oFri&- "-*-1- <3)
We refer to Ref.6 for details. From Eq.(3), we recognize that the dark as well as
phantom regime is unavailable. Concerning the one loop total energy, we get the
expression
+ oo 2
ETT {rt, e; /i) = 4tt <J 2 / dr . [(Pl (e) + p2 (e))] } , (4)
where the factor 47T comes from the angular integration, while the factor 2 in front
of the integral appears because we have come back to the original radial coordinate
r: this means that we have to double the computation because of the upper and
lower universe. p% (e) represents the regularized energy density and the renormalized
the self consistent equation becomes
A (to) 1
Go (/i0) 167T
r2 + r2 [n{ ^ )
(5)
where we have used a renormalization group-like equation to eliminate the
dependence on the arbitrary scale p, and where the coefficients a (to) and 6 (u>) come from
the integration over the r coordinate. In order to have only one solution, we find
the extremum of the r.h.s. of Eq.(5). Note that in the paper of Khusnutdinov and
Sushkov,7 to find only one solution, the minimum of the ground state of the
quantized scalar field has been set equal to the classical energy. In our case, we have no
external fields on a given background. This means that it is not possible to find a
minimum of the one loop gravitons, in analogy with Ref.7 Moreover the
renormalization procedure in Ref.7 is completely independent by the classical term, while in our
2177
case it is not. Indeed, thanks to the self-consistent equation (3), we can renormalize
the divergent term. Results are summarized into the following plot, where we have
made the following choice Go (no) = l„- It is visible the presence of a minimum for
1.35—1
1.3-
1.25—
1.2-
1.15—
1.1-
r(co)
I I I I I I I I I I I I I I I I I I I I
5 10 15 20
Fig. 1. Plot of the wormhole throat rt as
Go (/"<))•
function of uj in the positive range with a fixed
u> = 3.35204, where ft (u>) = 1.11891. As we can see, the radius is divergent when
uj —► 0. At this stage, we cannot establish if this is a physical result or a failure of
the scheme. When u> —> +oo, ft approaches the value 1.15624ZP, while for to = 1,
we obtained ft = 1.15882Zp. It is interesting to note that when u) —► +oo, the shape
function b (r) approaches the Schwarzschild value, when we identify ft with IMG.
In this sense, it seems that also the Schwarzschild wormhole is traversable.
References
Riess A G et al. 1998 Astron. J. 116 1009
Morris M S and Thome K S 1988 Am. J. Phys. 56 395. Morris M S, Thome K S and
Yurtsever U 1988 Phys. Rev. Lett. 61 1446. M. Visser, Lorentzian Wormholes (AIP
Press, New York, 1995) 64.
Lobo F S N 2005 Phys. Rev. D 71 124022 (Preprint gr-qc/0506001). Lobo F S N 2005
Phys. Rev. D 71 084011 (Preprint gr-qc/0502099)
Kuhfittig P K F 2006 Class. Quant. Grav. 23 5853 (Preprint gr-qc/0608055)
Sushkov S 2005 Phys. Rev. D 71 043520 (Preprint gr-qc/0502084)
Garattini R 2005 Class.Quant.Grav. 22 1105 (Preprint gr-qc/0501105)
Khusnutdinov N R and Sushkov S V 2002 Phys. Rev. D 65 084028 (Preprint hep-
th/0202068)
CLASSICAL AND QUANTUM WORMHOLES IN A COSMOLOGY
WITH DECAYING DARK ENERGY
FARHAD DARABI
Department of Physics, Azarbaijan University of Tarbiat Moallem, Tabriz, 53714-161 Iran
f. darabi@azaruniv. edu
We study the classical and quantum wormholes for a FRW universe filled with an
ordinary matter density plus a term playing the role of dark energy density.
1. Introduction
Wormholes are usually considered as Euclidean metrics that consist of two regions
connected by a narrow throat. They have been mainly studied as instantons, namely
solutions of the classical Euclidean field equations.1 In general, Euclidean worm-
holes can represent quantum tunneling between different topologies. Most known
solutions of general relativity which allow for wormholes require the existence of
exotic matter , a theoretical substance which has negative energy density. However , it
has not been proven mathematically this is an absolute requirement for wormholes.
It is well - known that wormhole like solutions occur only for certain special
kinds of matter that allow the Ricci tensor to have negative eigenvalues. Non -
existence of instantons for general matter sources , motivated Hawking and page to
advocate a different approach. They regarded wormholes typically as the solutions
of quantum mechanical Wheeler-DeWitt equation.2 These wave functions have to
obey certain boundary conditions in order that they represent wormholes. The main
boundary conditions are :
1) the wave function is exponentially damped for large tree geometries,
2) the wave function is regular in some suitable way when the tree-geometry
collapses to zero.
An open and interesting problem is whether classical and quantum wormholes
can occur for fairly general matter sources. Classical and quantum wormholes with
standard perfect fluids and scalar fields have already been studied.3 The study
of A-decaying cosmology in this framework has not received serious attention. In
the present work , we shall consider such a cosmology and study its classical and
quantum wormhole solutions.
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2. Classical wormholes
We consider a (FRW) universe filled with perfect fluid
" dr2
ds2 = -dt2 + a2(t)
Einstein equations then reads
1 — kr2
+ rz{d6z +sinz 6d<j)z)
a a
a, az
-P-
There is also a conservation equation
p + 3-(p + p) = 0.
By analytic continuation, t —> it we obtain
a2 a?
3'
In FRW models, wormholes are described by a constraint equation of the form
1
const
(1)
(2)
(3)
(4)
(5)
(6)
An asymptotically Euclidean wormhole requires a2 > 0 for large a , So n > 2.
We assume the total density as
A0
a
Substitution for p and k = 0 leads to
1 /A,
Po
a37
(7)
(8)
By defining R ■
f~a We obtain
An
R2
_ _L ctp
l¥~ R2~ R3-r
aQ
Po
- Ao
)37/2
(9)
This equation has the form of the constraint describing an Euclidean wormhole with
the correspondence 37 = n. Therefore, classical Euclidean wormholes are possible
for the combined source with any 7 > |. By substitution for p in the conservation
equation we obtain the equation of state
P = pm+Pv = Pm{l- 1) - -Pv,
(10)
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3. Quantum Wormholes
Quantum mechanical version of the classical equation for R is given by4
We set q = 0, and study the occurrence of Euclidean domain at large R by
considering the sign of the potential
'2
lm+U^
*(i?) = 0, U(R) = a0R4-3^ - R2, (12)
For U(R) > 0, oscillating solutions occur which represent Lotentzian metrics. For
U(R) < 0, wormhole solutions can occur. For 7 > § and po > 0 the potential is
negative. So, quantum wormholes can occur. But asymptotically Euclidean property
of the wave function is not sufficient to make it a wormhole. It also requires regularity
for small R. We ignore R4 term as R -> 0 when 7 > 2/3. The Wheeler-DeWitt
equation (for 7 7^ 0 ) simplifies to a Bessel differential equation with solution
*(i?) - R^-i)'2
*j'G&M*")+«i''G&M'/"
(13)
Where v = (1 — q)/3(2 — 7). The wormhole boundary condition at small R is satisfied
for Bessel function of the J kind. In the case of 7 = 4/3 which represents radiation
( or a conformally coupled scalar field ) dominated FRW universe, WheelerDeWitt
equation for q = 0 is written as
(J^ + «o-^2)*(^) = 0, (14)
which is a parabolic equation with solution in terms of confluent hypergeometric
functions5
*(JR)~exp(-JR72)[c3.iF1(l/4(l-ao); 1/2; i?2)+04.1^(1/4(3-a0);3/2;jR2)].
(15)
For cto > 1 and C4 = 0 we obtain a regular oscillation at R —> 0, and a Euclidean
regime for large R. Therefore, we have a quantum wormhole.
References
1. S. W. Hawking, Phys. Rev. D. 37, 904 (1988).
2. S. Hawking, D. N. Page, Phys. Rev. D. 42, 2655 (1990).
3. A. Carlini, D. H. Coule, and D. M. Solomons, Mod. Phys. Lett. A. 11, 1453 (1996).
4. J. J. Halliwell, in Quantum Cosmology and Baby Universes (World Scientific, 1991).
5. M. Abramowitz and I. A. Stegun, Handbook of Mathematical functions (Dover, 1965).
NARIAI-BERTOTTI-ROBINSON SPACETIMES AS A BUILDING
MATERIAL FOR ONE-WAY WORMHOLES WITH HORIZONS,
BUT WITHOUT SINGULARITY
NIKOLAI V. MITSKIEVICH
Department of Physics, CUCEI, Universidad de Guadalajara
Guadalajara, Jalisco, Mexico,
Apartado Postal 1-2011, C.P. 44100, Guadalajara, Jalisco, Mexico
mitskievich03@yahoo.com.mx
MARIA GUADALUPE MEDINA GUEVARA and HECTOR VARGAS RODRIGUEZ
C. U.Lagos de la UdeG, Enrique Diaz de Leon S/N,
Lagos de Moreno, Jal., C.P. 47460, Mexico
hv-8 ©yahoo, com
We discuss the problem of wormholes from the viewpoint of gluing together two Reissner-
Nordstrom-type universes while putting between them a segment of the Nariai-type world
(in both cases there are also present electromagnetic fields as well as the cosmological
constant). Such a toy wormhole represents an example of one-way topological
communication free from causal paradoxes, though involving a travel to next spacetime sheet
since one has to cross at least a pair of horizons through which the spacetimes' junction
occurs. We also consider the use of thin shells in these constructions. Such a "material"
for wormholes we choose taking into account specific properties of the Nariai—Bertotti—
Robinson spacetimes.
In general relativity, the problem of wormholes is not more exotic than that of black
holes. In this talk we consider a simple toy model which is still far from perfection
which could however be useful in better comprehension of the magnitude of the
wormhole problem.
The Nariai-Bertotti-Robinson (NBR) solution2'7-10 (about the result of
Robinson9 see however Ref. 3) can be described as ds2 — e2a(-r^dt2 - e~2a(rW2 -
\2{d'd2 + sin2 tfdcp2) where e2a = (k2 - A)r2 + Br + C and A = ^=p, B
and C being arbitrary constants, A the cosmological constant, and k, the
(constant) electromagnetic field intensity. The electromagnetic sources in Einstein's
equations correspond to the four-potential A = J^ ( akr dt + (fc2fc+A) cos$d(pY.
Tem = £. (0(O) ® e(0) _ Q(i) ® 0(1) + g(2) ® 0(2) + 0(3) ® 0(3)) with a = sin^; b =
cosip, ip being an arbitrary constant, while (see a general discussion in Ref.
4) E = *(0<°) A *F) = Ao,r0{1\ B = *(0<°) A F) = ^-A3,^(1) where
0(°) = eadt, 6>« = e~adr, 0<2) = \dd, 6^ = Xsmddp.
We consider pieces of NBR solutions with two horizons (null compact hyper-
surfaces along whose generatrices ds2 = 0, while |^| —> oo on the horizons).
When k2 > A > — k2, the two horizons are at r = ±r0 = ±\j\Jk? — A with a
non-stationary band between them and static regions outside. Alternatively, when
k2 < A, the horizons are at r = ±ro = ±l/-\/A — k2 and spacetime is static between
them and non-stationary outside. We now write these solutions in synchronous co-
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ordinates (see the definition in the footnote on p. 62 of Ref. 4):
ds2 = dT2 - [£2 - goo(r(T, R))]dR2 - 1 [dti2 + sin2 tidy2}. (1)
Here £ is energy per unit mass of the geodesically moving test particle identified
with the observer. On horizons where goo = 0, no singularities and degeneracy
appear in the metric coefficients. (This makes it unnecessary to apply the intrinsic
prescription in the Barrabes and Israel formalism. At the horizon there is then used
a thin null shell.1'5'6) This description also gives a unique junction of spacetimes
and enables the standard causal treatment of an infinite sequence of universes in
the Penrose diagram. In the case k2 > A > — k2. goo(r) = (k2 — A)r2 — 1; in the
case k2 < A, goo(r) = 1 - (A - k2)r2.
As the outside worlds we consider the Reissner-Nordstrom-Kottler (RNK)
solutions10 (those of Reissner-Nordstrom, but with the cosmological term),
ds2 = dT2 - [£2 - g00(r)] dR2 - r2 (d$2 + sin2 tidy2) (2)
with g00 = 1 - *2i + el - lAlr2 and g00 = 1 - ^ + f| - lA2r2, r = r(T, R).
They are to be joined via wormholes which belong to the NBR spacetimes, (1),
with g00(r) = (f^f) [(A + k2) r2 - l] (there is also r(T,R), but with another
dependence than in RNK), when the cases A > k2 and k2 > A > — k2 are unified
via a scales change in r, so that the horizons correspond to r = ±A = ± /A\k2
(the minus sign does not spoil our considerations since it can be inverted when we
consider the junction of the NBR-wormhole with the 'second' RNK world at this
horizon). At the horizons in synchronous coordinates we put in (2) goo = 0 and
substitute instead of r, f = f(mi, ei, Ai) = f(m2,e2, A2), corresponding to anyone
of the (three) horizons of RNK, while in NBR (1) the only change at the horizon is
to put goo = 0. Hence we conclude that
[«1=0 - 7tki=f- ,3)
The electromagnetic stress-energy tensors read T"nbr = ^r {dT ® dT — £2dR ® dR
+ P^A (dd ® dd + sin2 dd<p ® d<p)} and TRNK = ~^ {dT <g> dT - £2dR <g> dR
+f2 (d3 ®di!) + sin2 "ddip ® dip)}, thus
[V]=0 => k2=Gj£. (4)
-2_ 2 2
Taking into account (3) and (4), we see that A = ——^ and e2 2 = e2 = (fc2+A-,4,
thus the charges observed from opposite entrances of the wormhole coincide up to
the sign.
The interior of such wormholes is a non-stationary region if k2 > A > — k2 or
a static region if A > k2. These types of wormholes are observed in one universe
as black holes, in another universe (or on another spacetime sheet of the former
universe) as white holes, though there is no singularity which should correspond
2183
to usual black holes, since they belong here to NBR. Observers in these two
adjacent RNK universes would conclude that the wormhole has an electric charge with
the same absolute value, but opposite signs in different universes (or the similar
situation with the magnetic charge); they would also measure a non-zero positive
mass of the wormhole, but this mass in general will be different for observers in
different universes (together with the different values of the cosmological constant
corresponding to the respective worlds).
It is comparatively easy to construct examples of Penrose diagrams'
hybridization resulting in a connection of two RNK worlds via a NBR-wormhole. They show
that the wormholes under consideration are traversable only in one direction
(oneway wormholes) taking the traveller to another sheet of spacetime (behind the
future infinity of the abandoned world); this is also visualized by diagrams using
synchronous coordinates. Of course, the Penrose diagrams' hybridization cannot be
simply shown on one piece of paper since the RNK singularities and adjacent
sectors require more space than there is at one's disposal on one sheet so that one has
to identify some boundaries of these sectors without mixing them with those
pertaining to the NBR-wormhole. Therefore we do not show such hybridized diagrams
here.
Naturally, the junction of RNK worlds via static part of NBR-wormhole can be
also done not on horizons, but in the outside parts of RNK worlds and of NBR
spacetime, thus permitting to consider construction of two-way wormholes; in this
case it is natural to glue together only static regions of both space/times. This
requires the use of more complicated prescriptions for junction, and we do not
come in these details leaving them to another publication. The NBR solution is
chosen in this talk as a convenient tool to construct wormholes since it already has
the necessary properties for modelling them due to the angular part of the NBR
metric.
References
1. C. Barrabes and W. Israel, Phys. Rev. D43, 1129 (1991).
2. B. Bertotti, Phys. Rev. 116, 1331 (1959).
3. A. Krasinski, Gen. Rel. Grav. 31, 945 (1999).
4. N.V. Mitskievich, Relativistic Physics in Arbitrary Reference Frames (Nova Science
Publishers, 2006). See also the early book preprint gr-qc/9606051.
5. P. Musgrave and K. Lake, Class. Quantum Grav. 13, 1885 (1996).
6. P. Musgrave and K. Lake, Class. Quantum Grav. 14, 1285 (1997).
7. H. Nariai, Sci. Rep. Tohoku Univ., Ser. 7 34, 160 (1950); more available in Gen. Rel.
Grav. 31, 951 (1999).
8. H. Nariai, Sci. Rep. Tohoku Univ., Ser. 7 35, 46 (1951); more available in Gen. Rel.
Grav. 31, 963 (1999).
9. I. Robinson, Bull. Acad. Polon. Sci., Ser. Mat. Fis. Astr. 7, 351 (1959).
10. H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, Exact Solutions
of Einstein's Field Equations, Second Edition. (Cambridge University Press, 2003).
COSMIC TIME MACHINES AND GAMMA RAY BURSTS
FERNANDO DE FELICE
Dipartimento di Fisica Universita di Padova, 1-35131 Padova, Italy and
I.N.F.N. Istituto Nazionale di Fisica Nucleare, Universita di Padova, 1-35131 Padova, Italy
defelice@pd. infn. it
If a curvature singularity is globally naked then the space-time may be causally future ill-
behaved admitting closed time-like or null curves which extend to asymptotic distances
and generate a Cosmic Time Machine (de Felice (1995) Lecture Notes in Physics 455,
99). I conjecture that Cosmic Time Machines give rise to high energy impulsive events
like the Gamma Ray Bursts.
1. Introduction
If a naked singularity existed it would be legitimate to invoke the validity of a
theorem due to Clarke and de Felice (1984) which states that a generic strong-
curvature naked singularity would give rise to a Cosmic Time Machine (CTM). A
Cosmic Time Machine is a space-time which is asymptotically flat and admits closed
non-spacelike curves which extend to future infinity. Here I shall conjecture that a
Cosmic Time Machine may be source of fast varying and highly energetic events
like Gamma Ray Bursts (de Felice, 2004).
2. A cosmic burst
The connection between a naked singularity and a Cosmic Time Machine has been
established in general by Clarke and de Felice (1984) with a theorem (theorem II
of that paper). The main result of that theorem states: if there is a naked
singularity which satisfies Newman's strong curvature condition (Newman, 1983) and
exists arbitrarily far into the future of a set of initial regular data, then violation of
strong causality occurs arbitrarily close to future null infinity. Thus a Cosmic Time
Machine is naturally implied.
Nearby a naked singularity then light cones permit non space-like trajectories to
run backwards with respect to the coordinate time causing the local causal future
to overlap with what would have been the causal past in a flat space-time. Let a
coordinate time t be chosen so to coincide with the proper-time of an observer at
a positive infinity. Consider two events in the domain of time inversion, being one
to the (causal) future of the other (two subsequent flashes from the same light gun,
say); then there exist light rays from these events which propagate backwards with
respect to the local time coordinate untill they leave the time inversion domain and
escape to positive null infinity. If we allow for the existence of photon orbits which
spatially loop around the singularity before leaving the time inversion domain, it
may well happen that these light rays leave that domain at about the same value of
the t coordinate and therefore reach infinity at about the same value oft as well. But
at flat infinity, the t coordinate is also the proper-time of a stationary observer hence
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the latter would see the two events almost simultaneously on her (his) clock. If we
extrapolate this example to all the events which are to the future of any given one
in a CTM domain, we infer that in a Cosmic Time Machine the entire causal future
development within the time inversion domain may be seen by a distant observer
at the same time. Evidently this property makes a CTM potentially a source of an
arbitrary strong burst.
Impulsive cosmic events combine two main puzzling features, namely an
extremely short time of emission (order of a second) and a very high energy fluence.
The main challenge therefore is to find a unique mechanism which allows at once
for both properties. The most impressive examples of the above type of events are
the Gamma Ray Bursts (Kluzniak and Ruderman, 1998; van Putten, 2001; Piran,
2004 and references there in). The total energy emitted can be as high as 1054 ergs,
mostly concentrated in a pulse as short as a second. This amount of energy appears
much more stunning if we think to it as being the energy emitted in a second-long
pulse by 1010 galaxies each made of 1011 Sun-like stars, each emitting at a rate of
~ 1033 ergs/sec, concentrated in a region probably smaller than a galactic core!
Here I envisage a scenario based on the hypothesis that what we believe to be a
black hole is on the contrary a generic strong curvature naked singularity sitting
inside a time inversion domain. Since Cosmic Time Machines involve astronomical
objects, they allow one to make predictions which could in principle be confronted
here-and-now with observations. In the time inversion domain the coordinate time
decreases so when it reaches the value when the singularity first formed the
conditions for a time trap did not yet develop therefore all the photons could only
propagate to the coordinate future again (coordinate time t increasing) leaving the
region nearby the singularity just formed and leading to a burst of radiation as seen
at far distance.
We can plausibly think of a situation where an accretion disk sits around a
(spinning) naked singularity. Let a substantial part of the emitted radiation enter
the time inversion domain and be funneled, at least part of it, into spatially quasi-
circular orbits along which light cones allow for local time reversed time-like or null
trajectories. Furthermore let accretion cause an energy output of about 1040ergs /sec
corresponding to a moderate quasar-like object shining for some 109 years (~ 1016
seconds) untill the naked singularity decays close to a black hole state becoming
invisible to distant observers.
If a thiny fraction of the emitted radiation, ( = 1% say, propagates to the local
future along the time-reversed orbits it will likely reach the condition when all the
radiation leave it at the same value of the t coordinate as result of the local time
inversion. Then an observer at infinity would see the integrated energy of 1054ergs
almost at the same time.
Evidently the survival of the above conjecture about the nature of impulsive
sources depends on the possibility to be falsified by more definite observational
constraints; this however is a challenge for the future.
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References
1. Clarke C.J.S. and de Felice F. 1984 Gen. Rel. & Grav. 16 139
2. de Felice F. 2004 Cosmic Time Machines and Gamma Ray Bursts
http://www.mate.polimi.it/bh/
3. Newman R.P.A.C. 1983 Gen. Rel. Grav. 15 641
4. Kluzniak W and Ruderman M 1998 Astrophys. J 505 LI 13
5. Piran T. 2004 Rev. Mod. Phys. 76, 1143
6. van Putten M.H.P.M. 2001 Physics Reports 345, 1
STATIC AND DYNAMIC TRAVERSABLE WORMHOLES
JAROSLAW P. ADAMIAK
Department of Mathematical Sciences,
University of South Africa,
P. O. Box 392, Unisa 0003
jaroslaw@telkomsa- net
The aim of this work is to discuss the effects found in static and dynamic wormholes that
occur as a solution of Einstein equations in general relativity. The ground is prepared
by presentation of faster than light effects, then the focus is narrowed to Morris-Thorne
framework for a static spherically symmetric wormhole. Two types of dynamic worm-
holes, evolving and rotating, are considered.
Keywords: Traversable wormholes.
The immenseness of the interstellar void implies that even if we could accelerate
the starship to almost light speed, the exploration of nearby stars, distanced from
us by a few light years, would take a few human lifetimes as seen from Earth. The
exploration of the Milky Way which includes over 200 billion stars and is about
100,000 light years across, would involve almost-geological time scales. The nearest
large galaxy to our own, Andromeda, is estimated to be 2 million light years away.
Although the starship crew would be able to survive the trip because of the slowing
down of clocks aboard the starship after the return they might find nobody to report
to back on Earth. Definitely the traditional space travel technology will not allow
us for efficient space exploration so if we want to conquer space we have to look for
more sophisticated than currently existing travelling means. There are chances that
this can be achieved by utilization of gravitational physics, in particular Einstein
theories.
Curved spacetime gives rise to effects that may result in faster than light (FTL)
travel not contradicting special relativity limitations. One of such solutions is a
wormhole - a hypothetical shortcut for travel between points in the universe, or
even between two different universes. It has two entrances/exits called "mouths"
that are connected to each other by a tunnel called the "throat". The throat may
be very short, but the wormhole traveller may be able to cover very large distances
from the point of view of the outside observer.
The usual method of solving the Einstein equations would be to assume an
existence of matter for the source of the stress-energy tensor. Then the equations of
state would be derived for the tension and pressure as a function of the energy
density. These together with the field equations would provide the geometry of the
spacetime described by the metric. Morris and Thorne approach1 differs
substantially from this procedure. Firstly they provided a list of properties the traversable
wormhole should have. Secondly a diagonal stress-energy tensor was assumed and
by use of Einstein field equations its components were found.
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We set the static and spherically symmetric metric as
ds2 = _e2*(r)df2 + dJ^_ + r2(^2 + ^2 ^2) (1)
1 — o(r)/r
Here b(r) determines the spatial shape of the wormhole so we shall call it the
"shape function" and $(r) determines the gravitational redshift so we shall call it
the "redshift function". In order to avoid the horizon we set a condition on the
redshift function as $(r) < oo. The radial coordinate r covers the range [ro,+00)
where ro defines the wormhole's throat radius.
We solve the Einstein equations
Gab = Rab ~ ~^9abR = 8lrTab (2)
with the stress-energy tensor being in the form
Tab = diag(p(r), -T(r),p(r),p(r)) (3)
where p[r) is the total density, r(r) is the radial tension, p(r) is the lateral pressure,
and all mixed values of Tab are null. This way we obtain the equations of state
>=8^ (4)
T = ^-2 Mr ~ 2(r - 6)$'] (5)
P=\[{P- t)& -A-t (6)
Analysis of an embedding diagram of the wormhole together with the requirement
for a throat that connects two asymptotically flat regions of spacetime leads to the
conclusion that the wormhole has to be supported by the matter with property
^<0 (7)
I A) I
where the index 0 indicates that we are operating in immediate throat
surroundings. Above result is central to the wormhole analysis since it indicates that the
tension has to be greater than the mass-energy density and this undermines the
physical reasonability of stress-energy tensor. The generalization of this statement
is called "energy condition" and the name "exotic" is given to the matter that
exhibits property (7). If energy conditions strictly hold we would have no hope to ever
construct a traversable wormhole. However there exist a number of theoretical and
experimental examples on both classical and quantum level indicating that energy
conditions can be broken in some cases. These involve scalar fields,2 non-zero cosmo-
logical constant, Casimir effect,3 inflation,4 Hawking evaporation,5 Hawking-Hartle
vacuum,6 and a couple of others.7
There exists a possibility that wormhole construction is easier on the
quantum level than on the classical one.8 This wormhole would not be traversable since
2189
quantum fluctuations in spacetime metric live in distances of Planck length order
which is much too small comparable to any macroscopic object. However one could
imagine that with the presence of sufficiently advanced technology the quantum
wormhole can be pulled out of the spacetime foam, enlarged and adjusted to the
size and shape adequate for interstellar travel.1 Three popular approaches to
radially evolving wormholes, all of them related to cosmological physics are: conformal
transformation of wormhole metric, inflation of its spatial part and embedding the
traversable wormhole metric into Friedmann-Robertson-Walker (FRW) universe.
Conformal transformation of the wormhole was considered by Kar9 in order
to find out if within classical general relativity a class of nonstable not violating
energy conditions wormholes could exist. It was found that evolving geometry can
support a wormhole and the WEC violation can be avoided for arbitrarily large
intervals of time. Roman10 analyzed Morris-Thorne type wormhole embedded in
an inflationary background, with all non-temporal components of the metric tensor
multiplied by a factor of the form e2x*, where x is related to cosmological constant. It
was proven there that a wormhole can be enlarged to traversable size but violation
of WEC could not be avoided. At most the exotic matter needed for wormhole
maintenance can be minimized at the later stage of inflation. After inflation the
universe undergoes an evolution that is usually described by one of FRW models.
We checked the possibility of wormhole existence against a number of those models.
None of them allowed suspension of the need for exotic matter.
General analysis of the energy conditions near the throat of rotating wormhole
contained in11 and12 gives two major results:
(1) There is always a violation of energy condition, so rotation does not alleviate
the need for exotic matter.
(2) The exotic matter can be moved around the throat, so that some class of in-
falling observers would not encounter it.
I am grateful to Nigel Bishop for helpful discussion. This work was supported
by the National Research Foundation under GUN 2053724.
References
1. M.S. Morris and K.S. Thorne, Am. J. Phys. 56, 395, (1988).
2. M. Visser and C. Barcelo, Talk at COSMO 99, (1999).
3. H.G.B. Casimir, Proc. Kon. Ned. Akad. Wet. B 51, 793, (1948).
4. A. Borde and A. Vilenkin, Phys. Rev. D 56, 717 (1997).
5. S.W. Hawking, Commun. Math. Phys. 43, 199 (1975).
6. M. Visser, Phys. Rev. D 54, 5103 (1996).
7. M. Visser, Lorentzian Wormholes: Prom Einstein to Hawking (Springer-Veriag, 1995).
8. C.W Misner and J.A Wheeler, Ann. Phys. 2, 525 (1957).
9. S. Kar, Phys. Rev. D 49, 862 (1994).
10. T.A. Roman, Phys. Rev. D 47, 1370 (1993).
11. P.K.F. Kuhfitting, Phys. Rev. D 67, 064015 (2003).
12. E. Teo, Phys. Rev. D 58, 024014 (1998).
WORMHOLES IN THE ACCELERATING UNIVERSE*
GONZALEZ-DIAZ, PEDRO F.; MARTIN-MORUNO, PRADO
Colina de los Chopos, IMAFF, CSIC,
Serrano 121, 28006 Madrid (SPAIN) p.gonzalezdiaz@imajf.cfmac.csic.es
We discuss different arguments that have been raised against the viability of the big trip
process, reaching the conclusions that this process can actually occur by accretion of
phantom energy onto the wormholes and that it is stable and might occur in the global
context of a multiverse model. We finally argue that the big trip does not contradict any
holographic bounds on entropy and information.
1. We shall consider in more detail first how the big trip1 can be derived when
a simple non static Morris-Thorne metric is used for a wormhole, i. e.
dv
ds2 = -dt2 + ^r + r2 (dO2 + sm20d<j)2), (1)
r
where we have taken the shift function to be zero and we let1 the shape function
K to also depende on time. If dark energy is regarded to be a perfect fluid with
Tfn, — {p+p)unuv+pgnv, with p and p the pressure and energy density, respectively,
and u^ = dx^ /ds is the four-velocity, u^u^ — — 1, the conservation law for the time-
component of the energy-momentum tensor, T" = 0, can be integrated over r to
give
urV-2 (x _ ^M) _1 (x _ ^M + u2j V2 (p + p)eJL a™ = C(t)> (2)
in which we have introduced the exotic mass factor m~2 to provide the r.h.s. function
C(t) with the dimension of an energy density, and
a=d0T° doK(r,t) n-T;
TS 2r (l - ££*!) TS • { >
Integrating then over r the conservation law for energy-momentum tensor projected
on four-velocity, u^T.^ = 0, we have
"I/O
r\(l-*l?£\ eILirffceJ~d'-f) = A(t), (4)
in which A(t) is a function of time having the dimension of a squared mass satisfying
that A(t) = linv^^ ru2 does not depend on the radial coordinate and does on t
only through the mass m, so that A[t) = A'm2, A' being a dimensionless positive
*This research has been partially supported by Research Project FIS2005-01181. P.M-M
acknowledge CSIC and ESF for a I3P grant.
2190
2191
(w > 0) constant; finally
/? =
1 -
K(r,t)
1/2
W 1
K{r,t)
1/2
<V
d0K(r,t)
P + P 2r [ 1 - X(r'f)
%
1/2N
From Eqs.(2) and (4) we get
(p + p)[l ^U.
#M)
1/2
eJpo
irrfrfeeJ'i'M"-/?)
(5)
B(t),
(6)
with J5(t) = C(t)/A' = p[poa{t)] + Poo(t)- The rate of exotic mass due to phantom
energy accretion should be given by integrating over dS = r2 sin 9d9d<j) the nonzero
component T£, fa = f dSTfi, the sign being chosen to account for accreating
negative energy. Taking into account Eqs.(4) and (6), we obtain
1/2
fa = -An{p + p)A'mr I 1 -
It follows that in the asymptotic limit r —>
vanishes, this rate reduces to
K(r,t]
- fT dra
(7)
c«, in which the exponent in Eq. (7)
fa = -4:7rA'm2(p + p). (8)
Inserting the energy density for a general quintessence fluid with p = wp,2 for
w < — 1 (phantom energy) in Eq.(8) we finally derive for the time-dependent exotic
mass
4ttA'(\w\ - l)p0mo(t-t0)~
m = mo
1
1
(9)
fC(H-i)(*-*o)
1 /2
where the "0" subscripts mean current values and C = (87rp0/3) . Hence, a big-
trip where the wormhole throat diverges will take place before the occurrence of the
big rip singularity, at a time
tbr — to
t* — to
<thr,
(10)
in which tbr = to
l + {87TPo/3)1/2A'mo
is the big rip time. So, during a given time interval
3(\w\~l)C
before t* the size of the wormhole throat will exceed that of the universe.
Formally speaking, the above procedure does not take into account the feature
that we are not dealing with a vacuum solution, such as Faraoni has recently pointed
out.3 However, all our calculations are finally referred to the asymptotic case r —>
l) /r = On, which is obtained from the Einstein
—dt2 + exdr2 + r2dn?,, vanishes because ©n =
constant/r4 for solution (1), where we can still keep 0OO = 0. It follows that Eq.
(8) is correct if the big trip is defined for an asymptotic observer.
oo, where the r.h.s. of A' (e
equations for an ansatz ds2
2192
However, the most serious argument against the occurrence of the big trip in
the universe most recently raised by Faraoni3 is that the accretion of phantom
energy with a perfect fluid equation of state is characterized by a radial velocity
vr ~ a3(1+w)/2 which strictly vanishes at the big rip singularity and in any event
quickly decreases with time for w < — 1. Thus, according to Faraoni, also at the
time where the big trip would occur, accretion of phantom energy would be largely
prevented and the big trip phenomenon would not take place at all. Besides the
feature that the size of the wormhole throat equalizes that of the Universe before
it diverges, what matters here is not the fluid velocity but its flow (as expressed as
phantom energy per unit surface per unit time) which can be roughly given by Vrp,
that is ~ a-3(1+w)/2, which in fact increases with time and consistently diverges at
the big rip. Then the argument by Faraoni does not apply to the case and the big
trip can not be dismissed due to it.
On the other hand and even more importantly, what we are dealing here with
is no longer accretion of usual energy concentrated on given regions of space, but
vaccum energy which isotropically and homogeneously pervades the whole space,
even the regions ocuppied by physical objects. Hence, accretion of phantom energy
is not based on any fluid motion but on increasing more and more space filled with
phantom energy inside the throat. The big trip phenomenon would then appear
when one superposes to this effect the feature that the phantom energy density
increases with time.
2. Since the wormhole spacetime is asymptotically flat the big trip process has
debatably been considered to take place in the framework of the multiverse where
the mouth of a grown up wormhole can still be inserted in larger universes.
3. Wormholes undergoing a big trip process are quantum-mechanically
stable because the parameter £ characterizing the regularized Hadamard function,
(0^)reg ~ const/£4 should necessarily be nonvanishing during the process.
4. The Bekenstein bound on information and entropy could pose a further
problem if the final time for the phantom universe is taken to be that for the big rip.
However, in the neighborhood of the big rip, small wormholes would crop up and
be connected to the region after the big rip in such a way that any amount of
information is actually allowed to be transferred in the big trip.
The big trip process is a rather weird phenomenon which shows some paradoxical
consequences. Actually, one would expect such consequences and even the big trip
itself to be avoided by a quantum gravity treatment.
References
1. P. F. Gonzalez-Diaz, Phys. Lett. B 635, 1 (2006); Phys. Rev. Lett. 93, 071301 (2004)
2. P. F. Gonzalez-Diaz and C. L. Siguenza, Nucl. Phys. B 697, 363 (2004)
3. V. Faraoni, arXiv:gr-qc/0702143.
TRAVERSABLE WORMHOLES SUPPORTED BY COSMIC
ACCELERATED EXPANDING EQUATIONS OF STATE
FRANCISCO S. N. LOBO
Centra de Astronomia e Astrofisica da Universidade de Lisboa,
Campo Grande, Ed. C8 1749-016 Lisboa, Portugal
flobo@cosmo.fis.fc.ul.pt
We explore the possibility that traversable wormholes be supported by specific
equations of state responsible for the present accelerated expansion of the Universe, namely,
phantom energy, the generalized Chaplygin gas, and the van der Waals quintessence
equation of state.
Keywords: Traversable wormholes; dark energy.
We shall explore the possibility that traversable wormholes1 be supported by
specific equations of state responsible for the late time accelerated expansion of the
Universe, namely, phantom energy, the generalized Chaplygin gas, and the van der
Waals quintessence equation of state. Firstly, phantom energy possesses an equation
of state of the form to ~ p/p < — 1, consequently violating the null energy condition
(NEC), which is a fundamental ingredient necessary to sustain traversable worm-
holes. Thus, this cosmic fluid presents us with a natural scenario for the existence of
wormhole geometries.2~4 Secondly, the generalized Chaplygin gas (GCG) is a
candidate for the unification of dark energy and dark matter, and is parametrized by an
exotic equation of state given by pch = —A/p"h, where A is a positive constant and
0 < a < 1. Within the framework of a flat Friedmann-Robertson-Walker cosmology
the energy conservation equation yields the following evolution of the energy density
pch = [A + Ba~3(1+a)] , where a is the scale factor, and B is normally
considered to be a positive integration constant to ensure the dominant energy condition
(DEC). However, it is also possible to consider B < 0, consequently violating the
DEC, and the energy density is an increasing function of the scale function.5 It is
in the latter context that we shall explore exact solutions of traversable wormholes
supported by the GCG.6 Thirdly, the van der Waals (VDW) quintessence equation
of state, p = 7p/(l — j3p) — ap2, is an interesting scenario for describing the late
universe, and seems to provide a solution to the puzzle of dark energy, without the
presence of exotic fluids or modifications of the Friedmann equations. Note that
a, ft —> 0 and 7 < —1/3 reduces to the dark energy equation of state. The existence
of traversable wormholes supported by the VDW equation of state shall also be
explored.7 Despite of the fact that, in a cosmological context, these cosmic fluids
are considered homogeneous, inhomogeneities may arise through gravitational
instabilities, resulting in a nucleation of the cosmic fluid due to the respective density
perturbations. Thus, the wormhole solutions considered in this work may possibly
originate from density fluctuations in the cosmological background.
2193
2194
The spacetime metric representing a spherically symmetric and static wormhole
geometry is given by (with c = G = 1)
ds2 = -e2*(r) dt2 + [1 - b{r)/r}-1 dr2 + r2 {d62 + sin2 6d<j>2), (1)
where $(r) and b(r) are arbitrary functions of the radial coordinate, r.1 The latter
has a range that increases from a minimum value at ro, corresponding to the worm-
hole throat, to infinity. One may also consider a cut-off of the stress-energy tensor
at a junction radius a. The fundamental properties of traversable wormhole are:1
The flaring out condition of the throat, given by (b — b'r)/b2 > 0, which reduces
to b'(ro) < 1 at the throat b(ro) = ro; the condition 1 — b(r)/r > 0, i.e., b(r) < r,
is imposed; and the absence of event horizons, which are identified as the surfaces
with e2* —> 0, so that $(r) must be finite everywhere.
Using the Einstein field equation, G^v = BitT^, we obtain the relationships
b' = Snr2p, $' = lt^l]P,rr) » Pr=2-(Pt-Pr)-(P + Pr)&, (2)
where ' = d/dr. p(r) is the energy density, pr(r) is the radial pressure, and pt(r)
is the tangential pressure. The strategy we shall adopt is to impose an equation of
state, pr = pr(p), which provides four equations, together with Eqs. (2). However,
we have five unknown functions of r, i.e., p(r), pr(r), Pt(r), b(r) and $(r). Therefore,
is
to fully determine the system we impose restricted choices for b(r) or $(r). ' ' It
also possible to consider plausible stress-energy components, and through the field
equations determine the metric fields.3
Now, using the equation of state representing phantom energy, pr — top with
to < —1, and taking into account Eqs. (2), we have the following condition
$'(r) = b + UJrb' (3)
[ ' 2r2 (1 - b/r) l '
For instance, consider a constant 3?(r), so that Eq. (3) provides b(r) = r0(r/ro)-1/w,
which corresponds to an asymptotically flat wormhole geometry. It was shown that
this solution can be constructed, in principle, with arbitrarily small quantities of
averaged null energy condition violating phantom energy, and the traversability
conditions were explored.2 The dynamic stability of these phantom wormholes were
also analyzed,4 and we refer the reader to2,3 for further examples.
Relative to the GCG gas equation of state, pr = —A/pa, using Eqs. (2), we have
the following condition
Jp(1_i)*w._iW(5=!),w + J. (4)
Solutions of the metric (1), satisfying Eq. (4) are denoted "Chaplygin wormholes".
To be a generic solution of a wormhole, the GCG equation of state imposes the
following restriction A < (87rr2)~(1+Q), consequently violating the NEC. However,
for the GCG cosmological models it is generally assumed that the NEC is
satisfied, which implies p > A1/{1+a\ The NEC violation is a fundamental ingredient in
2195
wonnhole physics, and it is in this context that the construction of traversable worm-
holes, i.e., for p < Al/(l+a\ are explored. Note that as emphasized in5 , considering
B < 0 in the evolution of the energy density, one also deduces that pch < Al^1+a\
which violates the DEC. We refer the reader to 6 for specific examples of Chaplygin
wormholes, where the physical properties and characteristics of these geometries
were analyzed in detail. The solutions found are not asymptotically flat, and the
spatial distribution of the exotic GCG is restricted to the throat vicinity, so that
the dimensions of these Chaplygin wormholes are not arbitrarily large.
Finally, consider the VDW equation of state for an inhomogeneous spherically
symmetric spacetime, given by pr = 7p/(l — (3p) — ap2. Equations (2) provide the
following relationship
( h\ */ b ^b' ab'2
2r 1 --$'=_ + '-^r- - -— . 5
V r I r i - #%. 8irr2
It was shown that traversable wormhole solutions may be constructed using the
VDW equation of state, which are either asymptotically flat or possess finite
dimensions, where the exotic matter is confined to the throat neighborhood.7 The
latter solutions are constructed by matching an interior wormhole geometry to an
exterior vacuum Schwarzschild vacuum, and we refer the reader to7 for further
details.
In concluding, it is noteworthy the relative ease with which one may
theoretically construct traversable wormholes with the exotic fluid equations of state used
in cosmology to explain the present accelerated expansion of the Universe. These
traversable wormhole variations have far-reaching physical and cosmological
implications, namely, apart from being used for interstellar shortcuts, an absurdly
advanced civilization may convert them into time-machines, probably implying the
violation of causality.
References
1. M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395 (1988).
2. F. S. N. Lobo, Phys. Rev. D 71, 084011 (2005).
3. S. Sushkov, Phys. Rev. D 71, 043520 (2005).
4. F. S. N. Lobo, Phys. Rev. D 71, 124022 (2005).
5. M. Bouhmadi-Lopez and J. A. J. Madrid, JCAP 0505, 005 (2005).
6. F. S. N. Lobo, Phys. Rev. D 73 064028 (2006).
7. F. S. N. Lobo, "Van der Waals quintessence stars," [arXiv:gr-qc/0610118].
ON WORMHOLES OF MASSLESS K-ESSENCE
J. ESTEVEZ-DELGADO
Facultad de Ciencias Fisico-Matemdticas, Universidad Michoacana de San Nicolas de Hidalgo,
Morelia, Michoacdn, MEXICO
joaquin@fismat. umich. mi
T. ZANNIAS
Ins. de Fisica y Matemdticas, Universidad Michoacana de San Nicolas de Hidalgo,
A.P. 2-82, 58040 Morelia, Michoacdn, MEXICO
zannias ©ginette. ifm. umich. mx
We show that a i-C-essence model involving a massless scalar field <3> minimally coupled
to Einstein gravity admits a family of toroidal wormholes i.e., non singular, globally
static, spacecetime possesing two asymptotically flat ends connected via a throat which
is topologically a two-torus.
We consider Einstein gravity coupled to a K-essesnce massless scalar field $
according to:
Gap = -k(ya$Vp$ - iffQ/3VT$V7$), VaVa$ = 0, k>0, (1)
where conventions for signature and curvature are as in.1 Let (<?, 3?) a static axisy-
metric solution of (1). Relative to a set of oblate-spheroidal coordinates g takes the
form:
[A2 + Ag l-n2_
A 6 (0,oo), n 6 [-1, l],0<ip< 2?r, A0 ^ 0 where ($,[/) satisfice L$ = LU = 0
with L the flat Laplacian while V satisfies:
Vx
A (M>2-2[/2)- tj, (feA#2 _ 2 At/2 _ 2feAt*A#M +4Att7AC/#i)
2(A2+A2/x2)L V /* „; l_fl
V,
(!-^)2 U^2 or^ A2 + Ag
1-/X2
/z(fc$2 _2C/2) _ 1__^ (fe/i#2 _2/i[/2 _2fcA$A$M+4AC/AC/M)
2 i \2\2
(A2+A2)
" 0(\2 i \2,,2
2(A2+A2/z2)
2
A2 + A2
As it was shown in2 any C3 static axially symmetric wormholes of those eqs is
generated by the pair (U, $) described by:
C/(A) = Ci+C2arctan(A/Ao), $(\) = D1+D2 arctan(A/A0), Ae(0,oo), (2)
and for such choice, the resulting V has the form:
ipcl-^M-f
V(A,(.)=;(2C|-fcJi)lnC\2'|"^ j. (3)
Thus (2) generates the following family of metrics:
ff = -e2^^2 + e"2^)[(A2 + A2)(l - /z2)V + (^2++A|2)^+1
2196
2197
x (dX2 + (A2 + X20)dn2)] , A 6 (0, oo), (4)
where for convinience we have set: A = C|/2 — kD\j\. Since:
lim [/(A) = C1 + Jc2-^ + 0(A-3), lim $(A)=D1 + ^D2- ^ + 0(A~3),
A—>-oo Z A A—*oo Z A
lim F(A,/i) = 1+0(A~2),
A—>oo
the choice C\ + ~C2 =0 and D\ + \D2 =0 implies that (4) is asymptotically flat as
A —► oo. Moreover the scalar curvature R and Kretchman scalar K = Ra^lSRap1s
have the form:
with G{\) and F^A,^) smooth functions and H(X) = e2Ci+2C2arctan(^) Therefore
depending upon the value of the exponent A the curvature may have smooth limit
as the (A —> 0,/x —> 0) coordinate ring is approached. For the particular choices
j4 = — 1 or j4 = —l/2 the invariants i? and K are regular as (A —► 0,/x —► 0)
and therefore those metrics are extendable through the (A —> 0, /i —> 0) coordinate
ring. The spacetimes described by (4) for the choice A — — 1 or ^4 = — 1/2 are
incomplete and their completion can be accomplished as follows. At first we extend
the range of the A-coordinate to the domain (—00,00), and analytically continue
(C/(A), $(A)) over the extended domain. The function V(X,/j) over the extended
domain is described by (3). Moreover
lim U(X) = -nC2-(^+0(X~3), lim ^(X) = -nD2 ^ ^^ + 0(X~3),
A—> — 00 A A—>—00 A
and thus g is asymptotically flat at the end defined by A —► —00. For the case where
A = —1/2, and for all A 6 (—00, 00), we obtain from (4):
9 =
_e2U(X)dt2+e-2U(X)
X2 + X2
(Xz+Xz0)(l-^)d^ + dX' + T—^dfi
C/(A) = -^C2 + C2arctan( —), $ (A) = ~D2 + D2 arctan( —),
Z Aq Z Aq
this (M, g, $) describes a two parameter regular family of spherical wormholes (for
more details see3). For the choice A = — 1 we obtain from (4):
9 =
__e2U(X)dt2 + e-2U{X)
(A2 + A20)(l-^)V + ^i|_[dA2 + ~M
which exhibits a coordinate singularity across the (A = 0, fi = 0) ring. It is suficient
to analyze the extendability of the 2-dim Riemannian metric
g2=.f,+.f02[dA2 + ^±^V], A 6 (-00,00), ^[-i,0)u(0,i],
2198
across (A = 0, \i = 0) ring. Defining Cartecian like coordinates (x, y) via:
x(X, //) = (A + y/\20 + A>, y(\, n) = (\ + V/A2+A2)(l - M2),
<72 is transformed into:
(x2+y2 + A2)4[^2+V] , /n ,
g2=4(x2+y2)2[(x2+,2-A2)4+4A2xT 3;e^00'00)' ^^°°)
and in those coordinates the singularity appear at x = 0, y = Ao. Shifting the origin
to (x = 0,y = Ao) via x = x,y = y — Ao, it follows that g^ takes the form:
A2{x,y) 2 2 i2^,*?) .2 -2 -
52 =
[dx2 + dy2] =—V-^[dfl2 + fl2d02], R£(0,e), 06 [0,24
(x2+y2)2L * J #2
where x = RcosO, y = Rsin§ and yi2(-R, 6>) is a smooth with yi2(0,0) ^ 0. However
the singularity at R = 0 is removable. Indeed defining a new "radial" coordinate r
via: R = e~r^r° casts g in the form:
ff = A2(r,9)[dr2+d§2}, fe(e,oo), 0 G [0, 24
which is manifestly regular at x = 0, y = 0. It follows from this representation that
the induced metric g on R = 0 takes the form:
g = -e-°27rdt2 + eC27T\%dip2 + ec'27r\ld§2, ip 6 [0, 2tt], 0 6 [0, 2tt], (5)
and thus any t =const, two spaces represent a two torus, of area A = 47rec'27rA0!.
Moreover by defining a new coordinate p = 2R show that g is asymptotically flat as
p —► 00. On the other hand in terms of the coordinate p = —\q/2R it also follows
that g is asymptotically flat as R —► 0. In view of (5) the two ends are connected
via a throat that is topologically a two-torus. Further properties of those toroidal
wormholes will be discussed elswhere4
Acknowledgments
This work was partially supported by a grant of Coordination Cientifica - UM-
SNH. J. Estevez- Delgado acknowleds partial support from; SEP-PROMEP project:
PTC74.
References
1. R. M. Wald General Relativity (Chicago. Univ. Press.) (1984).
2. J. Estevez-Delgado and T. Zannias: Report, unpublished 2007
3. J. Estevez-Delgado and T. Zannias, Contribution in ERE 2006 in press Journal of
Physics C, 2007.
4. J. Estevez-Delgado and T. Zannias: On the structure of toroidal wormholes (Submitted
2007)
DYNAMIC WORMHOLE SPACETIMES COUPLED TO
NONLINEAR ELECTRODYNAMICS
AARON V. B. ARELLANO
Facultad de Ciencias, Universidad Autonoma del Estado de Mexico,
El Cerrillo, Piedras Blancas, C.P. 50200, Toluca, Mexico
vynzds @y ahoo.com. mx
FRANCISCO S. N. LOBO
Centro de Astronomia e Astrofisica da Universidade de Lisboa,
Campo Grande, Ed. C8 1749-016 Lisboa, Portugal
flobo@cosmo.fis.fc.ul.pt
We explore the possibility of dynamic wormhole geometries, within the context of
nonlinear electrodynamics. The Einstein field equation imposes a contracting wormhole
solution and the obedience of the weak energy condition. Furthermore, in the presence
of an electric field, the latter presents a singularity at the throat, however, for a pure
magnetic field the solution is regular. Thus, taking into account the principle of finiteness,
that a satisfactory theory should avoid physical quantities becoming infinite, one may
rule out evolving wormhole solutions, in the presence of an electric field, coupled to
nonlinear electrodynamics.
Keywords: Traversable wormholes; nonlinear electrodynamics.
Pioneering work on nonlinear electrodynamic theories may be traced back to Born
and Infeld,1 where the latter outlined a model to remedy the fact that the
standard picture of a point charged particle possesses an infinite self-energy. Therefore,
the Born-Infeld model was founded on a principle of finiteness, that a satisfactory
theory should avoid physical quantities becoming infinite. Recently, nonlinear
electrodynamics has found a wide range of applicability, namely, as effective theories at
different levels of string/M-theory, cosmological models, black holes, and in worm-
hole physics, amongst others (see2,3 and references therein).
Relatively to wormhole physics it was found that static spherically symmetric
and stationary axisymmetric traversable wormholes cannot exist within nonlinear
electrodynamic, mainly due to the presence of event horizons, the non-violation of
the null energy condition at the throat, and due to the imposition of the principle
of finiteness.3'4 In this work, we shall explore the possibility that nonlinear
electrodynamics may support time-dependent traversable wormhole geometries. This is of
particular interest as the energy conditions are not necessarily violated for evolving
wormhole spacetimes.0
The action of (3 + 1)—dimensional general relativity coupled to nonlinear
electrodynamics is given by (with G = c = 1)
s= V^g
ii+i(F)
d4.T, (1)
2199
2200
where R is the Ricci scalar. L(F) is a gauge-invariant electromagnetic Lagrangian,
depending on a single invariant F given by F = F^F^/A, where F^v is the
electromagnetic tensor. Note that in Einstein-Maxwell theory, the Lagrangian takes the
form L(F) = -F/4n.
Varying the action with respect to the gravitational field provides the Einstein
field equations GM„ = 87rTM„, with the stress-energy tensor given by
T^iv = g^,u L(F) — F^aFva LF , (2)
where Lp = dL/dF.
We shall consider that the spacetime metric representing a dynamic spherically
symmetric (3 + 1)—dimensional wormhole, which is conformally related to the static
wormhole geometry,6 takes the form
ds2 = n2(t)
***(r)di? + df + r^dQl + sin2 0^
1 — b(r)/r
(3)
where $ and b are functions of r, and fi = Q(t) is the conformal factor, which is
finite and positive definite throughout the domain of t. To be a wormhole solution,
the following conditions are imposed: $(r) is finite everywhere in order to avoid the
presence of event horizons; b(r)/r < 1, with b(ro) = ro at the throat; and the flaring
out condition (b — b'r)/b2 > 0, with b'(ro) < 1 at the throat.
For this particular case, the weak energy condition, which is defined as
T^VU^UV > 0 where Ufi is a timelike vector, is satisfied,2 contrary to the static
and spherically symmetric traversable wormholes.3'4
Through the Einstein field equation, we obtain the following relationship
h-^f^- = - |2(fi/?2)2 - n/n] , (4)
which provides the solutions
6(r)=r[l-aV-r02)], Q(t) = ^ ^ ^ &_at , (5)
where a is a constant, and Cx and Ci are constants of integration. Now, fl(t) —> 0
as t —► oo, which reflects a contracting wormhole solution. This analysis shows that
one may, in principle, obtain an evolving wormhole solution in the range of the
time coordinate. A fundamental condition to be a solution of a wormhole, is that
b(r) > 0 is imposed.7 Thus, the range of r is r0 < r < a = r0\/l + l//?2, with
(3 = aro- If a » ro, i.e., (3 ~ ro/a 4C 1, one may have an arbitrarily large wormhole.
Note, however, that one may, in principle, match this solution to an exterior vacuum
solution at a junction interface R, within the range ro < r < a.
The electromagnetic field equations take the following form
(F^LF);At=0, (*FH;Ai = 0. (6)
Taking into account the symmetries of the geometry, the non-zero compatible terms
for the electromagnetic tensor are F)lv = 2E(xa) 6^ 5rJ +2B(xa) 6^ Si , where Ftr =
2201
E is the electric field, and Fe</> = B, the magnetic field. From the electromagnetic
field equations, we deduce the following
EM = 32^r»(l-6/r)V2 ' B(9) = g* *m9, (7)
with / = (b'r — 36), and qe and qm are constants related to the electric and
magnetic charge, respectively. From this solution we point out two observations: (i) the
requirement of /2fi2r2 > (32Trqeqm)2; (ii) and E <x (1 — b/r)~1'"2, showing that the
E field is singular at the throat, which is in contrast to the principle of finiteness.
An interesting case arises considering a pure magnetic field, E = 0, from which
we obtain the Lagrangian and its derivative
L = ~ 8^ ib'/r2 + 3("/fi)2] ' Lf = 16^fi2r(6V ~ 36) • (8)
These equations, together with B = gmsin#, F = g2n/(2fi4r4) and solutions (5)
provide a regular wormhole solution at the throat, with finite fields. We emphasize
that this result is in close relationship to the regular magnetic black holes coupled
to nonlinear electrodynamic found by Bronnikov.4
In conclusion, we have explored the possibility of evolving time-dependent worm-
hole geometries coupled to nonlinear electrodynamics. It was found that the Einstein
field equation imposes a contracting wormhole solution and that the weak energy
condition is satisfied. In the presence of an electric field, a problematic issue was
verified, namely, that the latter becomes singular at the throat. However, regular
solutions of traversable wormholes in the presence of a pure magnetic field were found.
Another point worth noting is that we have only considered that the gauge-invariant
electromagnetic Lagrangian L(F) be dependent on a single invariant F. It would
also be worthwhile to include another electromagnetic field invariant G ~ *FtiV F^,
which would possibly add an interesting analysis to the solutions found in this work.
References
1. M. Born, Proc. Roy. Soc. Lond. A143, 410 (1934); A144, 425 (1934); M. Born and
L. Infeld, Proc. Roy. Soc. Lond. A147, 522 (1934)
2. A. V. B. Arellano and F. S. N. Lobo, Class. Quant. Grav. 23 5811-5824 (2006).
3. A. V. B. Arellano and F. S. N. Lobo, Class. Quant. Grav. 23 7229-7244 (2006).
4. K. A. Bronnikov, Phys. Rev. D 63, 044005 (2001).
5. S. Kar, Phys. Rev. D 49, 862 (1994).
6. M. Morris and K.S. Thorne, Am. J. Phys. 56, 395 (1988).
7. J. P. S. Lemos, F. S. N. Lobo and S. Q. de Oliveira, Phys. Rev. D 68, 064004 (2003).
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Exact Solutions
(Mathematical Aspects)
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ROBINSON-TRAUTMAN SPACETIMES IN HIGHER
DIMENSIONS
MARCELLO ORTAGGIO
Dipartimento di Fisica, Universita degli Studi di Trento and INFN, Gruppo Collegato di Trento,
Via Sommarive 14, 38050 Povo (Trento), Italy
marcello.ortaggio AT comune.re.it
JIRI PODOLSKY
Institute of Theoretical Physics, Faculty of Mathematics and Physics,
Charles University in Prague, V Holesovickdch 2, 180 00 Prague 8, Czech Republic
Jiri.Podolsky@mff.cuni.cz
We investigate metrics which admit a hypersurface orthogonal, non-shearing,
expanding geodesic null congruence in D > 4 dimensions. Einstein's equations are solved
for vacuum spaces (with an arbitrary cosmological constant) and aligned pure radiation.
1. Geometrical assumptions
During the Golden Age of theoretical studies of exact gravitational waves, Robinson
and Trautman investigated spacetimes that admit a geodesic, shear-free, twist-free,
expanding null congruence.1,2 The Robinson-Trautman family is by now one of the
fundamental classes of exact solutions to Einstein's field equations in D = 4.3 In
the particular case of vacuum spacetimes, the Goldberg-Sachs theorem3 implies that
these geometries are algebraically special. Here we discuss D > 4 extensions.4
Given a _D-dimensional spacetime (D > 4), let us consider a family of null
hypersurfaces u{x) = const, i.e. with normal ka = —u<a satisfying ga/3kak,0 = 0.
This implies that the null congruence of integral curves of the vector field ka =
gal3kl3 is twistfree and geodesic with an affine parameter. One can thus express the
associated optical scalars. shear and expansion,5,6 as
° = ^k% ^ = k{a,0)k^-^—2{k^f. (i)
It is then convenient to take the function u itself (constant along each ray) as one
of the coordinates, so that ka = -8% and guu = 0. As for the remaining coordinates,
we use the affine parameter r along the geodesies generated by ka, and "transverse"
spatial coordinates (x1, x2,..., xD~2) which are constant along these null geodesies.
This further implies ka = 5?, that is gur = -1 and guu = 0 = gu\ so that
ds2 = gij (dxi + gridu) (dx? + grjdu) - 2 dudr - grrdu2, (2)
where the metric functions can depend arbitrarily on all the coordinates (x,u,r)
(from now on, x stands for all the transverse coordinates xt and lowercase latin
indices range as i = 1,..., D — 2). Further useful relations are
gri = glJ9uj, grr = ~-guu + giJgui9uj, gUi = gr39ij, (3)
while grr = 0 = gri- It is also easy to see that ka-p = 29a/3,r-
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2206
Now, requiring that the congruence ka be shear-free (i.e., a2 = 0) leads to4
9ij = P~2lij, with 7yiT. = 0, det jij = 1, (4)
6 = -(lnp),r. (5)
2. Integration of Einstein's equations
We now integrate Einstein's equations Rap — \Rgap + A.gap = 8irTap for the line
element (2), (4). We focus on the case of vacuum spacetimes {Tap = 0) and of aligned
pure radiation {Tap = &2kakp), while the cosmological constant A is arbitrary.
The Sachs equation governing the rate of change of the expansion7 can be
generalized to D-dimensions.4'8,9 For a twistfree shearfree congruences it simplifies to
Rrr = —(£) — 2)(9>r + 62). In the expanding case 6 ^ 0, with eq. (5) (and a suitable
coordinate transformation) one finds from the Einstein equation Rrr = 0 that
e=-. (6)
r
It is thus convenient to factorize p = r~lP(x,u) (P is an arbitrary function) and
rescale the transverse metric 7^ by defining htj = P~2^fij. so that eq. (4) becomes
9ij = P~2lu = r2P~2Hj = r2htJ(x,u). (7)
The explicit integration of all other Einstein's equations is lengthy. We refer to4
for details, and summarize here the main results. First, one finds that
9" = 0. (8)
Then, at at any given u = Uq = const each spatial metric hij(x,uo) must be an
Einstein space (in D = 5 this implies that /iy is a 3-space of constant curvature);
also, the independence of hij can be factorized out in a conformal factor. Namely,
Ti
h^ = P~2(x,u)^ij(x), where detjij = l, (10)
in which IZij is the Ricci tensor associated with the metric h^, and 1Z = 1Z(u) the
corresponding Ricci scalar. In addition to the above equations for h^, one has to
solve an equation which controls the u-dependence of P, i.e.
167T Ti
(D-l)^(lnP),u-/x,u=-p-^, (11)
where n = n(x. u) and \i = fj,(u) are arbitrary functions. The former characterizes
the pure radiation term via Tuu = $2 = r2~Dn2(x, u), whereas the latter enters the
remaining metric coefficient grr = —guu = 2H, given by
2H= ^ 2r(lnP) - ^ r2 - ^ (12)
(D-2)(D-3) Z7[[nr)'u (D_2)(D-l)r rD~3 ' [ >
Robinson-Trautman solutions in D > 4 with aligned pure radiation thus read
ds2 = r2p-2 jij dxW - 2dudr - 2Hdu2, (13)
with eqs. (9)-(12). They are algebraically special of type D (or O) in the sense of.10
'Hj ~ n o '•?' ' )
2207
3. Vacuum solutions
Vacuum Robinson-Trautman spacetimes are given by n = 0, and they split into two
subcases /i ^ 0 and fi = 0. When /j, ^ 0, it can be set to a constant by a coordinate
rescaling4 so that P (and thus hij) must be independent of u (cf. eq. (11)). One can
also normalize 1Z = ±(D — 2)(D — 3) or 1Z = 0. Hence in eq. (13) one has
2H = K-(D-2)A(D-l)r2-^ ^ = °^ M
When the Einstein metric hij(x) = P~2^fij is compact, this family describes various
well-known static black hole solutions11-14 in Eddington-Finkelstein coordinates.
When [i = 0, P need not be independent of u. One can still rescale lZ(u) to a
constant value 1Z = K(D — 2){D — 3), K = 0, ±1, so that in eq. (13) one now has
2A
2H = K-2r0nPU-w^KJrT)r>. (15)
Some of the geometrical properties of these metrics will be elucidated elsewhere.15,16
The solutions studied here display fundamental differences4 with respect to their
D = 4 counterpart. In particular, D > 4 Robinson-Trautman spacetimes can be only
of type D or O. This is in agreement with the result6 that multiple principal null
congruences of D > 4 type N and type III vacuum spacetimes must have non-zero
shear if expanding (remember that the Goldberg-Sachs theorem does not hold in
D > 45'6'16). For the possible inclusion of an aligned Maxwell field in D > 4 see.17
References
1. I. Robinson and A. Trautman, Phys. Rev. Lett. 4, 431 (1960).
2. I. Robinson and A. Trautman, Proc. R. Soc. A 265, 463 (1962).
3. H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt, Exact Solutions
of Einstein's Field Equations, second edn. (Cambridge University Press, Cambridge,
2003).
4. J. Podolsky and M. Ortaggio, Class. Quantum Grav. 23, 5785 (2006).
5. V. P. Frolov and D. Stojkovic, Phys. Rev. D 68, 064011 (2003).
6. V. Pravda, A. Pravdova, A. Coley and R. Milson, Class. Quantum Grav. 21, 2873
(2004).
7. R. Sachs, Proc. R. Soc. A 264, 309 (1961).
8. J. Lewandowski and T. Pawlowski, Class. Quantum Grav. 22, 1573 (2005).
9. M. Ortaggio, V. Pravda and A. Pravdova, Class. Quantum Grav. 24, 1657 (2007).
10. A. Coley, R. Milson, V. Pravda and A. Pravdova, Class. Quantum Grav. 21, L35
(2004).
11. F. R. Tangherlini, Nuovo Cimento 27, 636 (1963).
12. D. Birmingham, Class. Quantum Grav. 16, 1197 (1999).
13. G. W. Gibbons, D. Ida and T. Shiromizu, Phys. Rev. Lett. 89, p. 041101 (2002).
14. G. Gibbons and S. A. Hartnoll, Phys. Rev. D 66, 064024 (2002).
15. M. Ortaggio, Proceedings of the XVII SIGRAV Conference, Torino, September 4-7,
2006 [gr-qc/0701036].
16. V. Pravda, A. Pravdova and M. Ortaggio, in preparation.
17. M. Ortaggio, J. Podolsky and M. Zofka, in preparation.
SOLUTIONS OF SEIBERG-WITTEN AND EINSTEIN-MAXWELL-
DIRAC EQUATIONS IN EUCLIDEAN SIGNATURE
CIHAN SAgLIOGLU
Faculty of Engineering and Natural Sciences,
Sabanci University, Tuzla, 814-74 Istanbul, Turkey
saclioglu@sabanciuniv. edu
The existence of infinite numbers of inequivalent smooth structures on 4-manifolds
suggests that the statistical odds for ending up in a 4-dimensional spacetime are
overwhelmingly large. We exhibit signature (++++) spacetimes that are simultaneous solutions
of Einstein-Maxwell-Dirac and Seiberg-Witten equations, which are used for classifying
inequivalent smooth structures.
1. Introduction
All known 4-manifolds admit infinitely many distinct smooth structures. For
compact 4-manifolds, the known examples are countably infinite in number, while in the
non-compact case, most notably RA ,there is an uncountable infinity of such
structures. There is no known 4-manifold with a finite number of smooth structures,1
and the number of distinct smooth structures in all other dimensions is finite. In the
light of recent suggestions2 that our universe is but one of 10500 possible universes,
perhaps existing in parallel, it is tempting to speculate that the reason we are in 4
dimensions because of these overwhelming statistical odds.
The more precisely defined problem of constructing invariants to distinguish
homeomorphic but non-diffeomorphic manifolds was first treated in Donaldson
theory3 by examining the moduli spaces of self-dual Yang-Mills fields on the manifold.
This was later considerably simplified in Seiberg-Witten (SW)theory4whose degrees
of freedom consist of a Weyl spinor ip representing a massless monopole, a U(l)
connection A^, and a Euclidean (++++) signature metric gap- The Seiberg-Witten
monopole equations (SWME) relating them read
P>A1P = 0, (1)
F^ = \{F^ + \^aBFn = -\i>]b^iM> ■ (2)
In the above pair, the first one is the usual Dirac equation with a derivative
covariantized with respect to both the U{\) and the spin connection lj°^. In the
second equation, only the self-dual part of F appears.
If one now asked whether these SW fields could also be physical (in the sense
of obeying the Einstein-Maxwell-Dirac field equations), the answer would appear
to be negative because the second equation in the SWME pair makes the whole
system overdetermined. Remarkably, however, Euclidean signature allows
simultaneous solutions because the source terms T^in Einstein's equations can be made to
vanish. For A^, this happens for self-dual or anti-self-dual F^, while Weyl spinors
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2209
not only have identically vanishing X^'s, but also vanishing vector and axial vector
bilinear currents for (++++) signature. The vanishing of these two currents means
that both F^v and its dual are sourceless and hence covariantly constant; thus self
duality guarantees a simultaneous solution of the SWME and the Einstein-Maxwell-
Dirac equations. In the following, we will present solutions of the SWME of the form
T,Pl x TiP2, where T,p is a Riemann surface of genus p. Non-singular solutions of the
SWME require p\ + p2 > 2, which means the 4-manifold has constant negative
curvature and the Einstein field equations must include a cosmological constant.
"Physicality", i.e., self-duality forces px = p2 = p.This means the solutions are a
Euclidean version of the Bertotti5-Robinson6 solution, with p magnetic vortices on
one 2-manifold and p electric ones on the other.
2. SWME Solutions of the form SPl X SP2
The SWME are solved by a product manifold EPl x Ep2 (j>\,P2 > 2), where the scalar
curvature of the Riemann surfaces are — 2|0|2 and —2(|?/;i|2 — |0|2), respectively.
The spinor consists of a single non-zero constant component ipi; <p is an additional
parameter. The spin connection-one forms for the two manifolds will be denoted
below by uj\ and uj\; the 1/(1) connection one-form by A, the corresponding £7(1)
curvature by F, while the manifold's curvature two-forms will be indicated by R\
and R\. Using complex dimensionless coordinates z1 = x + iy = \/2|0|(a*1 + i%2)
and z2 = s + it) = \^2(\ipx\2 — |0|2)1/2(a;3,a:4), these geometrical quantities are
parametrized in terms of special automorphic7 Fuchsian functions8 g(z\) and g{z2)
which tesellate9the constant negative curvature surfaces |<?(^i)| < 1 and |<?(^2)| < 1
by 4pi and 4p2-gons with geodesic edges, respectively. In terms of these functions,
we have the Kleinian metric for, say the second Riemann surface, in the form
ds2(M^) = e^dz2dz-2= nd92%2 (3)
(i -g292Y
and a similar one with g{z{) for the first one. The connections and curvatures are
then given by
R\
.,1„ ,dgxdzi^ iqidg-i - g-idgA ,
-%{-d\n (-p1—) + Kynyi _, },
2 dzidgi' (1-ffiffi)
X2 {dz2dg2> (l-<?2ff2) ^
A
-M+wl),
dgx/\ dgl 3
-2«7j —r^, R4
-2i
■ dg2Adg2
(1-S2ff2)2'
(4)
(5)
(6)
(7)
2210
_ dgx Adgl dg2 A dg2 _ l^i ^ ^
(i-ffiSi)2 (l-5232) 2
3. Simultaneous solutions of the SWME and the
Einstein-Maxwell-Dirac equations
It is not difficult to check that when |0|2 = j|^i|2, F becomes self dual, the two
manifolds become identical, all the source terms in the Einstein-Maxwell equations
i / i2
vanish, and the Einstein tensor becomes equal to Kg^v with A = ^y"-. Thus the
massless monopole condensate serves as a cosmological constant for this constant
negative curvature space. More details can be found in10 and.11
References
1. A. Scorpan, "The wild world of 4-manifolds", AMS Providence, Rhode Island (2005).
2. M. Tegmark, in "Science and ultimate reality: from quantum to cosmos", J. D. Barrow,
P. C. W. Davies and C. L Harper, eds., Cambridge University Press (2003).
3. S. K. Donaldson, J.Differential geom. 18279 (1983).
4. E. Witten, Math. Res. Lett. 1769 (1994).
5. B. Bertotti, Phys. Rev. 116, 1331 (1959.
6. I. Robinson, Bull. Acad. Pol. Sci. 7 , 351 (1959).
7. L. R. Ford, Automorphic Functions, Chelsea, NY (1951).
8. A. Dubrovin, T. Fomenko, and S. P. Novikov, Modern Geometry Vol. II, Springer-
Verlag, NY (1985).
9. Z. Nehari, Conformal Mapping, Dover, NY (1952).
10. C. Saghoglu, Class. Quantum Grav. 17, 485 (2000).
11. C. Saghoglu, Class. Quantum Grav. 18, 3285 (2001).
EULER NUMBERS ON COBORDANT HYPERSURFACES
TINA A. HARRIOTT
Department of Mathematics and Computer Science, Mount Saint Vincent University, Halifax,
Nova Scotia B3M 2J6, Canada
E-Mail: Tina.Harriott@msvu.ca
J.G. WILLIAMS
Department of Mathematics and Computer Science, and the Winnipeg Institute for Theoretical
Physics, Brandon University, Brandon, Manitoba R7A 6A9, Canada
E-Mail: williams@brandonu.ca
When two hypersurfaces are mediated by a Lorentz cobordism, a homology selection rule
restricts the number of general relativistic kinks that can occur on the hypersurfaces. In
2+1 dimensions, this selection rule translates to the requirement that the difference in
the number of kinks on the two hypersurfaces be balanced by a corresponding difference
in Euler number. This is explored for a particular spacetime by using a tetrad-based
Jacobian integral formula for calculating the kink number.
If a region of spacetime is bounded by two hypersurfaces, Si and E2, mediated
by a Lorentz cobordism, then the numbers of Finkelstein-Misner kinks [1] on the
hypersurfaces are governed by a selection rule. In 3+1 dimensions, Gibbons and
Hawking [2] have expressed this selection rule using homology theory and the Ker-
vaire semi-characteristic. In 2+1 dimensions. Low [3] has shown that the selection
rule can be written in terms of Euler number x(^):
kink(£2)-kink(£i) = i[x(£2)-x(£i)].
This present paper explores Low's result for a region of (2+l)-diinensional
Minkowski spacetime, ds2 = -dt2 + dx2 + dy2, bounded internally by a torus,
Ej = S1 x S1, and externally by a sphere, T,2 = S2.
The kink number for the torus can be found by expressing the Minkowski
metric in terms of toroidal coordinates, (T,£, ip), and suppressing the confornial factor
a (cosh T - cos^)-1 which becomes irrelevant when considering light cone
configurations:
ds2 = cos2adT2 + 2 sin2adTd£ — cos 2ad£2 + sinh" Tdip",
where the angle a(T,£) is denned by sina = sinhT sin £ (cosh T - cos£)-1. Solving
the eigenvalue equation gllvV1' = \h^vVu, with ds\ = + dT2 + d£,2 + sinh" T dip2
as the choice for the arbitrary positive-definite metric h^, leads to the normalized
(with respect to h^v) eigenvector V = (— sina, cosa. 0), with eigenvalue A = —1.
Being timelike, the vector V tracks the tipping of the light cones and feeds, through
the introduction of tetrads, V1 = <pa e%, and a covariant derivative D^0- = d^<j)a+
^ °6 06) (given in terms of a fiat-space spin connection one-form, w °6), into the
2211
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following formula for counting kinks [4]:
kink(E) = [vol^™)]"1 / det
/ 0° ... 4>n \
r dul A ... Adun.
\Dn<jP ... Dn<l>n)
In 2+1 dimensions, vol(5") = vol(52) = 4tt.
The spin connection one-forms, ioIJab, are all zero. It follows that there are only
two nonzero covariant derivatives, D^(f>T = d^<pT and D^<jfi = d^cjfi, which leads to
kink(i;1) = kink(51 x Sl) = 0.
Using spherical coordinates, (T, 6, tp), one can perform the analogous calculation
for the sphere. The Minkowski metric can be written
ds2 = -cos29dT2+ 2Tsin 29 dTdO + T2 cos 29 d92+T2 sin2 9 dip2.
Using ds2+ = + dT2 + T2(d92 + sin2 9 dip2) leads to the normalized eigenvector
V = (cos#, — T-1sin#, 0), with A = —1. Putting T = a to choose a specific
spherical hypersurface then gives V = (cos#, — a_1sin#, 0). The nonzero spin
connection one-forms are then found to be
This gives Dgcj)T = Dv<j)v = —sin9 and Dg(f)e = —cos9, and it follows that
kink(E2) = kink(52) = 1. The Euler numbers for Ej and £2 and are known to
be x(^i) = xiS1 x S1) = 0 and ^(^2) = x(S2) = 2, thereby agreeing with Low's
equation.
Acknowledgements
This work was supported by the Dean of Arts and Science, Mount Saint Vincent
University.
References
1. D. Finkelstein and C.W. Misner, Ann. Phys. (NY) 6, 230 (1959).
2. G.W. Gibbons and S.W. Hawking, Phys. Rev. Lett. 69, 1719 (1992).
3. R.J. Low, Class. Quantum Grav. 9, L161 (1992).
4. T.A. Harriott and J.G. Williams, Nuovo Cimento B 120, 915 (2005).
SYMMETRIES OF THE WEYL TENSOR IN BIANCHI V
SPACETIMES
A. R. KASHIF*
Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24 3UE, Scotland,
UK
kashmology@yahoo.com
K. SAIFULLAHt
School of Mathematical Sciences, Queen Mary, University of London, London, UK
saifullah@qau.edu. pk
G. SHABBIR
Faculty of Engineering Sciences, GIK Institute of Engineering Sciences and Technology, Topi.
Swabi, NWFP, Pakistan
shabbir@giki. edu.pk
Symmetries of geometrical and physical quantities in general relativity provide important
information about the curvature structure of the spacetimes. Symmetries of the curvature
and the Weyl tensors, known as curvature and Weyl collineations respectively, are two
of such important symmetries. Some results on these symmetries for Bianchi type V
spacetimes are discussed.
Symmetries of tensors in general relativity - Killing vectors and collineations -
play an important role in understanding not only the geometric structure of the
underlying spaces but their physical properties as well. They have been used in
finding new solutions of Einstein's Field Equations (EFEs), classifying these
solutions and by virtue of Noether's theorem constructing the conservation laws for the
spacetime. Thus, invariance under the Lie transport of the metric, Ricci and
energy momentum tensors define Killing vectors (KVs), Ricci collineations (RCs) and
matter collineations (MCs), respectively.1'2 Curvature collineations (CCs) which
are symmetries of the Riemann tensor are significant for studying the curvature
structure of spacetimes.2,3 The Weyl tensor, C, is fundamental in understanding
the purely gravitational field for a spacetime with the matter content removed.4 Its
local symmetries, Weyl collineations (WCs),3,5~7 are of particular interest since it
is conformally invariant.8 Mathematically, WCs are given by
.£XC = 0,
where £x is the Lie derivative along the vector field X. In component form this
* On leave from: College of EME, National University of Sciences and Technology, Rawalpindi,
Pakistan.
t On leave from: Centre for Advanced Mathematics and Physics, National University of Sciences
and Technology, Rawalpindi, Pakistan, and Department of Mathematics, Quaid-i-Azam University,
Islamabad, Pakistan.
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becomes
CbcdjX + CfcdXb + CbfdX<c + C%cfXd - CbcdXj = 0,
where comma denotes the partial derivative. This is a system of 20 nonlinear partial
differential equations as compared to the collineations of rank two tensors (RCs and
MCs, for example) which are systems of 10 equations.
On account of its algebraic symmetries we can write the 4th rank Weyl tensor
(and the curvature tensor) of 4-dimensions in the form of a 6 dimensional matrix,
whose rank gives the rank of the tensor. Further, while the metric tensor cannot be
degenerate, the other tensors can be and hence give way to the possibility of infinite
degrees of freedom (i.e. infinite dimensional Lie algebras) as well. The KVs of a space
form a subset of all other collineations but the inclusion relationship between the
symmetries of two fourth rank tensors, CCs and WCs, when both are finite is yet
to be established. While it is known5 that CCs can be properly contained in WCs
when both are finite, no spacetime is known to the present authors which admits
CCs which are not WCs and yet both are finite. On the other hand, there is no
proof available that this is not possible. The Schwarzschild interior spacetime, for
example, is Petrov type O1 and thus every vector field is a WC while CCs are finite.
The Reissner-Nordstrom spacetime is of Petrov type D and both the WCs and CCs
are finite and equal. But when we take pressure as constant in the Schwarzschild
interior we see that the WCs are properly contained in infinitely many CCs. For
vacuum spacetimes with zero cosmological term, however, the Ricci tensor, R, is
zero and WCs and CCs coincide because the Weyl tensor reduces to the curvature
tensor.
Enumeration of all Lie groups is useful in mathematics as well as in physics.
The G3, for example, were originally enumerated by Bianchi which were divided
into nine types, Bianchi I to Bianchi IX.1
Let us consider the Bianchi type V spacetimes which admit three KVs given by1
K1
K2
K3
These spacetimes in (t. x, y, z,) coordinates can be written as
ds2 = -dt2 + A{tfdx2 + B(tfdy2 + (C{tf + x2B(tf)dz2 - 2xB{tfdydz.
Following a well known procedure9'10 the components Cat,cd of the Weyl tensor
for these spacetimes can be written as the 6x6 matrix
d_
dy '
d_
dz
Odd
dx dy dz
2215
'-'abed —
/CWo 0 0 0 0 Cl023\
0 C2020 C2030 0 C1320 0
0 C2030 C3030 C1230 C1330 0
0 0 C1230 C1212 C1213 0
0 c, 320 330 Cl213 Cl313 0
C1023 0 0 0 0 C2323
V )
Similarly the curvature tensor can also be written as a 6 x 6 matrix. If its rank is
greater than or equal to 4 then the Lie algebra of CCs is finite dimensional.11 Now,
the rank of the Weyl matrix is always even and if it is 6 or 4 the Weyl symmetry
trivially reduces to the conformal symmetry.9 Further, we note that6 it cannot have
rank 2 . Thus we conclude that these spacetimes do not admit non-trivial WCs.
Acknowledgments
ARK and KS acknowledge a research grant from the Higher Education Commission
of Pakistan. They are also thankful to the National University of Sciences and
Technology, Pakistan for travel support to participate in MG11, Berlin, 2006.
References
1.
9.
10.
11.
H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers and E. Herlt, Exact
Solutions of Einstein's Field Equations (Cambridge University Press, 2003).
G. S. Hall, Symmetries and Curvature Structure in General Relativity (World
Scientific, 2004).
G. H. Katzin, J. Levine and W. R. Davis, J. Math. Phys. 10, 617 (1969).
R. Penrose and W. Rindler, Spinors and Spacetime (Cambridge University Press,
1986).
I. Hussain, A. Qadir and K. Saifullah, Int. J. Mod. Phys. D 14, 1431 (2005).
G. Shabbir and A. R. Kashif, A note on proper Weyl collineations in Bianchi V
spacetimes, (Submitted for publication).
G. S. HaB, Gen. Rel. Grav. 32, 933 (2000).
S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Spacetime (Cambridge
University Press, 1973).
G. S. Hall, Gravitation & Cosmology 2, 270 (1996).
G. Shabbir, Class. Quantum Grav. 21, 339 (2004).
A. H. Bokhari, A. R. Kashif and A. Qadir, Gen. Rel. Grav. 35, 1059 (2003); A. R.
Kashif, Ph.D. Thesis, Quaid-i-Azam University, Islamabad (2003).
CLASSIFICATION OF SPACETIMES ACCORDING TO
CONFORMAL KILLING VECTORS
K. SAIFULLAH*
School of Mathematical Sciences, Queen Mary, University of London, London, UK
saifullah@qau. edu.pk
Conformal Killing vectors (CKVs) preserve the spacetime metric up to a factor. Homoth-
etic vectors and Killing vectors are special cases of CKVs. Classification of some classes
of spacetimes on the basis of CKVs give interesting results showing how homothetic and
Killing vectors which form subsets of the set of CKVs can be recovered as a result of the
above classification.
Einstein's theory of general relativity is based on the realization that geometry,
represented by the Riemann curvature tensor R^cd of the spacetime can be related
to the distribution and motion of matter, denoted by the stress-energy tensor Tat,.
This relation is explained by Einstein's field equations (EFEs),
Rab - -^Rgab = nTab (a, b = 0,1,2,3). (I)
Here gab is the metric tensor, Rab the Ricci tensor, R the Ricci scalar and k = ^^,
where G and c are the gravitational constant and the speed of light respectively. (We
have ignored the term with the cosmological constant.) Metric, gab, is the
dynamical quantity in EFEs which varies over the spacetime. EFEs (1) break down into
ten highly non-linear differential equations and so far very few exact solutions have
been discovered by imposing certain restrictions.1 One of such restrictions could
be to allow a spacetime to admit certain symmetry properties. For example, the
isometry group Gm of (M, g) is the Lie group of smooth maps of manifold M onto
itself leaving g invariant. The subscript "m" is equal to the number of generators
or isometries of the group. It is the Lie algebra of continuously differentiable
transformations Kad/dxa where Ka = Ka (x6) are the components of the vector field K
known as a Killing vector (KV) field. In other words, a KV field K is a field along
which the Lie derivative of the metric tensor g is zero i.e. £a (gab) = 0.
In addition to isometries there are other types of motions which are even more
restrictive and therefore could be more useful as far as the solution of Eqs.(I) and
their properties are concerned. For example, the study of homothetic vectors (HVs)
and conformal Killing vectors (CKVs) are significant in general relativity.2 CKVs
are motions along which the metric tensor of a spacetime remains invariant up to a
scale i.e.
* On leave from: Centre for Advanced Mathematics and Physics, National University of Sciences
and Technology, Rawalpindi, Pakistan, and Department of Mathematics, Quaid-i-Azam University,
Islamabad, Pakistan.
2216
2217
££,9ab = gab„AC + 9ac€% + 9bc^a = 24>9ab ■ (2)
Conformal motions are determined by the arbitrary constants appearing in the
vector field £ = £ad/dxa when <f> = <p (t, x, y, z). In the above equation, ", "represents
derivative with respect to coordinates xa. If <f> is constant £ represents HVs and if
it is zero, we simply get the KVs. It is clear from the definition that HVs and
KVs are special cases of CKVs. The study of the symmetry groups of a spacetime
is a useful tool not only in constructing spacetime solutions of EFEs but also for
classifying the known solutions according to the Lie algebras, or structure generated
by these symmetries. Previously, CKVs have been studied for various spacetimes
like Minkowski,3 Robertson-Walker4 and pp-waves.5
Important results regarding the dimensionality of these symmetries include (see,
for example, Refs. 2, 6):
1. Riemannian space Vn admits a group of motions Gm where m < n (n + 1) /2.
2. A Riemannian space Vn cannot admit a maximal group of motions Gm where
m = n (n + 1) /2 — 1. If a spacetime admits a Gm as the maximal group of isometries
then the HVs group Hr is at the most of order r = m + 1.
3. The set of conformal vector fields on M is finite-dimensional and its dimension
is less then or equal to 15. If this maximum number is attained, the spacetime is
conformally flat. If it is not conformally flat then the maximal dimension is 7.
Let us consider, for example, the class of spherically symmetric spacetimes which,
in the usual coordinates, with v (t, r), A (t, r) and \i (t, r) as arbitrary functions, can
be written as
ds2 = -e"^dt2 + ex^dr2 + e"<''r> (d62 + sin2 6dp2) . (3)
These spacetimes admit 3 KVs
d d
Kl = sin0—- + cos 0cot^—- ,
80 ocp
2 9 . d
K = cos 07— — sin <*cot 0—- ,
86 ocp
o d
K3 = — .
Ocp
In the static case these admit a timelike KV, K4 = d/dt, also. The classification
of HVs of spherically symmetric spacetimes admitting maximal isometry groups
larger than SO (3) was obtained along with their metrics6 by using the homothety
equations and without imposing any restriction on the stress-energy tensor. The
possible maximal homothety groups Hr for these spacetimes are of the order r =
4,5,7,11; for r = 11, the only spacetime is Minkowski. The general solution and
classification of conformal motions for these spacetimes7 shows that the group of
CKVs is G4+„ where n, the number of CKVs, is either 2 or 11. In the case n = 2,
2218
both CKVs are necessarily proper. For the conformally flat case, up to 6 of the 11
CKVs may be improper.
For the plane symmetric metric
ds2 = -eu^x)dt2 + eW'^dx2 + e^*'x) [dy2 + dz2) , (4)
the minimal symmetry is given by
d 9 d o d d
K = 7T - K = 7T - K = Z7T ~ yjT ■
ay az ay az
In the static case the spacetimes admit a timelike KV, K4 = d/dt, in addition to
the KVs given above. The orders of the isometry groups for the associated metrics
are 4, 5, 6, 7 and 10; 8 and 9 are not admissible.8 Hence the possible groups for
HVs9 are of the order 5, 6, 7 or 11. Classification of these spacetimes according to
CKVs10 is also in accordance with the established results.
Acknowledgments
The author is grateful to George Alekseev for helpful comments. A research grant
from the Higher Education Commission of Pakistan is gratefully acknowledged.
The author is also thankful to the National University of Sciences and Technology,
Pakistan for the travel support to deliver this talk at MGll, Berlin, 2006.
References
1. H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers and E. Herlt, Exact
Solutions of Einstein's Field Equations (Cambridge University Press, 2003).
2. G. S. Hall, Symmetries and Curvature Structure in General Relativity (World
Scientific, 2004).
3. Y. Choquet-Bruhat, C. Dewitt-Morrette and M. Dillard-Bleick, Analysis, Manifolds
and Physics (North-Holland, 1977).
4. R. Maartens and S. D. Maharaj, Class. Quantum Grav. 3, 1005 (1986).
5. R. Maartens and S. D. Maharaj, Class. Quantum Grav. 8, 503 (1991).
6. D. Ahmad and M. Ziad, J. Mtah. Phys. 38, 2547 (1997).
7. R. Maartens, S. D. Maharaj and B. O. J. Tupper, Class. Quantum Grav. 12, 2577
(1995).
8. A. Qadir and M. Ziad, Static plane symmetric spacetimes, in Proc. 6th Marcel Gross-
mann Meeting, eds. T. Nakamura and H. Sato, p. 1115 (Scientific Publishing Co.,
1993).
9. S. Kiran, Classification of Homotheties of Plane Symmetric Static Spacetimes, M.Phil.
Dissertation, Quaid-i-Azam University, Islamabad (1997).
10. Shair-e-Yazdan, Classification of Conformal Motions in Plane Symmetric Static
Spacetimes, M.Phil. Dissertation, Quaid-i-Azam University, Islamabad (2005).
EXACT SOLUTIONS FOR RADIATING RELATIVISTIC STAR
MODELS
S. S. MISTHRY*
Astrophysics and Cosmology Research Unit, School of Mathematical Sciences, University of
KwaZulu-Natal, Private Bag X54OOI, Durban, ^OOO, South Africa
misthrys@dut.ac.za
S. D. MAHARAJ
Astrophysics and Cosmology Research Unit, School of Mathematical Sciences, University of
KwaZulu-Natal, Private Bag X54.OOI, Durban, ^000, South Africa
maharaj @ukzn .ac.za
We study realistic models of relativistic radiating stars undergoing gravitational collapse
which have vanishing Weyl tensor components. Previous investigations are generalised by
retaining the inherent nonlinearity at the boundary. Several classes of infinite solutions
exist.
1. Introduction
The evolution of a radiating star undergoing gravitational collapse, in the context of
general relativity, has occupied the attention of researchers in astrophysics in recent
times. In a recent treatment Herrera et al1 proposed a model in which the form of
the Weyl tensor was highlighted when studying radiative collapse. This approach
has the advantage of simplifying the Einstein field equations. However, Herrera et
al1 were not able to solve the junction conditions; only an approximate solution was
found. Maharaj and Govender2 showed that it is possible to solve the field equations
and the junction conditions exactly. The exact solutions in Maharaj and Govender2
depend upon the introduction of a transformation that linearises the boundary
condition. The purpose of this paper is to demonstrate that it is possible to obtain
other models by transforming the boundary condition to an Abel's equation which
is necessarily nonlinear.
2. The Model
We consider a spherically symmetric radiating star undergoing shear-free
gravitational collapse. The line element for shear-free matter interior to the boundary of
the radiating star is given by
ds2 = -A2dt2 + B2[dr2 + r2(d62 + sin2 6d<j>2)} (1)
where A = A(t, r) and B = B(t, r) are the metric functions. This has to be matched
across the boundary r = b to the exterior Vaidya spacetime
dg2 = _ / _ 2m^\ ^ _ 2dvdR + R2^&2 + s.n2 &d(f)2^ (2)
'Permanent address : Durban University of Technology
2219
2220
The vanishing of the Weyl tensor components leads to the condition
A=(Cl(t)r2 + l)B
and pressure isotropy gives
B
1
(3)
(4)
c2(ty + c3(t)
where C\{t),C2{t) and C3(t) are functions of time. The forms for the metric
functions A and B given above generate an exact solution to the Einstein field equations.
For our model the junction conditions reduce to the following nonlinear ordinary
differential equation
C2b2 + C3
+2
3 (C2b2 + C3)2 CtfiCztf+Ca)
2 C2b2 + C3
c'b2 + 1 [ft(C2
C2b2 + C3
Cxb2 + 1
2C1C3)&2+C3(C1C:
2(C3d - C2)b
2C2
(5)
resulting from the (nonvanishing) pressure gradient across the hypersurface r = b.
To complete the description of this radiating model we need to solve the junction
condition (5).
3. Abel Equation
Here we consider a particular nonlinear transformation which leads to exact
solutions. It is convenient to replace the function C\{t) with U = C\b2 + 1. Then the
governing equation (5) may be written with some rearrangement as
U{C2b2 + C3) + U
+2U2
b
3 {C2b2 + C3)2
2 C2P + C3
1
'-{C2b2 + C3)-(C2b2 + C3)
r2
C2b* + C3{C2b V]
+ 2U-
2C2b — C3 C3
C2b2 + C3 '^
0
(6)
The transformed equation (6) is an Abel's equation of the first kind in the variable
U. Abelian equations are difficult to solve in general. Several classes of solutions may
be generated with specific constraints. We present here two cases of exact solutions.
3.1. Case 1: C2b2 + C3 = 0
The restriction immediately gives C2b2+C3 = a where a is a constant of integration.
Two cases arise: U = 0 or U ^ 0. We easily find:
j_
' b2
4C3 | 3Cj
b2 ~*~ ab2
C2
c3
{ C3{2a-3C3) \b2
a-C3
b2
arbitrary function of time
£7 = 0
(7a)
(7b)
(7c)
2221
This solution is particularly attractive since we have an infinite choice of C3 and no
integration is required.
3.2. Case 2: 2°2£ 7%3 ■ % = 0
We have two possibilities: either 2C2b2 - C3 = 0 or C3 = 0. With 2C2b2 - C3 = 0
we have the solution
(7=1 f QA^e^ \
C = % (8b)
C3 = arbitrary function of time (8c)
This is an infinite class of solutions depending on C3. For C3 = 0 we have the
solution
<7, - X ( <*<Zm*t'b r\ (9a)
C2 = arbitrary function of time (9b)
C3 = 0 (9c)
Again we have generated an infinite class of solutions depending on C2 ■
These simple exact solutions may be used to study the physical features of the
gravitating star.
Acknowledgments
SSM thanks the National Research Foundation and the Durban University of
Technology for financial support. SDM acknowledges that this work is based upon
research supported by the South African Research Chair Initiative of the Department
of Science and Technology and National Research Foundation.
References
1. L. Herrera, G. Le Denmat, N. O. Santos and G. Wang, International Journal of
Modern Physics D 13, 583 (2004)
2. S. D. Maharaj and M. Govender, International Journal of Modern Physics D 14, 667
(2005)
AN EMP MODEL OF BIANCHI 1 COSMOLOGY
FLOYD L. WILLIAMS
University of Massachusetts
Department of Mathematics
Amherst, MA 01003, USA
williams@math.umass. edu
A new method of solving Bianchi 1 field equations is presented, which extends the
Ermakov-Milne-Pinney approach to solving Friedmann equations. We consider also the
possibility of a "Schrodinger" model of these equations.
1. Introduction
A striking paper of R. Hawkins and J. Lidsey1 established a connection between
flat Friedmann-Lemaitre-Robertson-Walker (FLRW) scalar field cosmology and
the classical Ermakov-Milne-Pinney (EMP) equation — an equation that also
occurs in areas such as non-linear optics, elasticity, and quantum field theory. This
connection has been extended to non-flat FLRW cosmology2'3, and to homogeneous,
anisotropic cosmologies4, where more general types of EMP equations occur.
Moreover, a complete formulation of FRLW scalar field cosmology (for arbitrary
curvature) in terms of a suitable time-independent, non-linear Schrodinger-type equation
was recently established5. The works1'2'3'4,5, in particular, provide for new methods
of obtaining exact solutions of Einstein field equations. We illustrate this for the
EMP equation set up in reference 4. by providing an explicit prescription (not given
there) for solving the field equation of a Bianchi 1 metric. This EMP equation we
feel should lead also to a Schrodinger formulation — details of which have not been
worked out yet.
2. Solution of Bianchi 1 Field Equations
We consider the Bianchi 1 metric
ds2 = -[a{t)b{t)c{t)]2dt2 + a{t)2dx2 + b{t)2dy2 + citfdz2 (1)
for a(t),b(t),c(t) > 0, with a(t)b(t)c(t) a non-constant function of t. For a time-
dependent scalar field 0 and potential function V, we work with the energy
momentum tensor
Tij = -<j)-i<j)-j + gtj
2222
2fffeV;fc0;A + ^o0
(2)
2223
which is the negative of that in4'6 by our sign convention. The corresponding
Einstein field equations are
ab at be (i) d>2 . , ,2 ,
- + - + - (=j ^r + (abef (V o 0)
ab ac be 2
a
'ib ac be b b2 c c2 (H) <j>
\2
ab + ^c + bc-b + ¥--e + 7i = t"(a6c) ^ ®
^ + ^ + ^-h- + a-2-^ + C-2^^-(abcf{Vot)
ab ac be a a2 c c2 2 '
ab ac be a a2 b b2 Uv) d>2 , , ,2 ,
^ + ^c + bc-a + ^-b + V2 " T-WV°fl-
Prom these equations one can deduce6
b=Ciexta, c = C2e^b, (4)
3^j + 2 (A + ») -a + A/z - C2C22a6e2«A+^ (Vo^)-^=0
for real numbers A,/x,Ci,C2 with C\,Ci > 0. Moreover, for any choice of real
numbers n ^ 0, 9 > 0, one can construct a solution4 y(x) > 0 of the generalized
EMP equation
V"(x) + Q{x)y{x) = U ^ , (5)
where for a function r{t) satisfying
f(t)=ee2(-x+^ty(T(t))6-^ (6)
f(x) is the inverse function t~1(x), and for 4>\{x) = <f>(f(x))
Q(x) = ~cj)[(x)2; y{x)=a{f{x))n'2. (7)
Note that for the choice n = 6, y~+l = y3 and equation (5) compares with the
classical EMP equation
y"(x)+Q(x)y(x) = ^, (8)
where C is a constant, though the numerator in (5) is x-dependent.
The main observation made here is a converse result. Namely, suppose real
numbers A, fi, Ci, C2, n, 8 are given, with C\,C2,6 > 0, n ^ 0, and functions
Q(x),f(x),y(x) (y(x) > 0) are given such that equation (5) holds. Suppose f(x) has
an inverse function r{t) that satisfies equation (6); one can usually solve equation
(6) by a Maple program. Motivated by equations (4), (7) we now define
a(t) d=f y(r(t))2/n, b(t) d=f Ciexta(t) (9)
2224
where (p[ (x)2 = -Q(x). Then for a function V(x) that satisfies the second equation
in (4) (which one could use to define V when (f>~1 exists), the quintet (a, b, c, <f>, V)
given by (9) solves the system of field equations (i), (ii), (iii), (iv) in (3).
Although tt, = 6 is a simple choice, as we have noted, examples indicate that it
is good to have the flexibility of other choices as well — similar to the situation
regarding the Schrodinger model5. The derivation of equation (5) does make use of
the initial assumption that a{t)b(t)c(t) is a non-constant function oft. This equation
can be exploited further to derive a Schrodinger model of the system (3). In fact,
such a model has recently been set up by Miss Jennie D'Ambroise, with some
assistance from the author, for the conformally equivalent version
ds2 = -dt2 + Aitfdx2 + B(tfdy2 + Citfdz2 (10)
of (1), where A(t), B(t), C(t) > 0. Namely, she has shown (for the same energy
momentum tensor TV, in (2)) that solutions of the Einstein field equations for (10)
correspond exactly to solutions of a linear Schrodinger equation
u"(x) + [E- P{x)]u(x) = 0 (11)
with constant energy E < 0, where u(x),E,P(x) are given explicitly in terms of
the data A(t), B(t), C(t), <j),V — and vice-versa.
We point out, in closing, that one can employ with X^ in (2) a second energy
momentum tensor T\- that involves density and pressure functions6 pit), p{i), and
thus generalize the field equations (3). These more general field equations can be
solved, in principle, by a corresponding generalization of equation (5). Namely, one
shows that T>,' contributes to (5) precisely the extra term
-^C2C2y(x)^e~2^f(*XP + P)f(x).
References
1. R. Hawkins, J. Lidsey, Phys. Rev. D 66 (2002) 023523.
2. F. Williams, Internat. J. of Modern Physics 20 (2005) 2481.
3. P. Kevrekidis, F. Williams, Class. Quantum Gravity 20 (2003) L177.
4. T. Christodoulakis, C. Helias, P. Kevrekidis, G. Papadopoulas, F. Williams, from
Progress in General Relativity and Quantum Cosmology Research, Nova Science Pub.
(2005).
5. J. D'Ambroise, F. Williams, arXiv:hep-th/0609125 (2006); to appear in Internat. J.
of Pure and Applied Math.
6. T. Christodoulakis, Th. Grammenos, Ch. Helias, P. Kevrekidis, A. Spanou, J. Math.
Phys. 47 (2006) 042505.
EXACT STATIC SOLUTIONS FOR SCALAR FIELDS COUPLED
TO GRAVITY IN (3+1)- DIMENSIONS *
AYSE H. BILGE
Istanbul Technical University, Faculty of Science and Letters,
Mathematics Engineering Department, Maslak, TR-344&9 Istanbul, TURKEY
bilge@itu.edu.tr
DURMUS DAGHAN
Istanbul Technical University, Faculty of Science and Letters,
Mathematics Engineering Department, Maslak, TR-34469 Istanbul, TURKEY
daghand@itu. edu. tr
Einstein's field equations for a spherically symmetric metric coupled to a massless scalar
field are reduced to a system of second order in terms of the variables /j = m/r and
y = (ct/ra), where a, a, r and m are as in [W.M. Choptuik, Physical Review Letters,
70(1993)]. Solutions for which /t and y are time independent may arise either from scalar
fields with <fit = 0 or with <j>s = 0 but <j> linear in t, called respectively the positive
and negative branches. For the positive branch we obtained an exact solution. For the
negative branch, we prove that /j = 0 is a saddle point for the linearized system, but
the non-vacuum solution /j = 1/4 is a stable focus and a global attractor for the region
/js + /j > 0, /j < 1/2.
1. Introduction
The initial value problem for Einstein's equations with massless scalar field was
studied analytically by Christodoulou in1 and2 where it was shown that "small
initial data" disperses while for "large initial data" the end state is a black hole
surrounded by vacuum. The work of Choptuik3 on the numerical search for "critical
initial data" that would separate these two types of behavior led to the discovery of
the "threshold phenomenon". A detailed overview of the literature on the threshold
phenomena can be found for example in.4
It was pointed out by D. Grumiller in a private communication that the solution
for the positive branch dates back to Fisher,5 as discussed in detail in6 and,7 Fisher's
solution have been rediscovered in8-11 and errors of the original equations 28-29 in5
were corrected by Grumiller.6 As opposed to the popularity of the positive branch,
the second class of static solutions that we call the negative branch is first noticed
by Wyman11 where it is studied perturbatively.
2. Reduction of the field equations
Let M be a four dimensional Lorentzian manifold and g be a spherically symmetric
metric on M. In the coordinate system adopted in.3 The metric is given by
ds2 = -a2(t,r) dt2 + a2(t,r) dr2 +r2 d62 +r2sin29 dip2 (1)
*This research is partially supported by the Turkish National Council for Scientific and
Technological Research.
2225
2226
The complete set of field equations coupled to a scalar, static particle with
k = 8n are given below.
at A A A a
— = Airrcpt (pr, -
a a
— 4- — = 4irr
a a
€ + °^A
a
■ (
1
= 0,
1
t =
>a
a
(2)
(3)
'a
^a /1
Defining the variables z = ^, a2 = [l — 2jr] then, using the s = ln(r), z = ry
and m = r/j, we obtain an autonomous system.12 Then, we shall eliminate <p from
the autonomous system and obtain a system of equations for /x and y.
y
Us
Vs
±2
4>l + (4>t/y?
{Us +/i)2
2tt1
(1-2/x)'
(/Lis +a*)2 + (a<s + y)n
1/2
(4)
/*?
y2
o,
20s(04/y)
1 vt/y
(5)
(6)
2/Li -°^"'<" 27T1-2/L*
3. Static metrics: Exact solution for the positive branch
In this section we shall study solutions for which the metric is independent of t,
hence static. Putting /it = 0 in equations (5-6). Solutions for which /j, and y are
time independent may arise either from scalar fields with 0t = 0 or with 0S = 0 but
<p linear in t, called respectively the positive and negative branches.
For the positive branch <f>t = 0 and we write ip = <f>s. The field equations reduce
to
y
l
1-2/Li'
Us
-/Li + 2tt(1 - 2/l*)i/l2
i>s_
1
1-2/Li
(7)
(8)
We will obtain an analytic solution12 for this system as given below.
i/>2
^
IV
Aix
^'h
ya
M I^-pIBi W+q\B\- = IVI-1 l^-p|Cl IV»+g|
C2
where ip = <f>s, r = es, y0 and r0 are constants,
2227
4. Static metrics: Phase plane analysis for the negative branch
For the negative branch <ps = 0, the equation 5 can be written as
2
HSS + fis+ {ns + fi)(fis + 2/i) - 2(/is + fj.) = 0. (9)
Writing v = [iSl we can express equation (9) for jj, as a dynamical system
v
v + 2fi- j^-(v2 + 3/xi/ + 2fx2)
(10)
1-2^
There are two critical points in this system, (0,0) and (1/4,0). Origin is a saddle
point and the point (1/4,0) is a stable focus. We proved that12 (1/4,0) is a global
attractor for solutions in the open half plane fis + [i > 0, fj, < 1/2.
Proposition 4.1. Let (/i(s), v(s)) be a solution of equation (10) and D be the region
bounded by fi+ v > 0 and jj, < 1/2. // (/i(0),f(0)) belongs to D, then the solution
curve (/i(s), v(s)) remains in D for all s and
lim (//(*),!/(*)) = (1/4,0). (11)
Acknowledgments
The authors would like to thank Dr. D. Grumiller for the pointing out references.5~n
References
1. Christodoulou, D., The problem of a self-gravitating scalar field, Common. Math.
Phys. 105, 337 (1986).
2. Christodoulou, D., Global existence of generalized solutions of the spherically
symmetric Einstein- scalar equations in the large, Cornrnun. Math. Phys. 106, 587 (1986).
3. Choptuik, W.M., Universality and scaling in gravitational collapse of massless scalar
field, Physical Review Letters 70, 9 (1993).
4. Gundlach, C, Critical phenomena in gravitatinal collapse, Phys. Rept, 376, 339
(2003), gr-qc/0210101.
5. Fisher, I.Z., Scalar mesostatic field with regard for gravitational effects, Zh. Eksp.
Teor. Fix. 18, 636 (1948), gr-qc/9911008.
6. Grumiller, D., Quantum dilaton gravity in two dimensions with matter, PhD thesis,
Technische Universiiat, Wien (2001), gr-qc/0105078.
7. Grumiller, D., Mayerhofer, D., On static solutions in 2D dilaton gravity with scalar
matter , Class, and Quant. Grav. 21, 5893 (2004), gr-qc/0404013.
8. Bergmann, O,, Leipnik, R., Space-time structure of a static spherically symmetric
scalar field, Phy. Rev. 107, 1157 (1957).
9. Buchdahl, H.A, Reciprocal static metrics and scalar fields in the general theory of
relativity, Phy. Rev. 115, 1325 (1959).
10. Janis, A.I., Newman, E.T., Winicour, J., Relativity of the Schwarzchild singularity,
Phys. Rev. Lett. 20, 878 (1968).
11. Wyman, M., Static spherically symmetric scalar fields in general relativity, Physical
Review D 24, 839 (1981).
12. Bilge, A.H., Daghan, D., Exact solutions for scalar fields coupled to gravity in (3+1)-
dimensions, gr-qc/0508020.
THERMODYNAMIC DESCRIPTION OF INELASTIC COLLISIONS
IN GENERAL RELATIVITY
GERNOT NEUGEBAUER and JORG HENNIG
Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universitat Jena,
Max-Wien-Platz 1, D-07743 Jena, Germany,
neugebauer@tpi.uni-jena.de, J.Hennig@tpi.uni-jena.de
1. Introduction
We discuss head-on collisions of spherically symmetric neutron stars and disks of
dust by comparing initial and final equilibrium states. This thermodynamic
approach avoids the description of the dynamical transition processes and leads to
a "rough" picture of the collision process. Starting with bodies separated by a
large ("infinite") distance we may model the initial situation by a quasi-equilibrium
configuration. As an always present damping mechanism, gravitational emission
provides again for the formation of an equilibrium configuration after the collision.
To decide for which initial parameters the collision of two stars/disks leads to a
new star/disk, we make use of the conservation of baryonic mass Ma and angular
momentum J. In this way we find relations between the initial and final
parameters and calculate the energy loss due to gravitational radiation. For a detailed
description see Ref.1
2. Example: Colliding neutron stars
As a first example we study the merger of two identical non-rotating neutron stars,
described by a completely degenerate ideal Fermi gas of neutrons (there is, however,
no problem to involve more realistic equations of state). Due to the emission of
gravitational waves the two initial stars will merge into one neutron star ("inelastic
collision"). By solving the Einstein equations in the TOV form (numerically) one
obtains the baryonic mass Ma, the gravitational mass M and the radius ro of a
star as functions of the central pressure. The resulting mass-radius relations M0(ro)
and M(ro) are shown in Fig. 1 (first graph). Now, the conservation equation for the
baryonic mass,
M0 = 2M0, (1)
allows us to calculate the radius fo and the masses M and Mo of the final star as
functions of initial parameters. Fig. 1 shows the ratio fo/ro (second graph) and the
efficiency (the relative energy loss) r) = 1 — M/2M (third graph) as functions of
the initial mass-radius radio 2M/tq. Due to our EOS, r\ cannot exceed a maximum
value of 2.3% and the formation of a new neutron star is only possible for initial
stars with radii of ro > 18 km and masses of M0 < 0.37 MQ. Beyond this limit the
collision must lead to other final states, e.g. to black holes.
2228
2229
0 0,01 0.02 0.03 0.04 0.05 0.06
2M/r0
0.01 0.02 0.03 0.04 0.05 0.059
2M/r0
Fig. 1. Parameter relations for the collisions of neutron stars (mass-radius relations for the bary-
onic mass Mo and the gravitational mass M, the ratio ro/ro and the efficiency 77 as functions of
the initial mass-radius ratio 2M/ro).
3. Systematic treatment of inelastic collisions
A helpful tool for the systematic treatment of the equilibrium states before and
after the collision is the variational principle
6E\
S,M0,J
0, E
2k0
« 1- £ ) v^d3.! + ttJ +
M
(2)
t=t0
(R: Ricci scalar, M: gravitational mass, Mo: baryonic mass, S: entropy, fl: angular
velocity, J: angular momentum) which yields the field equations as well as the
parameter thermodynamics of the system.2 Moreover, studying the minima and
maxima of E one can perform a stability analysis.1
We have applied this principle to the analysis of head-on colliding rigidly rotating
disks of dust. Due to the emission of gravitational waves ("damping mechanism")
the colliding disks merge into one differentially rotating disk. While the initial
configuration is explicitely known (as a superposition of rigidly rotating disks3) we may
calculate the final disk by the following considerations: For each of the rings forming
the disk (see Fig. 2) the baryonic mass and the angular momentum are conserved,
dM0 = 2dM0, dJ = 2d J. (3)
These equations together with the matter/vacuum junction conditions form a
complete set of boundary conditions for the final disk. For the numerical solution we
have used a spectral method in a compactified space-time. It turns out that the
angular velocity of the final disk is almost constant, cf. Fig. 2 (and precisely
constant in the Newtonian limit). For the efficiency r\ we find a maximum value of
r/max = 23.8%.
4. Summary
We have presented a thermodynamic way for the analysis of head-on collisions.
Applying it to neutron stars and disks of dust we found conditions for the formation
of final stars/disks (cf. Figs. 1, 2) and efficiencies of conversion of mass into
gravitational radiation. A summary of efficiencies for several collision scenarios (including
Hawking's and Ellis' upper limit for colliding Schwarzschild black holes) is given in
the table.
2230
fi
-——i£
V-
V-
= 0
25
= 0
= 0
10
05
p/po
0.4
0.3
Qpo
0.2
0
=_
"
0.2
__p, = 0.3
__JJ_= 0.7
ft = 0.9
H = 1.1
ft = 1.3
^i = 1.5
M = 1 .7
// = 1.9
0.4 0.6
--—--
0.8
p/po
Fig. 2. Illustration of the local conservation laws (first picture). The other pictures show rotation
curves f2(/3) for different values of the relativistic parameter fj, of the initial disks (po: coordinate
radius of the final disk). While ft, for rigidly rotating disks, can take all values in the interval
[0, 4.629 . .. ], colliding disks have to obey the condition fj, < 1.954 ... to merge into a final disk.
colliding objects
Vn
Schwarzschild black holes 29.3%
Rigidly rotating disks of dust (parallel angular momenta) 23.8%
Schwarzschild stars 19.7%
Rigidly rotating disks of dust (antiparallel angular momenta) 4.2%
Neutron stars (ideal fermi gas) 2.3%
Acknowledgments
This work was supportet by the Deutsche Forschungsgemeinschaft (DFG) through
the SFB/TR7 "Gravitationswellenastronomie". We would like to thank Marcus An-
sorg for his essential contribution to the numerical part.
References
1. J. Hennig and G. Neugebauer, Phys. Rev. D 74, 064025 (2006); J. Hennig, G. Neuge-
bauer and M. Ansorg, in preparation.
2. J. B. Hartle and D. H. Sharp, ApJ 147, 317 (1967); G. Neugebauer, in Relativity
Today, Proceedings of the 2nd Hungarian Relativity Workshop, Budapest, 1987, edited
by Z. Perjes (World Scientific, Singapore, 1988), p. 134.
3. G. Neugebauer and R. Meinel, Astrophys. J. 414, L97 (1993); Phys. Rev. Lett. 73,
2166 (1994); Phys. Rev. Lett. 75, 3046 (1995).
DISTORTED KILLING HORIZONS AND ALGEBRAIC
CLASSIFICATION OF CURVATURE TENSORS*
V. PRAVDA
Mathematical Institute, Academy of Sciences, Zitna 25, 115 67 Prague 1, Czech Republic
pravda@math. cas. cz
O. B. ZASLAVSKII
Department of Mechanics and Mathematics, Kharkov V N Karazin National University,
Svoboda Sq. 4, Kharkov 61077, Ukraine
ozaslav@kharkov.ua
We consider generic static spacetimes with Killing horizons and study properties of
curvature tensors in the horizon limit. It is determined that the Weyl, Ricci, Riemann
and Einstein tensors are algebraically special and mutually aligned on the horizon. It
is also pointed out that results obtained in the tetrad adjusted to a static observer in
general differ from those obtained in a free-falling frame. This is connected to the fact
that a static observer becomes null on the horizon. It is also shown that finiteness of
the Kretschmann scalar on the horizon is compatible with divergence of some Weyl
components in the freely falling frame. We call such new objects truly naked black holes.
We consider generic static spacetimes with the metric which in Gauss normal
coordinates takes the form
ds2 = -dt2N2 + dn2 + -fabdxadxb, (1)
where xl = n, a = 2,3. The Killing horizon surface corresponds to N = 0.
Our goal is to elucidate to what extent the presence of the Killing horizon
restricts the Petrov type of the gravitational field on the horizon, and find which
types are possible there (further details can be found in1). Our determination of the
Petrov type is based on studying curvature invariants I, J and coefficients K,L,N in
certain covariants (see chapter 9.3 in2), constructed from so-called Weyl scalars: The
algorithm for determining the Petrov type of the Weyl tensor is based on whether
or not equalities I3 = 27J3, / = J = 0, K = TV = 0, K = L = 0. Our strategy
can be described as follows. (1) We choose the complex tetrad frame and define
Weyl scalars; (2) We use 2+1 + 1 splitting of the metric of the static spacetime
(1) and find general expressions for Weyl scalars; (3) The regularity conditions on
the horizon impose severe restrictions on the asymptotic form of the metric; we
substitute this asymptotics into the formulae for Weyl scalars and find their near-
horizon values; (4) Compare the result with the conditions that define the Petrov
type; (5) Carry out this procedure for the static observer (SO) for non-extremal and
(ultra)extremal horizons separately; (6) Repeat it for a tetrad that corresponds to
a free-falling observer (FFO).
We construct the complex null tetrad from a usual orthonormal frame u^, eM,
aM, &'' where u11 is the 4-velocity of an observer, e^ is a vector aligned along the n-
*This work is supported by etc, etc.
2231
2232
direction, a^ and VL lie in the x2-x3 subspace. We define P = " t^e , n^ = " ^ ,
m^ = 2—y^—.We use the standard definition of the Weyl scalars. For example,
V2
ip0 = Ca^slam^Pms where CQ/j7^ is the Weyl tensor.
We found that there exist only two possibilities on the horizon: (i) ip0 = ipi =
ip3 = ip4 = 0, ip2 7^ 0, (ii) all components of the Weyl tensor vanish. Case (i)
corresponds to the Petrov type D and case (ii) to the Petrov type O. We must
make a reservation here. As the static frame becomes singular on the horizon, by
the Petrov type on the horizon we simply mean the type obtained by taking the
horizon limit from the outer region.
As an example of case (ii), we can mention the Bertotti -Robinson (BR) metric
which is of type O. However, quantum backreaction of massless conformally
invariant fields on spacetimes of the type AdS2xS2 (which the BR metric belongs to)
violates this condition.3 In contrast, backreaction of massive fields retains its
validity.4 Thus, as far as the role of quantum backreaction is concerned, conformal fields
change the Petrov type of the metric on the horizon from O to D, whereas massive
fields leave it intact.
Now consider the FFO. Then one can show that Weyl scalars are transformed
according to -0o —*• z2ip0, ipi —► ztpi, t/j2 —*• '02, ^3 —*• z_1V->3, ipi —*• z~2ip4, z = e~a,
cosh a = J|, E is the energy per unit mass. Usually, the parameter z is finite and
non-vanishing, so that classification criteria are not affected by the boost and all
timelike observers agree that the field belongs to the same type which is an invariant
characteristic of a spacetime at a given point. The situation is qualitatively different
on the horizon since z —► 0 and thus, in general, some of the quantities K, L, N that
vanish in the static frame may or may not vanish in the freely falling one. This
is obviously related to the fact that the SO becomes null on the horizon and the
corresponding null frame is singular there. Consequently, only the results obtained
in FFO's frame should be considered as physically relevant.
Thus, in general, there exists a variety of situations depending on the relationship
between invariants. SO registers types 1) D or 2) O in the vicinity of the horizons,
while FFO finds them, correspondingly, 1) II or D, 2) III, N, O on the horizon.
The essential role of the horizon in transformations between SO and FFO
reveals itself also in the following property. It was observed earlier for spherically-
symnietrical metrics that transformation to FFO leads to enhancement of the
curvature components although they remain finite ("naked black holes").5 It turns
out, however, that for distorted horizons some curvature components may become
infinite, although the Kretschmann invariant is finite ("truly naked black holes"
- TNBH). The reason why the finiteness of the Kretschmann invariant does not
guarantee by itself finiteness of all curvature components is that different terms can
enter this expression with different signs due to the Lorentz signature. Thus, the
horizon may look regular from the viewpoint of SO but singular from the viewpoint
of FFO. This reveals itself for non-spherical horizons only as a combined effect of
non-sphericity, extremality and presence of infinite tidal forces for FFOs. For the
2233
extremal TNBH, the block-diagonal structure typical of the stress-energy tensor on
the horizon6 fails and, apart from this, inevitably patches with the phantom
equation of state py + p < 0 (py is the longitudinal pressure, p is the energy density)
appear on the horizon.
We also considered properties of the Ricci tensor and found that Segre types
of the Ricci tensor on the horizon are [112], [(11)1, 1], [11(1, 1)] and [(11)Z Z] and
more special. On the horizon all components of the Weyl, Ricci and also Riemann
and Einstein tensors with positive boost weight vanish. This implies that all these
tensors are aligned, algebraically special on the horizon in the sense of7 and of the
alignment type (2).
We thank Org. Committee which demonstrated how effective may be so " large-
scale" conference. O. Z. acknowledges with gratitude Org. Committee and
especially H. Kleinert for support that made it possible for him to attend it.V. P.
was supported by institutional research plan #AV0Z10190503 and research grant
KJB1019403.
References
1. V. Pravda and O. B. Zaslavskii, Class. Quant. Grav. 22 (2005) 5053-5071.
2. Stephani H, Kramer D, Maccallum M, Hoenselaers C and Herlt E 2003 Exact Solutions
of Einstein's Field Equations (Cambridge: Cambridge University Press)
3. Zaslavskii O B 2000 Class. Quantum Grav. 17 497
4. Matyjasek J and Zaslavskii O B 2001 Phys. Rev. D 64 104018
5. Horowitz G T and Ross S F 1997 Phys. Rev. D 56 2180.
6. Medved AJM, Martin D and Visser M 2004 Class. Quantum Grav. 21 3111
7. Milson R, Coley A, Pravda V and Pravdova A 2005 Int. J. Geom. Meth. Mod. Phys.
2 41.
QUASI-STATIONARY ROUTES TO THE KERR BLACK HOLE
REINHARD MEINEL
University of Jena, Theoretisch-Physikalisches Institut,
Max-Wien-Platz 1, 07743 Jena, Germany
meinel@tpi.uni-jena.de
Quasi-stationary (i.e. parametric) transitions from rotating equilibrium configurations of
fluid bodies to rotating black holes are discussed. For the idealized model of a rotating
disc of dust, analytical results derived by means of the "inverse scattering method" are
available. They are generalized by numerical results for rotating fluid rings with various
equations of state. It can be shown rigorously that a black hole limit of a fluid body
in equilibrium occurs if and only if the gravitational mass becomes equal to twice the
product of angular velocity and angular momentum. Therefore, any quasi-stationary
route from fluid bodies to black holes passes through the extreme Kerr solution.
1. Introduction
The exterior metric of a spherically symmetric star, even in the case of collapse,
is always given by the Schwarzschild metric. This is a consequence of Birkhoff's
theorem. Therefore, the collapse of a sufficiently massive, non-rotating star at the
end of its life leads quite naturally to a Schwarzschild black hole, as in the idealized
case of the Oppenheimer-Snyder dust collapse. On the other hand, a continuous
quasi-static transition from stars (modelled as perfect fluid spheres) to black holes
is not possible (cf. Buchdahl's inequality). Without rotation, the black hole state
can only be reached dynamically.
For rotating stars, the situation is different in both previously mentioned
respects. Firstly, the exterior metric is not the Kerr metric in general. (There is no
analogue to Birkhoff's theorem in this case.) It is generally believed, based on the
cosmic censorship conjecture combined with the black hole uniqueness theorems,
that the collapse of a rotating star leads asymptotically to the Kerr black hole,
i.e. to the Kerr metric outside the horizon. This has not yet been proved however.
But secondly, a continuous quasi-stationary transition from rotating perfect fluid
bodies to rotating black holes is possible. In the following, this will be demonstrated
by reviewing analytical results for a rotating disc of dust as well as numerical results
for rotating fluid rings. Moreover, necessary and sufficient conditions for a black hole
limit of rotating fluid bodies in equilibrium will be discussed.
2. From rotating discs and rings to black holes
Bardeen and Wagoner1 solved the general relativistic problem of a uniformly
rotating disc of dust approximately. They found evidence that in a certain parameter
limit the solution approaches the extreme Kerr metric outside the horizon. This
has been confirmed by the exact solution to the disc problem2"4 derived by means
of the "inverse scattering method". The solution depends on two parameters, say
the gravitational mass M and the angular momentum J. Other parameters, such
2234
2235
as the disc's angular velocity fl (as seen from infinity3-), are then functions of M
and J. In the black hole limit, the relations J = M2 and M = 2QJ holdb. It
should be mentioned that a "separation of spacetimes" occurs in the parameter
limit. From the "exterior point of view", the extreme Kerr metric outside the
horizon emerges, whereas from the "interior point of view" a non-asymptotically flat
spacetime containing the rotating disc emerges, which approaches the extreme Kerr
throat geometry ("near-horizon geometry") at infinity. More details can be found
in Refs. 1 and 5.
The separation of spacetimes mentioned above, which turns out to be a universal
feature in the limit, allows for the existence of a black hole limit independent of
the fluid body's topology. Indeed, such a limit was found numerically for bodies
of toroidal topology, the "relativistic Dyson rings"6 and their generalizations.7'8
So far, these ring solutions with various equations of state are the only known
examples of genuine fluid bodies permitting a black hole limit. For a review of
relativistic equilibrium configurations of constant mass-energy density — including
the relativistic Dyson rings — see Ref. 9.
3. Conditions for a black hole limit
It can be proved that the parameter relation
M = 2SU (1)
is necessary10 and sufficient11 for a (Kerr) black hole limit of rotating fluid bodies in
equilibrium. This shows once again that such a limit is impossible without rotation.
Moreover, since fl must become equal to the "angular velocity of the horizon" of
the Kerr black hole,
qH = J ^ (2)
2M2 \m + y/M2 - (J/M)2
the relation
J = M2, (3)
characteristic of an extreme Kerr black hole, must hold in the limit. Therefore, any
quasi-stationary route from fluid bodies to black holes passes through the extreme
Kerr solution. Note that, in contrast to (1), the relation (3) alone is not sufficient
for a black hole limit of a fluid body in equilibrium. Indeed, there exist normal
fluid configurations with J < M2, J = M2 as well as J > M2 (the disc and ring
solutions discussed above, however, have J > M2 except for the black hole limit
where J = M2). But fluid configurations always satisfy M > 20,.J, and M = 2ttJ
(= 2fiH J) is approached precisely in the black hole limit. Non-extreme Kerr black
holes (characterized by J < M2) again satisfy M > 2fiH J.
aWe assume asymptotic flatness.
bWe use units in which G = c = 1.
2236
4. Outlook
It is an open question, whether there are sequences of stable equilibrium fluid
configurations approaching a black hole limit continuously, i.e. whether quasi-stationary
routes to the Kerr black hole as discussed here are to be expected in the real world.
It may well be that a configuration which is already close to the black hole limit
will dynamically evolve towards a slightly sub-extreme Kerr black hole as a result
of small perturbations. Investigations in this direction may lead to further
interesting insights concerning questions of gravitational collapse, black hole formation and
cosmic censorship.
Acknowledgments
I would like to thank Marcus Ansorg, Andreas Kleinwachter, Gemot Neugebauer
and David Petroff for valuable discussions. This research was supported by the
Deutsche Forschungsgemeinschaft (DFG) through the SFB/TR7 "Gravitations-
wellenastronomie".
References
1. J.M. Bardeen and R.V. Wagoner, Astrophys. J. 167, 359 (1971).
2. G. Neugebauer and R. Meinel, Astrophys. J. 414, L97 (1993).
3. G. Neugebauer and R. Meinel, Phys. Rev. Lett. 75, 3046 (1995).
4. G. Neugebauer and R. Meinel, J. Math. Phys. 44, 3407 (2003).
5. R. Meinel, Ann. Phys. (Leipzig) 11, 509 (2002).
6. M. Ansorg, A. Kleinwachter and R. Meinel, Astrophys. J. 582, L87 (2003).
7. T. Fischer, S. Horatschek and M. Ansorg, Mon. Not. R. Astron. Soc. 364, 943 (2005).
8. H. Labranche, D. Petroff and M. Ansorg, Gen. Rel. Grav. 39, 129 (2007).
9. M. Ansorg, T. Fischer, A. Kleinwachter, R. Meinel, D. Petroff and K. Schobel,
Mon. Not. R. Astron. Soc. 355, 682 (2004).
10. R. Meinel, Ann. Phys. (Leipzig) 13, 600 (2004).
11. R. Meinel, Class. Quantum Grav. 23, 1359 (2006).
CLASSIFICATION RESULTS ON PURELY MAGNETIC PERFECT
FLUID MODELS
LODE WYLLEMAN* and NORBERT Van den BERGH
Department of mathematical analysis, University of Ghent,
Galglaan 2, Gent 9000, Belgium
lwyllema@cage.ugent.be, norbert.vandenbergh@ugent.be
A non-conformally flat perfect fluid model for which the electric part of the Weyl tensor
w.r.t. the fluid 4-velocity field vanishes, is called a purely magnetic perfect fluid (PMpf).
Recently we showed that algebraically special PMpf's are necessarily locally rotation-
ally symmetric, and hence are all known. Secondly, the class of algebraically general,
non-accelerating PMpf's was explored. The dust case is conjectured to be inconsistent
because of a particular mathematical feature in the governing equations. The remaining
irrotational subclass contains a physically plausible and essentially unique member.
1. Introduction
Interchanging Ea(, and Hat, in the introduction of the proceeding contribution to
these proceedings, one gets the definition of a purely magnetic perfect fluid (PMpf).
The remarks and definitions given for PEpf's in the first paragraph are also valid
for PMpf's.
PMpf's are elusive. The purely magnetic condition seems to lead to severe in-
tegrability conditions. In particular, the non-existence of purely magnetic vacua
(fj, +p = 0) has been conjectured,1 a proof of which has only been given for Petrov
type D,1 Petrov type I (M°°)2 and in some kinematic subcases.3~10 In this
proceeding we give a survey of the literature and recently obtained results concerning
non-vacuum PMpf's.
2. Petrov type D
We recently showed11 that any PMpf is locally rotationally symmetric (LRS). class
I or III in the Stewart-Ellis classification.12 General coordinate expressions for the
metrics of LRS PMpf's had been determined previously13 up to one third-order
ordinary differential equation in the LRS I case and in closed form in the LRS III
case. Herewith all Petrov type D PMpf's are now fully classified. The only examples
of non-vacuum PMpf's previous to these two results are the axistationary rigidly
rotating PMpf's with circular motion14 in the LRS I case, and the p = /x/515
and Taub-NUT-like16 PMpf's in the LRS III case. A more detailed survey of all
investigations involving PMpf's may be found in.17
3. Petrov type I non-accelerating PMpf's
In the case where the acceleration ua of the perfect fluid vanishes, Eaf, is precisely the
part of the Weyl tensor that appears in the corresponding geodesic deviation equa-
*LW, the presentator of this talk, is a Ph. D. aspirant researcher of the Research Foundation -
Flanders (FWO).
2237
2238
tion. Thus this tensor may be seen as the general relativistic generalization of the
tidal tensor in Newtonian theory. Hab has no Newtonian analogue18 but determines
the acceleration of a spinning test particle initially comoving with the fluid.1 '20
Geodesic PMpf's may therefore be termed 'anti-Newtonian'.21 By the momentum
conservation equation and the Frobenius theorem they are either irrotational or
dust.22
3.1. Dust case (p = —A)
For PM dust and within the 1+3 covariant formalism based on u",23,24 the 'divE'
Bianchi constraint equation reads
[a,H]a-3HabLub + ^Dafi = 0, (1)
where [a, H]a is the vector spatially dual to the commutator of tensors <7°{, and Hat,
and Daf is the covariant spatial derivative of a function /.27 At the same time, the
covariant time evolution equations for /i, 9, u>a, aab and Ha(, form an autonomous
system of first order ordinary differential equations. Thus PM dust space-times
are 1+3 covariant 'silent universes'25 in the generalized sense of.26 Moreover, the
evolution equation for Haf, is decoupled from that of /i and the kinematic
quantities. Because of these facts, repeated covariant time evolution of (1) and
projection w.r.t. any orthonormal triad (eaa) leads to an in principle infinite chain
of linear and homogeneous equations in the components of da/i,da9,daU2,daU3
and Hap, parametrized by A,/j,9,uia and crap, where U% = o'at)aab — 2uauja and
U3 = <Jabat,caca + 3ctabU>au!b. These integrability conditions are very restrictive and
lead to the following
Conjecture 1: Purely magnetic dust-filled space-times do not exist.
This generalizes the conjecture stated in21 for the subcase of zero vorticity, which
has been proved28 for general Petrov type and cosmological constant. PM dust of
Petrov type D is also not allowed.17 Examples of further subcases which support
conjecture 1 are those where Dan = 017 and where the shear tensor is degenerate.29
3.2. Irrotational case (u>a = 0^
Whereas the restrictions tua = 0 or Da/i = 0 are inconsistent for purely magnetic
dust, we found the first and so far the only example of an algebraically general PMpf
model, emerging from the following theorem:17
Theorem 1. Up to a constant rescaling, the line element
ds2 = exp(-2e-4)(-d£2 + eUx2) + e\e~xdy2 + exdz2). (2)
represents the unique algebraically general PMpf which satisfies any two of the
three conditions ua = 0, wa = 0, Da/i = 0. This space-time is orthogonally spatially
2239
homogeneous (OSH)30 of Bianchi type VIq. The Petrov type is I{M°°) in the
extended Arianrhod-Mclntosh classification.31 while cjab commutes with Hab and is
degenerate in the plane orthogonal to the O-eigendirection of Hab- The equation of
state reads
V+P= 77<>-.P)m
I^-P)
(3)
The space-time starts off with a stiff matter-like big-bang singularity at a finite
proper time in the past and expands indefinitely towards an Einstein space.
References
1. C.B.G. Mcintosh, R. Arianrhod, S.T. Wade and C. Hoenselaers, Class. Quantum Grav.
11, 1555 (1994)
2. C.H. Brans, J. Math. Phys. 16, 1008 (1975)
3. B.M. Haddow, J. Math. Phys. 36, 5848 (1995)
4. M. Triimper, J. Math. Phys. 6, 584 (1965)
5. N. Van den Bergh, Class. Quantum Grav. 20, LI (2003)
6. N. Van den Bergh, Class. Quantum Grav. 20, L165 (2003)
7. J.J. Ferrando and J.A. Saez, Gen. Rel. Grav. 36, 2497 (2004)
8. E. Zakhary and J. Carminati, Gen. Rel. Grav. 37, 605 (2005)
9. J.J. Ferrando and J.A. Saez, Class. Quantum Grav. 20, 2835 (2003)
10. J.J. Ferrando and J.A. Saez, J. Math. Phys. 45, 652 (2004)
11. N. Van den Bergh and L. Wylleman, Class. Quantum Grav. 23, 3353 (2006)
12. J.M. Stewart and G.F.R. Ellis, J. Math. Phys. 9, 1072 (1968)
13. C. Lozanovski and J. Carminati, Class. Quantum Grav. 20, 215 (2003)
14. G. Fodor, M. Marklund and Z. Perjes, Class. Quantum Grav. 16, 453 (1999)
15. C.B. Collins and J.M. Stewart, Mon. Not. R. Astron. Soc. 153, 419 (1971)
16. C. Lozanovski and M. Aarons, Class. Quantum Grav. 16, 4075 (1999)
17. L. Wylleman and N. Van den Bergh, Phys. Rev. D 74, 084001 (2006)
18. G.F.R. Ellis and P.K.S. Dunsby, Astrophys. J. 479, 97 (1997)
19. A. Papapetrou, Proc. Roy. Soc. Lond. A209, 248 (1951)
20. C. Hillman, Electrogravitism versus Magnetogravitism, http://math.ucr.edu/home/
baez/PUB/electromagneto
21. R. Maartens, W.M. Lesame and G.F.R. Ellis, Class. Quantum Grav. 15, 1005 (1998)
22. J.L. Synge, Proc. Londom. Math. Soc. 43, 376 (1937)
23. G.F.R. Ellis, General Relativity and Cosmology, edited by R. K. Sachs (Academic,
New York, 1971)
24. R. Maartens, G.F.R. Ellis and S. Siklos, Cass. Quantum Grav. 14, 1927 (1997)
25. S. Matarrese, O. Pantano and D. Saez, Phys. Rev. Lett. 72, 320 (1994)
26. H. van Elst, C. Uggla, W.M. Lesame, G.F.R. Ellis and R. Maartens, Class. Quantum
Grav. 14, 1151 (1997)
27. R. Maartens and B.A. Bassett Class. Quantum Grav. 15, 705 (1998)
28. L. Wylleman, Class. Quantum Grav. 23, 2727 (2006)
29. L. Wylleman, PhD thesis, University of Ghent (2007)
30. G.F.R. Ellis and M.A.H. MacCallum Commun. Math. Phys. 12, 108 (1969)
31. R. Arianrhod and C.B.G. Mcintosh, Class. Quantum Grav. 9, 1969 (1992)
PURELY ELECTRIC PERFECT FLUIDS OF PETROV TYPE D
LODE WYLLEMAN*
Department of mathematical analysis, University of Ghent,
Galglaan 2, Gent 9000, Belgium
lwyllema@cage.ugent.be
The classification scheme for the complete class of purely electric perfect fluids (PEpf's)
of Petrov type D has been worked out and the main results are presented. The Bianchi
identities imply a subdivision of the solutions into three classes, with some remarkable
characteristic properties. Already known PEpf solutions of Petrov type D are categorized.
The scheme encloses previous classification results by Carminati, Wainwright, Barnes,
Rowlingson and Collins.
1. Introduction
A space-time for which the metric gab is a solution of the Einstein field equations
with perfect fluid source term,
Rab--Rgab = (n+p)uaub+pgab, (1)
and for which the magnetic part Hab of the Weyl tensor Cabcd w.r.t. the fluid 4-
velocity ua vanishes whereas the electric part Eab does not,1,2
Hab = ^TlacmnCmnbd UC U*, Eab = Cacbd UC U* + 0, (2)
is called a purely electric perfect fluid (PEpf). In (1) Rab is the Ricci trensor, R =
H — 3p is the Ricci scalar and a possible cosmological constant A is absorbed in the
fluid's energy density jj, and pressure p; in (2) r)abcd is the space-time permutation
pseudo-tensor. The covariant derivative of ua determines the kinematic quantities:
acceleration ua, vorticity tua, shear aab and expansion rate 6. PEpf's are either of
Petrov type I (algebraically general) or D (algebraically special). The latter form a
subclass of the so called 'aligned'3 perfect fluids of Petrov type D, whereby ua lies
in the plane of principal null direcions E.
Petrov type D PEpf's with an equation of state p = p(fi), \dp/d/j,\ < 1, were
investigated in.3 The main result was that one has either dp/d/j, = 0 (dust with
cosmological constant) or dp/dfx = 1 (stiff fluid-like space-time). At some places
the analysis heavily depended on the assumption p = p((i) such that the question
remained which differences arise and which parts may be recovered when this
assumption is dropped. In this proceeding the full classification scheme for Petrov
type D PEpf's is sketched; important solution families of this type occuring in the
literature are hereby naturally embedded. The formalism used for this analysis is
first introduced.
*LW is a Ph. D. aspirant researcher of the Research Foundation - Flanders (FWO).
2240
2241
2. ONP, a mixed real orthonormal/ complex null tetrad formalism
Aligned Petrov type D perfect fluids possess, in each space-time point, a one-
parameter family of canonical tetrads {m°, m°, e3°, e4a}. The fluid 4-velocity ua
plays the role of the basis vector e^a, while intersection of the orthogonal
complement u1- with £ determines the essentially unique unit spacelike vector e3a.
Complexification of T,1- finally yields two unique null directions, leaving a phase
freedom e2* for the complex null vector m realising gab m°mb = 1. The
formalism based on such a tetrad is denoted by ONP and the action of m on a scalar
function / by Sf. The set of independent connection variables may be subdivided
into four groups, according to the behaviour under the base change m —> el*m:
U,V (multiplication with e2**), A.X^W.Y^Z (multiplication with e1*), m, r, #3,^3
(invariant), and finally B, 77.3,^3 ('badly behaving') which might be absorbed into
new derivative operators by using a formalism which stands to ONP as GHP stands
to NP. Further details, such as the relations with the connection coefficients of an
orthonormal tetrad, where m = (ei — ie2)/\/2, are given in.4
Within ONP an aligned Petrov type D perfect fluid is locally rotationally
symmetric (LRS) if and only if U = V = W = X = Y = Z = A = 0.5 All members of
LRS class II in the Stewart-Ellis classification6 are PEpf's, but the reverse question
in which respect Petrov type D PEpf's deviate from being LRS II so far has only
been partially answered.3,7,8
3. Main results
Working in ONP, the remaining curvature variables are fj,,S = fj, + p and \&r =
—E33/2. The Bianchi equations readily yield U = 0 and m real. Together with four
Ricci equations they further lead to a natural division of the solutions into three
subclasses, which are denoted by IC, £- and £+. K. is characterized by Y = V =
X = A = 0 and r real, that is, its members are vorticity-free while aab and Eab
commute, both tensors being degenerate in E^; £- and £+ are characterized by
6*,- — 5 = 0 and 6\&r + 5 = 0, respectively. The most characteristic results for each
class are summed up below.
(1) K, consists of the non-conformally flat members of three fully known families:
the shearfree non-rotating Barnes family9 (r = #3), the non-accelerating, non-
rotating Szafron family10 [Z = 113 = 0, including the Szekeres dust inhomoge-
neous space-times11,12) and the LRS II class (Z = W = 0).
Non-rotating PEpf's in general were investigated in.13 It follows that the case 4.1.3
occuring there (corresponding to possible members of /C satisfying U3 = 0, r ^
63, Z ^ 0) is actually inconsistent.
(2) For £_ one has U = V = Y = Z = A = 0 and d0p = Sp = 0. For d3p = 0
(i.e. dust) the vector field e3° is orthogonal to hypersurfaces with zero extrinsic
curvature and one recovers the rotating dust solutions with conformally flat
2242
slices found by Stephani14-15 and Barnes16 (for zero and non-zero cosmologi-
cal constant, respectively), providing an alternative characterization of these
metrics.
(3) Any member of £+ satisfies U = W = X = Y = A=:0(in particular aab and
Eat, commute). It is either LRS II or it has a stiff fluid-like equation of state
dp/d/i = 1. In the latter case and when moreover Z = 0 (acceleration parallel
to £), we get (possibly rotating) generalizations of the Allnutt metrics.17 When
moreover V = 0 (<r06 degenerate in T,±) the space-time belongs either to /C or
to the shear-free, expansion-free, rotating families found by Collins.18
The reader is referred to19 for the calculations and a more detailed discussion.
References
1. A. Matte, Canadian J. Math. 5, 1 (1953)
2. L. Bel, Cah. Phys. 16, 59 (1962) (English translation Gen. Rel Grav. 32, 2047 (2000))
3. J. Carminati and J. Wainwright, Gen. Rel Grav. 17, 853 (1985)
4. L. Wylleman, Invariant classification of aligned Petrov type D purely electric perfect
fluids, to be submitted to Class. Quantum Grav.
5. S. W. Goode and J. Wainwright, Gen. Rel. Grav. 18, 315 (1986)
6. J. M. Stewart and G. F. R. Ellis, J. Math. Phys. 9, 1072 (1968)
7. J. Wainwright, Gen. Rel Grav. 8, 797 (1977)
8. J. Wainwright J. Math. Phys. 18, 672 (1977)
9. A. Barnes, Gen. Rel Grav. 4, 105 (1973)
10. D. A. Szafron, J. Math. Phys. 18, 1673 (1977)
11. P. Szekeres, Comm. Math. Phys. 41, 55 (1975)
12. J. D. Barrow and J. Stein-Schabes, Phys. Lett 103A, 315 (1984)
13. A. Barnes and R. Rowlingson, Class. Quantum Grav. 6, 949 (1989)
14. H. Stephani, Gen. Rel Grav. 14, 703 (1982)
15. H. Stephani, Class. Quantum Grav. 4, 125 (1987)
16. A. Barnes, Class. Quantum Grav. 16, 919 (1999)
17. J. A. Allnutt, 1982, PhD thesis, University of London
18. C. B. Collins, J. Math. Phys. 25, 995 (1984)
19. L. Wylleman, PhD thesis, University of Ghent (2007)
SELF-DUAL FIELDS ON THE SPACE OF A KERR-TAUB-BOLT
INSTANTON
ALIKRAM N. ALIEV
Feza Giirsey Institute, P.K. 6 Cengelkoy,
34-684 Istanbul, Turkey
aliev@gursey.gov. tr
CIHAN SAQLIOGLU
Faculty of Engineering and Natural Sciences,
Sabanci University, Tuzla, 81474 Istanbul, Turkey
saclioglu@sabanciuniv. edu
We discuss a new exact solution for self-dual Abelian gauge fields living on the space
of the Kerr-Taub-bolt instanton, which is a generalized example of asymptotically fiat
instantons with non-self-dual curvature.
1. Introduction
Gravitational instantons are complete nonsingular solutions of the vacuum Einstein
field equations in Euclidean space. The first examples of gravitational instanton
metrics were obtained by complexifying the Schwarzschild, Kerr and Taub-NUT
spacetimes through analytically continuing them to the Euclidean sector.1,2 The
Euclidean Schwarzschild and Euclidean Kerr solutions do not have self-dual
curvature though they are asymptotically flat at spatial infinity and periodic in imaginary
time, while the Taub-NUT instanton is self-dual. There also exist Taub-NUT type
instanton metrics3,4 which are not self-dual and possess an event horizon ("bolt").
Other examples of gravitational instanton solutions are given by the multi-centre
metrics.2 These metrics are asymptotically locally Euclidean with self-dual
curvature and admit a hyper-Kahler structure. The hyper-Kahler structure becomes most
transparent within the Newman-Penrose formalism for Euclidean signature.5
One of the striking properties of manifolds with Euclidean signature is that they
can harbor self-dual gauge fields. In other words, solutions of Einstein's equations
automatically satisfy the system of coupled Einstein-Maxwell and Einstein-Yang-
Mills equations. The corresponding solutions for some gauge fields and spinors that
are inherent in the given instanton metric were ontained in papers.6,7 In recent
years, there has been some renewed interest in self-dual gauge fields living on well-
known Euclidean-signature manifolds. For instance, the gauge fields were studied by
constructing a self-dual square integrable harmonic form on a given hyper-Kahler
space.8 The similar square integrable harmonic form on spaces with non-self-dual
metrics was found only for the the Euclidean-Schwarzschild instanton.9 As a
generalized example of this, we shall consider the Kerr-Taub-bolt instanton and construct
a square integrable harmonic form to describe self-dual Abelian gauge fields
harbored by the instanton.
2243
2244
2. The Kerr-Taub-bolt instanton
The Kerr-Taub-bolt instanton4 is a Ricci-flat metric with asymptotically flat
behaviour. It has the form
ds2 = ^(^-+d62\ +S^J-(adt+Prdvf+ ^{dt + Pgdvf , (1)
where the metric functions are given by
A = r2 - 2Mr - a2 + N2 , 2 = Pr - aPg = r2 - (N + acos6)2 ,
Pr = r2-a2- ** , Pg =-asm2 9 + 2N cose- °N\ . (2)
N2 — a2 ivz — az
The parameters M, N, a represent the "electric" mass, "magnetic" mass and
"rotation" of the instanton, respectively. The coordinate t in the metric behaves like
an angular variable and in order to have a complete nonsingular manifold at values
of r defined by equation A = 0 , t must have a period 2tt/k . The coordinate
(p must also be periodic with period 2ir (1 — Q/k), where the "surface gravity"
k = (r+ — r_)/2r2 , the "angular velocity" of rotation il = a/r2 and
r± = M± y/M2 - N2 + a2 , r2Q = r\ - a2 - N4/(N2 - a2) . (3)
As a result one finds that the condition k= 1/(4|_/V|) along with S > 0 for r > r+
and 0 < 6 < n guarantees that r = r+ is a regular bolt in (1) . Clearly, the
isometry properties of the Kerr-Taub-bolt instanton with respect to a U(l) - action
in imaginary time imply the existence of the Killing vector field dt = &•, d^ .
3. Harmonic 2-form
We shall use the above Killing vector field to construct a square integrable harmonic
2-form on the Kerr-Taub-bolt space. We start with the associated Killing one-form
field £ = £(t)M dx^. Taking the exterior derivative of the one-form in the metric (1)
we have
d£ = ^2 { iMr2 + (aM cos ° ~ 2Nr + MN) (N + a cos e)\ e1 A e4 (4)
- [N (A + a2 + a2 cos2 6) + 2 a{N2 - Mr) cos 6] e2 A e3 } ,
where we have used the basis one-forms satisfying the simple relations of the Hodge
duals: * (e1 A e4) = e2 A e3 , * (e2 A e3) = e1 A e4 . Straightforward calculations
show that the two-form (4) is both closed and co-closed, that is, it is a harmonic
form. However the Kerr-Taub-bolt instanton does not admit hyper-Kahler structure,
and the two-form is not self-dual.
4. Stowaway fields
To describe the Abelian "stowaway" gauge fields we define the (anti)self-dual two
form
F=~(d^±*dO, (5)
2245
where A is an arbitrary constant related to the dyon charges carried by the fields.
Using in this expression the two-form (4) and its Hodge dual we obtain the harmonic
self-dual two-form10
F=MM__N)_ {r + N + acosef(ei Ae4 + e2 Ae3) ^ (6)
which implies the existence of the potential one-form
A= -X(M-N)
r + N + acos9
cos 0 dip H — (at + Pg dip))
(7)
From equation (5) one can also find the corresponding anti-self-dual two-form. The
square integrability of these harmonic two-forms can be shown by explicitly
integrating the Maxwell action. For the self-dual two-form we have
Since this integral is finite, the self-dual two-form F is square integrable. The total
magnetic flux
S>=HF = 2X(M-N)(l-^-), (9)
must be equal to an integer n because of the Dirac quantization condition. We see
that the periodicity of angular coordinate in the Kerr-Taub-bolt metric affects the
magnetic-charge quantization rule in a non-linear way. It involves both the " electric"
and "magnetic" masses and the "rotation" parameter.
References
1. S. W. Hawking, Phys. Lett. 60A, 81 (1977).
2. G. W. Gibbons and S. W. Hawking, Phys. Lett. B 78, 430 (1978).
3. D. N. Page, Phys. Lett. 78B, 249 (1978).
4. G.W. Gibbons and M. J. Perry, Phys. Rev. D 22, 313 (1980).
5. A. N. Aliev and Y. Nutku, Class. Quant. Grav. 16, 189 (1999).
6. S. W. Hawking and C. N. Pope, Phys. Lett. B 73, 42 (1978).
7. C. Sa§hoglu, Class. Quantum Grav. 17, 485
8. G. W. Gibbons, Phys. Lett. B 382, 53 (1996).
9. G. Etesi, J. Geom. Phys. 37, 126 (2001).
10. A. N. Aliev and C. Saclioglu, Phys. Lett. B 632, 725 (2006).
THE KERR THEOREM, MULTISHEETED TWISTOR SPACES
AND MULTIPARTICLE KERR-SCHILD SOLUTIONS*
ALEXANDER BURINSKII
Gravity Research Group, NSI Russian Academy of Sciences,
B. Tulskaya 52, Moscow 115191, Russia, bur@ibrae.ac.ru
Kerr-Schild formalism is generalized by incorporation of the Kerr Theorem with
polynomials of higher degrees in Y £ CP1. It leads to multisheeted twistor spaces and
multiparticle solutions.
1. Introduction
The Kerr-Newman solution displays many relationships to the quantum world. It
is the anomalous gyromagnetic ratio g = 2, stringy structures and other features
allowing one to construct a semiclassical model of the extended electron1 3 which
has the Compton size and possesses the wave properties.
One of the mysteries of the Kerr geometry is the existence of two sheets of
space-time, (+) and ( —), on which the dissimilar gravitation (and electromagnetic)
fields are realized, and fields living on the (-f)-sheet do not feel the fields of the (—)-
sheet. Origin of this twofoldedness lies in the Kerr theorem, generating function F
of which for the Kerr-Newman solution has two roots which determine two different
twistorial structures on the same space-time.
The standard Kerr-Schild formalism is based on a restricted version of the Kerr
theorem which uses polynomials of second degree in Y, and, in fact, produced only
the Kerr geometry. The use of Kerr theorem in full power is related with the
treatments of polynomials of higher degrees in Y. On this way we obtain the multisheeted
twistor spaces and corresponding multiparticle Kerr-Schild solutions.4,5 The case of
a quadratic in Y generating function of the Kerr Theorem F{Y) was investigated in
details in.6'7 It leads to the Kerr spinning particle (or black hole) with an arbitrary
position, orientation and boost. Choosing generating function F(Y) as a product of
partial functions F{ for spinning particles i=l,...k, we obtain multi-sheeted, multi-
twistorial space-time over M4 possessing unusual properties. Twistorial structures
of the i-th and j-th particles turn out to be independent, forming a type of its
internal space. However, the exact solutions show that gravitation and electromagnetic
interaction of the particles occurs via the connecting them singular twistor lines.
The space-time of the multiparticle solutions turns out to be covered by a net of
twistor lines, and we conjecture that it reflects its relation to quantum gravity.
Recall that the Kerr-Newman metric can be represented in the Kerr-Schild
form g^v = r]^v + 2hkfikv, where r]^v is metric of auxiliary Minkowski space-time,
and h = (rnr — e2/2)/(r2 + a2 cos2 9). kfl(x) is a twisting null field, which is tangent
to the Kerr principal null congruence (PNC) which is geodesic and shear-free.7'9
"Talk at the GT6 session of the MG11 meeting, supported by RFBR grant 07-08-00234 and by
travel grant from J.Sarfatti.
2246
2247
PNC is determined by the complex function Y(x) via the one-form
e3 = du + Yd( + Yd(-YYdv = Pk^dx^ (1)
where u, v, £, ( are the null Cartesian coordinates. Here P is a normalizing factor
for k^ which provide k$ = 1 in the rest frame. The null rays of the Kerr congruence
are twistors.
The Kerr theorem8,9 allows one to describe the Kerr geometry in twistor
terms.7
It claims that any geodesic and shear-free null congruence in Minkowski space-
time is denned by a function Y(x) which is a solution of the equation
F = 0, (2)
where F(Y, Ai,A2) is an arbitrary holornorphic function of the projective twistor
coordinates
Y, \l=(-Yv, A2 = u + YC (3)
In the Kerr-Schild backgrounds the Kerr theorem acquires a more broad content,
allowing one to determine the normalizing function P and complex radial distance
f = r + ia cos 8,
P = d\t F — Yd\2F, f = PZ~X = — dF/dY.and therefore, restore all the
necessary characteristics of the corresponding solutions, including the electromagnetic
field of the corresponding Einstein-Maxwell equations up to an arbitrary function.
The position of singular lines, caustics of PNC, corresponds to f = 0, and is
determined by the system of equations F = 0; dF/dY = 0 .
Multi-twistorial space-time. Selecting an isolated i-th particle with
parameters qi, one can obtain the roots Yi (x) of the equation Fi{Y\q{) = 0 and express
Fi in the form F%{Y) = Ai{x){Y -Y+)(Y -Y~). Then, substituting the (+) or (-)
roots Yi (x) in the relation (1), one determines congruence k^(x) and consequently,
the Kerr-Schild ansatz for metric g^l = j]^ + 2h^k^'kv\
and finally, the function h^(x) may be expressed in terms of fi = —dyFi.
What happens if we have a system of k particles? One can form the function
F as a product of the known blocks Fi(Y), F(Y) = ]J^=1 Fi(Y). The solution of
the equation F = 0 acquires 2k roots Yi , and the twistorial space turns out to be
multi-sheeted. The twistorial structure on the i-th (+) or (—) sheet is determined
by the equation Fi = 0 and does not depend on the other functions Fj, j ^ i.
Therefore, the particle i does not feel the twistorial structures of other particles.
Similar, the condition for singular lines F = 0, dyF — 0 acquires the form
k k k
IIF' = 0' Y,IiFidY^ = 0 (4)
1 = 1 i=l Ijti
and splits into k independent relations Ft = 0, [\i=£i FidyFi = 0.
2248
One sees, that i-th particle does not feel also singular lines of other particles. The
space-time splits on the independent twistorial sheets, and therefore, the twistorial
structure related to the i-th particle plays the role of its "internal space". It looks
wonderful. However, it is a direct generalization of the well known twofoldedness of
the Kerr space-time which remains one of the mysteries of the Kerr solution for the
very long time. The negative sheet of Kerr geometry may be treated as the sheet of
advanced fields. In this case the source of spinning particle turns out to be the Kerr
singular ring (circular string,2'3) with the electromagnetic excitations in the form of
traveling waves which generate spin and mass of the particle (microgeon model1'3).
Multi-particle Kerr-Schild solution. Using the Kerr-Schild formalism with
the considered above generating functions Yli=i FiO^) = 0, one can obtain the exact
asymptotically flat multi-particle solutions of the Einstein-Maxwell field equations.
Since congruences are independent on the different sheets, the congruence on the
i-th sheet retains to be geodesic and shear-free, and one can use the standard Kerr-
Schild algorithm of the paper.8 One could expect that result for the i-th sheet will
be in this case the same as the known solution for isolated particle. Unexpectedly,
there appears a new feature having a very important consequence.
In addition to the usual Kerr-Newman solution for an isolated spinning
particle, there appears a series of the exact 'dressed' Kerr-Newman solutions which take
into account surrounding particles and differ by the appearance of singular twistor
strings connecting the selected particle to external particles. This is a new
gravitational phenomena which points out on a probable stringy (twistorial) texture of
vacuum and may open a geometrical way to quantum gravity.
References
1. A.Burinskii, Sov. Phys. JETP, 39(1974)193., W.Israel, Phys. Rev. D2 (1970) 641;
2. A. Burinskii, Grav.&Cosmol.lO, (2004) 50; hep-th/0403212.
3. A. Burinskii Phys.Rev. D 70, 086006 (2004); hep-th/0406063.
4. A.Burinskii, Grav.&Cosmol.ll, (2005) 301; hep-th/0506006.
5. A.Burinskii, Grav.&Cosmol.l2,(2006) 119; gr-qc/0610007;
Int.J.Geom.Meth.Mod.Phys.,iss.2 (2007)(to appear); hep-th/0510246.
6. A. Burinskii and G. Magli, Phys. Rev. D 61(2000)044017; gr-qc/9904012.
7. A. Burinskii, Phys. Rev. D 67 (2003) 124024; gr-qc/0212048.
8. G.C. Debney, R.P. Kerr, A.Schild, J. Math. Phys. 10(1969) 1842.
9. D.Kramer, H.Stephani, E. Herlt, M.MacCallum, "Exact Solutions of Einstein's Field
Equations", Cambridge Univ. Press, 1980.
ELECTRICAL FORCE LINES OF A 2-SOLITON SOLUTION OF
THE EINSTEIN-MAXWELL EQUATIONS
M. PIZZI
ICRA, Rome University "La Sapienza", p.le Aldo Mora 5, 00185 Rome, Italy
We briefly summarize the main features of a 2-soliton solution which describes an exact
(nonlinear) superposition of a Schwarzschild black hole near a Kerr-Newman (KN) naked
singularity. Then we give the force lines of the electrical field showing that also the black
hole has a charge in the resulting solution (parameter-mixing phenomenon). At the same
time we suggest that the plotting of the force lines can be a useful tool to understand
complicated solutions of the Einstein-Maxwell, whose deep understanding is still lacking
in literature.
Keywords: electric force lines, soliton method, exact solutions
1. Introduction; the soliton method
The coupled Einstein-Maxwell equations are a very hard-to-solve non-linear
problem, however in the last twenty-years there were developed some different techniques
which admit to face the problem in an exact way at least in some special cases (see
e.g. Refs. 1,2,3). The solution which we refer at here has been obtained15 with the
soliton method (Belinski and Zakharov4 , and Alekseev5 ; for a self-consistent review
see Ref.ll), which allows to find solutions of the form:
ds2 = gtt{p, z)dt2 + gtip(p, z)dtd<p + gvv,(p, z)dip2 + f(p,z)(dp2 +dz2), (1)
with gu9ipip — (fft<p)2 = —p2, and for the electromagnetic potential
'At = At{p,z) . _ ()
Then we considered a particular case of the 2-soliton solution, which has been
constructed adding one soliton to the Schwarzschild background —this is a different
way from the one that adds 2-solitons on the Minkowski background12'14 , and it
allows to find a solution with horizon. A great unpleasant feature of such solutions
is that it is very difficult to extract physical informations, since they are very
complicated. However we give an example how to get easy-to-see informations plotting
the force lines of the electrical field in the Hanni-Ruffini way.7
2. The main features of the solution; the electric force lines
The solution is stationary, axial-symmetric and asymptotically flat, and has five
physical parameters: mi, m2, Qtot, Atot and I, which are respectively mass of the
first and second source, total charge and total angular momentum, and the distance
on the z-axis of the two singularities. The physical interpretation we give is that
it describes a KN naked singularity linked by a 'strut' to a charged black hole.
Indeed, on the axis, between the two bodies, it is present an anomaly region which
2249
2250
consists of a conic singularity (i.e. for a small circumference L surrounding the axis
linip^o ^~ ^ 1) anc^ a "tube-singularity" (i.e. gvv < 0 near the axis, which means
that the angle <p becomes timelike). Unfortunately that anomaly, which hampers to
give an easy physical interpretation, is unavoidable;9 people usually call it 'strut' or
'string'. However outside that region the solution has a good behavior; furthermore
when it will be found such a regular solution10 it will be interesting to see how much
that 'string' modifies the gravitational and electric fields. We focused our attention
on the case in which the naked source has a much more smaller mass respect to the
black hole. In that case the anomaly region will be very small, practically coincident
with the segment of the axis between the two sources. The resulting force lines are
given in the Fig. 1.
3. Conclusions
In spite of the mathematical construction which suggest a neutral black hole near a
KN source, we found that also the black hole presents a charge. This seems to be a
typical non-linear effect of the superposition of two solitons for which the relations
(a) (b)
Fig. 1. Force lines of a small charge near a charged black hole at different distance: at r = 4m in
(a), and at r = 'Am in (b). The semicircle is the Schwarzschild horizon; the dotted line represent
the strut. The presence in both the graph of a separatrix (bold line) means that the black hole
has a charge of opposite sign.
2251
between the physical and the mathematical parameters, which are direct when the
sources are separated enough, become mixed and much more complicated; we called
it parameter-mixing phenomenon.15
Finally we want to stress that the method of plotting the force lines, which
has not yet been used quite at all in literature, can be indeed usefully applied to
understand the physical meaning of such complicated solutions of the Einstein-
Maxwell equations.
Acknowledgments
I wish to thanks prof. Belinski for the help along all the work, and prof. Alekseev
for the enlighting discussion. Finally I am grate to prof. Ruffini and the ICRA
institution for the supervision and the financial support.
References
1. A. Tomimatzu. Prog. Theor. Phys. 71, 409, 1984.
2. G. P. Perry and F. I. Cooperstock. Class. Quantum Grav. 14 (1997) 1329-1345.
3. G.A. Alekseev. Annalen der Physik (Leipzig), v.9, Spec. Issue, p.SI-17- SI-20 (2000),
and arXiv:gr-qc/9912109 vl 27 Dec 1999.
4. V.A. Belinski, V.E. Zakharov. Sov. Phys. JETP 50, 1 (1979)
5. G.A. Alekseev. JETP Lett, 32 277 (1980).
6. E. T. Copson. Roy. Soc. Pro., A, vol. 116, p. 720, 1927.
7. R. Hanni, R. Ruffini. Phys. Rem. D, 8 (10), pagg 3259-3265, 1973.
8. D. Bini, A. Geralico, R. Ruffini. Phys. Rev. D 75, 044012.(2007).
9. V. Belinski, J. of the Korean Phys. Soc, Vol 49, No.2, Aug. 2006.
10. G.A.Alekseev, V. Belinski. Equilibrntm static configuration of two charged masses in
general relativity.(To be published), 2006.
11. V.A. Belinski, E. Verdauger, Gravitational Solitons, (Cambridge University Press,
2001).
12. G.A. Alekseev. Proceedings of the Steklov Institute of Mathematics (Providence, RI:
American Mathematical Society) vol.3, page 215, 1988.
13. A.D. Dagotto, J. Gleiser, CO. Nicasio. Two-soliton solutions of the Einstein-Maxwell
equations, Class. Quantum Grav 10, pagg. 961-973, 1993.
14. A. Garate, J. Gleiser, CO. Nicasio. Cylindrical-spherical Einstein-Maxwell solitons.
Class. Quantum Grav. 11 (1994) 1519-1533.
15. M. Pizzi. Gravitational and electric fields of a 2-soliton solution. (Accepted by
IJMPD), 2007.
MONODROMY TRANSFORM APPROACH IN THE THEORY OF
INTEGRABLE REDUCTIONS OF EINSTEIN'S FIELD
EQUATIONS AND SOME APPLICATIONS*
GEORGE ALEKSEEV
Steklov Mathematical Institute, Gubkina 8, Moscow 119991, Moscow, Russia
G.A.Alekseev@mi.ras.ru
A brief sketch of the formulation of the monodromy transform approach and
corresponding integral equation methods as well as of various applications of this approach
for solution of integrable symmetry reductions of Einstein's field equations is presented.
1. Introduction
For various nonlinear systems integrable by the well known Inverse Scattering
Method (called sometimes also the Scattering Transform), the spaces of solutions
are parameterized in terms of the scattering data of the corresponding potentials in
the associated Schrodinger-like equation (associated spectral problem). The
scattering data consist of a set of coordinate independent functions of a spectral parameter
which characterize uniquely every potential (solution) and which can serve as the
"coordinates" in the space of solutions of a given completely integrable system.
In some physically important cases of the symmetry reduced Einstein equations,
the spaces of local solutions also can be parameterized by a finite set of coordinate
independent functions of a complex ("spectral") parameter w, which determine the
branching (monodromy) properties of a fundamental solution of associated linear
systems. These data exist for any local solution and thus, in the infinite-dimensional
space of local solutions we have two systems of " coordinates" - the sets of functional
parameters whose particular values characterize every local solution uniquely:
the monodromy data:
u±(w), v±(w), ...
the field components:
gik(xl,x2), Ai(x\x2), ...
The key difference between these " coordinates" is that the field components should
satisfy the field equations, while the space of monodromy data functions is uncon-
straint: for arbitrarily chosen set of these functions there exists a uniquely
determined local solution of the field equations. The "coordinate transformation" from
the monodromy data to the field components effectively solves the field equations.
That is why we call the approach using this transformation for solution of symmetry
reduced Einstein equations as the "monodromy transform" approach.
The construction of the monodromy transform1 provides a unified general base
for solving of various integrable symmetry reductions of Einstein's field equations
"This research has been partially supported by the Russian Foundation for Basic Research (grants
05-01-00219, 05-01-00498, 06-01-92057-CE) and the programs Mathematical Methods of
Nonlinear Dynamics of the Russian Academy of Sciences, and "Leading Scientific Schools of Russian
Federation" (grant NSh-4710.2006.1).
2252
2253
including the Einstein equations for vacuum, the Einstein - Maxwell and the Einstein
- Maxwell - Weyl equations for gravitational, electromagnetic and classical neutrino
fields as well as for the Einstein equations in higher dimensions which determine
the low-energy dynamics of the bosonic sector of some string gravity models.2
A large variety of physically different types of field configurations can be
considered in the framework of this approach. These include the stationary axisymmetric
fields of compact sources or asymptotically non-flat fields describing the interaction
of these sources with various external fields, the fields of accelerated sources with
boost-rotation or boost-translation symmetries, various wave fields such as colliding
and nonlinearly interacting waves with smooth profiles or some discontinuities on
the wavefronts and having plane, spherical, cylindrical, toroidal or some other forms
of the fronts, as well as different inhomogeneous cosmological models with two
commuting spatial symmetries. Below we outline some key-points of the monodromy
transform approach and mention some its applications.
2. Parameterization of the solution space by monodromy data
For electrovacuum Einstein - Maxwell fields depending on two coordinates, any local
solution with the complex Ernst potentials £ and $, is characterized uniquely by
the monodromy data which consist of the four functions of the spectral parameter
w holomorphic in some local regions of the spectral plane:
{^(a;1,*2),^1,*2)} <—> {u±H,v±H} (1)
For vacuum fields ^(x1,^;2) = 0 <-> v±(w) = 0 and the space of solutions is
parameterized by the monodromy data which consist of two arbitrary holomorphic
functions u±(w). For the structure of the monodromy data for other fields see1'2.
To determine the monodromy data for given solution of Einstein equations, one
should solve an overdetermined linear system of differential equations whose
coefficients depend on the field components of a given solution and their first derivatives.
3. Constructing solutions for arbitrary monodromy data
All components and potentials of a general local solution of electrovacuum Einstein
- Maxwell equations can be expressed in quadratures in terms of the monodromy
data (1) and of the corresponding solution of a master system of linear singular
integral equations whose kernels and rhs are expressed algebraically in terms of the
monodromy data. In particular, given monodromy data, the Ernst potentials are
£{x\x2) =e0- J{\}(k(0<pW(()d(, $>{x\x2) =y"[A]cfc(C)<PM(CK (2)
L L
where e0 = ±1 is the value of £ at some initial point; £ 6 L and the contour L on
the spectral plane consists of two disconnected parts L+ and L_ with the endpoints
(£o,£) and (VoyV) depending on the coordinates a;1, x2 and coordinates of a chosen
initial point; the value [A]^ is a jump at the point ( 6 L of a "standard" branching
function A = ^(C - 0(C - *?)/(C - &)(C - *fe) and the "weight" (7r/2)fc(C) = 1 +
2254
koCut(C), with ut(C) = u(C); the functions q>[u](Q = <pH(xl ,x2,(), cp[v,(0 =
cp W(xl ,x2,Q should satisfy the linear singular integral equations with the same
scalar kernel and different rhs, both depending on the monodromy data (t, ( e L)1:
1 JK,{x\x\tX)
L
Actually, each of these equations in general is a coupled pair of two integral
equations, because each function on the disconnected parts L± of the contour is
represented by two indpendent functions, e.g. u(t) should be understood as u(t) = u+ (t)
for t e L+ and u(t) = u_(t) for t e L_ and the same is for v(t), cp '"'(C), V 'v'(0-
For different problems the master integral equations (3) admit useful
modifications. For stationary axisymmetric fields, the regularity axis condition implies
u+(t) = u_(t), v+(t) = v_(t), and therefore, cp^C) = cpLu](C), <p£](0 =
(p - (0- This allows to merge L+ and L_ and reduce (3) to a simple scalar form
similar to Sibgatullin's modification of the Hauser-Ernst integral equations. For
the hyperbolic case, (3) can be reduced to the quasi-Fredholm integral "evolution
equations" well adapted for solving of the characteristic initial value problems.3
4. Applications
For all of gravitationally interacting fields and for each type of field configurations
mentioned in the Introduction, the developed approach suggests the effective tools
for analysis of the structure of the whole space of local solutions, a comparison of
different solution generating techniques (see, e.g.,4), construction of infinite
hierarchies of exact solutions with arbitrary finite number of free parameters including
multi-parametric generalizations and analytical continuations of many known
solutions in the space of their parameters5-7, analysis of asymptotical behaviour of
some classes of fields, solution of the Cauchy and characteristic initial value
problems for hyperbolic cases3,8 as well as of the boundary value problems for elliptic
cases of integrable reductions of Einstein's field equations. It is clear, however, that
in all of the directions outlined above a further work is necessary for the searches
of new interesting developments of these methods and their practical applications
for solving of various physically interesting problems.
References
1. G.A.Alekseev, Sov.Phys.Dokl. 30, 565 (1985); Proc. Steklov Math. Inst, American
Math. Soc, 3, 215 (1988); Theor. Math. Phys. 143, 720 (2005); gr-qc/0503043.
2. G.A. Alekseev, Theor. Math. Phys. 144, 1065 (2005); hep-th/0410246.
3. G.A. Alekseev, Theor. Math. Phys. 129, 1466 (2001); gr-qc/0105111.
4. G.A. Alekseev, Physica D 152, 97 (2001); gr-qc/0001012.
5. G.A. Alekseev, Abstracts of GR13 Int. Conf., Cordoba, Argentina, 3 (1992).
6. G.A. Alekseev and A.A. Garcia, Phys.Rev. D53, 1853 (1996).
7. G.A. Alekseev and J.B. Griffiths, Phys.Rev.Lett. 84, 5247 (2000).
8. G.A. Alekseev and J. B. Griffiths, Class. Quantum Grav. 21, 5623 (2004).
^{x\x\Q\
<pM(x\x2,0'
u(t)
v(t)
(3)
CLOSED TIMELIKE CURVES AND GEODESICS
OF GODEL-TYPE METRICS
OZGUR SARIOGLU
Department of Physics, Faculty of Arts and Sciences,
Middle East Technical University, 06531, Ankara, Turkey
sarioglu@metu. edu. tr
It is shown that the spacetimes described by Godel-type metrics with both flat and non-
flat backgrounds and with constant Ufc always have CTCs or CNCs. The geodesic curves
of these spacetimes are characterized by a lower dimensional Lorentz force equation for
a charged point particle in the relevant Riemannian background. An explicit example is
given for which timelike and null geodesies can never be closed.
1. The Godel metric1 in General Relativity
ds2 = -(dx0)2 + (dx1)2 - l- e2xl (dx2)2 + (dx3)2 -2exl dx° dx2 (1)
describes a stationary, homogeneous, uniformly rotating rigid universe full of an
incoherent pressureless perfect fluid. It admits closed timelike (CTCs) and closed
null curves (CNCs) but contains no closed timelike or closed null geodesies and is
geodesically complete.2 The existence of CTCs and CNCs in the Godel spacetime
can be best inferred by transforming (1) to the cylindrical coordinates as1,2
ds2 = -dr2 + dr2 + dz2 - sinh2 r (sinh2 r - 1) dip2 + 2\fl sinh2 rdipdr .
It follows that the curve C = {(t,r,ip,z)\t = to,r = ro,z = zo,ip 6 [0,2n}},
where to,ro and zq are constants, is a CTC for r0 > ln(l + \/2) and a CNC for
r0 = ln(l + y2).1'2
The Godel metric (1) can be thought of cast in the form g^v = h^v — u^uv in
two inequivalent ways: In the first, the 'background' h^v is a non-flat 3-metric and
u^ = 5° + ex S2 is a timelike unit vector; whereas in the second, the 'background'
h^u = diag(l, 1, 0,1) describes a flat 3-dimensional spacetime and the new u^ =
V25° + (l/\/2)ex <52 is once again a timelike unit vector.
The Godel-type metrics,3'4 which provide new solutions to various gravitational
theories in diverse dimensions, are of the form g^v = h^v — u^uv, where the
background hpu is the metric of an Einstein space of a (D — l)-dimensional Riemannian
geometry in the most general case and u^ is a timelike unit vector. One also
assumes that both h^ and u^ are independent of the fixed special coordinate xk with
0 < k < D — 1 and, that hk^, = 0. A detailed analysis has already been given3'4
corresponding to the two distinct cases Ut = const and Uk ^ const. The Godel-type
metrics with Uk = const provide solutions to the Einstein-Maxwell equations with a
dust distribution in D dimensions, for which the only essential field equation is the
source-free 'Maxwell's equation' in the relevant background.3 When Uk ^ const,
the confonnally transformed Godel-type metrics can be used in solving a rather
general class of Einstein-Maxwell-dilaton-3-form field theories in D > 6 dimensions,
2255
2256
for which all the field equations reduce to a simple 'Maxwell equation' in the
corresponding (D — l)-dimensional Riemanuian background.4 In fact, the Godel-type
metrics can be used in obtaining exact solutions to various supergravity theories,
in which case Uk may be considered as related to a dilaton field.3'4
The discussion of the CTCs in the literature seems to be restricted to an
investigation of the curves parametrized as the curve C above. However, it is obvious that
there can be other classes of curves that can be a CTC or a CNC. This contribution
gives a brief summary of a detailed analysis5 of these special curves in geometries
described by Godel-type metrics with u^ = const.
2. Let us assume that the fixed special coordinate xk equals x° = t, the background
h^v describes a flat Riemannian geometry, ho^ = 0 and uq = 1. We will take D = 4
but what follows can easily be generalized to higher dimensions.5 Then the Godel-
type metric with the line element
ds2 = dp2 + p2d<j)2 + dz2 - (dt + s(p, 0) dzf (2)
solves the charged dust field equations provided s(p, <p) is a harmonic function in
two dimensions.3 Consider the most general curve C = (£(r?), p(r/), 0(r/), z(r?)), where
the arc-length parameter r\ 6 [0, 2ir]. Normalizing the tangent vector of C to unity
in the geometry described by (2), one finds
I— <*<+Va+(£)'+(I),+"(3)'"=±i- (3)
where A = 0 for null and A = 1 for timelike curves. Now let the parametrizatioiis of
p, <p and z be all periodic functions in r\. Then the terms in the square root in (3)
can be expanded in a Fourier series in the interval [0, 27r] and it is clear that this
Fourier series expansion has a non-negative constant term in it which looks like
Since B ^ 0, i(r/) naturally picks up a non-periodic piece Br\ from the second term
on the right hand side of (3) and, for no CTCs or CNCs, it must be that
f2lT dz
J^ 8(p(ri),(l>(Ti))—dri = 0 (4)
for all arbitrary periodic functions z(r]). However, since p(rj) and 0(r/) are periodic
functions of 77, it follows that s(p(r]), 0(r/)) is also periodic in r/. In this case, one can
expand s(p(r}), <f>(r})) in a Fourier series in r\ as
00
s(p(v), <t>(v)) = ao+ g(v) =ao + ^2 (aP cosP'rl + bv sinpr/),
p=i
where clq, a,k and bk are the usual Fourier coefficients. Now for no CTCs, (4) implies
that J0 n g(rj) jf- drj = 0. If one chooses the periodic function z{rj) so that dzjdr\ =
(7(77), then Jq* (<?(r/))2 di] = 0, which is possible only if g(rj) = 0, (or s{p,4>) =
2257
const.) Therefore, unless s(p,<f>) = const (for which (2) becomes flat with no CTCs
or CNCs), one can always cancel out the Br] term above and find a CTC or a CNC
in the spacetime described by (2). Thus, one can always find a CTC or a CNC in
the geometry of (2) given an arbitrary non-constant harmonic function s(p,(p).
This discussion can also be generalized to Godel-type metrics with non-flat
backgrounds but with constant Uk'-5 The spacetimes described by Godel-type metrics with
both flat and non-flat backgrounds always have CTCs or CNCs, provided that at least
one of the Ui(xe) ^ const.
3. The geodesies of Godel-type metrics are described by the analogous (D — 1)-
dimensional Lorentz force equation for a charged point particle written in the
corresponding Riemannian background.5 As an example, consider the geodesies of the
spacetime described by (2), written using the Cartesian coordinates (x, y, z) instead
of the cylindrical coordinates. Then, the geodesic curve x^(t) must satisfy5
i + s(x(r),y(r)) i =—e = const, and z + es(x(r), y(r)) =£ = const, (5)
x — e(ds/dx) (I - es(x,y)) = 0, and y - e (ds/dy) (I - es(x, y)) = 0, (6)
subject to the constraint x2 + y2 = A + e2 — (i — es(x, y))2, where A = —1, 0 for
timelike and null geodesies, respectively, and a dot denotes derivative with respect
to the affine parameter r. Consider s(x, y) to be a linear function of its arguments
as a simple example. Then one finds that t(r) = — § (l — -^s) t + g(r), where g(r)
contains all the parts periodic in r. Hence for closed geodesies, one needs that
A = e2! However, this is not possible and one concludes that there are no closed
timelike or null geodesies in the spacetime described by (2) when s(x,y) is linear in
x and y. [When e = 0, x(t), y(r) and z(t) become linear functions in r and t(r)
becomes a quadratic function in r, which obviously do not describe closed geodesies
then.] It is conjectured that this result can also be generalized to the case of more
general harmonic functions s(x,y).5
4. It would be interesting to examine the existence of CTCs in spacetimes described
by the most general Godel-type metrics with non-flat backgrounds and non-constant
uk-
References
1. K. Godel, Rev. Mod. Phys. 21, 447 (1949).
2. S. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time (Cambridge,
Cambridge University Press, 1977).
3. M. Gurses, A. Karasu and O. Sarioglu, Class. Quant. Grav. 22, 1527 (2005) [arXiv:hep-
th/0312290].
4. M. Gurses and O. Sarioglu, Class. Quant. Grav. 22, 4699 (2005) [arXiv:hep-
th/0505268].
5. R. J. Gleiser, M. Gurses, A. Karasu and O. Sarioglu, Class. Quant. Grav. 23, 2653
(2006) [arXiv:gr-qc/0512037].
CONFORMAL SYMMETRIES IN SPHERICAL SPACETIMES
S. D. MAHARAJ*
Astrophysics and Cosmology Research Unit, School of Mathematical Sciences, University of
KwaZulu-Natal, Private Bag X54001, Durban, ^000, South Africa
maharaj@ukzn. ac.za
S. MOOPANAR
Astrophysics and Cosmology Research Unit, School of Mathematical Sciences, University of
KwaZulu-Natal, Private Bag X54OOI, Durban, ^000, South Africa
moopanar@ukzn. ac.za
We obtain the general conformal symmetry for spherically symmetric spacetimes and
regain the static solution. The general inheriting conformal symmetry is found by using
the condition that fluid flow lines are mapped conformally.
1. Introduction
The study of conformal symmetries in general relativity is important as they
provide a deeper insight into the spacetime geometry and assist in the generation
of exact solutions of the Einstein field equations. Unfortunately very few
conformal symmetries are known even in spacetimes of high symmetry. Here we describe
the conformal geometry of spherically symmetric spacetimes without specifying the
form of the matter distribution. We obtain the general conformal Killing vector and
conformal factor subject to integrability conditions that restrict the metric
functions. The results pertaining to static spherically symmetric spacetimes obtained
by Maharaj et al.1 are shown to be a special case of our solution. The inheriting
conformal symmetry vector, which maps fluid flow lines conformally, is obtained.
2. Conformal Equation
The line element for the general spherically symmetric spacetimes is
ds2 = -eW^dt2 + e2X^dr2 + Y2(t, r)(d02 + sin2 6dtf) (1)
We substitute the metric (1) in the conformal equation
£xffo6 = 2ipgab
to obtain the conformal Killing vector X = (A"°, X1, X2, Xs) and the conformal
factor ip = ip{xa):
X° = Y2e~2vA\r), + A0 (2a)
X1 = -F V2A^ + AA (2b)
X2 = Al{r]i)e - 03 sin0+ 04 cos 0 (2c)
Xs = esc2 6A1 (r]i)^ - cot6»(a3cos0+a4sin0) + a6 (2d)
iP = Yra [Ye~'2uAit + (2Yt - Yvt)e~2vA\ - Ye~2XvrA\\ + A°t + vtA°
+vrAA (2e)
2258
2259
where A1 = (A1, A2, A3), A0 and A4 are functions of t and r, r]i — (771,772,773) =
(sin 6 sin 0, sin 0 cos <p, cos 0) and 03-05 are constants. This solution is subject to the
integrability conditions
YA\r + (Yr - Yvr)A\ + (Yt - Y\t)Al = 0 (3a)
Ye-2vA\t + Ye~2XA\,r + {2Yt - Y\t - Yut)e~2uA
+ {2Yr - Y\. - Yur)e~2XAlr = 0 (3b)
Y2e-2vA\t + Y(Yt - Yvt)e-'lvA\ + Y(Yr - Yvr)e-'lxAlr + A1 = 0 (3c)
e2XAi _ e2uAo = 0 (3d)
-A° + (£ - „t) A* + (| - „P) A4 = 0 (3e)
-A°t + (At - vt) 4° + (A,. - i/r) .44 + 4? = 0 (3f)
Note that in our solution the angular dependence in 6 and <f> is known explicitly.
This is expected as the spacetime is spherically symmetric. There is freedom only
in the t and r coordinates. We now consider particular cases of our solution.
3. Static Spacetimes
For static spherically symmetric spacetimes
ds2 = -e2v^dt2 + e2X^dr2 + r2(d92 + sin2 6 dtf)
the components of the conformal Killing vector and the conformal factor (2) become
X° = r2e-2vA\m + A0 (4a)
X1 = -r2e-2XAlril + A4 (4b)
X2 = Al{rn)e - 03 sin 0 + 04 cos 0 (4c)
X3 = esc2 OA1^)^ - cot 9(a3 cos 0 + a4 sin 0) + a6 (4d)
i> = r2Vt (e~2vA\t - e"2V^) + AQt + v'A4 (4e)
and the integrability conditions (3) simplify to
rA\r + {l-ri/)A\ = Q (5a)
re-2vA\t + re-2XAlr + (2 - r\' - rv')e-2XA\. = 0 (5b)
r2e~2vA\t + r(l - rv')e-2XAlr + A1 = 0 (5c)
e2XA4 - e2uA°r = 0 (5d)
-A°t + (-- iA A4 = 0 (5e)
-A°t + (A' - v') A4 + A4r = 0 (5f)
where primes denote differentiation with respect to r. Note that the equations (4)-
(5) are equivalent to the corresponding system obtained by Maharaj et al.1 in their
analysis of static spacetimes.
2260
4. Inheriting Vectors
Coley and Tupper2 called vectors X satisfying the condition
inheriting conformal Killing vectors as fluid flow lines are mapped conformally. The
inheriting equation with the fluid 4-velocity ua = e~l'5% and the general conformal
vector (2) yield
X° = A°(t)
X1 = A\r)
X = — a^smcf) + 0,4 cos</>
X3 = — cot 6(0,3 cos (p + a-4 sin <fi) + ag
$ = A0 + vtA°(t) + vrA\r)
The consistency conditions (3) may be completely integrated to give
\nY -v = F(u) +\n A0
\-v = F(u) - G(u) +lnA°- In A4
/dt f dr
—- — / -jj and F, G are arbitrary functions.
A4
Acknowledgments
SM thanks the University of KwaZulu-Natal for financial support. SDM
acknowledges that this work is based upon research supported by the South African
Research Chair Initiative of the Department of Science and Technology and National
Research Foundation.
References
1. S. D. Maharaj, R. Maartens and M. S. Maharaj, International Journal of Theoretical
Physics, 34, 2285 (1995)
2. A. A. Coley and B. O. J. Tupper, Classical and Quantum Gravity, 7, 1961 (1990)
A THEOREM OF BELTRAMI AND THE INTEGRATION OF THE
GEODESIC EQUATIONS
DINO BOCCALETTI
Department of Mathematics - University of Rome "La Sapienza"
Piazzale Aldo Moro 5, 00185 Rome, Italy
boccaletti@uniromal. it
FRANCESCO CATONI, ROBERTO CANNATA and PAOLO ZAMPETTI
ENEA - C.R. Casaccia
Via Anguillarese 301, 00060 S. Maria di Galeria (Rome), Italy
cannata@casaccia. enea. it, zampetti@ casaccia. enea. it
We revisit a not widely known theorem due to Beltrami, through which the integration
of the geodesic equations of a curved manifold is accomplished by a method which is
purely geometric although inspired by the Hamilton-Jacobi method. The application of
the theorem to Schwarzschild and Kerr metrics leads straight to the general solution
of their geodesic equations. As a consequence, we re-obtain the results of Droste and
Schwarzschild and of Carter and Walker-Penrose in a simpler way.
1. Introduction
In GR we have some important spacetimes which are exact solutions of the Einstein
equations and whose metric tensor components are known explicitly in a given
system of coordinates. Starting from these components, one can write the geodesic
equations
^+rlkd4^ = o, (i)
ds2 lk ds ds '
and then try to integrate them to determine the paths of test particles. The
Schwarzschild spacetime (whose timelike geodesies can be used to calculate the
advance of the perihelion of Mercury) and the Kerr metric (representing the
gravitational field outside a rotating body or of a mathematical black hole) are two
important examples whose geodesies can yield important physical results.
Two methods are typically used to integrate the geodesic equations. Either one
starts with the Lagrangian equations of motion (obtained from a Lagrangian £
given by £ = g^. {dx%/ds) (dxk/ds), where for timelike geodesies s may be identified
with the proper time) or with the corresponding Hamilton-Jacobi equations, in both
cases representing a mechanical system governed only by a kinetic energy term, in
which the effects of the gravitational field are represented by the curvature of the
spacetime associated with the metric which determines this kinetic energy function.
Now one is dealing with a mechanical system again instead of pure geometry. In our
eyes, this approach seems to be a step backwards with respect to the spirit of GR.
The motion of a test particle in a gravitational field is interpreted as the motion
of a free particle in a curved spacetime which turns out to follow a geodesic. On
the other hand, a completely "geometric" integration of the geodesic equations can
2261
2262
be performed without referring to the equivalent point particle mechanical system.
Once the geometric problem has been solved, the constants of integration can be
interpreted as physical constants that are the first integrals of the motion in the
classic approach.
Let us consider an n-dimensional semi-Riemannian manifold Vn whose metric is
represented by
ds2 =gihdxldxh (i, h = 1,2, ...n). (2)
If U, V are any real functions of the xl (i = 1,2, ...n), the invariants denned by
^-•rgg^W* (3)
* W 10 = »*i?5F ■•"<'.<<'.» «>
are called Beltrami's differential parameters of the first order. Since the equations
U = const, V = const represent (n — l)-dimensional hypersurfaces in Vn, A\U
represents the squared length of the gradient of U as well as of a vector orthogonal
to the hypersurface U = const ; for the same reason, if A(/7, V) = 0, then the two
hypersurfaces U = const and V = const are orthogonal.
Beltrami's theorem states
Let us consider the equation
A1C/=1, (5)
the solution of this equation depends on an additive constant and oniV-1 essential
constants o>i? Now, if we know a complete solution of Eq. (5), we can obtain the
equations of the geodesies from the following theorem [2, p. 299], [3, p. 59]: when a
complete solution of Eq. (5) is known, the equations of the geodesies are given by
£;-«• (6>
where pi are arbitrary constants, and the geodesic arclength is given by the value of
U.
2. The geodesic equations for the Schwarzschild and Kerr metrics
2.1. The Schwarzschild metric
If we start from the standard form of the so-called Schwarzschild metric
ds"2=r-^-dt2 — dr2-r2(d82+ sin2 9d62); a = 2MG (7)
r r - a
and apply to this metric Eq. (5), we obtain in a "geometrical way" the same results
of the standard approach. For instance, in the limit a/r <S 1 (as is the case for
planetary motion), we obtain for <f>
dr
A3 I ; + const (8)
1 r* y/(Al-l)-Ai/r* { }
2263
which, compared with the relevant Newtonian equation (see Boccaletti-Pucacco,5
Eq. (2.14), p. 131), allows one to immediately identify A% with the angular
momentum per unit mass. We refer to J1] for the relevant calculations.
2.2. The Kerr metric
The method based on Beltrami's theorem that we have applied so far to study
geodesic motion in the Schwarzschild spacetime can clearly be applied to the Kerr
spacetime as well. We start from the Kerr metric4
ds2 = ^dt2-%(d4>-^^dt\\m2e-^dr2-p2de\ (9)
where p2 = r2 + a2 cos2 9, A = r2 + a2 - 2 Mr, Y? = (r2 + a2)2 - a2 A sin2 0; M
and a are constants that in the Newton limit represent the mass and the angular
momentum per unit mass.
The equations for the geodesies can be obtained by the standard procedure of
Eq. (6) following from Beltrami's theorem and we can also obtain the relations
analogous to those obtained for the Schwarzschild metric (for the calculations see
[1], where A\ = K is the constant introduced by Walker and Penrose6 who solved
the relevant Hamilton-Jacobi equation in a different way with respect to Carter4).
For a readable account, see Chandrasekhar4 pp. 344-347.
References
1. Boccaletti, D., Catoni, F., Cannata R., Zampetti, P., Gen. Relativ. Gravit. 37, 2261-
2273 (2005)
2. Luigi Bianchi, Lezioni di Geometria Differenziale, 2 edition, Vol. I, II, (Spoerri,
Pisa, 1902)
3. L.P. Eisenhart, Riemannian Geometry (Princeton University Press, Princeton, 1964).
4. S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press,
1983); see also:
R.P. Kerr, Phys. Rev. Letters 11, 237-8 (1963);
B. Carter, Phys. Rev. 174, 1559-71 (1968)
5. D. Boccaletti and G. Pucacco, Theory of Orbits, 3rd corrected printing (Springer-
Verlag, 2004), Vol. I
6. Walker, M., Penrose, R, Commun . Math. Phys. 18, 265-74 (1970) (Dover, 1976),
Chap. XIV
GRAVITATIONAL COLLAPSE AND HORIZON FORMATION IN
2+1 - DIMENSIONAL GRAVITY
DIETER R. BRILL
Department of Physics, Univeristy of Maryland
College Park, MD 20782, USA
brill@physics.umd.edu
PUNEET KHETARPAL
Rensselaer Polytechnic Institute
Troy, NY 12180, USA
bharat211 @gmail. com
1. Introduction
A number of physically interesting questions that cannot be treated exactly in
3+1 dimensional general relativity can be answered more simply and easily in 2+1
dimensional Einstein theory. In this paper we show how to distinguish initial
configurations that lead to collapse and black hole formation from those that do not,
and how the horizon develops in the former case.
2. Particles and black holes in 2+1 dimensions
In order to admit black holes at all in 2+1 dimensions there must be a negative
cosmological constant,1 and in order to stay within vacuum solutions (excluding
gravitational waves), we take the collapsing objects to be spinless point particles.
2+1-dimensional Einstein theory docs admit such particles; they are characterized
only by position and mass, and are represented by conical singularities, with a
spatial angle deficit in the particle's rest frame proportional to its mass.
Our first question, about the future formation of a black hole, is surprisingly easy
to answer because of the special feature that - unlike in the 3+1-dimensional case -
2+1-dimensional black holes and particles have a different asymptotic dependence.
In coordinates that exhibit explicitly the rotational symmetry, the metric of the
2+1 - dimensional, anti-deSitter vacuum (for unit negative curvature, A = — 1) can
be written as
ds2 = -(l + q2)dt2 + ^^+q2d62. (1)
A particle is constructed by identifying 6 = 0 and 6 = 2n — S, rather than giving
6 the usual periodicity of 2ir. By defining a re-scaled angular coordinate ip with
periodicity 2tt and a re-scaled radial coordinate we can put the metric (1) in the
form
ds2 = -(r2- m) dt2 + -£— + r2 cbp2 (2)
rl — m
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2265
with the parameter m = —(1 — 5/2-k)2 < 0, 5 being the the angular deficit of the
conical singularity. The metric of a black hole, on the other hand, has the same
form (2) with m > 0. Thus the asymptotic form on an initial surface distinguishes
eventual particles (mtotai < 0) from eventual black holes (mtotai > 0). For example,
for two particles of mass mi and m,2 and separation d, the angle deficits combine,
according to the hyperbolic geometry of a triangle on the initial surface, to a total
mass M satisfying
cos M = cos mi cos mi + sin mi sin 7712 cosh d. (3)
Generally M > mi + m-i, with equality holding in the limit of vanishing d. Thus
the presence of the factor cosh a! may be viewed as the effect of the gravitational
interaction energy between the particles. When the RHS of Eq.(3) exceeds unity,
the total "mass" corresponds to a black hole characterized by the circumference D
of the horizon that it will have after the particles collapse sufficiently,
coshD = — cos mi cosm,2 + sin mi sinm,2 cosh a! (4)
3. Horizon Development
In AdS space the metric outside any number of particles, collapsing to a black
hole, is exactly that of a single BTZ black hole. The horizon of that black hole is
a smooth, circular null surface propagating to infinity. Let us follow that surface
backward in time as it contracts and eventually comes to the "outermost" of the
particles. As it crosses the particle, it acquires a discontinuity in its tangent equal
to the particle's angle deficit. As we go farther backward in time this discontinuity
moves along a spacelike curve, since it is the intersection of two null surfaces. Similar
curves propagate backwards from the other particles. The curves join by pairs until,
in general, there is a single curve left with two discontinuities moving todards each
other. Where they come together is the origin of the horizon, a kind of center of mass
of all the particles. Let us consider this history forward in time for the two-particle
case.
For two particles in AdS space that will collapse to a black hole there is always
a moment of time-symmetry when the particles are at maximum separation. The
intrinsic geometry of that surface can be obtained from the funnel- or wormhole-
shaped spacelike geometry of the corresponding black hole by pinching it off along
the line joining the two particles. (Near the particles the surface then looks like
a Melitta coffee filter, whose two bottom corners are conical and represent the
particles.) If we cut this geometry in half along the line of symmetry, either half is
simply connected and can be drawn on a time-symmetric spacelike surface of full
AdS space. This is shown by the curve labeled by l's in Figure 1. The full curves,
including the dotted extensions, correspond to half of the single black hole. The
upper part, pinched off at the horizontal line, represents the two-wormhole initial
geometry, with the particles located at the two l's along the horizontal line. At this
time the black hole horizon Hi is in the pinched-off part of space.
2266
Fig. 1. Three time slices of two-particle collapse and associated horizon, superimposed in a
Poincare disk representation. The outer circle corresponds to infinity. Only the half space is shown;
the complete configuration at each time is obtained by reflecting the upper half about the
horizontal line and identifying the heavy curves that go to infinity. The two equal-mass particles are
indicated by black dots at successive times 1, 2, 3.
We describe the time development in the time coordinate of metric (1), which
has a finite lapse everywhere (unlike the Schwarzschild time coordinate of the BTZ
black hole). The spacelike metric is then time-independent, the only motion is in
the lines where the two halves of the complete spacelike surfaces are to be joined.
Curves labeled by 2's and 3's show these lines at two later times. The particles
approach each other, and their deficit angle (twice the angle between the curve
that reaches infinity and the horizontal line) increases, due to the particle's kinetic
energy. The horizon propagates generally upward and first enters the two-particle
spacetime when it touches the horizontal line at the center of the figure. From
there it expands with two slope discontinuities moving towards the particles with
spacelike "velocity." It reaches the particles at H2, becomes a smooth circle as it
crosses over them, continues expanding to H3, and reaches infinity at the same finite
time coordinate at which the particles collide. After H2 the horizon has constant
circumference as appropriate for the single black hole that has just been formed.
Features of the horizon development in more general cases are illustrated in
Figure 2 for the case the collapse of four equal particles starting from rest. In order
to show successive times on the same Poincare disk, the picture was successively
enlarged so that the particles remain at constant location. The particles' angle
deficits are indicated by the hatched regions, which are to be removed from the
space and their boundaries identified. The horizon starts lens-like near the center of
the diagram. As it expands it acquires further slope discontinuities that separate arcs
2267
Fig. 2. Development of the horizon in the collapse of four particles. The inner heavy lines show
the paths of the singular points. The lighter curves are stages of the horizon up to the time when
it reaches the particles.
of constant curvature. These discontinuities run along a tree-like graph (a spacelike
structure in spacetime) that ends at the particles. Each discontinuity disappears as
the horizon crosses the corresponding particle (not necessarily simultaneously as in
this case), until the horizon has the smooth and circular shape of the final black
hole.
If one (or several) particles in the above discussion is replaced by a black hole,
a similar analysis shows that the initial black hole horizon (at the moment of time
symmetry) already has a slope discontinuity, which disappears when it merges with
the infalling particle.
This pattern of horizon development indicates what may happen in the collapse
of more general matter distributions and in the more realistic, 3-dimensional case.
The horizon is expected to expand from a point at different (above-light) speeds in
different directions, so that its shape is anisotropic, the parts with largest extrinsic
curvature expanding fastest and acquiring the most new generators. The anisotropy
corresponds to that of the mass-energy it will later cross. As mass-energy falls into
the horizon, it becomes smoother and eventually spherical (in the case of vanishing
total angular momentum).
The case when the system has a total angular momentum is under investigation.
References
1. Banados, Teitelboim and Zanilli, Phys. Rev. Lett. 69, 1849 (1992); Bafiados, Hen-
neaux, Teitelboim and Zanelli, Phys. Rev. D48, 1506 (1993).
PURELY MAGNETIC SILENT UNIVERSES DO NOT EXIST
K. T. VU and J. CARMINATI
Mathematics and Computational Theory Group,
School of Information Technology,
Deakin University, Australia
We present a new Maple package called STeM (Symbolic Tetrad Manipulation). Using
STeM, we outline, using a formalism which is a hybrid of the NP and Orthonormal ones,
the proof of the nonexistence of purely magnetic silent universes.
1. Introduction
Electric silent universes are dust spacetimes in which the fluid four velocity vector,
ua is irrotational and the magnetic part of the Weyl tensor with respect to ua
vanishes (Hab = 0). In such spacetimes there are no sound waves and the condition
Hab = 0 precludes gravitational radiation. Hence the evolution of each fluid element
is determined by compatible initial data but not influenced by its environment so
that it proceeds like a separate universe. Since there are no propagating signals, the
resulting spacetimes are called silent. This concept was first introduced by Matarrese
et al1 Interestingly, spacetimes with Haf, ^ 0 do not have a Newtonian counterpart.
In the extreme case, the so called " anti-Newtonian" universe, which are those space-
times containing irrotational dust with a gravito-magnetic field (Ea(, = 0 ^ Hab),
are the ones which are the most non-Newtonian. Such spacetimes are also silent
due to the vanishing of Ea(, and are subject to integrability conditions which are
even more restrictive than in the Haf, — 0 case. Analysis shows that there exists
a nonterminating chain of integrability conditions and therefore one would suspect
that this class is quite restricted. In this paper, we outline how we established the
result that
Theorem 1.1. Anti-Newtonian silent universes do not exist.
This result has just recently been proven by Wylleman,2 as well, using the 1+3
covariant formalism. In contrast, we present, in an article to appear, a different
approach and a new Maple package which may be of use in similar problems
concerning perfect fluids. Essentially, we used a new formalism which is a hybrid of the
NP and Orthonormal formalisms. The working environment is established by
reading in our new Maple package called STeM (Symbolic Tetrad Manipulation). STeM
simultaneously makes available all three formalisms (NP , GHP, and Orthonormal
) for the user and is a major expansion of the GHPII package previously presented
by the authors.3 In particular, by allowing the construction of hybrid operators and
variables one may, in a transparent manner, "merge" the various formalisms. In
addition, new simplification routines, from those of GHPII, have also been included.
It is these combined features of our approach that have allowed us to construct the
proof with comparatively relative ease.
2268
2269
2. Setting up the STeM Environment
Consider a purely magnetic spacetime with a perfect fluid whose 4-velocity is ua.
Then it is possible to align the (canonical) null tetrad {l,n, m,m} so as to achieve
ua = 2-1/2(la + na) and
*00=*22 = 2*11 = ^,A= ^^ (1)
4 24 v
*01 = *02 = *10 = *12 = *20 = *21 = 0
*1 = *3 = 0, ^4 = "*0, ^0 = -*0, ^2 = -*2 (2)
where w is the non zero energy density and / is the fluid pressure which is constant,
by assumption.
In the chosen tetrad, the conditions for zero acceleration and vorticity are
7+7 + £ + £ = 0, 7T — T + J/-K = 0, (3)
2(a + /?) + V + k - t - ?f = 0, p - p + "p - [i = 0, (4)
respectively. In the first stage of our proof, the STeM environment was initialised
with the above conditions and we introduced "suitable" new operators and variables.
This was an appropriately "meld" of the Orthonormal formalism with the NP one,
where we replaced all spin coefficients with hybrid variables. The NP basic operators
{D, A, S, 5} were replaced by essentially the Orthonormal operators {eo, ex, e2, e3}.
Our choice of hybrid variables y = {j/i,..., y2o}, was
yi =a + 0 + a + (3, y2 =5 + fi ~ a - /?, y3 = a-0+a-/3,
y4 =5-/5-0; +/?, 2/5 = 7 + £ + 7 + £, y6 = 7 + £-7-£
2/7 = 7-£ + 7-£> 2/8=7-£-7 + £> 2/9 = P + P- + P + ~P,
yio = p + p-p-~P, yii = p - p- + p - ~P, 2/i2 = ~p - p - P + V>
yi3 = Tf + T+V+K,yU = Tf + T-V-K,yi5=Tf-T+V-K,
Vi6 =n -t ~V + k, yi7 = ct + X + a + A, yi8 = a + X~a-X
t/19 = ct - A + ct - A, j/20 = o:-A-o- + A
3. Outline of Proof and Conclusion
The proof was establised by carrying out a comprehensive investigation of the
entire system of NP equations which included the zero acceleration and zero vorticity
conditions. The function stemnormal(), in STeM, was the key tool that was used
to simplify resulting systems of equations, at various stages, by bring them into a
normal (simplest) form according to the given ordering of variables and operators.
StemnormalQ reduces the equations with respect to themselves until no further
reduction is possible. It includes two switches: 'algebraic' and 'factor'. The switch
'algebraic' presents to the user any algebraic equations derived during the reduction
2270
process and provides a control mechanism so as to include or exclude these
equations from the elimination/simplification processes. Any excluded algebraic
equations will be relegated to the system STEM.unused. The switch ''factor'' shows all
factored equations so that one can select a factor, as zero, with which to continue
the reduction process. The resulting analysis described above lead to many simple
polynomial equations in the new variables. Close inspection of these conditions gave
quite a number, though very simple, of special cases that were easily dealt with. All
branches were shown to quickly lead to contradictions without the use of resultants
or Groebner basis methods.
We would now like to make a few statements concerning the proof as presented
by Wylleman and ours. In his proof, Wylleman used an in-depth analysis of the inte-
grability conditions in the 1+3 covariant formalism . He presented two preparatory
lemmas and several technical observations relevent to the formalism. In addition
he had quite a few special cases to consider which required the use of resultants
and gcd analysis because of the size of some expressions. In our case, all equations
were straightforwardly derived. Even the form of the hybrid variables was strongly
suggested by the orthonormal ones and by "obvious" combinations appearing in
resulting equations. We were also presented with a fair number of special cases.
However, wc observed that all our special cases were very easily dealt with and did
not require resultants or Groebner basis methods. Indeed the relatively small size
of the resulting equations in the special cases suggests that this may have all been
done by hand. In conclusion, there is much to be said about both approaches: one
offering more geometrical insight due to its covariant nature and the other offering
more computational simplicity and ease of use due to its tetrad form being more
amenable to computer algebra manipulation.
References
1. Matarrese S, Pantano O and Saez D 1994 Phys. Rev. Lett. 72 320.
2. Wylleman, L. 2006 Class. Quantum Grav. 23, 2727.
3. Vu K T and Carminati J 2003 Gen. Rel. Grav. 35 263
Exact Solutions
(Physical Aspects)
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ZEEMAN-TYPE DRAGGING IN THE KERR-NEWMAN AND
NUT SPACETIMES
NIKOLAI V. MITSKIEVICH
Department of Physics, CUCEI, Universidad de Guadalajara
Guadalajara, Jalisco, Mexico,
Apart ado Postal 1-2011, C.P. 44100, Guadalajara, Jalisco, Mexico
mitskievich03@yahoo .corn, mx
LUIS I. LOPEZ BENITEZ
Mathematics and Physics Department,
Instituto Tecnologico de Estudios Superiores de Occidente A.C.,
Periferico Sur Manuel Gomez Marin 8585, Tlaquepaque, Jal, C.P. 45090, Mexico
In this communication we discuss two distinct Zeeman-type gravitomagnetic effects
deserving attention since they can be easily characterized in their exact form, not via
approximation procedures. Some observations are also made on gravitoelectric effects.
Gravitoelectromagnetism is an important part of general relativity, and is is
frequently characterized as dragging of local inertial frames when there is a motion
of sources in Einstein's equations which cannot be globally compensated by any
choice of a non-inertial frame co-moving with these sources (thus this is related to
rotation and/or luminal motion of sources). The corresponding spacetime is
usually stationary, but non-stationary cases should also lead to gravitoelectromagnetic
effects. There is a vast literature on these subjects, see Refs. 1, 3-12, 17. An
especially interesting aspect is discussed in Refs. 4, 9, first considered by B. DeWitt
and related to an interplay of gravitation and electroinagnetism in conductors and
superconductors, i.e. dragging of electromagnetic field (the usually studied cases of
gravitoelectric and gravitomagnetic effects are of general relativistic mechanical and
time-involving nature).
Let us first consider the case of circular motion of a neutral test particle
in the equatorial plane of the Kerr-Newman spacetime, ds2 = A~%sln ^ dt2 —
fdr2-£*?2-i
(r2 + a2) - a2A sin2 1? sin2 tfd<f>2-2ar +^+A sin2 tidtdfa where
Y, = r2 + a2 cos2 •d, A = r2 + a2 — 2Mr + Q2 (the Boyer-Lindquist coordinates),
while 1} = it/2. Since the Killing vectors are £} = dt and £H = d^, there are two
conservation laws (for energy and angular momentum around the z axis; we are
working outside the ergosphere), and the time and azimuthal angle coordinates
are determined unambiguously which gives them an objective meaning. However
we do not even need the corresponding two constants of motion to be evaluated
in this case of the dragging effect: it is sufficient to consider the r-component of
the geodesic equation, ^r = \gap^ruavP = 0 (dr/ds = 0 on a circular orbit), thus
9tt ri2 +9<t><t>,r4>2 +2gt<p,ri<i> = 0, F = ^j, which reads in our case as uj2 — 2aSu>—S = 0
where S = ' Mr^ ■• - d*
4-~a2Mr+a2Q2' ^ ~ df
There are two roots, w+ = -, , 1 y, or in terms of the revolution
o(l±0 + l/(o2S)j
2273
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period. T+ = 2ir I , r ±a , where the first term describes the "Newtonian"
V ' ± \y/Mr-Q* J
revolution period of a test neutral particle (the coefficients are exactly the same)
and the second one, the dragging effect due to rotation of the central body. An
analogous conclusion was drawn8'9'11 for motion of a test mass along a circular
equatorial orbit in the Kerr field. We see that in the Kerr-Newman spacetime the
result differs merely in the "Newtonian" term which now contains both the mass
M and electric charge Q of the central body, while dragging depends only on the
Kerr parameter a and is exactly the same as in the Kerr spacetime case (the results
are exact and not approximate ones). This effect is closely related to the Zeeman
effect (spin-orbital interaction).
The second effect occurs in the Taub-NUT spacetime. While the gravitational
mass may be called gravitoelectric charge, the NUT parameter / is similar (to certain
extent) to gravitomagnetic monopole charge (from the structure of Weyl's tensor
the differences are fairly obvious). The vacuum Taub-NUT metric is ds2 = ^(dt +
2Zcostfd0)2-f dr2-T, \d&2 + sin2 tidcj)2), where A(r) = r2-2Mr-l2, E(r) = r2+l2;
see for more details Refs. 2, 16.
It is clear that there should be an analogue of another case of electromagnetic
Zeeman-type effect (motion of an electrically charged point-like mass around a
centre possessing mass as well as electric and magnetic inonopole charges) if we consider
a circular motion of a (neutral) test mass about the Taub-NUT centre; like in the
electromagnetic case, the orbit has to be centred on the z axis and not on the origin
(central mass).
Then we have to use the conditions dr = 0 = dd, thus r- and ^-components of
the geodesic equation, £ {g^<*£) = \9ap,^^-, yield
""*=4/¥(?-8'2)
where gtt,r = 2Mr +2^2r~Ml . When I = 0, the orbit is centred on the origin (tantf =
oo), but in the Taub-NUT case proper, it lies above or under the origin depending
on the relative sign of / and the test particle's angular momentum, as one can see
from the last relation plus an elementary consideration of two conservation laws
(those of energy £ and angular momentum C, both taken per unit rest mass of the
test particle). Another form of i? then reads cos$ = — ^-8~10
Moreover, in Ref. 10 there was considered the energy (inertial mass) distribution
in the Reissner-Nordstrom field, and it was strictly shown that the electric part of
the gravitating mass density is precisely twice that of the respective inertial one
(electric energy). This point was treated there in terms of gravitoelectric concepts.
Let us recall the Sommerfeld-Lenz approach15 discussed from diametrically opposite
viewpoints,13,14 but now practically forgotten, primarily, since this approach during
decades worked merely in an intuitive "deduction" only of one — Schwarzschild's
— - solution. However it was later shown17 that it works astoundingly well in such
a deduction of the Reissner-Nordstrom, Kerr and Kerr-Newman solutions too, so
2275
that all famous eternal black holes can be intuitively reached in this elementary
way (nobody can clearly tell, for what reason). Here it is only worth mentioning
that for charged solutions this approach needs doubling the electromagnetic energy
density,10,17 precisely in the sense mentioned in the beginning of this paragraph.
Finally, we should emphasize that, in a contrast to the Sommerfeld-Lenz
approach, gravitoelectroiiiagnetism is not a hypothesis but a strict consequence of
Einstein's gravitation theory. It even is a paraphrase for a significant part of the
gravitation theory inside the general relativity, the latter having to be the whole
physics under the assumption that spacetime curvature is included in this picture of
universe. Similarly, the special relativity is not simply a theory of rapid motion but
also is the whole physics under the assumption of properly dealing with relativistic
objects such as any kind of electromagnetic field: in particular the static Coulomb
field is intrinsically relativistic since the spatial part of its stress-energy tensor is
endowed with the same worth as the temporal-temporal component of this same
tensor. Thus the problem is not so much to verify the theory from the experimental
viewpoint but to refine the experimental means in physics up to this new level.
We are studying general problems of general relativity to the end of better
understanding this theory; its most exotic features clearly and vividly show its profound
implications, its boundaries, and critical regions of growth of our knowledge.
References
1. D. Bini, Ch. Cherubim, R.T. Jantzen, and B. Mashhoon, Class. Quantum Grav. 20,
457 (2003).
2. M. Carmeli, Group Theory and General Relativity (World Scientific, 2000).
3. I. Ciufolini and J.A. Wheeler, Gravitation and Inertia (Princeton Univ. Press, 1995).
4. B.S. DeWitt, Phys. Rev. Lett. 16, 1092 (1966).
5. R.T. Jantzen, P. Carini, and D. Bini, Ann. Phys. (USA) 219, 1 (1992), see the article
with corrections in gr-qc/0106043.
6. B. Mashhoon, Gen. Relat. Grav. 31, 681 (1999).
7. Ch.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation (W.H. Freeman, 1973).
8. N.V. Mitskievich, Proc. Einstein Found. Internat. 1, 137 (1983).
9. N.V. Mitskievich, Relativistic Physics in Arbitrary Reference Frames (Nova Science
Publishers, 2006). See also the early book preprint gr-qc/9606051.
10. N.V. Mitskievich and L.I. Lopez Benitez, Gravitation & Cosmology 10, 127 (2004).
11. N.V. Mitskievich and I. Pulido Garcia, Doklady Akad. Nauk SSSR 192, 1263 (1970).
In Russian.
12. J.F. Pascual-Sanchez, Ed., Reference Frames and Gravitomagnetism (World Scientific,
2001).
13. W. Rindler, Amer. J. Phys. 36, 540 (1968).
14. L. I. Schiff, Amer. J. Phys. 28, 340 (1960).
15. A. Sommerfeld, Electrodynamics: Lectures on Theoretical Physics, Vol. 3 (Academic
Press, 1952).
16. H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, Exact Solutions
of Einstein's Field Equations, second edition (Cambridge Univ. Press, 2004).
17. Yu. Vladimirov, N. Mitskievich, and J. Horsky, Space, Tim.e, Gravitation (Mir
Publishers, 1987).
PHYSICAL IMPLICATIONS FOR THE UNIQUENESS OF THE
VALUE OF THE INTEGRATION IN THE VACUUM
SCHWARZSCHILD SOLUTION
ABHAS MITRA
Theoretical Astrophysics, Bhabha Atomic Research Centre,
Mumbai - 40085, India
* amitra@barc.gov.in
By using the principle of invariance of 4 -volume associated in any curvilinear coordinate
transformation, we show that, the integration constant ao appearing in the vacuum
Schwarzschild solution (VSS) has a unique value 0. This implies that the gravitational
mass of the neutral "massenpunkt" or "point mass" involved in the problem is zero
in exact agreement with the corresponding result by Arnowitt, Deser and Misner.1 It
also means that Schwarzschild Black Holes could only be the asymptotic solution of
collapsed objects which may approach the M —> Mq = 0 state by radiating away the
entire available mass energy.
Keywords: Black Holes
1. Introduction
The original vacuum Schwarzschild solution (VSS)
ds2 = -(1 - a0/R)dt2 + (1 - a0/R)-ldR2 + R2{d92 + sin2 4>d92) (1)
where 9 and <p are the polar angles and R is the radial coordinate, describes the
spacetime structure around a "point mass" Mq. It is this exact solution which is
believed to suggest the existence of Schwarzschild Black Holes (SBH). The mass
of the "massenpunkt" or the SBH Mo here arises through the integration constant
ao = 2Mo- Note that the mere identification of ao in terms of Mo is not really fixing
the value of this integration constant. But, we do fix the value of ao here by using the
principle of invariance of 4-volume element associated with the original VSS metric
and the extended Eddington Finkelstein metric; \/—g dx° dx1 dx2 dx3 = Invariant,
i.e., yJ—gdxP dx1 dx2 dx3 = \/—g* dx^ dx\ dx\ dx3, where dxls are infinetisimal
coordinate increments and g,g* are the corresponding metric determinants.
2. The Proof
The extended Eddington- Finkelstein metric which describes both interior and
exterior spacetimes of the SBH is
ds2 = - (1 - ^) dtl T ^dUdR + (1 + ^) dR2 + R2(de2 + sin2 4>d62) (2)
V R J R V R /
where the Finkelstein coordinates are
U = t T ao log ( 1 ) ; R* = R; 9* = 9; 0* = 0 (3)
\ao J
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The corresponding metric coefficients are
9ur = -(1 - a0/R), gRR = (1 + a0/R), 9ur = 9Rt, = a0/R (4)
In this case the determinant remains unchanged:
g* = -9eeg<p<p(gt,R ~ 9uu9rr) = -R4 sin2 9 = g (5)
Now let us apply the principle of invariance of 4-volume for the coordinate systems
(t, R, 9, <p) and (U, R, 9, 0):
I ( ( I \/=5* dt* dRd6 d<j>= f I J l 7=g dt dR d9 d<j> (6)
Since gt = g = — i?4 sin 9, we rewrite the foregoing equation as
[[j jR2 sin6 dtt dR d9 dcj> = f f f I R2 sin(9 dt dR d9 d<j> (7)
The integration over the angular coordinates can be easily carried out and cancelled
from both sides. Note that, here, neither t*, nor t nor R (or for that matter, 9 and
(j)) are vectors or any n-forms. On the other hand, they are just numbers. Therefore,
we can use Eq.(3) to find the following relationship:
dU = dt=f —^— dR (8)
R — ao
By using Eq.(7) in Eq.(7), we find,
f f R2 dtdR^ao f f ——dR dR = f J R2 dt dR (9)
and which leads to
ao I I—-—dRdR = 0 (10)
J J R- a0
Eq.(10) can be satisfied if and only if one has a0 = 0 implying M0 = 0.
3. Discussion
Note that, by virtue of Birchoff's theorem, Eq.(l) may represent the exterior vacuum
spacetime of a spherical object having R > 2M. In such a case, one would have
a — 2M. On the other hand, the since the "Tortoise" coordinate f* used in Eq.(3)
is obtained by integrating the full vacuum metric (1) from R = 0 to R = R, Eq.(3)
would cease to be valid in such a case of a spacetime filled with mass energy. In fact,
there would not be any need to invoke metric (2) in such a case. Correspondingly,
it would not be possible to constrain the value of a = 2M and M = f0 ° AttR2 p dR
would indeed be finite for a spacetime filled with mass energy where the radius
Ro > 0, where p is the mass energy density. On the other hand, we found that2
t-Ro
M0 = lim / AttR2 p dR = 0 (11)
2278
Therefore, realistic radiative (continued) collapse cannot result in formation of finite
mass SBHs. In fact, several recent papers on radiative collapse have shown that the
effect of outward heat flow can either stall the collapse or there can even be a
rebound. 3~5 It has been explained elsewhere that, as continued collapse would proceed
to high gravitational redshift regime, the collapse generated radiation quanta would
get virtually trapped by the strong gravitational field and a dynamical equilibrium
state is attained where outward heat flow force cancels the inward gravitational
pull.6-9 In a strict sense, however, continued collapse indeed continues
asymptotically towards the M —> Mq = 0 exact BH state suggested by the exact solution
(l).10'11 This would also be in exact agreement with the old result that the "clothed
mass" of a neutral "point particle" is M = 0.1
References
1. R. Arnowitt, S. Deser, & C.W. Misner, in Gravitation,: An Introduction to Current
Research, (ed. L. Witten, Wiley, NY, 1962), (gr-qc/0405109)
2. A. Mitra, Adv. Sp. Sc. 38(12), 2917 (2006)
3. L. Herrera and N.O. Santos, Phys. Rev. D70, 084004 (2004)
4. L. Herrera, A. Di Prisco, and W. Barreto, Phys. Rev. D73, 024008 (2006)
5. L. Herrerea, A. Di Prisco and J. Ospino, Phys. Rev. D74, 044001 (2006)
6. A. Mitra, MNRAS Lett. 367, 367 (2006), gr-qc/0601025
7. A. Mitra, MNRAS 369, 492 (2006), (gr-qc/0603055)
8. A. Mitra, New Astronomy 12, 146 (2006)
9. A. Mitra, Phys. Rev. D 74, 024010 (2006) (gr-qc/0605066)
10. A. Mitra, Found. Phys. Lett. 13, 543 (2000)
11. A. Mitra, Found. Phys. Lett. 15(5), 439 (2002)
SINGULARITY ANALYSIS OF GENERALIZED CYLINDRICALLY
SYMMETRIC SPACETIMES
D.A. KONKOWSKI
Department of Mathematics, U.S. Naval Academy, Annapolis, MD. S140S
dak@usna.edu
T.M. HELLIWELL
Department of Physics, Harvey Mudd College, Claremont, CA. 91711
helliwell@hmc.edu
Cylindrically symmetric spacetimes are generalized with the addition of disclinations
and dislocations (two types of quasiregular singularities). The resulting spacetimes are
studied to determine whether they contain quantum singularities as well as classical ones.
1. Introduction
This is a summary of an investigation [1] into the quantum singularity structure
of a class of spacetimes with and without classical (quasiregular and curvature)
singularities.
2. Generalized Cylindrically Symmetric Spacetimes
We study the general cylindrically symmetric static spacetime with a disclination
(5^1) and a dislocation {A 7^ 0) in the metrics
ds2 = e'2U[e2K{dp2 -dt2)+p2B2d<j)2}+e2U[dz + Ad<j)}2 (1)
where U,K,B.A are functions of p alone. The coordinate ranges are the usual [2,
3].
The classical singularity structure depends on U, K, B, A and can be determined
using the usual tests for each particular case under consideration.
3. Wave equations
To study the quantum singularity structure of these spacetimes we study wave
behaviour [1]. For the general cylindrically symmetric spacetimes the relativistic
Klein-Gordon equation □$ = M2$ can be separated in the coordinates t,r,8,z,
with only the radial equation left to solve. Mode solutions are given by
$ ~ e-iujteikzeim<pH(p) (2)
where
Hjpp +±Hfp +{w2 _ M2e-2Ue2K _ k2e-*Ue2K _ fl-2£2Kg-2 {m _ kA)2]H = Q (3)
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and where we restrict B to be a positive constant. With changes in both dependent
and independent variables, the radial equation can be written as a one-dimensional
Schrodinger equation. Explicitly,
rl>,xx+(E-V(x))T(, = 0 (4)
where E = uj2/B and
V(x) = ^e"-e- + ^e--e- + -^e™(m - kA)2 - JL_. (5)
This form allows us to use the Weyl limit point-limit circle criteria [4] described
in Reed and Simon [5] to determine essential self-adjointness.
4. Essential self-adjointness and Quantum Singularity
There are of course two linearly independent solutions of the Schrodinger equation
for given E. If V(x) is in the limit circle case at zero, both solutions are C2 at
zero, so all linear combinations are C2 as well. We would therefore need a boundary
condition at x = 0 to establish a unique solution. If V(x) is in the limit point
case, the C2 requirement eliminates one of the solutions, leaving a unique solution
without the need of establishing a boundary condition at x = 0. The whole idea
of testing for quantum singularities is that there is no singularity if the solution is
unique [1, 6], as it is in the limit point case. The critical theorem is due to Weyl
[4,5].
Theorem 4.1 (The Weyl limit point-limit circle criterion.). If V(x) is a
continuous real-valued function on (0, oo), then H = —d2/dx2 + V(x) is
essentially self-adjoint on Cq°(0, oo) if and only if V(x) is in the limit point case at both
zero and infinity.
Here a related theorem, Theorem X.8 of Reed and Simon [5], can be used to
establish the limit circle-limit point behavior at infinity. It is easy to show that V(x)
is limit point at infinity for these spacetimes. Similarly, Theorem X.10 of Reed and
Simon [5] can be used to help determine limit point behavior at zero. In particular
we can write V(x) as
1
4x
Then near zero we have the following results:
V{x) = V,{x)-—2. (6)
If V\{x) < 4^2-, then the theorem does not apply.
If V\{x) > x~2, then V(x) is in the limit point case at 0.
If 4^2" < V\(x) < *■ ~2e' for some e > 0, then V(x) is in the limit circle case
at 0.
2281
Usually, however, it is easiest just to solve the Schrodinger equation near zero
and test the resulting approximate solutions for square integrability.
5. Special Cases
In [1] we study the following spacetimes:
• Generalized Levi-Civita spacetimes with dislocation
• Chitre et al spacetimes
• Melvin universes
• Generalized Raychaudhuri spacetimes with disclinations and dislocations
We find that, generically, all classically nonsingular spacetimes are also non-
singular quantum mechanically and all classically singular spacetimes are singular
quantum mechanically.
6. Conclusions
The fact that classical and quantum analyses in these special cases give the same
results is interesting and a bit surprising. To examine a more general class of space-
times we decided to look at power-law spacetimes. Those results [7] are summarized
in another contribution to these proceedings.
Acknowledgments
One of us (DAK) thanks Queen Mary, University of London, where most of these
computations were done, for their hospitality.
References
1. Konkowski D A and Helliwell T M 2006 Gen. Relativ. Grav. 38 1069
2. Stephani H, Kramer D, MacCallum M, Hoenselaers C, and Herlt E 2003 Exact
Solutions of Einstein's Field Equations Vol. 2 (Cambridge: Cambridge University Press)
3. MacCallum M A H 1998 Gen. Relativ. Grav. 30 131
4. Weyl H 1910 Math. Ann. 68 220
5. Reed M and Simon B 1972 Functional Analysis (New York: Academic Press); Reed
M and Simon B 1972 Fourier Analysis and Self-Adjointness (New York: Academic
Press)
6. Horowitz G T and Marolf D 1995 Phys. Rev. D 52 5670
7. Helliwell T M and Konkowski D A 2007 "Quantum healing of classical singularities
in power-law spacetimes" submitted to Class. Quantum Grav. gr-qc/0701149
SOME PROPERTIES OF KERR GEOMETRY WITH A REPULSIVE
COSMOLOGICAL CONSTANT*
MARTIN PETRASEKt and STANISLAV HLEDlK*
Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava,
Bezrucovo nam. 13, Opava, CZ-746 01, Czech Republic
E-mail: tMartin.Petrasek@fpf.slu.cz, $ Stanislav.Hledik@fpf.slu.cz
We summarize general properties of the Kerr-de Sitter geometry, i.e., Kerr geometry in
the presence of a repulsive cosmological constant. An interesting difference between Kerr
geometry and Kerr-de Sitter geometry has been found — namely, the condition of free
fall (vanishing 4-acceleration) is satisfied for stationary observers located on the axis of
symmetry above the horizon.
Keywords: Stationary observers; Black hole; Kerr geometry; Cosmological constant;
Kerr-de Sitter geometry
1. Kerr—de Sitter Geometry and Stationary Frames
The Kerr geometry is a stationary axially symmetric vacuum solution to the
Einstein's field equation.1 Kerr-de Sitter geometry is generalization of this solution to
the Einstein's field equations for case of nonzero cosmological constant. In standard
Boyer-Lindquist coordinates, the line element reads
ds2 = - - ^—r (dt-asm26d6)
(1 + a)2p2 v ;
where
(l + a)V v ' rj Ar Afl
(r2 + a2)(l-^)-2Mr,
A9 = 1 + a cos2 6, p2 = r2 + a2 cos2 6, (2)
a = |Aa2 .
Using (1) and (2), one can derive all important properties which lead to the
clear description of stationary frames. Namely, we shall deal with tetrads and 4-
acceleration.2'3
In our contribution we restrict to those cases in which the cosmological constant
is positive and has a small value.
1.1. Stationary Frames
Stationary observer moves along a worldline of constant r and 9 with a uniform
angular velocity lo. Only such observer sees an unchanging geometry of the space-time
*This research has been supported by Czech grant MSM 4781305903 and grant IGS 33/2006.
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in his/her vicinity. Those observers are considered to be "stationary" in reference
to their local geometry.1 The most simple class of stationary frames are those with
zero angular velocity u> = dcfi/dt = u^/ul = 0 — these observers are called static
observers.
Orthonormal local frames of stationary observers (stationary frames — SnFs)
era) are defined in terms of 4-velocity u = (/(<9t + Lod^) and unit vectors pointing
in the direction of selected global coordinates. In contravariant notation these are
called tetrad 1-forms, while in covariant notation these are called tetrad vectors.
There are four important classes of stationary observers.2 Static Observers
(to = 0) mentioned above, Zero Angular Momentum Observers (ZAMO,
connected with Locally Non-Rotating Frames, LNRF, u> = wr), Carter's Observers (wco)
and Freely Orbiting Observers (wfoo±)-4 However, there can also be found another
class of stationary observers as follows.
2. Freely Falling Stationary Observers
One interesting class can be found using the definition of 4-acceleration. We look
for "freely falling" stationary observers — those stationary observers which 4-
acceleration is zero, in the equatorial plane and on the axis of symmetry.
We can find conditions similar as in the Kerr case for existence of stationary
observers in the equatorial plane, which is (at least for small value of cosmological
constant) almost undistinguishable from pure Kerr case. But on the axis of
symmetry there a new solution, which is not presented in Kerr geometry at all, arises. As
the condition for freely falling stationary observer on the symmetry axis of the Kerr
geometry is r = ±a (thus, under the outer horizon), in case of the Kerr-de Sitter
geometry it is split into a pair of solutions under the outer horizon, and one more
pair of solutions above the outer horizon, which is a new feature enabled by the
presence of the repulsive cosmological constant. Putting c = G = M = 1 and denoting
y = |A, the condition ar = 0 {a1 = ae = a^ = 0 hold implicitly) reads
(~a2 + r2)
ar {6 = {tt,0}) = \- ^t-ry,
(az +r^)
which immediately leads to
r [a1 + rz)
The plot of this function is in Fig. 1.
3. Conclusions
We presented new effect specific Kerr-de Sitter geometry — the existence of
stationary freely falling observers on the axis of symmetry above the outer black-hole
horizon.5'6 Unfortunately, for realistic values of cosmological constant, this point is
2284
Figure 1. The condition for existence of freely falling observers on the axis of symmetry of Kerr-
de Sitter black hole. If the cosmological parameter y = A/3 increases, the location of such observers
above the outer horizon lowers down to the outer horizon, but always remains below the
cosmological horizon. For realistic values of present cosmological constant, however, the position is too
far away from outer horizon.
in too distant from the outer horizon of the black hole, but not behind the
cosmological horizon. Only in cases of very massive black holes or in cases of very high
value of cosmological constant could shift this point as presented in Fig. 1 (as follows
from y = AA'I2/3). This could potentially lead to observable effects, for example
influence collimation of relativistic jets,7'8 which is subject to further investigation.
Bibliography
1. C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (Freeman, San Francisco,
1973).
2. O. Semerak, Gen. Relativity Gravitation 10 (1993).
3. Z. Stuchlik and S. Hledik, Classical Quantum Gravity 17, 4541(November 2000).
4. Z. Stuchlik and P. Slany, Phys. Rev. D 69, p. 064001 (2004).
5. Z. Stuchlik and J. Kovaf, Classical Quantum Gravity 23, 3935 (2006).
6. Z. Stuchlik, Modern Phys. Lett. A 20, 561(March 2005).
7. P. Slany and Z. Stuchlik, Classical Quantum Gravity 22, 3623 (2005).
8. J. Kovaf and Z. Stuchlik, Internat. J. Modern Phys. A 21, 4869 (2006).
SOLUTION GENERATING THEOREMS: PERFECT FLUID
SPHERES AND THE TOV EQUATION*
PETARPA BOONSERM*, MATT VISSERt and SILKE WEINFURTNER*
School of Mathematics, Statistics, and Computer Science,
Victoria University of Wellington, PO Box 600, Wellington, New Zealand
* Petarpa.Boonserm@mcs.vuw.ac.nz
^rnatt.visser@mcs.vuw.ac.nz
t silke.weinfurtner@mcs.vuw .ac.nz
We report several new transformation theorems that map perfect fluid spheres into
perfect fluid spheres. In addition, we report new "solution generating" theorems for the
TOV, whereby any given solution can be "deformed" to a new solution.
1. Introduction
Perfect fluid spheres, either Newtonian or relativistic, are the first approximation in
developing realistic stellar models.*~3 For our current purposes, the central idea is to
start solely with spherical symmetry, which implies that in orthonormal components
the stress energy tensor takes the form:
Tab —
p 0 0 0
0pr0 0
oofto
.0 0 0 jh_
(1)
and then use the perfect fluid constraint pr = pt- This simply makes the radial
pressure equal to the transverse pressure. By using the Einstein equations, plus
spherical symmetry, the equality pr = pt for the pressures becomes the statement
G§§ = Gff = G00-
2. Solution generating theorems
Start with some static spherically symmetric geometry in Schwarzschild (curvature)
coordinates
ds2 = -C(r)2*2 + ~+r2dfi2, (2)
B{r)
and assume it represents a perfect fluid sphere. Setting Gff = G§§ supplies us with
an ODE
[r{rQ']B' + [2r2C" - 2(rQ']B + 2( = 0, (3)
*This research was supported by the Marsden Fund administered by the Royal Society of New
Zealand. In addition, PB was supported by a Royal Thai Scholarship and a Victoria University
Small Research Grant. SW was supported by the Marsden Fund, by a Victoria University PhD
Completion Scholarship, and a Victoria University Small Research Grant.
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Solving for B(r) in terms of £(r) is the basis of the analyses in references.4,5 On the
other hand, we can also re-group this same equation as
2r2BC" + (r2B' - 2rB)C + (rB' -2B + 2)( = 0, (4)
which is a linear homogeneous 2nd order ODE for C(r)- Suppose we start with the
specific geometry defined by
ds2 = -Co(r)2 dt2 + -££- + rW (5)
B0{r)
and assume it represents a perfect fluid sphere. We will show how to "deform" this
solution by applying four different transformation theorems on {(q,Bq}.
2.1. Four theorems
The first theorem we present is a variant of a result first explicitly published in
reference.5
Theorem 1 Suppose {£o(r), Bo(r)} represents a perfect fluid sphere. Define
Ao(r) - ( ^ y r2 L [CM (o(r)-rgr) 1 ()
Then for all A, the geometry defined by holding Co(^) fixed and setting
J^-W'tftWm.M + w (7)
is also a perfect fluid sphere.
Theorem 2 Let {Co,^o} describe a perfect fluid sphere. Define
Zo(r) = a + e[ " ^^ ■ (8)
Then for all a and e, the geometry defined by holding Bo(r) fixed and setting
dr2
Bo(r)
rlr2
ds2 = -Co(r)2 Z0(r)2 dt2 + -f— + r2dtf (9)
is also a perfect fluid sphere.
Having now found the first and second generating theorems it is possible to
define two new theorems by composing them. Take a perfect fluid sphere solution
{Co,^o}- Applying Theorem 1 onto it gives us a new perfect fluid sphere {Co,^i}-
The new B\ is given in equation (6). If we now continue by applying Theorem 2,
again we get a new solution {£, B\}, where £ now depends on the new B\. For more
details regarding Theorem 3 and Theorem 4 see reference.6
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3. Solution generating theorems for the TOV equation
The Tolman-Oppenheimer-Volkov [TOV] equation constrains the internal structure
of general relativistic static perfect fluid spheres.7 In this analysis the pressure
and density are primary and the spacetime geometry is secondary. Using standard
results (see the explicit discussion in reference7) it is relatively simple to present
the following:
Theorem PI We derived Theorem PI by looking for changes in the pressure
profile with mo fixed. This theorem can also be viewed as a consequence of Theorem
2. The key difference now is that we have an explicit statement directly in terms of
the shift in the pressure profile.7
Theorem P2 A second theorem can be obtained by looking for correlated changes
in the mass and pressure profiles. In addition, we can also view this Theorem P2
as a consequence of Theorem 1 . The key difference now is that we have an explicit
statement directly in terms of the shift in the pressure profile.7
4. Discussion
Using Schwarzschild coordinates we have developed two fundamental
transformation theorems that map perfect fluid spheres into perfect fluid spheres. Moreover,
we have also established two additional transformation theorems by composing the
first and second generating theorems. Furthermore, we have also developed two
"physically clean" solution-generating theorems for the TOV equation — where by
"physically clean" we mean that it is relatively easy to understand what happens
to the pressure and density profiles, especially in the vicinity of the stellar core.
References
1. M. S. R. Delgaty and K. Lake, "Physical acceptability of isolated, static, spherically
symmetric, perfect fluid solutions of Einstein's equations," Comput. Phys. Commun.
115 (1998) 395 [arXiv:gr-qc/9809013].
2. M. R. Finch and J. E. F. Skea, "A review of the relativistic static fluid sphere", 1998,
unpublished. http://www.dft.if.uerj.br/usuarios/JimSkea/papers/pfrev.ps
3. H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, Exact solutions
of Einstein's field equations, (Cambridge University Press, 2003).
4. K. Lake, "All static spherically symmetric perfect fluid solutions of Einstein's
Equations," Phys. Rev. D 67 (2003) 104015 [arXiv:gr-qc/0209104].
5. D. Martin and M. Visser, "Algorithmic construction of static perfect fluid spheres,"
Phys. Rev. D 69 (2004) 104028 [arXiv:gr-qc/0306109].
6. P. Boonserm, M. Visser, and S. Weinfurtner "Generating perfect fluid spheres in
general relativity," Phys. Rev. D 71 (2005) 124037. [arXiv:gr-qc/0503007].
7. P. Boonserm, M. Visser, and S. Weinfurtner "Solution generating theorems for the
TOV equation," [arXiv:gr-qc/0607001].
SPHERICALLY SYMMETRIC GRAVITATIONAL COLLAPSE OF
PERFECT FLUIDS
P. LASKY and A. LUN
School of Mathematical Sciences, Monash University,
Melbourne, Victoria 3800, Australia
paul.lasky@sci.monash. edu. au
emailanthony.lun@sci.monash.edu.au
Formulating a perfect fluid filled spherically symmetric metric utilizing the 3+1
formalism for general relativity, we show that the metric coefficients are completely determined
by the mass-energy distribution, and its time rate of change on an initial spacelike
hypersurface. Rather than specifying Schwarzschild coordinates for the exterior of the
collapsing region, we let the interior dictate the form of the solution in the exterior, and
thus both regions are found to be written in one coordinate patch. This not only
alleviates the need for complicated matching schemes at the interface, but also finds a new
coordinate system for the Schwarzschild spacetime expressed in generalized Painleve-
Gullstrand coordinates.
Keywords: Gravitational Collapse, Perfect Fluid
The traditional approach to the analysis of gravitational collapse follows that
devised by Oppenheimer and Snyder,4 whereby the Einstein field equations are solved
for the interior, matter-filled region without consideration of the exterior. Whence a
solution to the interior is found, the Israel-Darmois matching conditions are utilized
to "glue" the interior spacetime to an appropriate exterior spacetime, commonly
Schwarzschild. Throughout the process the two regions are considered as separate
entities, mainly as they are described by different coordinate systems.
We introduce a different approach, whereby the spacetime is established as an
initial/boundary value problem, with the interface between the two regions of the
spacetime being a free-boundary. This enables us to describe both regions of the
spacetime under a single coordinate patch by simply letting the energy-momentum
variables go to zero at some finite coordinate radius. In this talk we use our
formalism to describe the gravitational collapse of a spherically symmetric perfect fluid.
As we are setting up an initial value problem, an ideal starting point is the ADM
system of equations. This will enable us to establish an initial spacelike hypersurface,
with perfect fluid for r < rg and vacuum for r > rg, where rg is some radius on the
initial slice. The system can then be evolved forward in time to describe the entire
collapse process. Furthermore, in order to describe both regions of the spacetime
utilizing a single coordinate system, one requires that the observer have a finite
radial velocity such that this observer will pass through the interface between the
two regions. In ADM language, this implies a non-zero, radial component of the
shift vector, fj(t,r). We therefore begin with an arbitrary, spherically symmetric
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w
line element expressed as
dS2 = -a2dt2 + (1 + Ey1 (fjdt + drf + rW, (1)
here a(t,r) is the lapse function, £(t,r) > —1 is an arbitrary function which
reduces to the energy function of the Lemaitre-Tolman (LT) metric,3'5 and dO,2 is
the metric of the two sphere.
By putting this line element through the ADM equations (for details see1'2),
one can derive the reduced field equations which are a coupled system of first order
differential equations. We define a "mass" function a
M:=4tt / p(t,s)szds, (2)
Js=0
where p(t, r) is the mass-energy density. The lapse function is related to the density
and pressure through Euler's equation
drP=-(p + P)drQna). (3)
The solution of this equation requires the specification of an equation of state (EoS)
which relates the density to the pressure, P. Thus, given an EoS, the system reduces
to the line element along with two equations,
dS2 = (!+£)
-l
-a
2/1 l jj-2
2M\ , , 2M
) dt2 + 2a J + £ dtdr + dr2 + r2dVl2
(4)
Ln£=2(1+*\—+£drP and £nM = AnPr2 J— + £. (5)
p + P V r V r
Here, £n denotes the Lie derivative with respect to the unit normal vector which,
when acting on a scalar, takes the form £nip = a~1 (dftp — (3dTip). Once an EoS
is specified, equation (3) is solved and hence the lapse function can be written in
terms of the density, and thus the mass. Therefore, equations (5) are two equations
for two unknown functions M and £.
As per our aim, these equations describe both the interior perfect fluid region
and the exterior vacuum region in the one coordinate patch. To see this we simply
let the pressure and density vanish external to some finite radius. Equation (2)
then implies that the mass function is constant, which further implies the right
hand equation in (5) is trivially satisfied. Equation (3) implies the lapse function is
simply a function of the temporal coordinate, and utilizing coordinate freedom the
lapse can be set to unity without loss of generality. The resulting system is what
we call the generalized Painleve-Gullstrand (GPG) line element as the special case
{£ = 0) is the Painleve-Gullstrand line element.
The GPG class of solutions comprise a family of coordinate systems for the
Schwarzschild spacetime. The coordinate transformation between this class of
solutions and Schwarzschild coordinates, (t,r,8,<f>j, is given by the solution to the
aWe note in the dust limit, M becomes the familiar mass of the LT solution, and in the vacuum
limit becomes the Schwarzschild mass.
2290
coupled differential equations
2 = l + £ and (l-™)drt = J™+£,
(Ofty = l + £ and ( 1 J drt = J + £, (6)
where t = t(i,r). One can show a solution exists to these equations, and thus the
coordinate transformation is always valid by utilizing the integrability conditions,
which are satisfied providing the left hand equation in (5) is satisfied (for details
see1).
Reverting back to the full system of equations in the interior with matter (2-5),
we can transfer these into diagonal coordinates, (t,R,6,<j>), such that the
generalization from the LT metric for dust becomes obvious. By letting r = r(T, R) such
that
the line element and equations (5) become
dS2 = -a2dr2 + ^-dR2 + rdtt2, (8)
l + £ I2M , „ „ „ , 0 ,, , „ 2 /2M
(dRr)(dT£) = ——-a\ + £ dRP and dTM = 4irPr'a\ +£, (9)
p+P V r V r
and equations (2) and (3) are suitably dealt with. The reduction to the LT dust
models is clear, again using the coordinate freedom that the lapse function becomes
a function of time, and can thus be set to unity without loss of generality.
A number of extensions of this work are currently under investigation:
• Searching for exact solutions of equations (2)-(5).
• The analysis and determination of shell-crossing singularities which exhibit
themselves as fluid shock waves in this coordinate system.
• The relaxation of the perfect fluid condition to allow for more realistic
matter sources, enabling the study of diffusion processes and anisotropic
stresses.
• A relaxation of the symmetries of the geometry to allow for quasi-spherical
symmetry or axial-symmetry.
References
1. P. D. Lasky and A. W. C. Lun. Generalized lemaitre-tolman-bondi solutions with
pressure. Phys. Rev. D, 74:084013, 2006.
2. P. D. Lasky, A. W. C. Lun, and R. B. Burston. Initial value formalism for dust collapse.
ANZIAM J., 49:53, 2007. arXiv:gr-qc/0606003.
3. G. Lemaitre. L'univers en expansion. Ann. Soc. Sci. Bruxelles A, 53:51, 1933.
4. J. R. Oppenheimer and H. Snyder. On continued gravitational contraction. Phys. Rev.,
56:455-9, 1939.
5. R. C. Tolman. Effect of inhomogeneity on cosmological models. Proc. Nat. Acad. Sci.
USA, 20-.169-76, 1934.
HIGH-SPEED CYLINDRICAL COLLAPSE OF TWO DUST FLUIDS
M. SHARIF* and and ZAHID AHMAD
Department of Mathematics, University of the Punjab,
Lahore 54-590, Pakistan
* msharif@math.pu. edu.pk
We discuss the gravitational collapse of cylindrically distributed two dust fluid system
using high-speed approximation scheme. This provides the generalization of the results
already given by Nakao and Morisawa for the dust fluid.
Keywords: High-Speed, Cylindrical Collapse, Two Dust Fluids.
General Relativity has solutions with singularities that can be produced by the
gravitational collapse of nonsingular, asymptotically flat initial data [1-3]. Nakao
and Morisawa investigated the gravitational collapse of a cylindrical dust fluid [4]
and of a cylindrical thick shell composed of a perfect fluid [5]. These studies have
provided strong motivation about the gravitational collapse. Here, we apply the
same procedure to discuss the gravitational collapse of cylindrical two dust fluids.
Our results reduce to the dust fluid case obtained by Nakao and Morisawa [4].
The spacetime (the whole-cylinder symmetry) is defined by the line element [6]
ds2 = e2^^\-dt2 + dr2) + e2i'dz2 + e~^ R2 d^2, (1)
where 7, ip and R are functions of t and r only. Einstein field equations yield
i = (R12 - R2)-1 {RR'(ip2 + y/2) - 2M#' + R'R" - RR'
-KV=g(#Ttt + RTrt)}, (2)
7 = -{R'2 - R2)-l{RR{ij2 +ip'2)- 2RR'ijriJj' + RR" - R'R'
-Ky/^(RTtt+R'Trt)}, (3)
7-7" = V/2-^2-^v/=5r%, R-R" = -Ky/^{Ttt + vr), (4)
t + |^ - V - ^' = -^v/=5(T44 + T\ - T\ + T%). (5)
The energy-momentum tensor for two dust fluid system [7] is given by
Tab = PlUaUh + P2VaVb- (6)
We define the new density variables D\, D2 for the two fluids as follows
V^gpiu* Re?-*Px V=gP2 = Rey-^P2 ()
l' ^U(2 - U) U(2-U)' 2- X/V(2-V) V(2-VY [>
The law of conservation of energy-momentum tensor, i.e., T6a;0 = 0 gives
du(D! +D2) = ~(D1U + D2V)' + +Dl{1~U){2du(i, - 7) - U{j> - 7)}
+ W~ V) {2du& - 7) - V(j) - 7)}, (8)
2291
2292
D&U+DiduV = (1 - U)duDl + (1 - V)duD2 + \{U(l - U)D1 + V(l - V)D2}
- ^-{2du^ - 7) - U{j> - 7)} ~ ;y{2a„(^ - 7) - ^ - 7)}.
(9)
where u = t — r is the retarded time and v = t + r is the advanced time.
The C-energy and the corresponding energy flux vector are defined as [6]
E=\{l+e-2\R2-R'2)}, y/=jJa = -(E',-E,0,0), (10)
where J° satisfies the conversation law. Using Eqs.(2), (3) and (10), the C-energy
flux vector can be expressed in component form as
—27
V^J* = {RR'(ip2 + ip'2) - 2RRrjfl/>' - K^/^JJT't + RTrM, (11)
K
—27
4^~gJr = {RRNj2 + 1P'2) - 2RR'^' - K^)(R'Trt - RT\)}. (12)
K
We want to use high-speed approximation scheme by introducing a small parameter
e and its linear perturbation analysis. The energy-momentum tensor takes the form
p3(V-7)
Tab = —1—[D1kakb + D2lJb}, ka = (1,-1 + U, 0,0), /° = (1, -l + v,0,0).
(13)
In the limiting case, U —> 0+, V —> 0+, the timelike vectors fc° and Z° become
null vectors. Keeping Di, D2 fixed with these limits, the energy-momentum tensor
coincides with the collapsing two null dusts. This implies that the two dust fluid
system is approximated by a two null dust system in the case of very large collapsing
velocities.
Assuming that tp vanishes initially, the solution for collapsing two null dusts is
</, = 0, 7=7b(«). R = r, Kpi + D2)e7 = ^ (14)
dv
which reduces to Morgain's [8] cylindrical null dust solution if either D\ = 0 or
D2 — 0. We take this solution as a background spacetime for the perturbation
analysis. Eqs.(13) and (14) indicate that the energy-momentum diverges at the
symmetry axis r = 0 if D\ and D2 do not vanish simultaneously and the same is
true for the Ricci tensor.
Using linear perturbation analysis by taking the large collapsing velocities, the
variables 7, R and D\, D2 become
e7 = e7B(1 + (57); jR = r(i + (5fi); D1=Db(1+6Di), D2 = Db(1+5d2),
(15)
where <57, Sr and 5d1, <5d2 are of O(e) and Db is given by
n 1 dlB na\
Db := -z ;—• (16)
2KCTB dv V '
2293
Expanding U, V, tp, <57, SR, SDl, SD2 up to first order w.r.t. e, Eqs.(2)-(5) and (8)
turn out respectively as
V = 2KDBe*> {S^^+ (^+foJ
2dv(rSR)} + (rSR)", (17)
<57 = 2KDBe^{51 - tf + {8d^+28d^ - i^±H - 2dv(r6R)} + (rSR)', (18)
S7 - &/' = 0, (19)
rSR - (rSR)" = 2Ke"tBDB(U + V), (20)
1 kp1b
$ - tf'--V =—DB(U+ V), (21)
plB J
du(SDl + SD2 + 251 - 2tf) = -~^~{(U + V)-^(DBe-">B) + (U + V)'e^BDB}.
(22)
The first-order expression for the C-energy w.r.t. e gives
£=§[*- e_27B + 2e-2^{<57 - (rSR)'}}. (23)
From Eq.(16), one can see that ^B is constant in the region where DB — 0. Eqs.(17)
and (18) imply that <57 — (rSR)' is also constant in the vacuum region. Thus the C-
energy is constant in the vacuum region, up to the first order w.r.t. e. This implies
that, up to first order in e, the C-energy flux vector J° vanishes but up to the
second-order in e, it is given in component form as
=^Jt = -(^2+^'2), (24)
K
—gJr = _^'. (25)
This corresponds to the massless Klein-Gordon field. It is verified that if we take
either p\ — 0 or p2 = 0 our results reduce to the cylindrical dust fluid case [4].
One of us (MS) would like to thank HEC for providing full grant to attend
MGM.
References
[1] Penrose, R.: Phys. Rev. Lett. 14(1965)57.
[2] Hawking, S.W.: Proc. R. Soc. London A300(1967)187.
[3] Hawking, S.W. and Penrose, R.: Proc. R. Soc. London A314(1970)529.
[4] Nakao, K. and Morisawa, Y.: Class. Quant. Grav. 21(2004)2101.
[5] Nakao, K. and Morisawa, Y.: Prog. Theor. Phys. 113(2005)73.
[6] Thorne, K.S.: Phys. Rev. 138(1965)B251.
[7] Hall, G.S. and Negm, D.A.: Int. J. Theor. Phys. 25(1986)405.
[8] Morgan, T.A.: Gen. Relativ. Grav. 4(1973)273.
SOME PHYSICAL CONSEQUENCES OF THE MULTIPOLE
STRUCTURE OF THE KERR AND KERR-NEWMAN SOLUTIONS
KJELL ROSQUIST
Stockholm University
AlbaNova University Center
10691 Stockholm, Sweden
kr@physto.se
We discuss physical aspects of the Kerr and Kerr-Newman solutions relating to the
multipole structure, especially its nonlinear nature in general relativity. It is argued that
the Kerr and Kerr-Newman multipole structure is likely to be important for general
macroscopic as well as microscopic systems.
1. Aspects of non-Newtonian self-gravitating systems from the
perspective of the multipole structure
In Newtonian gravity, gravitational moments are independent in the sense that
each moment is by itself a solution of the (linear) vacuum field equation, namely
the Laplace equation. Moments can therefore be added to give a new linear
superposition of solutions. By contrast, in general relativity, the nonlinearity of the field
equations implies that sums of moments do not correspond to solutions. One
consequence of this fact is that in an evolving system, the multipoles are not independent
and may interact with each other. This is especially important for collapsing
systems where the gravitational forces are strong. Because of the nonlinearities, one
expects that some relations between multipoles (i.e. relative sizes) will be more
likely than others. However, such multipole interactions must necessarily exist also
for self-gravitating systems in general, for galaxies for example, albeit weaker. The
question is: How strong are they and how do they act? Although we cannot give a
complete answer here, the above argument is an indication that the Kerr multipole
structure can be regarded as an attractor in "multipole space".
Provided the inequality M > a = J/M is satisfied (the underextreme casea),
the Kerr geometry represents the only possible (uncharged) black hole (often stated
as ''black holes have no hair"1).
The exact conditions under which a system becomes a black hole are not known
however. Thorne has formulated the hoop conjecture (Misner et al.,1 p.868). It says
that any system which can be encircled by a hoop which has a circumference which
is smaller than AirM will collapse to form a black hole. The conjecture has a certain
intuitive appeal but does not take into account the angular momentum. We know
that a black hole could not form if a > M, but there is no mention of this in the
aIn the literature, black holes with M = a are referred to as extreme since they saturate the
black hole inequality M > a. Solutions with a > M have often been referred to as hyperextreme
or sometimes as overextreme. In this note we will use the more neutral sounding nomenclature
underextreme and overextreme for a < M and a > M respectively.
2294
2295
Table 1. Spin values of some typical macroscopic and microscopic objects.
Object
Andromeda
Solar system
Sun
Neutron star
(1.5MSlm)
Earth
CD disk*
proton
electron
Object radius
R
26kpc = 8-10~*cm
700 000 km
10 km
6400 km
6 cm
~ 10 cm (= Ifm)
~2-10~'°em(=Ac)
Spin radius
J
a _ —
M
2-1018cm
300 m
400 m
3.3 m
3-10"8cn
l-10-,4c,
2-10~~nei
Extremal ity
a
11
.. i
11
■. ■ i ■
Rim
velocity
parameter
t'-jn, a
:-io-5
l ■ 10~7
0.04
MO4
-0.1
-0.1
?
Overextreme
a ■
X
%-.
•a
p
X
B
>
O
♦Pointed out for vinyl LP records (a/M ~ 1018) by Dietz and Hoenselaers3
conjecture. It follows that the conjecture cannot be true unless overextreme angular
momenta are excluded.
There is also the issue of cosmic censorship. Even though the black hole state is
■unique, there might also exist other final states without horizons. Such states would
then probably have naked singularities, the existence of which are precluded by the
as yet unproven cosmic censorship hypothesis. These arguments are all confined
to the classical non-quantum regime. In any case, our primary interest here lies in
what happens outside of the quantum regime.
From the physical point of view it is necessary to consider the overextreme
case a > M as well as the underextreme. Indeed, there are many physical systems,
including astrophysical ones, which are overextreme, e.g. the solar system which has
a/M « 40 (see Table I). Even though there is no counterpart of the no hair theorem
for the overextreme case, one would still expect that the multipole interactions work
in the same way, namely that they tend to force the system towards a Kerr-like
structure.
2296
Fig. L. The first few Kerr normalized moments. The light grey bars represent the gravitoelectric
moments and the dark grey bars the gravitomagiietic moments.
2. General relativistic multipoles4 7
The multipole moments of Einstein-Maxwell fields arc naturally divided in the
familiar electric and magnetic moments of electromagnetic fields and analogously
in gravitoelectiic and gravitomagiietic moments of gravitational fields. In analogy
with the electromagnetic case, gravitational multipoles can be collected in a
complex combination given by Q\ = ni[ + ij[. The gravitoelectiic part is given by mi
and the gravitomagiietic part by j;. The Kerr solution is very special in that it
has an infinite number of multipoles given by5 Qi = M(ia)1. The first nonzero
moments are the mass ron = M, the angular momentum ji = Ma and the quadrupole
■iri2 = —Ma2. To get a better feeling for this structure it is convenient to use spin
normalized (dimensionless) moments defined by
mj = _ = _jr_ (i>i), .,, = _ = ___ d>2). (i)
The nonzero normalized Kerr moments are given by Qi = il. They are displayed
graphically in Fig.2.
3. The weakness of the Kerr and Kerr-Newmari singularities as a
nonperturbative phenomenon
An important feature of the Kerr and Kerr-Newman solutions is that their
singularities are significantly weaker than the Schwarzschild and Reissner-Nordstrom
singularities. One may take this fact as an indication that the Kerr and Kerr-Newman
solutions are more physical. There are at least two ways to illustrate the weaker
nature of the singularities. The first is well-known and concerns the geodesic structure
of the Kerr solution. In the Schwarzschild geometry no observer, whether in free fall
or not, can escape the central singularity. The Kerr geometry is different in that
respect. Only geodesies which lie in the equatorial plane can hit the singularity.9
This implies that generic observers escape the singularity. Therefore, the singularity
can be said to be weaker in that sense. Unlike the Schwarzschild singularity, the
Kerr singularity doesn't pull everything into if.
The second indication that the Kerr and Kerr-Newman singularities are weaker
can be illustrated by examining the Kerr-Newman electromagnetic field in the limit
2297
G —► 0. It turns out the Kerr-Newman electromagnetic Lagrangian is finite, a result
which is in sharp contrast with the corresponding diverging Lagrangian associated
with the Coulomb field. This property depends crucially on the multipole structure.
In particular any perturbation involving a finite number of multipoles necessarily
destroys the convergence of the Lagrangian integral. We may conclude that this
improved behavior of the singularity depends on a delicate balance involving inifinitely
many multipoles. In other words, the weakness of the singularity is a fundamentally
nonperturbative phenomenon.
The fact that the finiteness of the Kerr-Newman electromagnetic Lagrangian
is intimately connected with its multipole structure as discussed above is likely to
have implications also for the gravitational field. The reason is that the gravitational
multipoles have exactly the same structure as the electromagnetic ones. This is an
indication that the gravitational field itself is also less singular in the same sense.
4. The general relativistic monopole-quadrupole
To illustrate the nonlinearities in the multipole structure we consider the monopole-
quadrupole system which has recently been given in exact form.8 In Weyl's
coordinates, the general static axisymmetric metric takes the form
g = ^e2udt2 + e2^"^(dR2 + dZ2) + R2e~2Ud<p2 , (2)
where U is invariantly defined as a certain function of the norm of the timelike
Killing vector. Einstein's equations imply that U satisfies the flat space Laplace
equation in the cylindrical coordinates (R, Z,<p). The general relativistic monopole-
quadrupole field is given by8
U(M,q) = -Mr- |(M3 + ?)P2(cos(9) f3 - T^M2(21M3 + 40?) P4 (cos (9)f5 + 0{r7)
where f = 1/y/R2 + Z2. Let us now compare the differences with respect to the
Schwarzschild monopole of the physical monopole-quadrupole U(M, q) vs. the
superposed potential U(M, q) = U(M, 0) + U(0, q)
U(M,q) - U(M,0) = -±qP2(cos6)r3 - £ P4 (cos (9)r5 + 0(f7)
U{M,q) - U(M,0) = U(0,q) = -i<zP2(cos#)f3 - ^q3P8(cos6)rg + 0(r15) .
From these expressions it is evident that the nonlinearities when putting together a
monopole and a quadrupole start at the fifth order (corresponding to I = 4) in the
expansion. We also see that the physical monopole-quadrupole field U(M, q) and the
superposed field U(M, q) differ from the Schwarzschild monopole at the third order
by the same amount but U(M, q) next differs at the fifth order while U(M, q) does
not differ until the ninth order. Therefore the superposed field may be considered
as being a smaller deformation of the monopole than the physically combined field.
This result can be taken as supporting the view that a Schwarzschild monopole is
more likely as a final state than a monopole-quadrupole field.
2298
5. Discussion
We have argued that the Kerr and Kerr-Newman solutions may be considered as
attractors in "multipole space". This would indicate that their very special multi-
pole structure is a natural classial ground state of Einstein-Maxwell systems (cf.
Rosquist10). This may seem to contradict the cosmic censorship hypothesis in the
case of overextreme (a > M) systems which have been proven to be unstable
for some parameter values.11 However, quantum effects may very well prevent the
formation of naked singularities. For microscopic systems this situation would be
analogous to the quantum mechanical stability of atoms vs. their classical
instability. Since the multipole structure of macroscopic self-gravitating systems must
inevitably approach the Kerr attractor, at least in the underextreme case,
individual multipoles will be forced towards the Kerr values. This kind of dynamical
behavior can be characterized as nonlinear multipole interactions. A particularly
important conclusion from the above considerations is that the special combination
of infinitely many multipoles present in the Kerr and Kerr-Newman geometries has
a regularizing effect, most strikingly illustrated in the finiteness of the Kerr-Newman
electromagnetic Lagrangian.
Acknowledgement
Part of this work has been carried out with support from the ICRANet network.
References
1. C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (Freeman, San Francisco,
USA, 1973).
2. R. M. Wald, Gravitational collapse and cosmic censorship, in Black Holes,
Gravitational Radiation and the Universe, eds. B. R. Iyer and B. Bhawal (Springer, 1998)
(related online version: gr-qc/9710068).
3. W. Dietz and C. Hoenselaers, Ann. Phys. (N.Y.) 165, p. 319 (1985).
4. R. Geroch, J. Math. Phys. 11, p. 2580 (1970).
5. R. O. Hansen, J. Math. Phys. 15, p. 46 (1974).
6. C. Hoenselaers, Prog. Theor. Phys. 55, p. 406 (1976).
7. W. Simon, J. Math. Phys. 25, p. 1035 (1984).
8. T. Backdahl and M. Herberthson, Class. Quantum Grav. 22, p. 1607 (2005).
9. B. Carter, Phys. Rev. 174, p. 1559 (1968).
10. K. Rosquist, Class. Quantum Grav. 23, p. 3111 (2006), (related online version:
gr-qc/0412064).
11. G. Dotti, R. Gleiser and J. Pullin, Instability of charged and rotating naked
singularities (2006), E-print: gr-qc/0607052.
VISUALIZING SPACETIMES VIA EMBEDDING DIAGRAMS*
STANISLAV HLEDlKt, ZDENEK STUCHLlK and ALOIS CIPKO
Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava,
Bezrucovo nam. IS, Opava, CZ-746 01, Czech Republic
E-mail: t Stanislav.Hledik@fpf.slu.cz
A simple but powerful method how to visualize curved spacetimes is via embedding
diagrams of both ordinary geometry and optical reference geometry 2D sections into 3D
Euclidean space. They facilitate to gain an intuitive insight into the gravitational field
rendered into a curved spacetime, and to assess the influence of spacetime metrics
parameters. Optical reference geometry and related inertial forces and their relationship
to embedding diagrams are particularly useful for investigation of test particles motion.
Embedding diagrams of static and spherically symmetric, or stationary and axially
symmetric black-hole and naked-singularity spacetimes thus present a useful concept for
intuitive understanding of these spacetimes' nature.
Keywords: Black holes; Naked singularities; Ordinary geometry; Optical reference
geometry; Embedding diagram.
1. Introduction
The analysis of embedding diagrams1^5 rank among the most fundamental
techniques that enable understanding phenomena present in extremely strong
gravitational fields of black holes and other compact objects. The structure of spacetimes can
suitably be demonstrated by embedding diagrams of 2D sections of the ordinary
geometry (t = const hypersurfaces) into 3D Euclidean geometry.
Properties of the motion of both massive and massless test particles can be
properly understood in the framework of optical reference geometry allowing
introduction of the inertial forces in the framework of general relativity. 6~9
The optical geometry results from an appropriate conformal (3 + 1) splitting,
reflecting certain hidden properties of the spacetimes under consideration through
their geodesic structure.10 Fundamental properties of the optical geometry can be
demonstrated by embedding diagrams of its representative sections.3'4'6
2. Optical geometry and inertial forces
The notions of the optical reference geometry and the related inertial forces are
convenient for spacetimes with symmetries, particularly for stationary (static) and
axisymmetric (spherically symmetric) ones. However, they can be introduced for a
general spacetime lacking any symmetry.9
Introducing a spatial positive definite metric hK\ giving the so-called ordinary
projected geometry, and the optical geometry hK\ by conformal rescaling9
hKX = e-2*/*KA , (1)
•This research has been supported by Czech grant MSM 4781305903.
2299
2300
the projection of the 4-acceleration aj: = h^u^V^Ux can be uniquely decomposed
into terms proportional to the zeroth, first, and second powers of v, respectively,
and the velocity change v = (e*7«)jA1 u^9
mai = GK(v°) + CK(vl) + ZK(v2) + EK(v) , (2)
where the terms on the r.h.s. correspond to the gravitational, Coriolis-Lense-
-Thirring, centrifugal and Euler force, respectively.
3. Embedding diagrams
The properties of the (optical reference) geometry can conveniently be represented
by embedding of the equatorial (symmetry) plane into the 3D Euclidean space with
line element expressed in the cylindrical coordinates (p,z,a). The embedding
diagram is characterized by the embedding formula z = z(p) determining a surface in
the Euclidean space. Requiring the line element of the Euclidean space to be
isometric to the 2D equatorial plane of the ordinary or the optical space line element,11
we arrive at parametric form of the embedding formula z(p) = z(r(p)) with r being
the parameter,
d7 = vhrr ~ (X) - p2 = h^- (3)
Because dz/dp = {dz/dr){dr/dp), the turning points of the embedding diagram,
giving its throats and bellies, are determined by the condition dp/dr = 0. The
reality condition hrr — (dp/dr)2 > 0 must be satisfied.
4. Example of embedding diagrams — Ernst spacetime
The static Ernst spacetime4'12
ds2 = A2[(l - 2Mr"1)dt2 + (1 - 2Mr-1)~1dr2 + r2 d(92] + r2A~2 sin2 6 d02 , (4)
where M = McgsG/c2 is the mass, B = BcgsG1/2/c2 is the strength of the magnetic
field, A = 1 + B2r2 sin2 9, is the only exact solution of Einstein's equations known
to represent the spacetime of a spherically symmetric massive body or black hole
of mass M immersed in an otherwise homogeneous magnetic field. If the magnetic
field disappears, the geometry simplifies to the Schwarzschild geometry. Therefore,
sometimes the Ernst spacetime is called magnetized Schwarzschild spacetime. Some
illustrative embedding diagrams are collected in Fig. 1.
5. Concluding remarks
Embedding diagrams of the optical geometry give an important tool of visualization
and clarification of the dynamical behaviour of test particles moving along equatorial
circular orbits: we imagine that the motion is constrained to the surface z(p)-3 The
shape of the surface z(p) is directly related to the centrifugal acceleration. Within
2301
B = 0.2 > Bc
B = 0.08 < B,
Figure 1. Left column: ordinary geometry, right column: optical geometry of Ernst spacetime.
It can be proved4 that a critical magnetic field Bc ~ 0.0947 exists. For B > Bc, neither throats
nor bellies and no circular photon orbits exist. For B < Bc, the throat and the belly develop,
corresponding to the inner unstable and outer stable photon circular orbit.
the upward sloping areas of the embedding diagram, the centrifugal acceleration
points towards increasing values of r and the dynamics of test particles has an
essentially Newtonian character. However, within the downward sloping areas of
the embedding diagrams, the centrifugal acceleration has a radically non-Newtonian
character as it points towards decreasing values of v. Such a kind of behaviour
appears where the diagrams have a throat or a belly. At the turning points of the
diagram, the centrifugal acceleration vanishes and changes its sign.
Bibliography
1. C. W. Misner, K. S. Thome and J. A. Wheeler, Gravitation (Freeman, San Francisco,
1973).
2. Z. Stvtchlik and S. Hledik, Phys. Rev. D 60, p. 044006 (15 pages) (1999).
3. S. Kristiansson, S. Sonego and M. A. Abramowicz, Gen. Relativity Gravitation 30,
275 (1998).
4. Z. Stuchlik and S. Hledik, Classical Quantum Gravity 16, 1377 (1999).
5. P. Slany, Some aspects of Kerr-de Sitter spacetimes relevant to accretion processes,
in Proceedings of RAGtime 2/3: Workshops on Mack holes and neutron stars, Opava,
11-13/8-10 October 2000/01, eds. S. Hledik and Z. Stuchlik (Silesian University in
Opava, Opava, 2001).
6. M. A. Abramowicz, B. Carter and J. Lasota, Gen. Relativity Gravitation 20, p. 1173
(1988).
7. M. A. Abramowicz, P. Nurowski and N. Wex, Classical Quantum Gravity 12, p. 1467
(1995).
8. M. A. Abramowicz and J. C. Miller, Monthly Notices Roy. Astronom. Soc. 245, p.
729 (1990).
9. M. A. Abramowicz, P. Nurowski and N. Wex, Classical Quantum Gravity 10, p. L183
(1993).
10. M. A. Abramowicz, J. Miller and Z. Stuchlik, Phys. Rev. D 47, 1440 (1993).
11. Z. Stuchlik and S. Hledik, Acta Phys. Slovaca 49, 795 (1999).
12. F. J. Ernst, J. Math. Phys. 17, p. 54 (1976).
CANONICAL ANALYSIS OF RADIATING ATMOSPHERES OF
STARS IN EQUILIBRIUM *
ZOLTAN KOVACStt, LASZLO A. GERGELY* and ZSOLT HORVATH*
f Max-Planck-Institut fur Radioastronomie,
Auf dem Hiigel 69, D-53121 Bonn, Germany
\ Departments of Theoretical and Experimental Physics, University of Szeged,
Dom ter 9, H-6720 Szeged, Hungary
zkovacs@mpifr-bonn.mpg.de, gergely@physx.u-szeged.hu, zshorvath@titan.physx.u-szeged.hu
The spherically symmetric, static spacetime generated by a cross-flow of non-interacting
null dust streams can be conveniently interpreted as the radiation atmosphere of a star,
which also receives exterior radiation. Formally, such a superposition of sources is
equivalent to an anisotropic fluid. Therefore, there is a preferred time function in the system,
defined by this reference fluid. This internal time is employed as a canonical coordinate,
in order to linearize the Hamiltonian constraint. This turns to be helpful in the canonical
quantization of the geometry of the radiating atmosphere.
Keywords: canonical gravity, spherical symmetry, null dust
The quantum theory of gravitational collapse motivated many authors to study
models with both in- and outgoing thin null dust shells in a spherically symmetric
geometry. Such models can equally apply for other phenomena, like radiative
domains around stars in thermodynamical equilibrium. The model of a radiative stellar
atmosphere composed of two null dust streams provides good prospects for carrying
out a complete canonical analysis and quantization. We present here an overview of
the Hamiltonian description of two cross-streaming radiation fields with spherical
symmetry and the first steps towards the Dirac quantization of this constrained
Hamiltonian system.
Letelier demonstrated that the energy-momentum tensor of two superimposed,
counter-propagating radiation fields is equivalent to the energy-momentum tensor
of a specific anisotropic fluid.1 Based on this algebraic equivalence we have recently
shown that the dynamics derived by extremizing the matter Lagrangians of these
two models are the same.2 For the purpose of canonical analysis the two cross-
flowing radiation fields can therefore be substituted with a single anisotropic fluid
(with radial pressure equaling the energy density and no tangential pressures).
The equivalence with the fluid model is crucial for our purposes since earlier
works on the Hamiltonian formalism of two cross-flowing radiation fields with
spherical symmetric geometry, although achieving important results, could not solve the
problem of the absence of an internal time.3 The possibility of replacing the two-
* Research supported by OTKA grants no. T046939 and TS044665, the Janos Bolyai Fellowships
of the Hungarian Academy of Sciences, the Pierre Auger grant 05 CU 5PD1/2 via DESY/BMF
and the EU Erasmus Collaboration between the University of Szeged and the University of Bonn.
Z.K. and L.A.G. thank the organizers of the 11th Marcel Grossmann Meeting for support.
2302
2303
component null dust with an anisotropic fluid raises the possibility to introduce the
proper time as an internal time in the Hamiltonian formalism, in analogy with the
case of the incoherent dust.4
We foliate the static and spherically symmetric geometry by the spherically
symmetric leaves £t labelled by the parameter time t:
ds2 = -{N - ANr2)dt2 - A2Nr2dtdr + A2dr2 + R2dtt2 , (1)
where A(t, r) and R(t,r) are the metric functions and N and Nr are the lapse
function and the non-vanishing component of the shift vector, respectively.5
A static, spherical symmetric space-time describing the cross-flow of two null
dust streams (or equivalently an anisotropic fluid) has been found:6
ds2 = -2aez2R-1{Z)[dT2 - R2{Z)dZ2} + R2(Z)dn2 , (2)
where T and Z are time and radial coordinates of the fluid particles and
-R{Z) = a
Motivated by this exact solution we chose the scalar fields A, R, T and Z appearing
in the metrics (1) and (2) as the canonical coordinates of the gravity and the matter
source. The proper time T of the fluid particles provides the internal time for the
colliding radiation fields, whereas the radial coordinate Z gives the Lagrangian
coordinate of the fluid particles for constant 9 and 0.
In order to provide the Hamiltonian description of this model, we perform the
Legendre transformation of the Lagrangian
S2ND[i4)9ab,p} = fd*Xy/W^bP(UaUa+VaVa) ,
describing two non-interacting null dust streams with time-independent energy
density p, which propagate along the null congruences ua and va. We perform the
transformation by decomposing the tangent vectors of the two null congruences
with respect to the gradients of the matter variables,
ua = WT<a + RWZta , va = WTia - RW Z\a , W = aez2 R ,
and introducing the momenta P and Pz canonically conjugated to T and Z,
P = gN-\Tt-NrTr), Pz = QR2N-\Z,t~NrZ,r) G = 2a^pW2.
The matter Lagrangian can be then rewritten in the "already Hamiltonian" from
L2ND =fP + zpz - NH\ND - NrH2ND ,
where the super-Hamiltonian and supermomentum constraints of the system
consisting of the two null dust streams are
H2ND = g-l(p2 + p2/R2) + £(T/2 + R2^ > H2ND = j,,p + #pz
2Z\ B +
r
dx
2304
By eliminating the comoving density p form the Hamiltoiiian constraint and
employing that the super-Hamiltonian and the supermomentum constraints of the total
system weakly vanish,
H± := Hi + N\ND « 0 , Hr := H? + N2rND « 0 , (3)
we are able to solve the constraints with respect to the momenta Pr and Pz. The
vacuum constraints H^ and Hff are expressed in terms of the preferred canonical
variables of spherically symmetric vacuum gravity,7 with A and its canonical
momentum is replaced with the Schwarzschild mass M and its canonical momentum
Pm- After solving the constraints with respect to the momenta we can introduce
a new set of linearized constraints, equivalent to Eq. (3), in which the momenta of
the matter variables are separated from the rest of the canonical data:2
Hr.= P + h[M, R, T, Z, PM, Pr] = 0 , H]z := Pz + hz[M, R, T, Z, Pm,Pr] = 0 .
The above linearized form of the constraints is advantageous for two reasons. First,
the Hamiltonian constraint H^ is resolved with respect to the momentum P canon-
ically conjugated to the internal time T. Second, the new constraints have strongly
vanishing Poisson brackets and as such form an Abelian algebra instead of the Dirac
algebra of the old constraints.
In the canonical quantization of gravity coupled to the two null dust streams
with spherically symmetric geometry the super-Hamiltonian constraint becomes
an operator equation on the state functional \& [Z, T, M, R] of gravity, restricting
the allowed states. Since classically the super-Hamiltonian constraint was resolved
with respect to the momentum P, the operator condition leads to the functional
Schrodinger equation
i — *[T, M, R] = h[M, R, T, Z, PM, Pr]*[T, M, R] . (4)
The operator version of the supermomentum constraint H^z applied on the state
functional ensures that the quantum states are independent of the dust frame Z.4
Besides the Hilbert space structure of the solutions to the Eq. (4), the other
advantage of the linearized constraints is that their Abelian algebra turns into a true Lie
algebra of vacuum gravity. These promising achievements point towards a possible
consistent canonical quantization of the presented superposed null dust system.
References
P.S. Letelier, Phys. Rev. D 22, 807 (1980).
Zs. Horvath, Z. Kovacs, G. A. Gergely, Phys. Rev. D 74, 084034 (2006).
J. Bicak, P. Hajicek, Phys. Rev. D 68, 104016 (2003).
J.D. Brown, K.V. Kuchaf, Phys. Rev. D 51, 5600 (1995).
B.K. Berger, D.M. Chitre, V.E. Moncrief, Y. Nutku, Phys. Rev. 1)8, 3247 (1973).
L. A. Gergely, Phys. Rev. D 58, 084030 (1998).
K.V. Kuchaf, Phys. Rev D 50, 3961 (1994).
Self-Gravitating Systems
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PLATONIC SPHALERONS IN EINSTEIN-YANG-MILLS AND
YANG-MILLS-DILATON THEORY *
BURKHARD KLEIHAUS*, JUTTA KUNZt and KARI MYKLEVOLL*
Institut fur Physik, Universitat Oldenburg, Postfach 2503
D-26111 Oldenburg, Germany
* kleihaus@theorie.physik.uni-oldenburg.de, tkunz@theorie.physik.uni-oldenburg.de,
* myklevoll@theorie.physik.uni-oldenburg.de
We here present new sphaleron solutions in EYM and Yang-Mills-dilaton theory. These
sphalerons have no continuous rotational symmetries at all, but have the symmetries
of crystals or of platonic bodies, and we therefore call them platonic sphalerons. Their
symmetries are related to certain rational maps of degree N. Since the gravitating
platonic sphalerons are static regular solutions without continuous symmetries, they belong
to a completely new kind of gravitating solutions, and most importantly these solutions
indicate the existence of static black holes with only discrete symmetries of the horizon.
Gravitating classical solutions in Yang-Mills theories have many suprising
properties.1 In SU(2) Einstein-Yang-Mills (EYM) theory for instance, globally regular,
spherically symmetric2 and axially symmetric3 solutions form sequences, which
converge to extremal Reissner-Nordstrom solutions.4 Moreover, static black holes with
only axial symmetry have been found.5 Motivated by the aim to demonstrate that
even gravitating solutions with only discrete symmetries exist we here consider
platonic solutions in Yang-Mills-dilaton (YMD) theory.6 These solutions may be viewed
as exact (numerical) solutions of scalar gravity, by considering the dilaton as a kind
of scalar graviton, or as approximate solutions of EYM theory, when the metric is
parametrised in the form
ds2 = -e^dt2 + e'^dsl , dsj = Sijdx'dx?
For static configurations, in this approximation the EYM action then agrees with
the action of YMD theory,
S= f{-\d'^d^- \eUMVv)\ #x ,
where F^v = d^Ay — duAfl +i [Afl,A„] denotes the SU(2) field strength tensor, and
Afl = Aara/2 the gauge potential. Variation of the action with respect to the gauge
potential and the dilaton field yields the field equations, which have to be solved
numerically. In order to obtain solutions with certain symmetries, it is convenient
to decompose the gauge potential with respect to the unit vector ur and its partial
derivatives, where Hr is related to a rational map R of degree N via7
2. 1
UR
l + \R\-
(R + R, ~i(R-R) , l-|i?i2)
"This research has been partially supported by by the DFG under contractKU612/9-1 and by the
Research Council of Norway under contract 153589/432
2307
2308
The nine profile functions of Afl involved in this parametrisation together with the
dilaton function have to be found numerically as solutions of the field equations.
Boundary conditions need to be imposed on the functions to ensure regular, finite
energy solutions.6 The residual gauge degrees of freedom are fixed by the gauge
condition diA1 = 0.
We focus on platonic solutions with cubic symmetry, related to a certain rational
map of degree N = 4. We constructed numerically the fundamental cubic: solution
(k = 1) and its first excitation (k = 2), which form the first two solutions of the
cubic N = 4 sequence. In Fig. 1 we present surfaces of constant total energy density
dot = \di<t>&4>+ l^Tv (FijF**)
and energy density of the gauge field
€V = \e2+Ti{FiiFii) .
2
The energy densities clearly reflect the symmetries of a cube. The energy density
of the gauge field for the excited solution reveals a cube within a cube.
Fig. 1. Isosurfaces of ftot for the fundamental solution (left) and the first excitation (middle),
and isosurface of cp for the first excitation (right).
The symmetry of the dilaton field is demonstrated in Fig. 2, where we show
isosurfaces of the function —.goo = e2^- Clearly, the dilaton reflects the symmetry of
the energy density.
The (dirnensionless) energies of the fundamental and first excited cubic solutions
are found to be Ek=i = 2.203 and Ek=2 = 3.11, respectively, which are below the
energies of the corresponding axially symmetric solutions.9 We conjecture, that
the sequence of cubic solutions converges to the extremal abelian solution with
magnetic charge P = 4, analogous to the sequences of spherically symmetric8 and
axially symmetric9 solutions.
By including the dilaton to approximate the effects of gravity, we have obtained
evidence for the existence of platonic EYM solutions, while avoiding the complexity
of the full set of EYM equations in the absence of rotational symmetry. Construction
2309
= 1,-g =0.07979
k=2»-g =0,019125
0.2
0
-0.2-
-o.;
0.2
-0.2
U.t
0.1-
-0.1-
-0.1
0.1
-0.1
0
0.1
Fig. 2. Surfaces of constant metric function —goo = e2^ for the fundamental cubic YMD solution
(left) and the first excitation (right).
of exact (numerical) platonic EYM solutions, however, remains a true challenge. The;
existence of gravitating regular solutions involving non-Abelian fields, is related to
the existence of black holes with non-Abelian hair.10 Thus the existence of
gravitating regular solutions with platonic symmetries strongly indicates the existence of a
completely new type of black holes: static black holes which possess only discrete
symmetries.11
References
1. M. S. Volkov and D. V. Gal'tsov, Phys. Rept. 319, 1 (1999).
2. R. Bartnik and J. McKiunon, Phys. Rev. Lett. 61, 141 (1988).
3. B. Kleihaus and J. Kunz, Phys. Rev. Lett. 78, 2527 (1997); Phys. Rev. D57, 834
(1998).
4. Note that in Schwarzschild like coordinates the limiting solutions of the sequences
possess a non-abelian part inside the event horizon.
5. B. Kleihaus and J. Kunz, Phys. Rev. Lett. 79, 1595 (1997); Phys. Rev. D57, 6138
(1998).
6. B. Kleihaus, J. Kunz, and K. Myklevoll, Phys. Lett. B638, 367 (2006).
7. C. J. Houghton, N. S. Manton and P. M. Suteliffe, Nucl. Phys. B 510, 507 (1998).
8. G. Lavrelashvili and D. Maison, Phys. Lett. B295, 67 (1992); P. Bteon, Phys. Rev.
D47, 1656 (1993) 1656; D. Maison, Commun. Math. Phys. 258, 657 (2005).
9. B. Kleihaus and J. Kunz, Phys. Lett. B392, 135 (1997).
10. M. S. Volkov and D. V. Galt'sov, Sov. J. Nucl. Phys. 51, 747 (1990); P. Bizon, Phys.
Rev. Lett. 64, 2844 (1990); H. P. Kiinzle, and A. K. M. Masoud-ul-Alam, J. Math.
Phys. 31, 928 (1990); B. Kleihaus and J. Kunz, Phys. Rev. Lett. 79, 1595 (1997);
Phys. Rev. D57, 6138 (1998).
11. S. A. Ridgway, and E. J. Weinberg, Phys. Rev. D51, 638 (1995); Phys. Rev. D52,
3440 (1995); Gen. Rel. Grav. 27, 1017 (1995).
COMMENT ON "GENERAL RELATIVITY RESOLVES GALACTIC
ROTATION WITHOUT EXOTIC DARK MATTER" BY
F.I. COOPERSTOCK & S. TIEU
B. FUCHS
Astronomisches Rechen-Institut am Zentrum fur Astronomie der Universildl Heidelberg,
Monchhofstrasse 12-14,
69120 Heidelberg, Germany
fuchs@ari.uni-heidelberg. de
S. PHLEPS
Max-Planck-Institut fur exlralerrestrische Physik,
Giessenbachslrasse,
8574-8 Garching, Germany
sphleps@mpe.mpg.de
Recently Cooperstock & Tieu1 (hereafter CT05) have proposed a new approach to
the interpretation of rotation curves of spiral galaxies, which is based on the theory
of general relativity. They argue that even in the case of such weak gravitational
fields as in galaxies certain non-linear terms in Einstein's field equations play an
important albeit hitherto neglected role. Their formalism is applied to concrete
examples, and CT05 provide quantitative fits of the rotation curves of the Milky
Way and three further external spiral galaxies and they derive mass models for these
galaxies. The resulting models are quite flattened and their total masses are typically
one order of magnitude lower than those of current models of spiral galaxies. In these
models the flat outer rotation curves are usually modelled by massive dark halos.
The low total masses estimated by CT05 can be accounted for by the baryonic mass
content of the galaxies alone. CT05 conclude that it is thus not necessary to invoke
"exotic dark matter" to model galactic rotation curves.
Although not yet in print, this spectacular result raised considerable interest
but was also met with scepticism in the astronomical community. For instance
CT05 have not dealt with the dark matter problem of galaxy clusters. A
conceptual problem arises from the non continuously differentiable shapes of the density
cusps of the vertical density profiles of the models at the galactic midplanes. This
seems to indicate that each galaxy would at least formally harbour at its mid-
plane a sheet of negative mass density,23 Other formal inconsistencies are discussed
in.4 In a rebuttal to these criticisms CT055 maintain the claim of their original
paper.
We have demonstrated6 how observations of the Milky Way can be used as an
empirical counter example against CT05's conjecture of the dynamics of galactic
disks.
According to CT05's formalism the distribution of mass in their galaxy models
is given by
2310
2311
p(r, z) = 8
•36-105((EfcnCne-fe"(^Jo(fcnr)j
(1)
where </0ji denote Bessel functions of the first kind. The coefficients fcn and Cn
have been determined by CT05 by fitting the corresponding model rotation curve
to the observed rotation curve of the Milky Way and are given in their Table 1.
Fig. 1 shows in the left panel the vertical mass density profile at the position of the
Sun, p(rQ,z), calculated with Eq. (I). The Sun lies close to the Galactic midplane,
z « 0, and the galactocentric distance of the Sun is about rQ = 8 kpc,7 but other
determinations are discussed in the literature as well. Thus density profiles assuming
r0 = 7 kpc and r0 =8.5 kpc, which bracket the literature values for r0, are also
shown in Fig. 1. Holmberg & Flynn8 have meticulously compiled an inventory of the
contributions by the various phases of the interstellar gas and the stellar populations
to the mass budget in the vicinity of the Sun and find a local mass density of /?(Vo, 0)
= 0.094 MQ/pc3 = 6.3-10-21 kg/in3. As described in8 this value is consistent with
dynamical measurements of the local mass density, if the gravitational force field is
calculated in Newtonian approximation. However, as can be seen from Fig. 1 the
mass model of CT05 predicts at the position of the Sun a density of about p(rQ, 0)
= 0.0f5 M©/pc3 = f.0-10-21 kg/m3. This amounts to only f6 percent of the mass
density actually observed in the form of baryons in the solar neighbourhood.
0.025
0.02
cL 0.015
G
2L o.oi
0.005
0
(
j , i i | , i , , | i i i i | i ri i |
- , , , , , , 1
) 1 2 3 4
z [kpc]
;
j
-
C
10
8xl07
CO
^exio7
^4xl07
^ 2xl07
0
1' 1
F}
; 11
j 0
'"' '
* ■ .
,,, i,
i
-Tr-pm-Tyrm-
> >
, ,\ , , , ,1 , , , i
, , , ,_
-
-
2 3 4 E
7 [kpc]
Fig. 1. Predicted versus observed vertical distribution of the mass density in the Milky Way at the
position of the Sun. Left panel: Vertical distribution predicted by the mass model of Cooperstock
& Tieu. The profiles are labelled by the assumed galactocentric distance of the Sun ranging from 7
to 8.5 kpc. Right panel: The observed distribution of stars perpendicular to the Galactic midplane.
Moreover, the predicted shape of the vertical density distribution looks totally
different from what is actually observed. In the right panel of Fig. 1 the observed
number density distribution of stars perpendicular to the Galactic midplane at the
2312
position of the Sun, v{tq, z), is shown. The number densities have been determined
with counts of K and M stars in five fields of the Calar Alto Deep Imaging Survey.9
Since the CADIS star counts suffer from severe Poisson errors near to the midplane
due to the conical counting volumes (cf. Fig. 1), the local normalization has been
determined by counting stars of the same spectral types in the Fourth Catalogue of
Nearby Stars,910 The CADIS fields point towards different galactic longitudes and
latitudes so that the scatter of the data points in the right panel of Fig. 1 reflects also
some mild variations of the vertical shape of the Galactic disk seen in the various
directions. We may add that the vertical density profile derived from CADIS data
is in perfect agreement with the results of Zheng et al.11 Early type stars and most
of the interstellar gas are distributed in a narrow layer at the Galactic midplane
so that the overall distribution of baryons is even more concentrated towards the
midplane than the late type stars stars, whereas the vertical distribution predicted
by CT05's model is extremely shallow compared to the observations. Indeed the
implied surface density of the disk at the position of the Sun is 179 MQ/pc2 =
0.37 kg/m2. Although the midplane density is much too low, the predicted surface
density is a factor of about four higher than the observed surface density of baryons
of 48 M0/pc2 = 0.1 kg/m2.8 As can be seen from Eq. (1) and Eq. (18) of CT05
any attempt to rescale the model by increasing the coefficients kn in order to obtain
a smaller scale height would alter also the radial shape of the predicted rotation
curve V(r, z = 0) and thus destroy the fit to the observed rotation curve.
This implies that the model of CT05 for the Milky Way, which was so constructed
that it gives an excellent fit of the observed rotation curve, has singularly failed to
reproduce the independent observations of the local Galactic mass density and its
vertical distribution. This one counter example casts, in our view, severe doubts on
the viability of Cooperstock & Tieu's theory of the dynamics of galactic disks in
general.
References
1. Cooperstock, F.I., Tieu, S., astroph/0507619 (2005a)
2. Korzynski, M., astro-ph/0508377 (2005)
3. Vogt, D., Letelier, P.S., astro-ph/0510750 (2005)
4. Cross, D.J., astro-ph/06011191 (2006)
5. Cooperstock, F.I., Tieu, S., astro-ph/0512048 (2005b)
6. Fuchs, B., Phleps, S., New Ast. 11 (2006) 608
7. Reid, N.I., ARA&A 31 (1993) 345
8. Holmberg, J., Flynn, C, MNRAS 313 (2000) 209
9. Phleps, S., Drepper, S., Meisenheimer, K., Fuchs, B., A&A 443 (2005) 929
10. Jahreifi, H., Wielen, R., in: B. Battrick, M.A.C. Perryman and P.L. Bernacca (eds.):
HIPPARCOS '97ESA SP-402 (1997) 675
11. Zheng, Z., Flynn, C, Gould, A. et al., ApJ 555 (2001) 393
SOLITONIC AND NON-SOLITONIC Q-STARS
Y. VERBIN
Department of Natural Sciences, The Open University of Israel,
P.O.B. 808, Raanana 43107, Israel
verbin@openu.acil
Q-balls1 are a simple kind of non-topological solitons which occur in a wide
variety of (theoretical) physical contexts2"9 like the supersymmetric Standard Model.2'3
Most of the Q-ball studies are based on the "original" flat space Q-balls.
However, it is evident that for a large enough mass scale, gravitational effects become
important and one needs to study Q-stars10 which are their self-gravitating
generalizations. That is, they are finite mass and charge solutions of the following U(l)
symmetric action:
S = y d^vlfff Q(VM$)*(V$) - J7(|$|) +
:R
where the potential function is usually taken to be:
U(\$\) =—\3>\2 |$|p + W- (2)
I p q
Two kinds of choices are popular in the literature: p = 3, q = 4 and p = 4, q — 6.
Actually, this system allows for a different kind of localized solutions already without
self-interaction (i.e. only mass term) or with an additional |<J>|4 term, namely, boson
stars.11""13 Boson stars have also a conserved global U(l) charge, but unlike Q-stars,
they do not have a flat space limit.
We will assume spherically-symmetric solutions with non-vanishing U(l) charge,
i.e. <J> = m/(r)eia;* and ds2 = A2(r)dt2~B2(r)dr2-r2(d92+sm2 9d<p2) so the charge
and mass are given by
/>0O nO
Q = 47rwm2/ drr2{B/A)f2, M = 4tt /
Jo Jo
drr
j2m2 f2 m2 f'2
Without loss of generality we will assume u> > 0 so we will have Q > 0 as well.
The existence of Q-stars was demonstrated by Friedberg et al14 and by Lynn10
together with a presentation of the basic properties of the solutions for the 2-4-6
potential. A discussion of 2-3-4 Q-stars appeared only quite recently.15 It was shortly
followed by a study16 which showed that gravity limits the size of Q-balls. On the
other hand, a recent analysis17'18 of spinning Q-balls and Q-stars is concentrated
in the 2-4-6 case.
We give here the main results of a systematic comparative study of both kinds
of Q-stars, including the dependence on the gravitational strength parametrized
by the dimensionless parameter 7 = 47rC?m2. A more detailed summary will be
presented elsewhere.19 We choose in both cases the parameters a = 2 and A = 1
so the potentials will have a similar form, and for 7 will take the following three
values: 7 = 0, 0.02, 0.2.
2313
2314
Fig. 1. Plots of log(Q) (solid line) and log(M/m) (dashed) vs. /(0) for 7 = 0 (Q-balls), 7 = 0.02
and 7 = 0.2. (a) 2-3-4 Q-stars; (b) 2-4-6 Q-stars.
Figure 1 summarizes the main results in the Q — /(0) and M — /(0) planes
namely, the general behavior of the charge and mass. Figure 2 depicts the binding
energy per particle (in dimensionless form), 1 — M/mQ as a "function" of Q which
is more instructive from a physical point of view. It is evident from this figure that
Q-stars are more strongly-bound than their non-gravitating counterparts with the
same Q. From our results one can draw the following observations and conclusions:
Already in flat space there is a very significant difference between the "thick
wall" (small /(0)) solutions of the two potentials: the thick wall Q-balls of the 2-3-4
potential are small and stable, i.e. Q and M vanish as lu —> m or /(0) —>• 0 while
M/m < Q. On the other hand, those of the 2-4-6 potential are large and unstable,
i.e. both charge and mass diverge while M/m > Q in the same limit.
Gravity introduces significant changes such as allowing solutions in regions where
flat space solutions do not exist and limiting the charge and mass of Q-stars. But
still the changes are quite small for weak scalar field 2-3-4 Q-stars, as seen in figures
la and 2a. On the other hand, gravity changes completely the nature of the weak
field 2-4-6 solutions even for a small 7 (say, 0.02) as figure lb shows: as /(0) —> 0
the charge and mass do not blow up, but on the contrary go to zero. Looking from
the other direction, one sees that the charge and mass start rising from zero, reach
a local maximum, decrease a little and then go to the thin wall region and beyond
as described below. Moreover, unlike the 2-4-6 Q-balls for /(0) << 1 which were
unstable, now there appears a region of stability (below the resolution of figures lb
or 2b) for small enough /(0). This is followed by a region of unstable solutions up
to a certain value of /(0). From this point on, all solutions are stable.
For larger values of /(0) we encounter for both potentials "thin wall" Q-stars
quite similar to the corresponding Q-balls, although the self-gravitating solutions are
not so well described by the thin wall approximation. The reason is that where the
thin wall approximation in flat space is accurate, gravity already causes deviations.
To push it to the extreme, the thin wall approximation becomes exact for Q —> 00,
but gravity keeps Q-stars away from this best region by introducing a maximal
2315
(a)
0.6
0.4
0.2
0
-0.2
2-3-4
y=0.2
J#
y=0J]^-
/>"
^.
/
1.5
2 2.5
log(Q)
3 .5
3 .5
Fig. 2. Plots of binding energy per particle (mQ - M)/mQ vs. ]og(<3) for 7 = 0 (dashed), 7 = 0.02
and 7 = 0.2 and for boson stars with 7 = 0.2 (dotted), (a) 2-3-4 Q-stars; (b) 2-4-6 Q-stars.
value of Q. For large values of 7 there are no thin wall solutions altogether.
Another new gravitational effect is the existence of solutions when the central
field becomes considerably larger than /*(0) which is the flat space critical field.
Unlike the Q-ball case, the mass and charge curves cross this point and there are
solutions as far as we were able to explore numerically. All small 7 solutions are
stable, but their nature for /(0) > /*(0) becomes quite different from the Q-balls
as we go further. Moreover, it is obvious that this kind of solutions cannot be
considered solitonic as they do not have a flat space limit: while the charge and
mass of the solutions with /(0) < /*(0) have a (finite) limit as 7 —> 0, those in the
other region blow up. In other words, it is only thanks to gravity that this kind of
solutions with /(0) > /*(0) exists.
References
1.
2.
3.
4.
5.
6.
7.
8.
S.
A
K
A
K
A
G
K
9. M.
10. B.
11. P.
12.
13.
14.
15.
16.
17.
18.
T.
A.
R.
A.
T.
M
B.
19. Y,
R. Coleman, Nucl. Phys. B 262, 263 (1985) [Erratum-ibid. B 269, 744 (1986)].
Kusenko, Phys. Lett. B 405, 108 (1997)
Enqvist and J. McDonald, Phys. Lett. B 425, 309 (1998)
Kusenko and M. E. Shaposhnikov, Phys. Lett. B 418, 46 (1998)
Enqvist and J. McDonald, Nucl. Phys. B 538, 321 (1999)
Kusenko, Phys. Lett. B 404, 285 (1997)
R. Dvali, A. Kusenko and M. E. Shaposhnikov, Phys. Lett. B 417, 99 (1998)
Enqvist and A. Mazumdar, Phys. Rept. 380, 99 (2003)
Dine and A. Kusenko, Rev. Mod. Phys. 76, 1 (2004)
W. Lynn, Nucl. Phys. B 321, 465 (1989).
Jetzer, Phys. Rept. 220, 163 (1992).
D. Lee and Y. Pang, Phys. Rept. 221, 251 (1992).
R. Liddle and M. S. Madsen, Int. J. Mod. Phys. D 1, 101 (1992).
Friedberg, T. D. Lee and Y. Pang, Phys. Rev. D 35, 3658 (1987)
Prikas, Phys. Rev. D 66, 025023 (2002)
Multamaki and I. Vilja, Phys. Lett. B 542, 137 (2002).
S. Volkov and E. Wohnert, Phys. Rev. D 67, 105006 (2003)
Kleihaus, J. Kunz and M. List, Phys. Rev. D 72, 064002 (2005).
Verbin, to be published, Phys. Rev. D , (2007).
ROTATING MONOPOLE-ANTIMONOPOLE PAIRS
AND VORTEX RINGS*
ULRIKE NEEMANN, JUTTA KUNZ and BURKHARD KLEIHAUS
Institut fur Physik, Universitat Oldenburg, Postfach 2503
D-26111 Oldenburg, Germany
neemann@lheorie.physik.uni-oldenburg.de
We discuss dyons and electrically charged monopole-antimonopole pairs and vortex rings
in Einstein-Yang-Mills-Higgs theory. The solutions are stationary, axially symmetric and
asymptotically flat. In monopole-antimonopole pair solutions the Higgs field vanishes at
two discrete points along the symmetry axis. In vortex ring solutions the Higgs field
vanishes on a ring, centered around the symmetry axis. The dyons represent non-static
solutions with vanishing angular momentum. In contrast to the dyons the monopole-
antimonopole pairs and vortex rings possess vanishing magnetic charge, but finite angular
momentum, equaling n times their electric charge. The dependence of the solutions on
the strength of gravity is studied.
The non-trivial vacuum structure of SU(2) Yang-Mills-Higgs (YMH) theory gives
rise to regular non-perturbative finite energy solutions, such as magnetic monopoles,
multimonopoles and monopole-antimonopole (MA) systems.
When gravity is coupled to YMH theory, gravitating monopoles and gravitating
MA systems arise.l In each branch of gravitating solutions emerges smoothly
from the corresponding flat space solution, and extends up to a maximal value of the
coupling constant, where, for vanishing Higgs self-coupling constant, it merges with
a second branch. For monopoles this second branch extends only slightly backwards
before it merges with the branch of extremal Reissner-Nordstrom black holes. For
MA systems, in contrast, this second branch extends all the way back to vanishing
coupling constant, where the solutions shrink to zero size.
The coupling constant a, entering the Einstein-Yang-Mills-Higgs (EYMH)
equations, is proportional to the Higgs vacuum expectation value v and the square root
of the gravitational constant G, a2 = 4ttGv2 . Variation of a may thus be considered
as variation of the gravitational constant G along the first branch and as variation
of the Higgs vacuum expectation value v along the second branch. Consequently,
the Higgs field vanishes in the limit a^Oon the second branch.
To any static solution of the YMH and EYMH equations there corresponds a
family of electrically charged solutions which are stationary. Since monopole
solutions carry magnetic charge they cannot possess finite angular momentum.2
Therefore dyons with higher magnetic charge cannot rotate either.
In MA pairs, on the other hand, the magnetic charge vanishes. When electric
charge is added the pair begins to rotate about its symmetry axis, yielding an
angular momentum proportional to its electric charge, J = n Q.2~4
We consider the SU(2) Einstein-Yang-Mills-Higgs action
"This research has been partially supported by the DFG under contract KU612/9-1.
2316
2317
S
= //
R
l-Tx (F^F^) -\^{D^ D^)
gd x
\16ttG
with curvature scalar R, su(2) field strength tensor F,iL, = dtlAv -d^A^+ilA^,, Av\ ,
gauge potential A^ = A^ra/2, and covariant derivative of the Higgs field <& = <J>ara
in the adjoint representation, D^<& = d^ + i[Afl, <f>] .
We employ the Lewis-Papapetrou form of the metric in isotropic coordinates
ds2
-fdt2 + ~{dr2+,
+
lr2 sin-" 9
f
dip ■
-dt
The gauge potential and the Higgs field are parametrized by
f
Andx*
$
^lTr(«,m) + B2r0(""m)) dt - nsin0 (h^t^ + (]
+ (H1/rdr + (l-H2)d0)T^ ,
H4)T^m)) dip
where n and m are integers, with ±n representing the magnetic charge of single
(anti)monopoles and m the total number of monopoles and antimonopoles in MA
systems.1 The su(2) matrices rr , T0n'm\ and t^ are defined as scalar products
of spatial unit vectors with the Pauli matrices
T(m,n) _ sin(mQ}T(n) + cos{m0) Tz } T(n) = Cos(n(j))Tx + sin(ncf>)Ty ,
T(m,n) _ cos^mQ^ T(n) __ sjn(m$) Tz ^ T^l> — —sin(n(j)) Tx + COs(n(j)) Ty .
Dyons with n > 2 show a similar a-dependence as singly charged dyons5 and
monopoles. They merge with the corresponding extremal Reissner-Nordstom
solutions. We exhibit the scaled mass and the electric charge in Fig. f. Rotating MA
pairs show an analogous a dependence as static MA pairs. A branch of gravitating
solutions emerges from the flat space solution and merges at an amax with a second
branch which extends all the way back to a = 0. Interestingly, for the scaled mass
of these solutions the two branches cross before they merge (Fig. 1).
0.6 0.66 0.72
m=2, n=3
~~—-—
__^V-
m=2, n=2 ^~~~x
~~~:
~S^S 7~^
— ""^
^ ——'
m=1,n=3/
m=1, n=2/
~-
1st branch -
2nd branch
/
-
1 2 1.6
Fig. 1. The scaled mass aM and the charge Q are shown as functions of the coupling constant
a for dyon solutions with n = 2,3 and for MA pair resp. vortex ring solutions with n = 1, 2, 3.
£L%J I O
a=1.40
0=0.67,1st branch
v
0.-"= ~
0
0.(
Fig. 2. The angular momentum density T*, for a monopole solution with n = 2 (left) and for ;
monopole-antimonopole pair solution with n = 2 (right).
z0,1 'branch
p0.1st branch
Po, 2"" branch • • -
1a branch, «<«,,
M
/"
" A
^
"~"~~ —^
1s'branch, 0 = 0,,
1st branch,
M |
T
"M
1st branch, a > a^
Fig. 3. The location of the nodes for MA pair resp. vortex ring solutions with n = 2 at 7 = 0.32
and 7 = 0 are shown as functions of the coupling constant a. The parameter 7 is related to the
electric charge.
From Fig. 2 we observe that the angular momentum density T* for dyonic
solutions is antisymmetric with respect to reflection z —> —z. Hence these solutions
possess vanishing angular momentum. In contrast T* is symmetric for MA pairs,
allowing for finite angular momentum.
The nodes of the Higgs field represent the locations of the magnetic poles. For
n=l the moiiopoles sit on the symmetry axsis, whereas for n = 3 the zeros of the
Higgs field form a vortex ring. However, for n = 2 the nodes change character. As
a is increased along the first branch the poles approach each other and merge at
avr, beyond avr they form a vortex ring. This ring increases in size, until it reaches
a maximum at amax, and then decreases to zero size on the second branch.
References
1. B. Kleihaus, J. Kunz, and Ya. Shnir, Phys. Rev. D71, 024013 (2005).
2. J. J. van der Bij and E. Radu, Int. J. Mod. Phys. A17, 1477 (2002); Int. J. Mod. Phys.
A18, 2379 (2003).
3. V. Paturyan, E. Radu, and D. H. Tchrakian, Phys. Lett. B609, 360 (2005).
4. B. Kleihaus, J. Kunz and U. Neemann, Phys. Lett. B623, 171 (2005).
5. Y. Brihaye, B, Hartinann, and J. Kunz, Phys. Lett. B441, 77 (1998),
SOURCES OF STATIC CYLINDRICAL SPACETIMES
MARTIN ZOFKA
Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University
V Holesovickdch 2, 180 00 Praha 8 - Holesovice, Czech Republic
zofka@mbox. troja. mjj. cuni. cz
We present various shell sources of vacuum, static cylindrical spacetimes with and
without the cosmological constant. The matter forming the sources can be interpreted as
geodetical (null) dust or perfect fluid. We give ranges of metric parameters admitting
such interpretations and find relations for the mass per unit length of the source.
1. Introduction
Cylindrical solutions of Einstein equations are important due to their implications
regarding the properties of gravitational fields in the vicinity of finite prolate bodies.
Although the assumed symmetry reduces the complexity of the field equations,
there is no analytical cylindrical analogue of the static, spherically symmetric star
of perfect fluid with an ordinary equation of state (e.g., constant density).
We thus restrict ourselves to studying exact solutions representing two vacuum
regions—sections of the Levi-Civita (LC) solution, possibly including a
cosmological constant—separated by an infinitely thin wall and we further require that there
be no curvature or conical singularity along the axis. Such a source is an
approximation to more realistic situations involving extended bodies. By imposing certain
physically plausible conditions on the matter forming the wall, we obtain intervals
for the parameters of the vacuum solutions. To achieve this, we employ Israel
formalism with the separating hypersurface denned by constant radial distance from
the axis of symmetry. With this choice, the intrinsic geometry on the hypersurface
is clearly fiat which simplifies the interpretation of its energy-momentum tensor.
The present contribution is an overview of our results and we refer the reader
to the respective articles for more details.
2. Levi-Civita: Static, cyllndrlcally symmetric vacuum spacetime
The metric written in cylindrical coordinates reads
ds2 = -p2mdt2 + p~
p2m\dz2 + dp2) + -^p2d^
where m (the LC parameter) is related to the mass per unit length of the source
as revealed by the behavior of geodetical test particles far away from the axis as
compared to the Newtonian case. Parameter C is a measure of conicity or deficit
angle of the spacetime and ensures the correct range ip € [0, 2n).
We now take an interior (m_,C_,/9 < p_) and exterior (m+,C+,p > p+) LC
spacetimes and join them together requiring the circumference of the junction
hypersurface to be the same from both sides. Using Israel formalism,1 we calculate
2319
2320
the 3-dimensional induced energy-momentum tensor, Sij, on the shell. We further
define mass per unit length of the shell source:
Mi = (Circumference of the shell) ■ Stt = 2ttp_Stt-
One of our goals is to replace the axis singularity of the original spacetime with
a shell source and we thus require m_ = 0 and C_ = 1. If we want to interpret
Sjj as due to perfect fluid or counter-streaming (zero total momentum and angular
momentum) massive particles or photons, we find m+ G [0,1] and Mi £ [0, |], see
Figure 1. If we relax the conditions imposed on the shell and only require the weak,
0.8
0.6
0.4
0.2
?w
^—-""^^
MY
0.2
0.4 0.6
Fig. 1. Left: Shells of counter-rotating, purely azimuthal massive particles. The mass per unit
length, Mi, and the velocity ti($) of the particles (measured by static observers; c = 1) as mono-
tonically increasing functions of the external LC parameter m+ with V(^) —> 1, M\ —> 1/4. Right:
Shells of perfect fluid. Mi and the magnitude of the surface pressure integrated along a ring,
Pc = (27rp_)p, as monotonically increasing functions of m^ with Pq —> +oo, M\ —> 1/4.
strong or dominant energy condition we can extend the range of m+ to [0, 2] but
we still find Mi < 1/4.2 This upper bound on Mi exists for any matter on the shell
and is in accord with the notion that a spacetime without singularities is free of
horizons if mass within a given region is bounded by a certain finite value.
3. Levi-Civita-A: Static cylindrically symmetric vacuum spacetime
with non-zero A
In cylindrical coordinates, the metric now reads3'4
ds2 -= Q(r)2/3{-P(r)-2(4*2-8* + l)/3Adf2 +p(r)2(8a2-4a-l
+P(r)-4(2a2+2a-l)/3A^2/c,2} + ^2 >
V3Adz2 +
where for A > 0 we have P(r) = tan(V3Ar/2), Q(r) = sin(V3Ar) and for A < 0 we
have P{r) = tanh(x/-3Ar/2), Q(r) = sinh(x/-3Ar), with A = 4a2 - 2a + 1. If we
take the limit A —> 0, the parameter a corresponds to m/2 of the LC metric.
If we want to avoid singularities apart from the axis, we must require A < 0.
We now proceed in analogy to the LC case, defining the junction surface and the
2321
induced energy-momentum tensor which can be again interpreted as that of counter-
streaming particles (see Figure 2) or perfect fluid. The resulting ranges of a+ and
Mi remain exactly the same as in the LC case.
We further find M\ < 1/4 for any cylinder without a singularity on the axis
or outside of the shell and satisfying A_ < A+ < 0 and r_ < r+.5 This is a
generalization of the analogous property of the LC spacetimes.
4-
ln(w)
0
.••■'*"" |
..=••"•""' ,***
■ y^^^ CT+
1
1
■-■' I
'Js'
0.5
Fig. 2. The proper velocity (left) and the unit-length mass (right) of particles within a shell for
purely azimuthal motion with <r_ = 0, A_ = A+ = A < 0, C_ = 1. The plotted curves correspond
to A = -10, -1,-1/10, —1/1000 (top to bottom in the left and bottom to top in the right) for a
shell of fixed radius r_ = 1. For a+ £ [0,1/2) the proper velocity is finite and positive. The graph
corresponding to A = -1/1000 approaches the graph for a LC shell (m = 2a).
4. Conclusions
We found several shell sources of various static, cylindrically symmetric, vacuum
solutions of Einstein equations for both zero and non-zero cosmological constant.
We established ranges of the metric parameters admitting a physical interpretation
of the sources and gave relations for their mass per unit length.
Acknowledgment s
This contribution resulted from collaboration with Jifi Bicak and Tomas Ledvinka
and was supported by grants GACR 202/06/0041 and 202/05/P127, by the Centre
for Theoretical Astrophysics, and by research project MSM 0021620860.
References
1. Israel W 1966 Nuovo Cirnento B 44 1 (erratum B 49 463)
2. Bicak J and Zofka M 2002 Class. Quantum Grav. 19 3653-3664
3. Tian Q 1986 Phys. Rev. D 33 3549
4. da Silva M F A, Wang A, Paiva F M and Santos A O 2000 Phys. Rev. D 61
5. Zofka M and Bicak J in preparation
GRAVITATING MULTI-SKYRMIONS *
BURKHARD KLEIHAUS
Institut fur Physik, Universitat Oldenburg, Postfach 2503
D-26111 Oldenburg, Germany
kleihaus@theorie.physik.uni-oldenburg.de
THEODORA IOANNIDOU
Mathematics Division, School of Technology
Aristotle University of Thessaloniki
Thessaloniki 54124, Greece
ti3@auth.gr
JUTTA KUNZ
Institut fur Physik, Universitat Oldenburg, Postfach 2503
D-26111 Oldenburg, Germany
kunz@theorie.physik.uni-oldenburg.de
Gravitating multi-Skyrmion configurations with either discrete platonic symmetry or
axial symmetry are investigated numerically. We use the rational map Ansatz for the
Skyrmion field and a simplified Ansatz for the metric to obtain approximate solutions of
multi-Skyrmions coupled to gravity. These solutions are static and asymptotically flat.
The symmetry of the solutions is imposed by the choice of the rational map. We present
axially symmetric solutions with baryon number B=2,3,4, as well as the tetrahedral B=3
and cubic B=4 solutions. We show that for fixed baryon number (and given symmetry)
two branches of gravitating multi-Skyrmions exist, which merge at a maximal value of
the coupling parameter.
Nonlinear field theories coupled to gravity lead to globally regular gravitating
configurations.1 In the Einstein-Skyrme model the nonlinear chiral field theory
describing baryons and nuclei in terms of solitons (so-called Skyrmions) is coupled to
gravity. Static spherically symmetric SU(2) gravitating Skyrmions and black holes
with Skyrmion hair2 exhibit a characteristic dependence on the coupling parameter:
two branches of solutions merge and end at a maximal value of the coupling
parameter. The same pattern of behaviour has also been observed for axially symmetric
Skyrmions.3
In this talk we consider gravitating Skyrmion configurations with only axial and
platonic symmetries.4 In particular, we focus on configurations with tetrahedral and
cubic symmetry, possessing baryon number B = 3 and B = 4, respectively.
The SU(2) Einstein-Skyrme action reads
4a 4 l ll ' 32
where R is the curvature scalar, a represents the coupling parameter, and the ST/(2)
Skyrme field U enters via Ktl = d^UU^1.
S-
I
+ 7Tr (Kti K») + -Tr ([K^ Kv\ [K»,K»\)
-gdAx ,
*This research has been partially supported by by the DFG under contractKU612/9-l.
2322
2323
While aiming at the numerical construction of exact platonic gravitating
Skyrmions, we will here restrict to simpler approximate solutions, based on the
rational map ansatz for the Skyrme field,5
U = cos(h)l + i sh\(h)nR ■ f .
Here the unit vector Ur specifies the spatial symmetry of the solution. It is related
to the rational map R via
ftR = 1+\R,2 {R + R> -i(R -R), i - \R\2) ■
An appropriate ansatz for the metric is given by
ds2 = -fdt2 + - (dr2 + r2d92 + r2 sin2 Odtf?)
where we allow to have only two metric functions / and I ("/ - /-approximation")
or one function /, I = 1 ("dilaton-approximation").
Substitution of the ansatz in the Lagrangian and subsequent variation with
respect to the Skyrme field function h and the metric functions / and I leads to a set of
coupled partial differential equations to be solved numerically. Boundary conditions
are imposed on the functions to ensure regular, asymptotically fiat solutions.4
0.5 ' ' ' ' ' ' ' ' ' 0 ! ' ' ' ' ' ' > '
0 0-005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 0 005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
a a
Fig. 1. The dimensionless mass per baryon number M/ B (left) and the value of the function /
at the origin (right) are shown as functions of the coupling parameter a for axial (B = 2) and
platonic (B = 3,4) Skyrmions in the "/ — ^-approximation" and the "dilaton-approximation".
We have constructed (approximate) gravitating Skyrmions with baryon number
B = 2,3,4 and studied their dependence on the coupling parameter a. When a
is increased from zero a branch of gravitating Skyrmions emerges from the
corresponding flat space Skyrmion solution. This first (lower) branch extends up to the
maximal value, where it merges with a second (upper) branch of solutions, which
extends back to a = 0. The mass per baryon number decreases with increasing a
on both branches, see Fig. 1 (left). But whereas the mass remains finite in the limit
a^Oon the lower branch, it diverges in this limit on the upper branch. Thus on
the upper branch the limit a —> 0 does not correspond to a flat space limit, where
2324
gravity decouples. We observe however that, on both branches the metric functions
at the origin take finite values in the limit a —> 0, as shown in Fig. 1 (right) for the
function /.
We exhibit in Fig. 2 surfaces of constant baryon density for tetraheclral B = 3
(left) and cubic B = 4 (right) Skyrmion solutions on the lower branch . For a
given rational map and coupling parameter a the Skyrmion on the upper branch
is confined in a smaller volume than the Skyrmion on the lower branch, while the
shape of the baryon density is primarily determined by the rational map, analogous
to the shape of the energy density.7
Fig. 2. Isosurface plot of the baryon density B° for the 6 = 3 Skyrmion (left) for the B = 4
Skyrmion (right) in the "/ — I-approximation" for a = 0.02
References
1. M. S. Volkov and D. V. Gal'tsov, Phys. Rept. 319, 1 (1999).
2. H. Luckock and I. Moss, Phys. Lett. B176, 341 (1986); S. Droz, M. Heusler and N.
Straumann, Phys. Lett. B268, 371 (1991); P. Bizon and T. Chmaj, Phys. Lett B297,
55 (1992).
3. N. Sawado and N, Shiiki,gr-qc/0307115; N. Shiiki, N. Sa,wado, T. Torii and K. Maeda,
Gen. Rel. Grav. 36, 1361 (2004); N. Shiiki and N. Sawado, gr-qc/0501025.
4. T, loannidou, B. Kleihaus and J. Kunz, Phys. Lett. B635, 161 (2006); Phys. Lett.
B643, 213 (2006).
5. C. J. Houghton, N. S. Manton and P. M. Sutclitfe, Nucl. Phys. B510, 507 (1998).
6. E. Braaten, S. Townsend and L. Carson, Phys. Lett. B235, 147 (1990); C. J. Houghton
and P. M. Sutcliffe, Commun. Math. Phys. 180, 343 (1996); R. A. Battye and P. M.
Sutclitfe, Phys. Rev. Lett. 79, 363 (1997); Phys. Lett. B416, 385 (1998); D. Yu. Grig-
oriev, P. M. Sutcliffe, D. H. Tchrakian, Phys. Lett. B540, 146 (2002).
7. B. Kleihaus, J. Kunz, and K. Myklevoll, Phys. Lett. B582, 187 (2004); Phys. Lett.
B805, 151 (2005); Phys. Lett. B832, 333 (2006); Phys. Lett. B638, 367 (2006).
A NEW EXACT STATIC THIN DISK
WITH A CENTRAL BLACK HOLE
GUILLERMO A. GONZALEZ
Grupo de Investigation en Relalividad y Gravitation
Escuela de Fisica, Universidad Industrial de Santander
A.A. 678, Bucaramanga, Colombia
guillego@uis. edu. co
A new exact solution of the Einstein equations corresponding to the superposition of an
annular static thin disk with a central black hole is presented. All the metric functions
of the superposition are explicitly computed and the obtained expressions are simply
written in terms of oblate spheroidal coordinates. The obtained solution represents an
infinite annular thin disk around the Schwarszchild black hole. The mass of the disk is
finite and the energy-momentum tensor agrees with all the energy conditions.
1. The Einstein Equations
The Weyl metric for a static axially symmetric spacetime is1
ds*
„2$,
d£2 + e-l9[rld^ + e2A(dr2 + dz2)]
(1)
with <& and A only depending on r and z. Thus, the Einstein vacuum equations are
r
A,r = r(<P2r - *fj ,
the well known Weyl equations2'3
We consider a solution of the form
$ = (j) + ip,
A[*]=A[0]+A[V>] + 2A[0,V],
where ip and A[i/j] are given by the Schwarszchild solution
i,
1
In
'u- 1"
u +1
AM
1
r „,2
In
u^-l
with the prolate spheroidal coordinates defined by means of
1)(1
muv
(2)
(3)
(4)
(5)
(6)
(7)
(8)
r = m (u
and 1 < u < oo, — 1 < v < 1.
Now, in order to obatin for 0 and A[0] an annular thin disk solution, we introduce
the oblate spheroidal coordinates by means of
= a\e+i)(i~v2
z = atr,, (9)
with -oo<f<oo,0<77<l. The disk is obtained by taking 77 = 0 and so is
located at z = 0, r > a. On crossing the disk, 5 changes sign but does not change
in absolute value, so that an even function of f is a continuous function everywhere
but has a discontinuous £ derivative at the disk.
2325
2326
2. The annular thin disk solution
The annular thin disk solution (p and A[4>] is given by
<P-
arj
a{e+r,2V
m =
a2{\ - 772)[£4(V - 1) + 2£V(r/2 + 3) + vHv2 ~ 1
(10)
(11)
4a2(£2+r/2)4
with a an arbitrary constant and a the inner radius of the disk, and the mixed term
A[<p,ip] in (6) is given by
A[<p, 4>]
£2+??2
(A/-A//),
where
A,
A
//
fflT?(l - t/)(1 + g2) - mg(l + n)(u - 1)(1 - «)
[a£ + m(l — u — v)]2 + a2(l — r/2)
a7;(l - r/)(l + e2) - m£(l + r/)(u + 1)(1 - v)
[a£ - m(l+ u-v)]2+a2(l - r/2)
The Surface Energy-Momentum Tensor of the disk can be written as
Sab = eVaVb,
where Va = e~®5a. The surface energy density is given by
4a
i2t3
exp
a2^3 i 4a2e
(12)
(13)
(14)
(15)
(16)
Fig. 1. Energy density e = ae as a function of r = r/a for the annular thin disks obtained by
taking a = a/a = 1,...,9.
2327
for £ > 0, and will be allways positive if we take a > 0. We then have a dust disk in
agreement with all the energy conditions. The total mass of the disk can be easily
computed and we obtain
M = 2tt r(l/4)v/2aa , (17)
so that the disk is of infinite extension but with finite mass. The behavior of the
energy density is shown at Figure 1.
Acknowledgments
The author wants to thank the finantial support from COLCIENCIAS, Colombia,
and Vicerrectoria de Investigaciones y Extension, Universidad Industrial de San-
tan der.
References
1. D. Kramer, H. Stephani, E. Herlt, and M. McCallum, Exact Solutions of Einsteins's
Field Equations (Cambridge University Press, Cambridge, England, 1980).
2. H. Weyl, Ann. Phys. 54, 117 (1917)
3. H. Weyl, Ann. Phys. 59, 185 (1919)
BIFURCATIONS OF NONLINEAR CURVATURE LAGRANGIANS
IN THE BOSON STAR MODEL
FRANZ E. SCHUNCK
Institut fur Theoretische Physik, Universitdt zu Koln, 50923 Koln, Germany
fs@thp.uni-koeln.de
If scalar fields exist in Nature, soltion-type configurations kept together by their
self-generated gravitational field can be formed, i.e., gravitational variants of Bose-
Einstein condensates. Such objects are called boson stars,8 boson halos6 or, more
general, scalar field halos.3'7 In the spherically symmetric case, we8 have shown
via catastrophe theory that boson stars have a stable branch on the so-called cusp
catastrophe.1 In this method, one has to calculate the integration constants (here
mass M and particle number N of the boson star) and constructs the bifurcation
diagram M(N) where an infinite number of cusps appear. Physically spoken, each
cusp can be connected with a perturbation frequency within the star. The cusps form
a curve in a way similar to a zigzag mountain road (if you turn the diagram by 45°).
The idea is now that at each new cusp (starting from the origin), a perturbation
frequency becomes unstable. This is due to the fact that each line in the bifurcation
diagram represents the projection of minima and/or maxima of so-called Whitney
surfaces.
In this Letter, we will investigate boson star models with a real scalar field8 and
show: (i) the boson star model can be represented by a nonlinear Lagrangian, (ii)
L(R) shows a catastrophic behavior.
If we consider a general Lagrangian density2'4'5 C = L(R)y/\ g | through the
conformal change
9a/3 -> 9a/3 = %a/3 with O = 2k —- , (1)
dti
of the metric, this Lagrangian can be mapped to the usual Hilbert -Einstein
Lagrangian with a particular self-interacting scalar field. We are interested into the
scalar field which will arise via
^y^lnO (2)
from the nonlinear parts of a higher- order Lagrangian in the scalar curvature R.
In the conformal frame (1), the boson star Lagrangian density for a complex
scalar field is
£BS = 2,
.g^(<9^*)(«9^)^2C7(|0|2)]}, (3)
where k = 8ttG is the gravitational constant in natural units, g the determinant of
the metric g^v, and R the scalar curvature of Riemannian spacetime with Tolman's
sign conventions, acquires the form
1 /r
2k
RQ. - 2k02C7(0) , (4)
2328
2329
where U(fl) := C7(0(O)) = U {^/iJ2K\n^l\ is the reparametrized potential. Thus
in our approach, the scalar will not be regarded as an independent field, but is
induced via (2) by the non-Einsteinian pieces of the general Lagrangian L = L(R).
Solving for the potential, we obtain
Rfl
C/(0) = H(R)/Q2 =
2k
L
/n2
(5)
If we identify the conformal factor with the field momentum via O = 2ndL/dR,
the bracket in (5) can be regarded as a Legendre transformation L —> H(R) =
RdL/dR — L from the original Lagrangian (3) to the general nonlinear curvature
scalar Lagrangian L = L(R). Then, the parametric reconstruction2
dUl
-R = 2Kexp(v/2K/30
2/7(0)+ V3/2K-
and
L = exp( 2^2^/30
(7(0) + y/3/2K
dU_
~d4
(6)
(7)
of the higher-order effective Lagrangian L(R) from the boson star potential8 U(4>) =
m2(j)2 arises where m is the mass of the scalar field. We changed now to a real scalar
field.
The boson star model provides us with an exact parametric solution of the
equivalent nonlinear Lagrangian L(R) for the free field5
3m2
R = Qm2xex(l + x)
L
2k
-xe
2x
(2 + x)
(8)
where x :— InO under the reality condition O > 0 which ensures that the scalar
field values are always real.
A series of L at the center R = 0 shows us the nonlinear behavior of the
Lagrangian explicitly5
R3 7R4
,D, R R
L{R)lH=0=2^ + 2^2^
+
-j T 0(RJ
(9)
144m4K5 2592m6K7
The dependence of L(R) given in Fig. 1 is rather surprising and represents the
bifurcation set of a swallow tail catastrophe1 associated with some higher
dimensional grand manifolds (the generalization of a Whitney surface). According to the
theory of singularities, this bifurcation set indicates that the Lagrangian manifolds
are associated with two local minima and one maximum (and saddle points at the
meeting points of the grand manifold). Each of the minima merges with the
maximum at the cuspoidal points A and B and then disappears.
We can derive four striking points: (a) cusp A at negative L and negative R,
(b) point R = 0 with L^O, (c) point L = 0 with positive R, and (d) cusp B. Let
us figure the values of the BS real scalar field8 0 along L(R) (0 in units of l/y/R).
We start from the center (0, 0) to B. At the center, the scalar field has the value
minus infinity. At cusp B, we find cp = —3.206, at (c) 0 = —y/Q = —2.449, at (b)
2330
0 .2
0 . 1
0
-0 .1
J-0 .2
-0.3
-0 .4
-0 .5
/ B
A
-0 .5
0.5
R
1.5
Fig. 1. The swallow tail behavior of the boson star Lagrangian L(R) (k = m = 1).
4> = —\JZ/2 = —1.224, and, finally, at cusp A,
the center, now with 0 = 0.
—0.467; then, again, we meet
Acknowledgments
We would like to thank Burkhard Fuchs, Fjodor V. Kusmartsev, and Eckehard W.
Mielke for helpful discussions.
References
1. V.I. Arnol'd: Catastrophe Theory (Springer-Verlag, Berlin 1992).
2. F.E. Schunck, F.V. Kusmartsev, and E.W. Mielke, "Dark matter problem and
effective curvature Lagrangians", Gen. Rel. Grav. 37, 1427-1433 (2005).
3. E.W. Mielke, B. Fuchs, and F.E. Schunck: "Dark matter halos as Bose-Einstein
condensates", Proc. of the Tenth Marcel Grossman Meeting on General Relativity,
Rio de Janeiro, 2003, M. Novello, S. Perez-Bergliaffa and R. Ruflini, eds. (World
Scientific, Singapore 2006), p. 39-58.
O. Obregon, L.A. Urena-Lopez, and F.E. Schunck, "Oscillatons formed by nonlinear
gravity," Phys. Rev. D 72, 024004 (2005).
F.E. Schunck and O. Obregon, "Self-gravitating complex scalar fields conformally
transformed into higher-order gravity: Inflation and boson stars", preprint (1997).
F.E. Schunck, "A scalar field matter model for dark halos of galaxies and gravitational
redshift", astro-ph/9802258.
F.E. Schunck, B. Fuchs, and E.W. Mielke, "Scalar field haloes as gravitational lenses",
Mon. Not. R. Astron. Soc. 369, 485-491 (2006).
F.E. Schunck and E.W. Mielke: "TOPICAL REVIEW: General relativistic boson
stars", Class. Quantum Grav. 20, R301-R356 (2003).
APPROXIMATE DYNAMICS OF DARK MATTER ELLIPSOIDS
GENNADY S. BISNOVATYI-KOGAN
Space Research Institute of Russian Academy of Science,
Profsoyuznaya 84/32, Moscow 117997, Russia,
Joint Institute Nuclear Research, Dubna, Russia and
Moscow Engineering Physics Institute, Moscow, Russia
gkogan@iki. rssi. ru
OLEG YU. TSUPKO
Space Research Institute of Russian Academy of Science,
Profsoyuznaya 84/32, Moscow 117997, Russia and
Moscow Engineering Physics Institute, Moscow, Russia
tsupko@iki. rssi.ru
Collapse of a non-collisional dark matter and formation of pancake structures in the
universe are investigated approximately. The collapse is described by a system of
ordinary differential equations, in the model of a uniformly rotating, 3-axis, uniform density
ellipsoid. Violent relaxation, mass, and angular momentum losses are taken into account
phenomenologically. The formation of the equilibrium configuration, secular instability
and transition from a spheroid to 3-axis ellipsoid are investigated numerically and
analytically in this dynamical model.
The study of the formation of dark matter objects in the Universe is based on
N-body simulations, which are very time consuming. In this situation a simplified
approach may become useful.
Let us consider a compressible 3-axis ellipsoid, consisted of non-collisional non-
relativistic particles, with semi-axes a ^ b ^ c and rotating uniformly with an
angular velocity O around the axis z. Let us approximate the density of the matter
p in the ellipsoid as uniform. The case of spheroid (a = b ^ c) was considered
by Bisnovatyi-Kogan1 where there are analytical formulaes for the gravitational
potential and forces.
The mass m and total angular momentum M of a uniform ellipsoid are connected
with density, angular velocity and semi-axes as m = ^- pabc, M = y fl(a2 + b2).
Assume a linear dependence of the velocity on the coordinates: vx = ax fa , vy =
by/b , vz = cz/c. The gravitational energy of the uniform ellipsoid is defined as:
oo
ug = -™*[ , du (i)
9 10 J ^(a2 + u)(b2 + u)(c2+u)
Consider a compressible ellipsoid with a constant mass and angular momentum,
a total thermal energy of non-relativistic dark matter particles Eth ~ V~2/3 ~
(abc)~2/3, and the relation between pressure P and thermal energy Eth as Eth =
|£. In absence of any dissipation the ellipsoid is a conservative system.
To derive equations of motion let write for it the Lagrange function
L = Uhin — Upot , Upot = Ug + Eth + Urot , Ukin = ~ (<i + b + C ) , (2)
2331
2332
Eth ~ ^F5 " wr1*' '2 J 2 ™(«2 +fo2)'
By variation of the Lagrange function we obtain Lagrange equations of motion.
Collapse in the dark matter are characterized by non-collisional relaxation,
based on the idea of a "violent relaxation" of Lynden-Bell.2 Therefore there
is a drag force, which is described phenomenologically by adding of the terms
—— , — , —— in the right-hand parts of equations of motions. Here we
Trel ' Trt,i ' Trt,i O f ~l
have scaled the relaxation time Trei by the Jeans characteristic time with a constant
value of arei : rrei = areiTj = 2tt arei\/-^^ . The process of relaxation is
accompanied also by energy, mass and angular momentum losses from the system. These
losses may be described phenomenologically by characteristic times rei, rmi, tmi ■
The entropy function e is constant in the conservative case, but variable in the
presence of dissipation.1 Furthermore, because of variability of mass, there are new
terms, proportional m, in the equations of motion.
To obtain a numerical solution of these equations we write the following non-
dimensional system of equations3,4
oo
a dm, 2>ma f du 10 e 25M2
a
m dt 2
o
f du 10 £ 25MZ a a_
J (a2 + u)A + 3^ (a&c)2/3 + m2 (a2 + &2)2 Trel ' ( )
o
oo
b dm 3mb f du 10 e 25M2 b _6_
m~dt~^2~ J (b2 + u)A + imb (abc)2/3 + m2 (a2 + b2)2 ~ ^ ' ( )
oo
r du 10 e c
J (c2 +u)A + 3m~c {abc)2^ ~ ^ ' ()
oo
c dm 2>mc f du 10 e c
m dt 2
o
(abc)2/3 Uk
2 J^ 2^\ Urot ( 2 1^\ Ukin
Trel Tel T-ml J Ug \TMl Tml J V'g Tml
(7)
mUkin M f m \tmi „ „ 9 „
m=jy~,Ug<07 —~= ( — ) , A2 = (a2 + u)(b2 + u)(c2+u). (8)
UgTml Min \mln J
The system was solved numerically for several initial parameters, until the
formation of stationary rotating figures in presence of the relaxation. For lower angular
momentum M we have a formation of the oblate spheroid, while at larger M we
follow the development of three-axial instability and formation of three-axial ellipsoid
(see Fig.l). The instability in this approximation happens at the bifurcation point
of the sequence of Maclaurin spheroids, where Jacobi ellipsoidal system starts.
Furthermore, the development of instability, connected with radial orbits,5'6 is found
for low-entropy, slowly rotating collapsing bodies.
2333
a-axis
b-axis
■ c-axis
0)
x
ro
120
time
Fig. 1. The development of a bar-like instability at large angular momentum, and the formation
of a stationary triaxial figure.
The bifurcation point coinciding with the point of loss of stability is found
analytically in the form of a simple formula, by static and dynamic approaches.4
We obtain the equation
arccosfc fc(13 — 10fc2)
yrrp! 3 + 8fc2 - 8fc4
k = c/a,
(9)
which solution k = 0.582724 (e = y'l - c2/a2 = 0.81267) determines the
bifurcation point at the sequence of the Maclaurin spheroids. At this point compressible
spheroids become secularly unstable to triaxial deformations. The position of this
point does not depend on the polytropic exponent n.
References
1. G. S. Bisnovatyi-Kogan, MNRAS 347, 163 (2004).
2. D. Lynden-Bell, MNRAS 136, 101 (1967).
3. G. S. Bisnovatyi-Kogan and O. Yu. Tsupko, Astronomical and Astro-physical
Transactions 24, 5, 377 (2005).
4. G. S. Bisnovatyi-Kogan and O. Yu. Tsupko, MNRAS 364(3), 833 (2005).
5. V. A. Antonov, in The dynamics of galaxies and stellar clusters, 139 (Nauka, Alma-
Ata, 1973).
6. A. M. Fridman and V. L. Polyachenko, Physics of Gravitating Systems (Springer
Verlag, Berlin, 1985).
NONEXTENSIVE STATISTICAL THEORY OF DENSITY
DISTRIBUTIONS IN GRAVITATIONALLY CLUSTERED
STRUCTURES
MANFRED P. LEUBNER
Institute for Astro- and Particle Physics, University of Innsbruck, A-6020 Innsbruck, Austria
manfred. leubner@uibk.ac.at
The radial profiles of dark matter (DM) and hot gas density distributions in galaxies
and clusters are commonly fitted by empirical functions. Vice versa, the fundamental
concept of entropy generalization in nonextensive statistics accounts for long-range
interactions and correlations in a system. We present a new theory where the underlying
entropy duality generates a bifurcation of the density distribution into a kinetic DM
and thermodynamic gas branch. The derived profiles, controlled by the mean energy
and degree of correlations of the system, reproduce accurately the radial dependences
known from observations and simulations. We suggest modeling density distributions of
clustered matter within the fundamental context of entropy generalization, accounting
for nonlocality and long-range interactions in gravitationally coupled systems.
To date only a few attempts provide physically motivated models for density
profiles of astrophysical clusters. Early analytical analysis1 for the collapse of density
perturbations was subsequently further studied2 and based on infall models.3'4
In practice, dark matter (DM) and hot plasma density profiles, as observed in
galaxies or clusters or generated in simulations, are widely modeled by empirical
fitting functions. The phenomenological (3—model,5 provides a reasonable
representation of the hot gas density distribution of clustered structures, further improved
by the double /3-model, with the aim of resolving the (3—discrepancy.6 Similarly,
the radial density profiles of DM halos are analyzed primarily with the aid of
phenomenological fitting functions, thus lacking physical support as well.7-9 Physically,
we regard the DM halo as an ensemble of self-gravitating, collisionless and weakly
interacting particles in dynamical equilibrium.
Since any astrophysical system is subject to long-range gravitational and/or
electromagnetic interactions, this situation motivates to introduce nonextensive
statistics as physical background for the analysis of DM and hot plasma density profiles. In
this context the entropy of the standard Boltzmann-Gibbs-Shannon (BGS) thermo-
statistics is generalized by a pseudo-additive term weighted by a single parameter
k, which mimics the degree of long-range interactions and correlations within the
system. The situation where gravity can be neglected was successfully analyzed by
Leubner et al.10-12 yielding a particular class of power-law distributions.
Here we retain long-range interactions and generalize the standard BGS
extensive thermo-statistics to nonextensive astrophysical systems. A generalization of the
BGS entropy for statistical equilibrium from basic principles was recognized first by
Renyi13 and later revived by Tsallis14 leading to a variety of profound
mathematical and physical consequences,15_18 including astrophysical plasma turbulence.12'19
2334
2335
10"1 10° 101 102 10"1 10° 101 102
log(r) [normalized] log(r) [normalized]
Fig. 1. Left panel: Nonextensive family of density profiles. The lower branch corresponds to the
DM (p~) and the upper branch to the plasma (p+) distributions. For increasing re—values both
sets of curves converge to the isothermal sphere solution (re = oo, dots). Right panel: Comparison
of the DM nonextensive density profile (re = —7, a — 1, solid) with the Burkert (dashed) and the
Navarro8 (dashed-dotted) profiles. The radial nonextensive gas distribution (re = 7) is compared
with a single j3—model (dashed line) and the decomposition of a double (3—model (dotted line).
The dual nature of nonextensive statistics provides also the physical maifestation
of entropy bifurcation in the theory of DM and plasma density distributions.20
The generalized entropy S(k) characterizing systems subject to long-range
interactions and couplings reads10'14 SK = nkB{^ZPi~ ~ 1) where pi is the probability
of the ith microstate, ks is Boltzmann's constant and the 'entropic index' k denotes
a coupling parameter quantifying the degree of nonextensivity, i.e. of statistical
correlations within the system, k = oo represents the extensive limit of statistical
independence recovering the classical BGS entropy as Sb = —kB^Pi ^nPi-W
Extremizing the generalized entropy with regard to conservation of mass and
energy in a gravitational potential Vl/ yields the energy distribution /±(i?r) =
B^ [l + (v2/2 — \I/)/(k<72)] . The superscripts refer to the positive or negative
intervals of the entropic index k, accounting for less (+) and higher (-) organized
states and thus reflecting the accompanying entropy increase or decrease,
respectively.20 a represents the mean energy of the distribution and B^ are normalization
constants.10 The density evolution of a system subject to long range interactions
in a gravitational potential p^ = p0 [l — \I//(k<72)] k is found after integration
over all velocities. Combining with Poisson's equation A\I/ = —4nGp± provides a
second order nonlinear differential equation to be solved numerically, determining
the density profiles of both, plasma and DM of clustered structures.
As natural consequence of nonextensive entropy generalization the standard
isothermal sphere profile bifurcates into two distribution families controlled by the
sign and value of the correlation parameter k. Consequently, the self-gravitating DM
component, a lower entropy state due to gravitational interaction, resides besides
the second branch, a thermodynamic plasma component of higher entropy.
2336
The left panel in Fig. 1 illuminates schematically the radial density profile
characteristics for some values of n for both, DM below and the plasma distributions
above the exponential solution. Increasing k values correspond to a decoupling
within the system and both branches merge simultaneously in the isothermal sphere
profile for k = oo, representing the extensive limit of statistical independence.
In Fig. 1, right panel, we compare one negative k nonextensive DM density
profile with the Navarro et al.8 model as well as one positive k plasma distribution
with a single j3—model. Changes of the variance a generates a radial shift of the
profile and the correlation parameter k controls the overall shape. The nonextensive
plasma distribution follows a single (3—model in the core but deviates in the halo
tail. The entire nonextensive profile is fitted accurately by a double f3—model (a
decomposition is included for visibility by the dotted lines), confirming that the
nonextensive theory provides naturally a context able to solve the f3—discrepancy.6
The dual nature of the nonextensive theory provides a solution to the problem of
DM and plasma density distributions of clustered matter from fundamental physics
where both parameters admit physical interpretation. Consistently, the theory
reproduces accurately also the density profiles generated by numerical simulations, as
well as integrated mass profiles available from observations.22 Consequently we
propose to favor the physical family of nonextensive distributions over empirical models
when fitting observed or simulated density profiles of astrophysical structures.
References
1. J. E., Gurni, J. R. I. Gott, Astrophys. J. 176, 1 (1972).
2. Y. Hoffman, Astrophys. J. 328, 489 (1988).
3. L. L. R. Williams, A. Babul, J. J. Dalcanton, Astrophys. J. 604, 18 (2004).
4. Y. Ascasibar, G. Yepes, S. Gottldber, V. Miiller, MNRAS 352, 1109 (2004).
5. A. Cavaliere, and R. Fusco-Femiano, Astron. Astrophys. 49, 137 (1976).
6. N. A. Bahcall, and L. M. Lubin, Astrophys. J. 426, 513 (1994).
7. A. Burkert, Astrophys. J. 447, L25 (1995).
8. J. F. Navarro, C. S. Frenk, and S. D. M. White, Astrophys. J. 462, 563 (1996).
9. B. Moore, F. Governato, T. Quinn, J. Stadel, and G. Lake, Astrophys. J. 499, L5
(1998).
10. M. P. Leubner, Astrophys. J. 404, 469 (2004).
11. M. P. Leubner, Phys. Plasmas 11, 1308 (2004).
12. M. P. Leubner, and Z. Voros, Astrophys. J. 618, 547 (2005).
13. A. Renyi, Acta Math. Hungaria 6, 285 (1955).
14. C. Tsallis, J. Stat. Phys. 52, 479 (1988).
15. A. R. Plastino, A. Plastino, and C. Tsallis, J. Phys. A: Math. Gen. 27, 5707 (1994).
16. R. Silva, A. R. Plastino, and J. A. S. Lima, Phys. Lett. A 249, 401 (1998).
17. M. P. Almeida, Physica A 300, 424 (2001).
18. I. V. Karlin, M. Grmela, and A. N. Gorban, Phys. Rev. E 65, 036128 (2002).
19. M. P. Leubner, Z. Voros, and W. Baumjohann, Adv. Geosci. 2, 43 (2006).
20. M. P. Leubner, Astrophys. J. 632, Ll (2005).
21. Y-J. Xue, and X-P. Wu, MNRAS 318, 715 (2000).
22. T. Kronberger, M. P. Leubner, and E. van Kampen, Astron. Astrophys. 453, 21
(2006).
GENERAL RELATIVISTIC ACCRETION WITH BACKREACTION
JANUSZ KARKOWSKI, BOGUSZ KINASIEWICZ, PATRYK MACH and EDWARD MALEC
M. Smoluchowski Institute of Physics, Jagiellonian University,
Reymonta 4, 30-059 Krakow, Poland
ZDOBYSLAW SWIERCZYNSKI
Pedagogical University, Podchorqzych 1, Krakow, Poland
The spherically symmetric steady accretion of polytropic perfect fluids onto a black hole
is the simplest flow model that can demonstrate the effects of backreaction. Backreaction
keeps intact most of the characteristics of the sonic point. For any such system the mass
accretion rate achieves maximal value when the mass of the fluid is 1/3 of the total mass.
Fixing the total mass of the system, one observes the existence of two weakly accreting
regimes, one overabundant and the other poor in fluid content.
Keywords: general-relativistic hydrodynamics, accretion, black holes
1. Introduction
Calculations of selfgravitating fluids onto a compact object are, in general, very
difficult. So it is not suprising that this problem has been solved in only a few
idealized cases. The spherical steady accretion of perfect fluids onto a Newtonian
gravitational center was investigated by Bondi in 1952 [1] and onto a Schwarzschild
black hole by Michell [2] and others [3, 4, 8]. The first fully general relativistic
model taking into account the backreaction was dealt with by Malec [5]. The effects
of backreaction of selfgravitating fluids on a spherical black hole was examined in [6].
The influence of backreaction of steadily accreting gases on the stability has been
studied in [7]. In this paper we will briefly present the main results of [6].
2. Formulation of the problem of quasistationary accretion
Let us consider the spherically symmetric cloud of an ideal gas falling onto a non-
rotating black hole. The general spherically symmetric line element is given by
ds2 = -N2dt2 + adr2 + R2dB2 + R2sin26d4>2,
where N, a and R depend on the asymptotic time variable t and the radius r. We
assume the energy-momentum tensor of perfect fluid T'^ = (p + g)ulluu + pg'1",
where u^ denotes the four velocity of the fluid, p is the pressure and g the energy
density in the comoving frame.
The conservation of the energy-momentum tensor V,iT>J-1' = 0 leads to the
continuity equation dtg = —NtrK(g + p) and to the relativistic version of the Euler
equation NdRp + (g + p)8rN = 0. Here the extrinsic curvature Krr = dta/(2Na)
and tvK = N~ldt In (^/aR2) (for details see [5]). The quasilocal mass m(R) is
defined by dR-m(R) = 4irR2g. We will assume that the accretion is steady and the
fluid satisfies the polytropic equation of state p = KgT with constant T € (1, 5/3].
More precisely:
2337
2338
i. the accretion rate, defined as m = (dt - (dtR)dR)m(R) for the given areal
radius R, is assumed to be constant in time;
ii. the fluid velocity U = (dtR)/N, energy density g, sound velocity a etc.
should remain constant on the surface of fixed R: (dt — (dtR)dp)X = 0,
where X = U, g,a,...
Strictly saying, a stationary accretion must lead to the increase of the central mass
and of some geometric quantities. This in turn means that the notion "steady
accretion" is approximate - it demands the mass accretion rate is small and the time
scale is short, so that the quasilocal mass m(R) does not change significantly.
One can show [5] that the accretion rate is independent of the surface (characterised
by a given R) for which it is calculated, i.e., Ortti = 0. Let us now define a sonic
point as such, where the length of the spatial velocity vector equals the speed of
sound 1171 = a. In the Newtonian limit the above definition coincides with the
standard requirement of the equality between the velocity of the fluid and the local
sound speed. In the following we will denote by the asterisk all values referring to
the sonic point.
3. The importance of backreaction
One of the two main results of [6] is the observation that significant information
about the full system with backreaction can be obtained through the investigation of
steady flows with the test fluid approximation. It appears that the characteristics
a-1, U%, m*/R* of the sonic point practically do not depend, for a given Y and
a^,, on the asymptotic energy density g^. One can get all parameters describing
the sonic point, with the exception of its location _R„ and mass m* simply from a
related polytropic model with the test fluid having the same index Y and the same
asymptotic speed of sound a^.
The second main result come from investigation of the mass accretion rate [6]
where x = rrif/m. This expression clearly demonstrates that the mass accretion rate
achieves a maximum at rrif = rn/3 and tends to zero when rrif —> 0 and rrif —> m
(Fig. 1). The factor 1/3 is universal - independent of the parameters _Roo, Y and
Gtoo. In the test fluid approximation the situation is quite different - the quantity
rh grows with g^.
The above result show the importance of the backreaction. We are convinced that
this qualitative features demonstrated by the spherically symmetric model will also
appear in the descriptions of accreting fluid onto a rotating black hole.
4. Summary
In conclusion, in the simple model of accretion with backreaction considered here,
one can get all parameters describing the sonic point, with the exception of its
2339
1.6
1.4
1.2
1.0
8.0
6.0
4.0
2.0
0.0 ■ 10°
Fig. 1. The dependence of m on the ratio m,f/m for T = 4/3 and a^ =0.1.
location R* and mass m» simply from a related polytropic model with the test fluid
having the same index T and the same asymptotic speed of sound Om . The main
result is that the mass accretion rate rh achieves a maximum at mf/tubh ~ 1/2.
Therefore, there exist two different regimes, nif/ttibh -C 1 and nif /ttibh ^> 1, with
low accretion.
This paper has been partially supported by the MNII grant 1P03B 01229.
References
[1]
[2]
[3]
[4]
[5]
[7]
[8]
H. Bondi, Mon. Not. R. Astron. Soc. 112, 192 (1952).
F. C. Michel, Astrophys. Space Sci, 15, 153 (1972).
S. Shapiro and S. Teukolsky, Black Holes, White Dwarfs and Neutron Stars, Wiley,
New York, 1983.
B. Kinasiewicz and T. Lanczewski, Acta Phys. Pol. B36, 1951(2005).
E. Malec, Phys. Rev. D60, 104043 (1999).
J. Karkowski, B. Kinasiewicz, P. Mach, E. Malec and Z. Swierczyiiski Phys. Rev.
D73, 021503(R) (2006).
B. Kinasiewicz, P. Mach and E. Malec, Preprint gr-qc/0606004, Proc. of the 4%
Karpacz School of Theoretical Physics, Ladek Zdroj, Poland 6-11.02.2006 to appear
in vol. 3 of the International Journal of Geometric Methods in Modern Physics (2006).
B. Kinasiewicz and P. Mach, gr-qc/0610040, to appear in Acta Phys. Pol. (2006).
NON-HOMOGENEOUS AXISYMMETRIC MODELS OF
SELF-GRAVITATING SYSTEMS
CHRISTIAN CHERUBINI* and SIMONETTA FILIPPlt
Facoltd di Ingegneria, Universitd Campus Bio—Medico, Via E. Longoni 83, 1-00155 Rome, Italy
ICRA, Universitd di Roma "La Sapienza," 1-00185 Rome, Italy
* c.cherubini@unicampus.it, *s.filippi@unicampus.it
REMO RUFFINI
ICRA, Universitd di Roma "La Sapienza," 1-00185 Rome, Italy
ruffini@icra.it
ALONSO SEPULVEDA* and JORGE I. ZULUAGA§
Department of Physics, University of Antioquia, A.A. 1226, Medellin, Colombia
ICRA, Universitd di Roma "La Sapienza," 1-00185 Rome, Italy
t h-alonsos@yahoo.com, 'jzuluaga@urania.udea.edu.co
A functional method developed for analyzing rotating self-gravitating fluid is discussed
in relation with selected velocity profiles. Specific numerical techniques developed in the
past for the solution of the problem are adopted.
We consider, from a rotating frame with constant angular velocity $1, a self-
gravitating, steady state, incompressible fluid, with flow lines of the velocity field
perpendicular to $1 = Qe3. Thus: v ■ 63 = 0. The fluid, having differential rotation,
is described by an inhomogeneous density, p = p(x,y,z). From the steady state
condition (dp/dt = dp/dt = 0) we may conclude v • V/? = 0; then, as the fluid is
incompressible (V ■ v = 0), there exists a velocity potential vector ■»/>, such that
v = CV x ■0/2. By taking into account that v • 63 = 0 and using an appropriate
gauge, the vector potential can be reduced to its third component: xj> = e^ij^
being ip the scalar hydrodynamical potential. Finally, we get from the rotating frame
(see1 and references therein): v = y S3 x Vi/>. From v • V/? = 0 it is possible to
demonstrate that p = p(ip,f(z)), while the continuity equation for steady state,
V ■ (pv) = 0, is identically satisfied. Hydrodynamics of a self-gravitating fluid as
seen from a rotating frame can be described by equation
+VU+-Orn-vVv-2Oxv = 0 (1)
where rc is the cylindrical radius of he axisymmetric configuration. The gravitational
acceleration can be defined in terms of the gravitational potential 0, as g = V0
and the potential satisfies the Poisson equation
V20 =-4ttGP. (2)
According to the integrability condition,1 the fluid is barotropic, taking the simplest
form in the case of a politrope, defined as P = ap1+1/n. Using relevant equations of
the theory, it can be shown that the gravitational potential inside the fluid is given by
0 = a(n+l)pl/n-^g(il))-\£l2rl + \v2-CQip+D where D is a constant and g(tj;) is
2340
2341
given by g{ij}) = f J'V2V>(iV- We manage now to cast equations in non dimensional
form. We recall that cartesian (xi), polar (rc,<p,z) o spherical (r,6,<p) coordinates
can be expressed in non-dimensional form, (Si), (£c, <£, 2), (£,p,,<p), respectively.
With the introduction of a parameter b, with dimension of (length)-1, writing also
p = /0C9™ (© is the non-dimensional density and n is the politropic index), the final
equation becomes2
v2e + e™ + —^-{v • [v vv + 2n x v] - 202} = 0, (3)
4ir&pc
where V2 = l/b2V2.
Regarding the gravitational potential, from the inertial frame (f2 = 0) we have:
(p = a(n+l)p1^-~g^) + \vl + D.
On the other hand, the gravitational potential at any point, into the the mass or
outside, satisfies the Poisson equation (2), whose solution is given by
= Gij£Ldy
I-
J lm >
E^^/0"(^')^V)|^' (4)
|r - r'l
i « * ii ii/ i /1 i i
= 47rG
lm
4irGpc ^Yim(6,<p) f 5
b2
where dV' is the spherical non-dimensional volume. The general problem we need to
solve is to find the mass distribution for which the adimensional critical quantities
satisfies the hydrostatic equilibrium equation:
£(£,0) = e(£,0) + A(o + 5 (5)
where A(£,0) = — ^f- J Q(£,c)2£,'cd£c in which the non dimensional differential
angular velocity is given by 0(£c) = ip(S,c)-
The solution of eq. (5) is found by using a procedure due to Eriguchi and Muller.3
We have to find the values of the adimensional density for which the equation (5)
is satisfied in all points inside the distribution including its boundaries (which are
defined by the condition O = 0).
In order to do this we have discretized the space where the bulk of the
distribution is supposed to be, as in figure (1) and followed the involved numerical
technique developed in the literature . Using this technique, equilibrium
configurations can be found for selected velocity profiles.4 In particular we have studied the
case f2(£) = O01e+mf2 (£ is the cylindrical radius) where a an m are free parameters
and fio is a constant. Figures 2 and 3 show the velocity profiles for selected values
2342
Figure 1. Mesh diagram.
Figure 2. Velocity profiles for a model with a = 0 and m=0.3, 0.8, each one for three values of
VQ.
FVRP = 101 HE/Rp-131 RE/RP = 1.61
Figure 3. Configurations for n = 1, a = 0.2 and m = 0.3.
of the parameters as well as preliminary results of the numerical integration of the
equations of the theory.
Bibliography
1. Filippi S., Ruffini R. and Sepulveda A., Phys. Rev. D 65, 044019 (2002).
2. Tassoul J., Theory of Rotating Stars, Princeton University Press, (1978).
3. Eriguchi, Y, Muller E, A & A, 146, 260 (1985).
4. Cherubim C, Filippi S., Ruffini R. and Sepulveda A., in preparation (2007).
GRAVITATIONAL WAVE DAMPING PROM A SELF
GRAVITATING VIBRATING RING OP MATTER AROUND
A BLACK HOLE
PRASAD BASU
Centre for Space Physics,
Chalantika-43, Garia Station Road, Kolkata-700084, India
pbasu@csp.res.in
S. K. CHAKRABARTI
S. N. Bose National Centre for Basic Science,
JD block, Sector-3, Salt Lake, Kolkata-700098,India
and Centre for Space Physics, Chalantika-43 Garia Station road,
Kolkata-700084, India
chakraba@bose.res .in
We consider the space time structure of a black hole-ring system1 in which a non-
rotating black hole is surrounded by a gravitating accretion disk, here simplified to
be a thin ring. We then consider the vertical oscillation of the disk about its equilibrium
position caused due to a small perturbation of the ring along the vertical direction. We
compute the gravity wave luminosity and loss rate of angular momentum to study how
the perturbation dampens after the continuous emission of the gravitational waves.
1. Introduction
Thick accretion disks are formed when the accretion rate is high and the
radiation emitted during the accretion process interacts with the matter dynamically,
resulting in puffing up of the disk. The description of such disks are available in
literature. It is well known that the matter is mostly concentrated near the centre
of the toroidal disk and thus the disk may be replaced by a thin ring.
Let us consider a system consisting of a Schwarzschild black hole, surrounded
by a massive ring. If the disk is perturbed (for instance by a nearby passing
compact object in highly eccentric orbit), the surrounding disk will start to oscillate
vertically. This vertically oscillating disk will emit gravitational wave which will
carry away energy and angular momentum from it resulting the disk to execute
a damped oscillating motion. We calculated the loss rates of the energy and the
angular momentum due to gravity wave emission and estimate the damping of the
amplitude of the vibration of the disk. We use full general relativistic treatment to
study the motion of the disk under some simplified assumption where the gravity
wave luminosity is computed using well known quadruple approximation.
2. Dynamics of the system
We assume that each particle of the ring is independently moving around the black
hole on geodesies. The geodesic of a given particle of the ring is determined by the
initial position and velocity of that particle. Let us consider an infinitesimal element
of the ring having mass Am. Let r, 6, <fi be the coordinates of that portion of the ring
in spherical polar coordinate system. Then the equations of motion of the element
2343
2344
are given by,
2\ dt ^ 9.2 *dd> ■ /i\
--)—=E; r2sm29-^=j (1)
rJdr dr
r*P +^=_f_ f.2+(l_2\(l+ J\ \=E\ (2)
sin2 (9 sin2 6»0 V rA r2 sin2 (V
where, E and j are the energy and angular momentum per unit mass of the
ring and we have assumed that at t = 0, 9 = do and ^| = 0. Here we are using the
system of unit in which G = c = Mbiackhole = 1- Solving the above equations we
obtain a relation between 9 and 0:
cot 9 = cot 6q sin 0 (3)
The above equation implies that each infinitesimal part of the ring will move in an
inclined orbit with same 9 and cp frequency. As a result the ring as a whole will
execute an oscillatory motion along the z axis with frequency same as the orbital
frequency of an element of the ring keeping its shape unchanged.
3. Gravity wave luminosity
The non vanishing quadrupole moment of the ring are
Qxx = -mr2 sin2 9 - -mr2; Qyy = Qxx; Qzz = -2QXX, (4)
where, r is the radius of the ring. Since the energy emitted per cycle due to gravity
wave are very small compared to the binding energy of the ring, we can assume that
during one orbit of an element of the ring (i.e., during one complete oscillation of
the ring) the radius of the ring remains constant. Then the average power emitted
per cycle by an element of the ring is given by,
12mAmZo^6
Ie= g , (5)
where w is the angular frequency of the oscillation of the ring. Suppose that the
particle is initially in an orbit, inclined with the polar axis with an angle a and the
azimuthal angle is 0 = |. The components of angular momentum per unit mass as
observed from infinity are, lz = I, lx = 0, ly = I tan a. The average rates of emission
of the components of angular momentum per unit mass in one cycle are given by,
/dlz\ /dl^\ (dly\ = Umzfiuj5 sec 90
V dt J avg. V dt J avg. V dt J avg. 5
where, zq is the height of the ring. From the change of the angular momentum
components we determined the change in the orbital inclination of the ring. From
this, we finally computed the change in the vertical height of the ring per oscillation.
The results are presented in the following figures.
2345
0 IC409 2e+09 1cm 4e*09 5«09 0 km 2cm 3e*09 4e*09 5e+09
time in year lime in year
Fig. 1. (left) The variation of vertical height and (right) radius of the ring (both in units of
GM/c2) with time.
^&-07 nic-m
frequency (hi)
Fig. 2. The metric perturbation is compared with the frequency of oscillation assuming the source
is at a distance 2.0 X 106ly from the Earth.
4. Discussion
In this paper, we show how the oscillation of a self-gravitating ring may gradually
dampen by emission of gravitational waves. These would also be a source to look into
through future gravitational wave detector systems. The details would be published
elsewhere. The work of PB is supported by a CSIR fellowship.
References
1. Chakrabarti, S.K., 1988, J. Astron. Astrophys., 9, 49
2. Chakrabarti, S.K., 1985, ApJ, 288, 1
VARIATIONAL PRICIPLES AND HAMILTONIAN FORMULATION
OF SPHERICAL SHELL DYNAMICS
JERZY KIJOWSKI
Center for theoretical physics,
Polish Academy of Sciences, Warsaw, Poland
and
College of Sciences,
Cardinal Wyszynski University, Warsaw, Poland
kijowski@cft. edu.pl
GIULIO MAGLI
Dipartimento di Matematica,
Politecnico di Milano, Milano, Italia
magli@mate.polimi.it
DANIELE MALAFARINA
Dipartimento di Matematica,
Politecnico di Milano, Milano, Italia
and
Center for theoretical physics,
Polish Academy of Sciences, Warsaw, Poland
malafarina@mate.polimi.it
A general approach to the hamiltonian description of thin shells of matter in General
Relativity is duscussed. The system composed of an ideal fluid self-gravitating spherical
shell is then analyzed and its lagrangian and hamiltonian functions are derived from
first principles. For this purpose the standard Hilbert action is modified by an
appropriate surface term at spatial infinity. Known results for the spherical dust shell are then
recovered as a special case.
1. Introduction
Thin matter shells were introduced by Werner Israel1 as the simplest model to
study gravitational collapse. The dynamics of a thin shell of matter is obtained
considering Einstein's equations concentrated on an hypersurface which tailors
together two different manifolds. The simplest case is that of a spherical dust shell
in vacuum whose dynamics was already exhaustively discussed in the pioneering
work by Israel.2 Spherical shells with more general equations of state have been
also investigated.3
The formulation of shell dynamics within the context of canonical gravity
however was developed only recently.4 In the spherically symmetric case (which means
tailoring of an internal Minkowski geometry to an external Schwarzschild) this leads
to a simple Hamiltonian system which has only one degree of freedom.5 Nevertheless
this Hamiltonian, as evaluated from the standard Hilbert action, does not coincide
with the total energy of the system for an observer at spatial infinity. The solution
to this problem is obtained when an appropriate boundary term at spatial infinity
is introduced to improve the Hilbert action.6
2346
2347
2. Tailoring, curvature tensor and variational principle
The history of a dynamical 2-dimensional matter shell is described by the tailoring of
two different vacuum space times, namely M+, the exterior, and M~, the interior,
along a common hypersurface S. The hypersurface S is therefore assumed to carry
the matter content of the shell which will be described by constitutive equation
m(u) depending on the specific volume v of the fluid (or, equivalently, on its local
density).7 The function m contains both the rest frame energy density, the dust
case will therefore be m = mo, and the interaction energy of the fluid particles.
Restriction to spherical symmetry suggests that the internal geometry must be
that of Minkowski, while the external is Schwarzschild with fixed mass parameter
M. The dynamical evolution of the shell will be described by a function tp(t), where t
is the Schwarzschild time and dotted quantities represent derivatives with respect to
t.8 The matching conditions of the two geometries across £ will give the constraint
equation:
sinhjit ip
coshM-^/l-2|f 1- —
Where ji is the hyperbolic angle between the surfaces {t = const.} on the
Schwarzschild side and the surfaces {t = const.} on the Minkowski side and can be
thought of as an implicit function of ip and ip.
With the use of the theory of distribution the entire dynamics of the gravitational
field interacting with the shell may be obtained performing the variation of an
appropriate Hilbert action consisting in a singular part concentrated on the shell
and a regular part outside the shell.9 However in this manner the variation of the
standard Hilbert action leads to an Hamiltonian which fails to represent the ADM
(Arnowitt-Deser-Misner) mass at infinity for the system. This is due to the fact
that the mass parameter M in the Schwarzschild metric represents the total mass
of the system for an observer at spatial infinity and therefore having it fixed a
priori (before performing the variation of the Hilbert action) does not lead to a true
Hamiltonian variational principle.
Analysis of boundary terms arising in the variational principle10 suggests that
a more general family of external fields, namely a Schwarzschild-like geometry with
a variable mass parameter M(t), might be considered, provided that the standard
Hilbert action is substituted with an improved one Atot consisting in a regular part
outside the shell, a singular part, concentrated on £ and a boundary part, evaluated
on a world tube external to the shell and whose radius will be shifted to infinity.6 In
this manner the total Lagrangian ~Ltot in the variational principle will be a function
on the dynamical variables \j}, ip, M and M. The improved Hilbert action takes the
form:
Atot = [ * UotW, <A, M, M)dt + F(t2) - F(h) (2)
Jti
2348
where the boundary terms F(tt) (i = 1, 2) may be neglected and
Uot =m(u)
f 2M\ ft 2McoshM
^ ^ !"T coshM-^l-Ml
- 2ih + V'V'M-
It is immediately evident that Ltot does not depend on M and therefore the fact
that M must be constant comes now as a consequence of the equations of motion
rather than as an imposed prerequisite for the system.
3. Hamiltonian
Evaluating the equation of motion for the variable M it is possible to solve explicitly
for M thus obtaining:
M(A^) = th I" coshM-WHr^+sinh2M ) (3)
where units were chosen such as the total amount of homogeneous fluid contained
in the shell equals Sit therefore giving the relation v = |i/j2. Equation (3) can be
substituted in Ltot to give Ltot = thip/j, ~~ M.
Now from the usual Legendre transformations it is easy to obtain the
Hamiltonian function as:
H(n,v) = M(»,v) . (4)
It can be proved that the hyperbolic angle \x is the momentum canonically
conjugated to the proper volume v.
References
1. W. Israel, Singular hypersurfaces and thin shells in general relativity, Nuovo Cimento
44B, 1-14 (1966).
2. W. Israel, Gravitational collapse and causality, Phys. Rev. 153, 1388-1393 (1967).
3. J. Kijowski, G. Magli, D. Malafarina, Relativistic dynamics of spherical timelike shells,
Gen. Rel. Grav. 38, 1697-1713 (2006).
4. P. Hajfcek, J. Kijowski, Lagrangian and Hamiltonian formalism for discontinuous fluid
and gravitational field, Phys. Rev. D 57, 914-935 (1998).
5. J. Kijowski, "True degrees of freedom" of a spherically symmetric, self-gravitating dust
shell , Ada Phys. Polon. B 29, 1001-1013 (1998).
6. J. Kijowski, G. Magli, D. Malafarina, New derivation of the variational principle for
the dynamics of a gravitating spherical shell, Phys. Rev. D 74, (2006).
7. J. Kijowski, G. Magli, Relativistic elastomechanics as a lagrangian field theory, Journal
Geom. Phys. 9, 207 - 223 (1992).
8. P. Hajfcek, J. Kijowski, Spherically symmetric dust shell and the time problem in
Canonical Relativity, Phys. Rev. D 62, 044025-1-044025-5 (2000).
9. J. Kijowski, E. Czuchry, Dynamics of a self-gravitating shell of matter, Phys. Rev. D
72, 084015-1 - 084015-12 (2005).
10. J. Kijowski, A simple derivation of canonical structure and quasi-local Hamilionians
in general gelativity, Gen. Relat. Grav. 29, 307-343 (1997).
Operating GW Detectors
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VIRGO COMMISSIONING PROGRESS
F. ACERNESE6, P. AMICO10, M. ALSHOURBAGY11, F. ANTONUCCI12, S.
AOUDIA7, P. ASTONE12, S. AVINO6, D. BABUSCI4, G. BALLARDIN2, F. BARONE6,
L. BARSOTTI", M. BARSUGLIA8, F. BEAUVILLE1, S. BIGOTTA11, M.A.
BIZOUARD8, C. BOCCARA9, F. BONDU7, L. BOSI10, C. BRADASCHIA", S.
BIRINDELLI", S. BRACCINI", A. BRILLET7, V. BRISSON8, D. BUSKULIC1, E.
CALLONI6, E. CAMPAGNA3, F. CARBOGNANI2, F. CAVALIER8, R. CAVALIERI2,
G. CELLA", E. CESARINI3, E. CHASSANDE-MOTTIN7, N. CHRISTENSEN2, A.C.
CLAPSON8,F. CLEVA7, C. CORDA11, A. CORSI12, F. COTTONE10, J.-P. COULON7,
E. CUOCO2, A. DARI10, V. DATTILO2, M. DAVIER8, M. DEL PRETE2, R. de ROSA6,
L. di FIORE6, A. di VIRGILIO", B. DUJARDIN7, A. ELEUTERI6, I. FERRANTE", F.
FIDECARO", I. FIORI", R. FLAMINIO1'2 , J.-D. FOURNIER7, S. FRASCA12, F.
FRASCONI", L. GAMMAITONI10, F. GARUFI6, E. GENIN2' A. GENNAI", A.
GIAZOTTO", G. GIORDANO4, L. GIORDANO6, R. GOUATY1, D. GROSJEAN1,
G. GUIDI3, S. HEBRI2, H. HEITMANN7, P. HELLO8, S. KARKAR1, S.
KRECKELBERGH8, P. La PENNA2, M. LAVAL7, N. LEROY8, N. LETENDRE1, B.
LOPEZ2, M. LORENZINI3, V. LORIETTE9, G. LOSURDO3, J.-M. MACKOWSKI5, E.
MAJORANA12, C.N. MAN7, M. MANTOVANl", F. MARCHESONI10, F. MARION1,
J. MARQUE2, F. MARTELLI3, A. MASSEROT1, M. MAZZONI3, F. MENZINGER2, L.
MILANO6, C. MOINS2, J. MOREAU9, N. MORGADO5, B. MOURS1, F. NOCERA2, C.
PALOMBA12, F. PAOLETTI2;", S. PARDI6, A. PASQUALETTI2, R. PASSAQUIETI",
D. PASSUELLO", F. PIERGIOVANNI3, L. PINARD5, R. POGGIANI", M.
PUNTURO10, P. PUPPO12, K. QIPIANI6, P. RAPAGNANI12, V. REITA9, A.
REMILLIEUX5, F. RICCI12, I. RICCIARDI6, P. RUGGI2, G. RUSSO6, S. SOLIMENO6,
A. SPALLICCI7, M. TARALLO", M. TONELLl", A. TONCELLl", E.
TOURNEFIER', F. TRAVASSO10, C. TREMOLA11, G. VAJENTE11, D. VERKINDT1,
F. VETRANO3, A. VICERE3, J.-Y. VINET7, H. VOCCA10 and M. YVERT1
Laboratoire d'Annecy-le-Vieux de Physique des Particules (LAPP), IN2P3/CNRS, Universite de Savoie,
Annecy-le-Vieux, France
European Gravitational Observatory (EGO), Cascina (Pi) Italia
INFN, Sezione di Firenze/Urbino, Sesto Fiorentino, and/or Universita di Firenze, and/or Universita di Urbino,
Italia
INFN, Laboratori Nazionali di Frascati, Frascati (Rm), Italia
LMA, Villeurbanne, Lyon, France
INFN, sezione di Napoli and/or Universita di Napoli "Federico II" Complesso Universitario di Monte
S.Angelo, Italia and/or Universita di Salerno, Fisciano (Sa), Italia
Departement Artemis - Observatoire Cote dAzur, BP 42209, 06304 Nice, Cedex 4, France
Laboratoire de VAccelerateur Lineaire (LAL), IN2P3/CNRS Universite De Paris-Sud, Orsay, France
9ESPCI, Paris, France
INFN Sezione di Perugia and/or Universita di Perugia, Perugia, Italia
1 INFN, Sezione di Pisa and/or Universita di Pisa, Pisa, Italia
1 INFN, Sezione di Roma and/or Universita "La Sapienza ", Roma, Italia
2351
2352
1 The Virgo project
Virgo is a 3-km gravitational wave interferometer, aimed to the detection of the
gravitational waves emitted by astrophysical sources and built near Pisa, by a French-
Italian Collaboration. Details about the gravitational waves sources, their detection
through interferometric techniques, and the scheme of the Virgo detector can be found in
the plenary session paper "The status of the Virgo gravitational wave detector"[l]. In the
following we will focus only on the commissioning aspects.
2 Commissioning general path
The Virgo commissioning started in September 2003, when a first laser light was
sent over one of the two 3-km long arms. Useful experience was acquired in 2001-2002
[2] during the commissioning of the central section of Virgo, even if the problems
connected with a kilometric scale interferometer are very different.
The commissioning was organized in steps of increasing complexity: first the two 3-
km Fabry-Perot cavities were studied independently (September 2003 - February 2004),
then a Fabry-Perot Michelson interferometer was commissioned (February 2004 -
October 2004), and finally the full Recycled-Fabry-Perot-Michelson interferometer
(since October 2004). For each step, once the longitudinal lock was achieved under
angular local controls, low noise robust operation was engaged by means of automatic
alignment of the mirrors, laser frequency stabilization and suspension hierarchical
control. The noise of the interferometer was then studied and reduced. Since several
noises depend on the optical configuration (Fabry-Perot cavity, recombined), a more
effective noise hunting phase started only when the full (recycled) interferometer was
locked, in October 2004. Obviously, the control strategy and the noise reduction are
strictly related. At low frequency the controls directly contaminate the sensitivity, while
at high frequency several laser noises affect the interferometer output through angular
and longitudinal accuracies. For this reasons, following the commissioning progress, the
control design is constantly upgraded.
3 Low power interferometer and backscattering problems
The first part of the recycled interferometer commissioning was carried out with a
reduced input power (about 0.8 W), obtained by attenuating the mode-cleaner transmitted
power by one order of magnitude. This choice was motivated by the presence of
backscattering fringes between the input mode-cleaner and the interferometer, due to the
absence of optical isolation between these two elements. These fringes were the origin of
large perturbations in the frequency of the laser and consequently of difficulties in the
interferometer control.
The lock of the interferometer was achieved in October 2004, through an original
technique, called variable finesse lock acquisition [3]. The commissioning with low
power lasted for about one year (from October 2004 to September 2005) and during this
period the automatic alignment was commissioned [4] as well as the frequency
stabilization and the suspension hierarchical control.
2353
Two data takings (C6, 2 weeks long and C7 5 days long), were performed between
August and September 2005 [5,6], with duty cycles respectively of 90% and 65%. The
sensitivity obtained is given in Fig. 1.
At the end of this phase the noise of the interferometer was almost completely
understood, being control noise below 200 Hz and read-out (shot) noise above 200 Hz.
4 Injection bench upgrade
After the run C7 (September 2005), the interferometer was shut-down in order to
allow the replacement of the injection bench and the installation of the Faraday isolator
between the mode-cleaner and the interferometer. Due to the limited space it was not
possible to install the Faraday isolator on the existing bench an4 a new one was built.
This was also the opportunity to redesign some optical elements of the bench, to enlarge
apertures and reduce diffused light, dump more effectively spurious beams and to host
the full input matching telescope.
Along with the replacement of the bench, the power recycling mirror, made with a
composite structure, was also replaced with a monolithic one, having a higher reflectivity
and flat substrate.
The commissioning of the new injection system and the recovering of the control due
to the power increase took about six months. In spring 2006 it was possible to relock the
interferometer, with about 9 W input power.
5 High power interferometer and thermal effects
The power increase by an order of magnitude revealed unexpected problems. The
mirrors are deformed by the heat absorbed, and this causes a change in the light
wavefront, mainly visible in the sidebands, which resonate only in the central recycling
cavity, that has nominal flat-flat geometry. The excess of absorption, with respect to the
nominal value is not fully understood, but it can be related to mirror cleaness.
The thermal effects highly complicate the lock acquisition. First of all, the
interferometer experiences a thermal transient, with a time constant of about 10 minutes.
During this period the feedback loops (both longitudinal and angular) must deal with
changes of the optical gains and phases of the interferometer signals, and react in order to
keep the interferometer locked.
Other consequences of the thermal effects inside the interferometer are under
investigation. Among them, the excess of noise due to wavefront deformation is not
excluded.
In general the thermal effects slow down all the commissioning activity, partially
because of the effort needed to achieve a robust operation of the interferometer, but also
because of the time needed to reach a steady state, at least 30 minutes.
In September 2006 the complete recovering of the sensitivity (after the two major
changes of the injection bench and power recycling) and robust operation, were
achieved. The stored power was -280 W, corresponding to 7 W input power and
recycling factor about 40.
2354
Substantial sensitivity upgrades have been performed between September 2006 and
March 2007 (see Fig. 1), due to several improvements. The main ones are: frequency
stabilization upgrades, reduction of longitudinal control noise, reduction of the oscillator
phase noise due to better alignment stability, reduction of many environmental noise
sources.
6 Week-end Science Run program
The Week-end Science Run program was started in September 2006, as a transitory
phase between the commissioning and a long science ran, planned for mid-2007.
The goal is to collect data for 56 hours during the week-end, and use them for data
analysis studies and detector characterization. The schedule of the science run is decided
following the commissioning activities, that remains the priority during this period.
During each WSR the activity is organized in 8-hours shifts, covered by one operator and
one scientist.
Nine WSRs have been performed from September 2007 to March 2007. The duty
cycle ranges from 65% to more than 90% for WSRS and WSR9. The longest lock was 55
hours (WSR9). The sensitivity evolution is given in Fig. 1.
10
10
!0'
10'
to
10
10'
HI'
10'
10
1
10'
14
1>1
16
1?
IS
30
•Jt
22
a
1
%
-
^
»A
It '' S
~k\ ) \ \
I - ''\ \
L '' \i
1
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:>..
=
w
-
C1 Nov 2003
C2 Feb 2004
C3 Apr 2004
C4 Jul 2004
CS Dec 2004
C6 Aug 2005
C7 Sep 2005
WSR1 Sop 308
WSR? Jan 2007
WSRS Tab 2007
Design
10*
10
C1 & C2: single mm ; 03 & C4: mcmnMned ; CS & after: resyclsct
«4
Frequency {HzJ
Figure ]. Virgo sensitivity evolution for the different commissioning runs and for some of the week-end science
runs.
The first WSRs and, more in general, the daily commissioning activity, were limited
by a strong sensitivity to bad weather conditions. Wind and sea activity increase the
seismic noise at very low frequency (10 mHz - 0.6 Hz). In order to deal with this
problem a large effort was done both on the suspension control and on the automatic
alignment. A substantial increase in the stability, duty cycle and data stationnarity was
achieved between the first WSRs and the last ones.
7 Current status
The best sensitivity achieved is -10'22 Hz""2, at a few hundreds Hz. Above -400 Hz
sensitivity is mainly limited by the shot noise; below ~50 Hz it is mainly limited by
control noise. In the central region of the spectrum the sum of all the known sources of
noise is still 2-3 times below the measured sensitivity. However, there are strong
evidences that the environmental noise, coupled to the interferometer output through
diffused light and spurious beams, is limiting the sensitivity in this region.
The reorganization of the optical benches and their acoustic isolation is on-going.
The commissioning activity and the WSR program will continue until mid May,
when a long science run will start for about 4 months.
References
1. Acemese et al., The status of the Virgo gravitational wave detector, MG11
proceedings
2. Acemese et al., The commissioning of the central interferometer of the Virgo
gravitational wave detector, Astroparticle Physics 21, 1-22 (2004)
3. Acemese et al, The variable finesse lock acquisition technique, Classical and
Quantum Gravity, 2006.23 (S85-89)
4. Acemese et al., The Virgo automatic alignment status, Classical and Quantum
Gravity, 2006.23 (S91-101)
5. Acemese et al, Virgo Status, Classical and Quantum Gravity, 2006.23 (S63-69)
6. Acemese et al, Virgo data analysis for C6 and C7 engineering runs, MG11
proceedings
RESULTS FROM LIGO OBSERVATIONS: STOCHASTIC
BACKGROUND AND CONTINUOUS WAVE SIGNALS
NELSON CHRISTENSEN, FOR THE LIGO SCIENTIFIC COLLABORATION
Physics and Astronomy,
Carleton College, Northfield, Minnesota, 55057 USA
nchriste@carleton.edu
The search for gravitational radiation has entered a new era as the Laser
Interferometer Gravitational Wave Observatory (LIGO) has reached its initial target sensitivity.
Other similar interferometric detectors are also approaching their design goals. There is
presently vigorous activity in the gravitational radiation community in the search for
signals. Here we review the status of the LIGO search for a stochastic background, and
continuous wave signals.
1. Introduction
The Laser Interferometer Gravitational Wave Observatory (LIGO)1'2 has achieved
its initial target sensitivity, and the detection of an event could come at any time.
The expected gravitational wave (GW) sources include supernovae, pulsars, the in-
spiral of binary systems with neutron stars and/or black holes followed by merger
and black hole ringdown phases, or even the stochastic background from the Big
Bang. Members of the LIGO Scientific Collaboration (LSC) are enthusiastically
working to make gravitational radiation detection a reality. LIGO and the LSC
have gone through a number of science runs where data was collected and
analyzed. So far, LIGO has completed four science runs (S1-S4) and is now in its fifth
science run, S5. Between these runs the interferometer performance was improved
through commissioning work. LIGO has more than met its design goal with a strain
sensitivity of h(f) < 3 x lCT23Hz^1/2 at 200 Hz, and hrms « 10^21 within a
bandwidth of 100 Hz. In S5 the LIGO 4 km interferometers have a sensitivity range for
optimally oriented 1.4M0-1.4M0 neutron star binary inspirals out to a distance of
33 Mpc for an SNR of 8. Here we summarize the LIGO results for searches for a
stochastic background, and for continuous wave signals.
2. The Stochastic Background Search
Various mechanisms during the Big Bang and in the early universe will produce a
stochastic background of GWs, analogous to the electromagnetic cosmic microwave
background. This would seem to be a background noise in each detector, but the
signal could be extracted through a correlation of the outputs of two detectors.4'5
A background could also be produced after the Big Bang, e.g. through the addition
of signals from binary systems or supernovae throughout the universe. LIGO is
actively searching for the stochastic background,6""8 and setting limits on its strength.
The magnitude of the stochastic background is usually described by the GW
energy density per unit logarithmic frequency, divided by the critical energy density
to close the universe, f2gw(/). Using the S4 data LIGO was able to set a limit on
2356
Frequency (Hz)
Fig. 1. As presented in Ref. 11, the upper curves are the ho amplitudes detectable from a known
generic source with a i% false alarm rate and 10% false dismissal rate for single detector analyses
and for a joint detector analysis. All the curves use typical S2 sensitivities and observation times.
HI and H2 are the 4 and 2 km detectors located in Hanford WA. LI is the 4 km detector situated
in Livingston LA. Lower curve: LIGO design sensitivity for 1 yr of data. Stars: upper limits for 28
known pulsars. Circles: spindown upper limits for the pulsars with frequency derivative values if
all the measured rotational energy loss were due to GWs (for a moment of inertia of 1048 gem2).
the stochastic GW energy density of ilsw(f) < 6.5 x 1CT5 in the frequency band
from 51 Hz to 150 Hz for a frequency independent GW spectrum.8 An important
benchmark in stochastic background sensitivity is the indirect bound set by
nucleosynthesis.9 If the energy density of GWs at the time of nucleosynthesis were too
large it would affect the ratio of light nuclei production. LIGO's S5 sensitivity and
data could allow it to set a limit below the nucleosynthesis level.
3. Continuous Wave Signal Searches
Rapidly spinning neutron stars, or pulsars, could be sources of GWs. In order for
radiation to be produced the neutron star would need to be non-axisymmetric in
shape. This type of gravitational radiation would be a nearly perfect sinusoidal
signal. One must still account for Doppler shifts due to the motion of the Earth,
and changes in the interferometers' response as the Earth rotates and orbits about
the sun. Radio observations can help the search as this provides sky location,
rotation frequency and spindown rate. Typically, GWs will be emitted at twice the
rotation frequency. In the absence of a signal it is still possible to produce
meaningful astrophysical results. An upper limit on the strength of a GW corresponds
to an upper limit on the ellipticity of the neutron star; an indirect limit can be set
from the star's spindown rate, and this is used as a benchmark of the sensitivities
of the direct limits. LIGO has published a series of results on the upper limits of
2358
signal strength for various known pulsar signals. 10~12 Using the S2 data 28 pulsars
were studied, and limits on the strain signal strength as low as 1.7 x 10~24 were
achieved, along with limits on pulsar ellipticity as low as 4.5 x 10-6.11 The pulsar
gravity wave signal limits set by LIGO with its S2 data are displayed in fig. 1.
LIGO all-sky searches can detect unknown periodic sources due to any emission
mechanism; for the S2 search12 the overall best upper limit on the GW amplitude
at the detector was 4.43 x 1CT23 for the 200-400 Hz band. For upcoming analyses
the detector sensitivity has increased by a factor of 20, we have looked for many
more known pulsars, and the frequency band of some of our unknown searches has
increased to 50-1500 Hz.
Acknowledgments
The authors gratefully acknowledge the support of the U.S. National Science
Foundation for the construction and operation of the LIGO Laboratory and the Particle
Physics and Astronomy Research Council of the United Kingdom, the Max-Planck-
Society and the State of Niedersachsen/Germany for support of the construction and
operation of the GEO600 detector. The authors also gratefully acknowledge the
support of the research by these agencies and by the Australian Research Council, the
Natural Sciences and Engineering Research Council of Canada, the Council of
Scientific and Industrial Research of India, the Department of Science and Technology
of India, the Spanish Ministerio de Educacion y Ciencia, The National Aeronautics
and Space Administration, the John Simon Guggenheim Foundation, the Alexander
von Humboldt Foundation, the Leverhulme Trust, the David and Lucile Packard
Foundation, the Research Corporation, and the Alfred P. Sloan Foundation.
References
1
2.
3
4.
5
6
7
8
9
10
12.
B. Barish and R. Weiss, Phys. Today 52, 44 (1999).
B. Abbott et al, Nucl. Instrum. and Methods A, 517, 154 (2004).
B. Abbott et al., Phys. Rev. D 69 082004 (2004)
N. Christensen, Phys. Rev. D 46, 5250 (1992).
B. Allen and J. Romano, Phys. Rev D 59, 102001 (1999).
B. Abbott et al, Phys. Rev. D 69, 122004 (2004).
B. Abbott et al, Phys. Rev. Lett. 95, 221101 (2005).
The LIGO Scientific Collaboration, astro-ph/0608606, Ap. J. in-press (2006)
M. Maggiore, Phys. Rep. 331, 283 (2000).
B. Abbott et al, Phys. Rev. D 69, 082004, (2004).
B. Abbott et al, M. Kramer, A.G. Lyne, Phys. Rev. Lett. 94, 181103 (2005).
B. Abbott et al, Phys. Rev. D 72, 102004 (2005).
EXPLORER and NAUTILUS GRAVITATIONAL WAVE
DETECTORS - A STATUS REPORT
P. ASTONE1, D. BABUSCI,3 M. BASSAN,45 P. CARELLI,6.5 G. CAVALLARI,8 A.
CHINCARINI,2 E. COCCIA,4-7 S. D'ANTONIO,5 M. Di PAOLO EMILIO,6 F. DUBATH,10
V. FAFONE,4.5 S. FOFFA,10 G. GEMME,2 G. GIORDANO,3 M. MAGGIORE,10
A. MARINI,3 Y. MINENKOV,7 I. MODENA,4'5 G. MODESTINO,3 A. MOLETI,4'5
G.V. PALLOTTINO,9*1 R PARODI,2 G. PIZZELLA,4'3 L. QUINTIERI,3 A. ROCCHI,5
F. RONGA,3 S. STANLIO,7 R STURANI,10 R. TERENZI,"1 G. TORRIOLI,12-1
R. VACCARONE,2 G. VANDONI8 and M. VISCO11'5
INFN, Sezione di Roma, Roma, Italy
INFN, Sezione di Genova, Genova, Italy
INFN, Laboratori Nazionali di Frascati, Frascati, Italy
4 Dip. Fisica, Universita di Roma "Tor Vergata", Roma, Italy
5 INFN, Sezione di Roma Tor Vergata, Roma, Italy
6 Universita dell'Aquila, Italy
7 INFN, Laboratori Nazionali del Gran Sasso, Assergi, L'Aquila, Italy
8 CERN, Geneva , Switzerland
9 Dip. Fisica, Universita di Roma "La Sapienza", Roma, Italy
10 Dep. de Phys. Theorique, Universite de Geneve, Geneve. Switzerland
11 INAF, Istituto Fisica Spazio Interplanetario, Roma, Italy
12 CNR, Istituto di Fotonica e Nanotecnologie, Roma, Italy
We review the state of operation of the two cryogenic resonant antennas of the ROG
Group, with updated statistics on observation time and data quality. We also mention
some preliminary results from joints searches with other gravitational detectors. Finally,
the present a brief overview of the development work into advanced readouts, that could
increase the peak sensitivity and the bandwidth of our apparata.
1. Introduction
The ROG group has been operating two cryogenic gravitational wave (g.w.) bar
detectors: EXPLORER (at CERN) and NAUTILUS (in Frascati).1
The ultra-cryogenic detector NAUTILUS is operating at the INFN Frascati
National Laboratory since December 1995. It is equipped with a cosmic ray detector
based on streamer tube technology. The present data taking started in 2003, with
a new bar tuned at 935 Hz, with a more sensitive readout chain, and a new
suspension cable, to provide a more stable position sotting. NAUTILUS is the only
resonant detector that showed capable of reaching a temperature as low as 0.1 K,
being equipped with a 3He-4He dilution refrigerator. This ultra-cryogenic
operational mode would result in a better sensitivity but also in a decrease of the duty
cycle. Up to now. priority was given to the observational time and so we keep the
standard operation at 3.5 K. The resulting strain spectral noise has a minimum
h ~ 1 -f 2 ■ Kr21 /VWz around 935 Hz, and h < 10~20 /VWz over about 30 Hz.
Integration over this bandwidth yields the minimum detectable pulse energy, or
2359
2360
noise temperature: it is less than 2 mK. This corresponds to a conventional (1 ms)
amplitude of GW bursts h = 3.4 ■ 1CT19.
The EXPLORER, antenna, in operation at CERN since 1986, is very similar to
NAUTILUS, but works at a fixed temperature of 2.6 K. Its noise temperature is of
the order of 2 mK, with a minimum spectral strain sensitivity h ~ 2-^3-10~21 /'VHz
around the two resonances at 904 Hz and 927 Hz, and h < 10~20 /\fWz over about
30 Hz. Also EXPLORER is equipped with a cosmic ray detector, based on a set of
long plastic scintillators.
Each detector consists of an aluminum cylindrical bar having a mass of ~ 2.3
tons, with a capacitive resonant transducer mounted on one of the bar faces. The
read-out systems installed in 2001 on EXPLORER and in 2003 on NAUTILUS,
mainly consisting of a large capacitance, small gap resonant transducer and a high
coupling, low noise dc SQUID, allowed us to obtain a larger bandwidth and
consequently an improved time resolution (now less than 10 ms).
Fig. 1. The sensitivity curves of the EXPLORER detector before and after the change of the
transducer bias voltage. Since April 2006 we are operating on the symmetric curve
When searching for impulsive signals, the data are filtered with an adaptive filter
matched to a delta-like signal. This search for bursts is suitable for any transient
GW with a nearly fiat Fourier spectrum in the sensitive bandwidth of each detector.
NAUTILUS has been kept in continuous observational mode since May 2003,
and EXPLORER since March 2004, both with a duty cycle close to 90%, mainly
limited by the unavoidable periodic maintenance operations: normally one day for
2361
refilling of cryogenic fluids every 3 weeks. Data taking also continued over Christmas
holidays, despite the shut down of the respective Laboratories.
In April 2006 we have changed the bias voltage in the transducer of Explorer:
this moves the resonant frequencies of the coupled system bar + auxiliary oscillator
by a few hertz, resulting in a more symmetric sensitivity curve.
2. Data Quality and Calibration
In the last year we devoted a large effort to ensure the longest time of data taking
with the best, possible sensitivity. To this purpose we have postponed operation at
ultra low temperature (0.1 K) of Nautilus, that would require daily maintenance
operation and therefore lower its up time. We repeated detector calibrations via
both hardware and software injections of brief pulses to tune the output of our
detectors and of the filters. Periodic calibration is standard practice, but it was also
motivated by the need of testing new filtering algortihms: beside new realizations
of simple delta-like burst previously considered, we have performed detailed studies
of the detectors response to other classes of signals.2 This was done also to prepare
for the upcoming joint analyses with the interferometric detectors.
15
m
£
3
z
EXPLORER 2006
l \ f i I 1 J I I i I !
rVo^fcwfea
sow
3000
2005
[
NAUTILUS 2Qm
i
I;
"""I
!
1
i
i
'
.l.l.-.M,
3 ■! c fi
hourly mean of H(«o) (1CTZ/Hz)
10 1! 12 n H I! 16 (S 1 2
■22,
f> ? 8 9 50 11 12 n U IS IS
hourly mean of H(») {10"22/Hz)
Fig. 2. Histograms of the noise level, averaged over one hour, in the whole year 2006, for
EXPLORER and NAUTILUS
In fig. (2) we show, for each detector, an histogram of the hourly average of
sensitivity during 2006, expressed in units of H(ur)(Hz^1), the Fourier transform
of the pulse at the antenna frequency: we see that Nautilus was for 86% of the year
2362
at, a sensitivity H{oj) < lQ~2lHz~1. For Explorer, the corresponding figure is 0070,
although its average noise level is slightly higher.
The cosmic ray detectors were originally installed as veto systems, but turned out
to be excellent calibrators for the antennas: indeed, a cosmic ray shower produces
in the bar a real burst signal, probably the closest excitation to a g.w.
7. h
ykJlitlLflflffl LOJBJlik
-0.05 -0.03 -0.01 0 .01
A x Is)
0.03 0.05
Fig. 3. Distribution of the time differences between events in coincidence at the antenna output
(EXPLORER) and in the cosmic rays detector. The gaussian fit yields a standard deviation (on a
single detector) a = 3.6ms, and an average systematic delay At0 = 1.3ms
The amplitude calibration relies on the so called Thermoacoustic Model,3 while
the excellent timing resolution of the shower detectors (better than 0.1 ms) has
allowed us to study in great detail the timing in the response of antennas and
filters. In fig. (3) we show, for instance, the time delay between (small) EXPLORER
events and cosmic ray signals, together with a gaussian fit. For the time being, we
have conservatively set. At = 30ms the conicidence window, including the delays
due to systematics, time of flight and other possible offsets.
3. Data Analysis
- Explorer and Nautilus The analysis of correlations and coincidences
between the outputs of EXPLORER and NAUTILUS is an ongoing project of our
collaboration, with periodic updates. In the period 2001- 2003 EXPLORER and
NAUTILUS were the only operating detectors. Some analyses relative to the data
gathered by both detectors in 2001 and 2003 were published,4 including a new
upper limit for the flux of bursts of g.w. The same analysis has been carried out on
2363
the data produced by our detectors in the following years 2004, with 218.5 days of
overlapped good data, and 2005, when the two detector were simultaneously on the
air for 182.1 days. The results of this ongoing search are still being refined, and will
be disclosed in the near future.
- IGEC-2 collaboration Since 2005, both the ALLEGRO detector at LSU
(Usa) and AURIGA at INFN Legnaro Labs, have resumed regular operation:
therefore we have restarted the IGEC collaboration under a new agreement (IGEC-2)
between the 4 bar detectors. As a first product of this agreement, six months of
data (May-Nov. 2005) were searched for triple coincidence, (the ALLEGRO data
are kept for further analysis in the case of positive results). A very low threshold
of accidental rate was set, namely 1 per century, and no triple coincidence was
found. Detailed results of this search will be released shortly,6 while a new analysis,
covering data of all 2006, is about to begin.
- Bars and interferometers A first joint data analysis between all the INFN
GW detectors (AURIGA, EXPLORER, NAUTILUS and VIRGO) has been
performed for the period of the VIRGO C7 run (September 2005). Since the period
of exchanged data was very short, the analysis has addressed more methodological
than scientific issues. The efficiency of each detector separately, and then of the
network, was extensively studied through a large number of software injections of
damped sinusoid signals.
- Search for periodic signals We also continued analysis of monochromatic
signals,5 both with the already tested coherent algorithms and a new non-coherent
one, currently under test. A non-coherent search is in principle less sensitive than
a coherent one: however, being much faster, it allows us to analyze, for a given
computing time and power, amounts of data more than 100 times larger, thus
providing at the end a better overall sensitivity.
- Triggered search The analysis of our data at the times of a large number of
Gamma-ray bursts allowed us to set upper limits on the amplitude of possible GW
signals associated to them.7 This kind of study is continuing and has been extended
to detailed analysis of the data collected in coincidence with some rare astrophysical
events, like the giant flares of 1998 and 2004.
4. Future Developments
The effort to improve the detectors performance is ongoing, and is mainly devoted
to the reduction of the so called minimum detectable energy change ksTeff-. this is
determined by the resonator thermal noise and by the readout noise and coupling.
Cooling of Nautilus to its design temperature of 0.1K is still in our agenda, although
the above mentioned caveat advices against it: all the required hardware is in place,
and the cooling operations requires few days, followed by a tune-up period of a few
weeks.
Two experimental programs are under way to produce an improved readout:
- We are continuing development of an improved version of the present capac-
2364
itive plus squid readout, characterized by better coupling (via a double electrode
transducer) and extremely low noise (via a double stage dc SQUID).8 This should
reduce the energy sensitivity of our readout to about 70h at 2 K and drop further
linearly with temperature, allowing a sensitivity for short burst of h = 2 • 10~20 i.e.
about 8 times better than present performance.
- A new transducer based on a microwave cavity whose resonant frequency
(around 5 GHz) is modulated by the antenna vibrations is also under
development, and has shown high potential and promising results on a room tempeture
prototype. It is a non contact, completely wide band readout, that can lead us to
quantum limited sensitivity with use of mostly commercial components.9
Either change in readout would require to stop the antenna operation for a
period of 3-6 months (including warm-up and cool-down): its feasibility and timing
will be discussed within the international network of g.w. detectors.
Acknowledgments
We thank F.Campolungo, G.Federici, M.Iannarelli, R.Lenci, R.Simonetti,
F.Tabacchioni, E.Turri and the CERN cryogenic service for their technical
support. Part of the developments for the new readout are supported by the European
Commission, in the FP6 project ILIAS, research activity JRA3.
References
1. P. Astone et al, Class. Quantum Grav., 23, S57 (2006).
2. P. Astone et al, Journal of Physics Conf.Ser. 32 192 (2006).
3. M.Bassan et al. Europhys. Lett., 76 (6), 987 (2006)
4. P. Astone et al Class. Quantum Grav., 19, 5449; (2002); Class. Quantum Grav., 20
S785; (2003); Class. Quantum Grav., 23, S169 (2006).
5. P. Astone et al, Class. Quantum Grav., 23, S687 (2006).
6. IGEC2 Collaboration- in preparation.
7. P. Astone et al, Phys. Rev. D66, 102002 (2002); Class. Quantum Grav., 21, S759
(2004).
8. M. Bassan, P. Carelli, V.Fafone, Y. Minenkov, G.V. Pallottino, A. Rocchi, F. Sanjust,
G.Torrioli, Journal of Physics Conf.Ser.,32 89 (2006).
9. R. Ballantini, M. Bassan, A.Chincarini, G.Gemme, R.Parodi and R.Vaccarone,
Journal, of Physics Conf.Ser., 32 339 (2006).
AURIGA ON THE AIR: SENSITIVITY, CALIBRATION,
DIAGNOSTICS AND OBSERVATIONS
A. ORTOLAN for the AURIGA Collaboration*
INFN - Laboratori Nazionali di Legnaro
Viale dell'Universita 2, 1-35020, Legnaro, Italy
ortolan@lnl.infn.it
We report on the present status of the AURIGA gravitational wave detector, which
entered its second scientific run on May 2005. Performances and sensitivity are given
together with some results on the data quality. Results on the upper limit on gw emissions
at the time of the Dec 27 2004 giant flare of SGR1806-20 are presented.
1. Introduction
We report on status and performances of the upgraded gravitational wave (gw)
detector AURIGA designed to look for gw bursts from sources in the Local Group
of galaxies. The diagnostic and pre-operational phases of the detector was concluded
on December 2004 and data taking begun after few months for new gw searches. On
May 2005, after the installation of 4 insulation stages for the low-frequency seismic
noise, the AURIGA duty cycle for gw bust searches reached the very good figure
of about 97 % with a sensitivity of 2 x 10"21 < S]/2^) < 5 x 10~2° Hz'1'2 over
the detection band 850 < v < 950 Hz, which translates into a gw burst sensitivity
of hmin ~ 1.4 x 10~22 Hz^1. Here S^ is the power spectral density of the intrinsic
noise, expressed in terms of gw amplitude fluctuations at the detector input, and
hmin represents the minimum amplitude of the Fourier transform of the gw burst
h 5(t) detectable at unitary signal-to-noise ratio (SNR).
To search for impulsive gw events, AURIGA joined a network of gw detectors1
either resonant (i.e. the IGEC collaboration2) or interferometric.3 The network
operation of gw detectors reduces by order of magnitudes the false alarm probability
by the simple requirement of arrival time consistency of candidate events produced
by each detector.2 The detection of gw bursts at SNR as low as 4 4- 5 requires the
careful description of the intrinsic noise properties (stationarity, gaussianity, etc.).
However, the performances of a gw detector depend also on non-modeled (or
spurious) noise sources, usually related to cosmic rays, environmental noise and human
activities. To get rid of the non-modeled noise, we implemented procedures to define
the epoch vetoes and anti-coincidence vetoes.
Finally, we point out that in the presence of "astrophysical triggers", e.g. the
arrival time of a 7-ray burst, AURIGA can set interesting upper limits on
concomitant emission of gw and 7-rays produced by the progenitor. In fact, we were able
to set an upper limit on gw emissions during the giant flare of SGR1806-20.13
*see http://www.auriga.lnl.infn.it
2365
2366
2. The AURIGA detector
The AURIGA detector consists of a resonant bar of 2300 Kg equipped with a
resonant capa.citive transducer read by a dc-SQUID amplifier. From June 1997 until
November 1999, AURIGA operated at the sensitivity Slh/2 ~5x 10-22 \fMz with a
bandwidth of ~ 1 Hz and the duty cycle was ~ 30 % of the data acquisition time.
Sensitivity and duty cycle were mainly limited by the readout system due to poor
dc-SQUID energy resolution e ~ 104/i, and by the mechanical suspension system
winch gave a mechanical attenuation of vibrational noise of -240 dB at 920 Hz,G
To improve the detector performances (sensitivity, bandwidth and duty cycle) we
re-designed the electromechanical transducer, the superconducting matching
transformer5 and the detector suspension system.6 Figure 1 shows a simplified scheme
of the AURIGA detector with the resonant capacitive transducer read by a
double SQUID amplifier. The 3.5 Kg mass and the 897 Hz resonant frequency of the
transducer were optimized for the best AURIGA sensitivity and bandwidth.4 The
matching transformer couples the output impedance of the bar and transducer
system to the input impedance of the first SQUID. The tuning the resonant frequency
of the LC circuit formed by the matching transformer and the transducer
capacitance crucially depended upon the development of Q ~ 106 electrical resonators.5'7
The resulting three-mode detection scheme allowed the band widening with a lower
bias electrical field of 7.5 x 10e V/in in the transducer. The current in the input
coil of the first SQUID (sensor) is pro-amplified and fed to the input coil of the
second SQUID (amplifier) which is equipped with room temperature standard
electronics. The measured sensitivity of the complete transduction chain turned out
to be ~ 650 h at 4.2 K. On the other hand, to improve the seismic and acoustic
resonant matching
Fig. 1. Electromechanical scheme of the AURIGA detector. Relevant electrical parameters for
the matching transformer arc L = 7.89 H and Ls = 3.48 \\,H for the primary and secondary coils
with a high coupling constant k = M/t/LL„ = 0.8C.
insulation of the defector, we decided also to re-design the mechanical suspension
system, cleaning up every resonance suspected to decrease its performance. In the
hope of overcoming the creep problem, the maximum load was kept lower than 25
% of yield stress of the material. The new suspensions were tested and showed no
2367
resonances inside the bandwidth [700-^1200] Hz and an attenuation up to —240 dB.
The last stage of the suspensions is a Cu-Be cable that supports the bar from its
center of mass and ensures an additional mechanical attenuation of —60 dB.6 The
bar cool down started on November 2003 and, one month after, the AURIGA
detector reached the set of parameters suitable for tests and operation. Data taking
began at 4.5 K for diagnostics and calibration showing that both noise floor and
bandwidth were in close agreement with the performance predicted by the
thermodynamic model of the detector. However, due to the presence in the detection
band of spurious noise lines of seismic origin, we decided on December 2004 to
install further insulation stages for the low-frequency seismic noise. On May 2005, the
upgrade of the low frequency suspensions was completely finalized, while keeping
the detector in data taking. AURIGA is now suspended on 4 commercial isolation
stages with a cut off frequency of about 1 Hz which ensure a sufficient insulation
from low frequency noise sources.
3. Noise estimate and detector calibration
The noise of the bar, transducer, matching and feedback lines, and the double dc-
SQUID amplifier is due to intrinsic noise sources ("small fluctuations") that scale
linearly with the temperature. This component of the noise has been fully
characterized.5'7 In fact, we have proved that, for the fluctuating component of the
detector output at small amplitudes, the fluctuation-dissipation theorem holds.7
This important result has been achieved by measuring, through the calibration
line in Fig. 1, the electrical admittance 1/Z{v) of the detector at the SQUID
input port and comparing the result to the power spectral density of the noise. In
fact, the fluctuation/dissipation theorem implies that the total noise at the
detector output is proportional to S(u) = 2kBTRe{l/Z(u)} + Sv\l/Z(u)\2 + Si, where
Si = 3.2 x 10"26 A2/Hz and Sv = 3.2 x l(r30V2/Hz are the double SQUID
additive and back-action noise spectra at 4.5 K.5 In addition, the thermal contribution
of each mode dominates around around the maxima of Re{l/Z}, i.e. the resonant
frequencies u^ (k — 1,2,3) of bar, transducer and superconducting matching
circuit. This contribution is proportional to the resistive part of the admittance, i.e.
S(Vfc) ~ 2kBTRe{l/Z(i/k)}; the estimated noise temperature T ~ 4.5 K turns out
to be in good agreement with the thermodynamic temperature of the detector. The
above procedure calibrates only the AURIGA noise energy. The calibration of
AURIGA to the amplitude of gw strain involves also the application of a calibrated
force pulse / 6(t) to the bar, with the aim of estimating amplitude and phase of
the detector transfer function H(is).8 We emphasize that our calibration procedure
makes use of the following assumptions: i) H(u) is a simple-pole transfer function;
ii) the mass of the fundamental longitudinal mode of the bar is half of the bar
mass (this assumption is supported by experimental tests performed on the bar
at room temperature); and iii ) the gw interaction with the bar can be calculated
by means of the geodesic deviation equation and it results in an equivalent force
2368
900
Frequency [Hz]
Fig. 2. The AURIGA strain sensitivity (gray) compared with the expected (black curve)
sensitivity. The dotted curve represents the prediction for the thermal noise of bar and transducer at
4.5 K, i.e. the limiting noise source of the actual detector setup. The AURIGA sensitivity during
the first run is also shown for comparison.
fg = AMLv1 h. where M and L are the physical mass and length of the bar.8 Once
we estimated H{u) in the detection band, the spectral strain sensitivity readily
follows from Sh(v) = S{v)/\H[v)\2. Figure 2 shows the one-sided spectral strain
noise predicted by a thermodynamical model of AURIGA at 4.5 K compared to its
experimental measurement. The curves agree within to 10 % in the detection band.
It is worth noticing that AURIGA performances have remained almost constant
from the beginning of its second run: in particular, i) the noise rms in the detection
band (calculated every 3 hours) fluctuates less than few %; ii) after the application
of an anti-coincidence veto (see Sect. 4), the rate of events at 4.5 < SNR < 6 is
~ 45/hour, in close agreement with a gaussian noise simulation; iii) the rate of large
SNR > 6 events is few per day.
4. Anti-coincidence and epoch vetoes
At some times, the AURIGA output is contaminated by unmodeled noise sources
which affect its duty cycle and/or capability in gw searches, for instance, during
maintenance operations such as liquid helium transfer or electronic failures. In
addition, we have to mention electromagnetic interferences (short spikes or power
supply lines) and up conversion of seismic noise which give rise to large fluctuations
and may be recognized as candidate events by the data analysis procedures. The
excess noise may be short lived (isolated events of few msec duration), or might
last for many hundreds of seconds as spurious spectral lines in the detection band.
Obviously, amplitudes of candidate events or energy contents of spurious lines do
not agree with the fluctuation/dissipation theorem. We expect that most of "large
fluctuations" in the detector output are due to detector environment and there-
2369
fore we need some data conditioning procedure to cope with these effects. We also
note that the gw transfer function limits the AURIGA sensitivity to gw signals
within the detection band. As a consequence, spectral energy excesses outside the
band [850 -=- 950] Hz and above the SQUID noise level can be identified as spurious
signals. On these bases, we have implemented three kinds of vetoing procedures:
• An anti-coincidence veto, which identifies very short transient events (mainly
wide-band electromagnetic spikes) on a time scale of about 10 rnses; The
anticoincidence veto is set by threshold crossing of the spectral energy of the sub-band
[600 -7- 800] Hz; the threshold is adapted to the slow variation of noise level.
• A spurious line veto, which consists in few notch filters tuned to spurious
frequencies in the detection band.
• An epoch veto, which discriminates the periods of time when spurious noise
appear in the detection band or identifies instrumental malfunctioning. The epoch
vetoes are set by thresholding the energy content of frequency sub-bands around
1.1 kHz or by a threshold on the curtosis index of a suitable data buffer.
The choices of sub-bands, buffer length and thresholds are empirical, and they are
based on experimentalist feedbacks and on results of dedicated analysis of
playground data. It should be noted that epoch vetoes reduce the live time of AURIGA
of about 3 % with no impact on its efficiency. The anti-coincidence veto has much
more efficacy as it reduce the rate of SNR > 4.5 impulsive events from 190/hour
to 45/hour, a value close to the prediction of gaussian noise simulations. The cost
of the anti-coincidence veto is a reduction of detection efficiency.
5. AURIGA and the giant flare of SGR1806-20
A huge 7-flare, with an energy spectral content up to several MeV, occurred in Soft
Gamma-ray Repeater SGR1806-20. It was reported9 that the gamma-ray
luminosity was much bigger than any previous transient event observed in our Galaxy: in
the first 0.2 s, the flare released 3.7 x 1046 erg, assuming isotropic emission and
a source distance of 15 kpc. Its power can be explained by a catastrophic
instability involving global crust failure on a magnetar (a neutron star with a huge
B ~ 1015_;~16 Gauss magnetic field).10 This scenario has been confirmed by the
resolution of three timescales (about 0.25, 4.9 and 70 ms) in the first spike.11
According to some theoretical models,12 such a crust fracture may cause concurrent
emissions of gravitational wave (gw) and 7-rays with comparable luminosity, due to
the excitation of normal mode of the magnetar. On 27 December 2004, AURIGA
was taking data and was performing around the arrival time of the flare with noise
fluctuations quite close to gaussian and stationary behaviour. In addition, AURIGA
was favourably oriented in respect to the direction of SGR1806-20 as its antenna
pattern, averaged over polarizations, gave maximal sensitivity at the time of the
flare. The arrival time of the gw busts tp was assumed to coincide with the arrival
time of the flare peak at the AURIGA site i.e. 21:30:26.68 UT of 27 December
2370
2004. The complete analysis to validate the gaussian model for the AURIGA noise
within ±100 s around tp and the search in the AURIGA data for gw emissions at
tp is published elsewhere.13 Here we report the main results based on the power
localized on rectangular tiles of amplitudes A/ = 5 Hz and At = 201.5 ms13 in
the frequency band [850 -V- 950]. After a careful analysis of the localized power,
we conclude that no excess of gw power is found at tp and therefore we derived
that the initial amplitude of the the neutron star normal modes ho is limited as
/io<4x 10~20 at 95 % CL in the most sensitive frequency tile centered at 930 Hz.
The best upper limit can be conveniently expressed as in terms of the total gw
energy egw emitted by the normal modes excitation during the peak of the giant flare
of SGR1906-20, i.e. egw < 3 x 10"6 MQc2; the upper limit in the [850 -=- 950] Hz
< 5 x 10~5 MQc2. These limits are of some astrophysical interest as they
invade part of the parameter region of existing models of neutron stars dynamics.13
6. Conclusions
In its second run, the AURIGA detector exhibits an improvement in bandwidth,
sensitivity and duty cycle tanks to the upgrade of suspensions, detection scheme
and SQUID amplifier. The search for gw sources require a careful modeling of noise
and the identification of candidate events at low SNR. In this respect, we can state
that AURIGA can join the worldwide network of gw detectors in operation with
an high duty cycle of ~ 95 % and a good sensitivity to gw bursts of hmin ~
1.4 x 10~22 Hz^1 for long observing campaigns. In the presence of astrophysical
triggers, the stationary operation of AURIGA allows relevant searches of specific
gw sources, even with a single detector, as demonstrated by our best upper limit
egw < 3 x 10~6 Mqc2 on the gw emitted by normal mode excitations during the
peak of the giant flare of SGR1806-20.
References
1. see http://www.auriga.lnLinfn.it/a,uriga/MoU/MOU.html.
2. Z.A. Allen et al. Phys. Rev. Lett. 85 5046 (2000); P. Astone et al., Phys. Rev. D 68
022001 (2003).
3. L. Cadonati et al., CQG 22 1 (2005); S. Poggi et al., J. of Phys. 32 198 (2006).
4. J.P. Zendri et al., GQC 19 1925 (2002).
5. L. Baggio et al., Phys. Rev. Lett. 95 081103 (2005).
6. M. Bignotto, et al., Rev. Sci. Instrum. 76 084502 (2005).
7. A. Vinante et al., Rev. Sci. Instrum. 76 074501 (2005).
8. J.P. Zendri et al, "Amplitude and phase calibration of the AURIGA gw detector", in
preparation.
9. K. Hurley et al., Nature 434 1098 (2005).
10. C. Thomson and R. Duncan, ApJ 561 L133 (2001); C. Thomson et al, ApJ 574 332
(2002) and refs. therein.
11. S.J. Schwartz et al., ApJ 627 L129-L132 (2005).
12. J.A. de Freitas Pacheco, A&A 336 397 (1998); K. loka, M.N.R.A.S. 327 639 (2001).
13. L. Baggio et al., Phys. Rev. Lett. 94 241101 (2005).
Advanced GW Detectors
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OPTICAL SPRING AT THERMAL EQUILIBRIUM
A. DI VIRGILIO*
INFN, Sez. di Pisa, Pisa, Largo B. Pontecorvo 3, 56127 Pisa,Italy
angela. divirgilio@pi. infn. it
An optica] spring effect has been observed in the motion of a Fabry-Perot cavity
suspended to the Low Frequency Facility (LFF). The experimental set-up consists of 1 cm
long cavity hanging from a mechanical isolation system, conceived to suppress seismic
noise transmission to the optical components of the VIRGO interferometer. The observed
radiation pressure effect corresponds to an optical stiffness kopt ranging between 2.5 X 104
and 6.5 X 104 N/m. The measured relative displacement power spectrum is compatible
with a system at thermal equilibrium within its environmental. The absorption
coefficient 7; = 5.8 1/s, associated to the longitudinal motion of AX, is found by fitting the
data selected around the optical spring resonance 65 — 80 Hz. The upper limits of 10~15
m/VHz at 10 Hz for seismic and thermal noise contamination of the Virgo test masses,
suspended by a SuperAttenuator, are deduced by the data.
The described measurement deals with two points, very important for present and
future interferometers: the optical spring produced by radiation pressure inside a
Fabry-Perot cavity, and thermal noise spectrum outside resonance. The radiation
pressure will play a crucial role in the next generation of laser interferoinetric
detectors of gravitational waves ].1_5 Thermal fluctuations of mechanical systems are
considered the most relevant limitation of ground based interferometers for
gravitational waves detection in the low frequency region, where several gravitational wave
signals are expected6 . In this short paper it will be shown that the relative motion
of the mirrors of a suspended Fabry-Perot cavity is compatible with the presence
of an optical spring due to the radiation pressure and is at thermal equilibrium.
In the following only the main points of the experimental apparatus are gives,
please see other papers7_10 for details. The last stage of the experimental apparatus
is sketched in fig. 1. The suspension system adopted to insulate from seismic noise
the high finesse (ranging between 4000 and 6000) 1 cm long Fabry-Perot cavity is
equal to the suspensions of the VIRGO interferometer (SuperAttenuator SA). The
flat mirror of the cavity (AX, auxiliary mirror) is hung to the last mechanical seismic
filter of the chain called Filter7, by means of an independent three-stage suspension.
The other mirror, called VM (Virgo mirror), is similar to the Virgo test masses. The
control of the longitudinal motion is done by acting only on the VM mirror, with
a scheme identical to the one implemented in the VIRGO interferometer. Fig. 1
shows the cavity, the input beam, the longitudinal control loop scheme and the
acquired signals. The feedback control loop is based on a Digital Signal Processor
(DSP). The mechanical model is confined to the study of the dynamical system
formed by the two mechanical branches hung to Filter7 (see fig. 1). The feedback
loop circuit has been included within the model, and the optical spring is modeled
as a spring constant acting between the two mirrors. The model predicts the con-
*Presente at Parallel section GW2
2373
2374
F'17FF> 7
Fig. 1. Sketch of the experiment set-up from Filter?. The optical circuit, and the control loop are
shown; gray boxes underline the components under vacuum. La laser beam is frequency stabilized
by a rigid reference cavity shown in figure. ERROR, COIL2 and PROBE are the acquired signals
tributions to the power spectrum coining from external noise sources (electronic
and seismic noise, from the Laser etc.) and from the thermal noise,9,1-1 ~13 using the
Fluctuation Dissipation Theorem (FDT). The evidence of an optical spring effect
emerged from the observation that it was possible to lock the cavity only for positive
de-tunings (cavity longer than the closest resonance), corresponding to positive kopt-
A static detuning, different from run to run, ranging between lO^11 and 10~t2 m,
was measured,8 corresponding to a stiffness constant kopt ranging between 70000
and 10000 N/rn. In different spectra, it has been observed a. resonance changing
its position in accordance with the change of the static detuning. The error signal
exhibits all the statistical characteristics of the displacement power spectrum of a
system at the thermal equilibrium. Figure 2 shows one of the measurements and
the thermal noise estimated by the model assuming an optical gain 1.56 x 1010 V/rn,
kb = 56000 N/rn (this parameter is found by fitting the data with the model), and
the typical working conditions. The region of the spectrum below 3 Hz, seismic
noise dominated, has been cut off by filtering the data applying a high pass filter.
As it is shown in figure 2; the result of the fit and the data well agree below 90
Hz, at higher frequency the higher order modes are relevant, and the model cannot
reproduce the data. The result of the fit gives quite large absorption coefficients, an
other paper9 all problems connected with this point are analyzed in details. Seismic
noise contamination and thermal noise are a very important points for the Virgo
suspensions. The present measurement with the help of the model gives the upper
limit10 of 10~15 m/VWz at 10 Hz for the seismic noise contamination and thermal
noise for the test masses of the Virgo mirrors.
2375
Continuus Line measurement
Dashed Line fit result
Fig. 2. Measured power spectrum, 10 mHz frequency resolution, compared with the thermal
noise estimated by the model, assuming an optical gain 1.56 X 1010 V/m, kb = 55000 N/m, and
the typical working conditions of the present set of runs, the losses, two parameters constant in
frequency, are associated to the AM longitudinal and rotational degree of freedom, their fitted
values are 5.8 - for the longitudinal and 6.5 -.
References
1. A. Buonanno and Y. Chen, Class. Quantum Gravity 18, L95 (2001)
2. V. B. Braginsky, M.L. Gorodetsky and F. Ya. Khalili, Phys. Lett. A 232, 340 (1997)
3. V. B. Braginsky, and F. Ya. Khalili, Phys. Lett. A 257, 341(1999)
4. D. Vitali et al. Physics Revew A, 65, 063803.
5. O.Arcizet et al, Nature, 444, 71-74, 2006.
6. K. Thorne, gr-qc/9704042 and B. Shutz, Clas. Quan. Grav., 16, 1999, A131-A156.
7. A. Di Virgilio et al, J. Physics: Conference Series, Vol. 32 (2006), 346-352.
8. A. Di Vigilio et al. Phys. Rev. A, 74, 13813 (2006);
9. A. Di Virgilio et al.displacement power spectrum measurement of a macroscopic
optomechanical system at the thermal equilibrium, preprint gr-qc/0612130
10. A. Di Virgilio Seismic and thermal noise upper limits at 10 Hz for the Virgo
suspensions, Virgo Note, VIR-NOT-PIS-1390-334
11. Callen H.B. And Welton T.A. , Phys. Rev. 83 34-40
12. R Kubo 1966 Rep. Prog. Phys. 29 255-284
13. P. Saulson, Phys. Rev. D 42, 2437 (1990).
MEASUREMENTS OF ELECTRICAL CHARGE DISTRIBUTION
VARIATIONS ON FUSED SILICA
L.G. PROKHOROV and V.P. MITROFANOV1
Physics Department, Moscow State University, Moscow 119992, Russia
Fused silica test masses (mirrors) of interferometric gravitational wave detectors may accumulate
electrical charges which interact with surroundings. Variations of the charge or of its distribution
create fluctuating force which acts on the test mass. To study this effect we have developed the high
sensitive electrometer which allowed us to search some factors determined charge distribution
variations on fused silica sample.
1. Introduction
In the last few years a number of long baseline laser interferometric gravitational
wave detectors have begun operation and the next generation of detectors is presently
developed. The test masses (mirrors) of the LIGO Project detectors are fabricated from
fused silica (Si02) [1]. Being suspended in vacuum chambers they can accumulate and
store electrical charges. These charges interact with surroundings. Variation of the value
of electrical charge located on the test mass or variation of the charge distribution may be
a source of additional fluctuating force acting on the test mass, which can reduce the
detector sensitivity [2, 3]. The effect of test mass charging associated with cosmic rays
has been analyzed in [3]. In this work we present results of experimental search for some
factors, which determine variations of charge distribution located on fused silica samples.
The measurements were carried out in air and in vacuum by means of capacitive probe
placed under the rotating sample. Such technique is used for measurements of surface
charges and potentials due to a high sensitivity and minimal influence on the charge
distribution [4, 5].
2. Experimental setup
A schematic layout of the setup is shown in Fig. la. The fused silica sample had a
mushroom shape (the disk diameter and height were 60 mm and 12 mm, the leg diameter
and height were 10 mm and 38 mm). Such a shape of the sample decreased the effect of
the collet clamp via which charges might leak to ground or be injected in the sample. The
probe consisted of a circular sensor plate with diameter 2 mm and an outer guard tube
with diameter 4 mm was placed under the sample. Electrical charge induced on the
sensor plate of the probe was proportional to the local electrical charge density over the
probe. The induced charge was detected using the high impedance preamplifier. An
f This work was supported by the LTGO team from Caltech and in part by NSF and Caltech grant No PHY-
0353775, by Russia Agency of Industry and Science, Contract No 02.445.11.7423
2376
2377
Fig 1. (a) Experimental setup, (b) Standard deviation of the probe voltage for different averaging time: 1 - probe
was moved away or in vacuum, 2 - in air.
additional optical sensor was used to identify the angle of the sample rotation and to
control the rotation speed Qra, ~ 120 ipm. The angular distribution of the charge density
a (<p) located on the scanning strip of the sample was transformed to the periodic function
of time a (Qm,t) when the sample was rotating. The sample and the probe together with
the preamplifier were placed in the vacuum chamber, which has been pumped oul to a
pressure of about 1(T5 Torr. Data acquisition and processing were performed with PC.
The Faraday cup technique was used for calibration of the capacitive probe in air.
The charge deposited at the small area was measured alternately by the probe and by the
Faraday cup so that we could bring in correspondence these measurements.
3. Results of measurements and discussion
To study relaxation of charge, an additional charge was deposited on the sample by
contact electrification. Contact electrification is interesting because the LIOO test mass
touches the earthquake stops from time to time. This may build up relatively big
electrical charge on the surface. Touching the sample resulted in a peak on the spatial
charge distribution, which decayed with time. Measurements carried out in atmospheric
air have shown that humidity of ambient air and the sample history (the way of its
cleaning and preparation for measurements, the duration of exposure to the humid
atmosphere) influenced significantly the evolution of the deposited charge because the
adsorbed water substantially determined the charge transport along the surface of fused
silica samples. The relaxation time from 102 s to 104 s has been observed in atmospheric
air. In vacuum, no changes of deposited charges have been found within the limits of the
measurement errors which were about 2%. The relaxation time may be estimated as more
than 8000 hours.
2378
Some distribution of electrical charge with the spatial variations of charge density of
about l(rl3C/cm2 was always observed on the fused silica sample. It was associated with
a history of the sample and with stray electric fields existing inside the chamber. These
fields were caused by different values of the work function of materials situated around
the sample. The mobile charges relocated on the sample in order to decrease the total free
energy of the system. In air this occurred mostly due to the surface conductivity of fused
silica associated with adsorbed water. In vacuum the process became much slower. If the
immovable sample was in air for a long time, the peak corresponded to the charge
accumulated over the probe appeared in the charge distribution. This peak was likely
associated with the image force.
To study time variations of the charge density at some point on the sample we
measured the probe voltages when the probe was under this point in the process of the
rotation. This resulted in a set of discrete voltage values VJ which were averaged over a
time interval r» Qrot~'. The standard deviation aT was calculated for the difference
Vt+i'-Vj between adjacent values. It is plotted as a function of r in Fig.lb
(ffr«5-103e/cm2 for r= 10s.). Curve 1 was obtained in the case when the probe was
moved away from the sample. It coincided with the curve obtained in the case when the
probe was under the rotating sample and the measurements were carried out in vacuum.
Curve 2 was obtained when the probe was under the sample in atmospheric air. The
increase in standard deviation observed for r> 100 s indicates existing of excess random
charge variations up to 10"l5C/cm2 which are likely associated with the sample
electrification by dust particles in the process of the rotation.
The considered sources of variations of charge located on fused silica samples may
appear to some extent in gravitational wave detectors. They need more detailed study.
The authors are grateful to V. B. Braginsky for fruitful discussions.
References
1. B. Abbott et al. (The LIGO Scientific Collaboration), Nucl. lustrum. Meth. A 517,
154(2004).
2. R. Weiss, LIGO technical note. Available athttp://www.ligo.caltech.edu/T/
T960137-00.pdf.
3. V. B. Braginsky, O. G. Ryazhskaya, S. P. Vyatchanin, Phys. Lett. A 350, 1 (2006).
4. D. K. Davies, J. Sci. Instrum. 44, 521 (1967).
5. P. Molinie, IEEE Trans. Dielectr. Electr. Insul. 12, 939 (2005).
DEVELOPMENTS TOWARD MONOLITHIC SUSPENSIONS FOR
ADVANCED GRAVITATIONAL WAVE DETECTORS
ALASTAIR HEPTONSTALL, CAROLINE CANTLEY, DAVID CROOKS, ALAN CUMMING,
JAMES HOUGH, RUSSELL JONES, IAIN MARTIN and SHEILA ROWAN
SUPA, Institute for Gravitational Research,
Department of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ
a. heptonstall@physics.gla. ac. uk
GIANPIETRO CAGNOLI
Istituto Nazionale di Fisica Nucleare Sezione di Firenze
via G. Sansone, 1, 1-50019 Sesto Firentino, Italy
The proposed upgrades to both the LIGO and Virgo gravitational wave observatories will seek to
improve detector sensitivity by reducing thermal noise. Based on technologies first implemented at
the GEO600 detector, the test mass mirrors will be suspended using fused silica fibres of either
circular or rectangular cross section to form monolithic suspensions. In GEO600 cylindrical fused
silica fibres were produced using a hydrogen-oxygen flame based machine. Here we report on a new
C02 laser based fibre pulling system under development in Glasgow designed to achieve higher
tolerances and reduce contamination of fibres. Preliminary testing of a laser welding process
suitable for constructing full scale monolithic suspensions for advanced detectors is described.
1 Introduction
In 2000 the first quasi-monolithic suspensions were installed in a long baseline
interferometric gravitational wave detector. While the LIGO, Virgo and TAMA
detectors use fused silica optics suspended using steel wire slings, the GEO600 optics are
hung from synthetic fused silica fibres welded to small silica attachment points, which in
turn are silicate bonded to the test masses1. Fused silica fibres reduce detector thermal
noise primarily because of a lower intrinsic dissipation, a reduction of approximately
three orders of magnitude compared to steel2"4. Monolithic suspensions also eliminate
any noise caused by slip-stick mechanisms associated with using wires.
Upgrades planned for the LIGO6 and Virgo7 detectors are proposing to use this
technology, pushing them closer to their limits, increasing the working load to 0.8GPa
compared to the 0.6GPa loading in GEO600, and demanding higher tolerances on
dimensions. In order to achieve these requirements a fibre production system based on
heating using a C02 laser has been developed. Heating of this form has previously been
used by other authors to produce thin fibres of 1 to 20um8. The large test masses
planned for the advanced detectors will require 'ribbon' fibre cross sections of closer to
1.3xl0"7m2, or cylindrical fibres with diameters of 200um, and the system developed in
Glasgow is capable of producing fibres up to millimeters in diameter.
The system is also capable of welding fused silica, and preliminary investigations of
suspension construction techniques are discussed below.
2379
2380
2 CO! laser production of cylindrical fused silica fibres
Compared to the fibre production system used at GEO600, the new BOW CO2 laser
based system was designed to have significantly improved mechanical systems, based on
recirculating bearing races, and a heat source that would be both constant and
reproducible. A 'feed and pull' system is used, whereby silica is slowly fed into the laser
beam from below and then pulled quickly out from the top. The ratio of these speeds
gives an easily calculable reduction in cross section. Computer control of the pull allows
the shape of the fibre to be carefully tailored to create fibres of either variable or constant
cross section and neck regions with specific shapes. A rendered drawing of the machine
is shown in Figure 1, while Figure 2 shows a photograph of the prototype.
ilscrcw unit
Figures 1&2 (left to right) Rendered image of pulling machine; Photograph of prototype machine in Glasgow.
A heating arrangement was developed where conical gold coated mirrors are used to
create an optical path that heats the silica uniformly from all sides, improving fibre
symmetry. The laser itself is power stabilised, giving a power variation of below 1%.
A prototype machine was developed in Glasgow, with a twin machine, funded by the
European Gravitational Wave Observatory, having now been delivered to the Virgo
detector site where it will form part of research toward installation of monolithic
suspensions at the Cascina site.
Preliminary measurements of mechanical loss and strength indicate that fibres
produced by this method meet the targets required for the Advanced LIGO suspension
fibres. Further research is now being conducted with a view to producing rectangular
cross-section fibres, which arc currently the baseline design for Advanced LIGO and
which we have previously produced using a hydrogen-oxygen flame.8
3 Suspension construction using a CO2 laser
Preliminary tests of C02 laser welding techniques suitable for installation of the
monolithic suspensions of advanced detectors have been made. Figure 3 shows an early
2381
design of Advanced LIGO attachment ear, silicate bonded to a silica plate, A silica slide
of the type suitable for fibre production was welded to the tip of the ear using a CO2
laser. Figure 4 shows the bonded, welded part under a 12.5kg test load.
Based on our initial loading tests a new design of ear has been produced to reduce
stress and improve weld access, shown in Figure 5.
Figures 3,4 & 5 (left to right) Early design of Advanced LIGO ear that has been bonded and weided; Bonded
and welded ear under test loading of 12,5kg; Rendering of the intermediate mass design for Advanced LIGO
from which the test mass is hung.
Acknowledgements
The authors would like to thank our colleagues in GEO600 and the wider LIGO
scientific collaboration. We are grateful for the financial support provided by the
University of Glasgow, the Particle Physics and Astronomy Research Council and the
European Gravitational Observatory. We would also like to thank Prof. J. Faller of JI LA,
Boulder, and Prof. K. Strain, University of Glasgow, for many useful discussions.
References
1. J. Smith el al., Class. Quantum Grav., 21 (2004) SI091.
2
3
4
5
6
A. Grelarsson, G. Harry, Rev. Sci. Inst. 70 (1999) 4081
G. Cagnoli et al., Phys. Lett. A 255 (1999) 230
G. 1. Gonzalez, P. R. Saulson, J. Acoust. Soc. Am., 96 (1994) 207
P. Fritschel, Proceedings of SPIE Vol. 4856, Bellingham, WA, (2003)
P. Amico et al., Class. Quantum Grav., 19 (2002) 1669
V.P. Mitrofanov et al., 1985 Sov. Phys. Tech. Phys. 30 (4) 454-6
A. Heptonstall et al., Phys. Lett. A 354 (2006) 353-359
CONCEPT STUDY OF YUKAWA-LIKE POTENTIAL TESTS
USING DYNAMIC GRAVITY-GRADIENTS WITH
INTERFEROMETRIC GRAVITATIONAL-WAVE DETECTORS
PETER RAFFAI1, SZABOLCS MARKA2, LUCA MATONE2 and ZSUZSA MARKA2
1 Eotvos University, Institute of Physics, 1117 Budapest, Hungary
2 Columbia University, Department of Physics, New York, NY 10027, USA
We present a technique to measure possible violations to Newton's 1/r2 law using a pair
of matched Dynamic gravity Field Generators (DFGs) in a null-experiment and taking
advantage of the exceptional sensitivity of modern suspended mass interferometric
gravitational wave detectors. The correct placement of the DFGs, i.e. rotating symmetrical
two-body masses, in proximity to one of the interferometer's suspended test masses,
allows future tests of composition independent non-Newtonian gravity beyond the present
limits. We give our calculation and simulation results in context of Yukawa-like potentials
in the 0.5 — 10 meter range.
Dynamic gravity fields generated by rotating masses have been used previously in
several experimental tests for both calibration of gravitational wave (GW)
detectors12 and testing Newton's inverse square law (ISL) in the laboratory scale.2'6 ISL
tests so far provided results confirming Newton's 1/r2 law within experimental
uncertainties. These previous experiments were based on a single Dynamic gravity
Field Generator (DFG), consisting of a symmetric rotating object with a
significant quadrupole moment, and implemented for use with bar type GW detectors.
In this paper we step beyond these and present a new concept. We study a pair of
well matched and symmetrical DFGs rotating with the same frequency (/o) but at
/3 = 90° out of phase. Such system can induce a detectable motion in Test Masses
(TMs) of current and future interferometric GW detectors1'3'11 and the expected
signal at 2/o would be dominated by a term related to deviations from Newton's
law. Here we emphasize the concept, the detailed simulation results are presented
elsewhere.10
Additionally, a single DFG can be used to directly validate/evaluate gravity
gradient noise generation and its coupling mechanisms to complex structures. A
DFG can also provide an alternative and independent sub-percent amplitude and
phase calibration of interferometric GW detectors. This is the subject of a separate
publication.8
A hypothetical design (see inset of Fig. 1) consists of two symmetric titanium
discs placed at a distance of dl and du from the TM. Both discs have two cylindrical
slots which can hold different materials at r\ 2 and r\l2 apart, respectively, from the
rotational axis. Placed in the slots, tungsten cylinders serve as rotating masses, with
m\ 2 and ml^2 effective mass, respectively.
Composition independent tests of ISL4 are customarily interpreted as adding a
Yukawa-term (Vy (r)) to the Newtonian potential (VN(r)):
m A/7
V(r) = VN(r) + VY(r) =-G [1 + ae~rlx\ (1)
where G is the gravitational constant, m and M are the interacting masses, r
2382
2383
denotes the distance between two point masses, a is the Yukawa interaction coupling
strength and A describes the length scale of the coupling.
In our proof-of-concept model, we consider the TM and the DFG fillings as
point masses (rI,n <C dl'U limit). The TM is treated as a damped pendulum (with
known parameters) driven by the net dynamic gravitational force induced by the
DFGs. By calculating the TM's induced acceleration, aN,Y, analytically, we can
express the induced displacement, xN'Y, by using the pendulum's transfer function.
The DFG pair gives rise to sharp features appearing in the xN'Y(f) spectrum at
multiples of the DFGs' operational frequency, /o. In case of two ideally symmetric
DFGs, peaks at odd times /o vanish in both Newtonian and Yukawa dynamics.
If the DFG radii and distances are chosen to be r11 = y/rj rl and d11 = ^/rj d1,
where r\ = mll/ml is the mass ratio, the effect of the two DFGs cancel each other
at 2/o in the Newtonian limit. In the ideal case (perfectly symmetric and matched
DFG pair), this leaves only the ISL violating gravitational terms to have effect on
the TM at 2/o. For a Yukawa-like potential violation this effect is proportional to
the coupling strength, a, and the quadrupole moment of the two-DFG system. In
practical systems, imperfections will limit the techniques' ability to measure a.
We performed simulations of the TM displacement for the setup shown in Fig.l
by computing xN,Y in the Fourier-domain. We studied the effect of uncertainties
associated with the DFG setup parameters (d, r, m and phase) via Monte Carlo
simulations. A large number (N) of hypothetical setups was generated with the
DFG parameters normally distributed around a mean for simplicity. The mean
of the parameters are chosen to maximize the response of the interferometer in
terms of \a\, while taking into account the technically achievable or future plausible
range of values and measurement precision. We maximized the integration time
of the measurement at T = 107s ~ 4 months and chose rj to be 2 for all setups.
Effective masses of DFG fillings were maximized for each case such as to keep the
kinetic energy of the first DFG constant. The means of DFG operational frequencies
were cast based on spectral sensitivites of interferometric detectors 1>3>11 to make
the situation more realistic. However, it is likely that advanced special purpose
interferometers will prove more advantageous for these tests.
The results of the simulations are shown in the \a\ vs. A plot of Fig. 1 together
with the current limits. Uncertainty values for DFG parameters used with case I (see
table) are within the limits of current state of the art machining and measurement
technologies while case II and III presents a metrology challenge. The practical
DFG desigii and geometry should be determined via finite element simulations for
various geometries followed by rigorous experimental investigations to mitigate the
metrologcal difficulty.
In conclusion, two DFGs in a null-experiment setup in conjunction with a
suspended mass interferometric detector promises studies of deviations from Newton's
1/r2 law in the meter scale. Simulation studies on composition independent Yukawa-
type violation measurements indicate a realistic opportunity to explore a below the
current limit in the A ~ 0.5 — 10 meter range. Further investigation of DFG geome-
2384
'o "4'
8»
s
«
«
6
*
f
f
Titanium Disc • , I
I
I »
Ttmgsten RIHngs. sur mt
0
if
*m*
1 He „
M /, iij"
!.■' !).«;;
,i' .,.
r' M
rfM
I
r.i • i. • to- ')
1 •. 2 10"'')
2". - •_• I0"»>
0.25 ± (2 x 10 "6)
r/2i(6x 10" TJ
It
16±(10-8)
15.25 ±( 10-r)
2.5 ± (10-°)
11.25 ±(10- 7)
t/2±(5x 10-")
111
ao___tr.
0.75 ± (4 X 10 "»)
2.5:1- (10"'1
(1.25 ± (4 X 10 -8)
rr/2±(Wv)
-1
to§10W [m]
Fig. 1. Bounds on the limits on Yukawa parameter |ev|. In practice the limits shall be at or
above the respective curves due to imperfections. Current limits4 (gray area) and a limit may
be achievable with aii ongoing experiment5 (thin grey line)) arc also shown. I,II and III refers
to hypothetical detectors, with sensitivities at 2/q close that of LIGO. VIRGO and AdLlGO
respectively. The inset shows the null-experiment geometry of two DFGs for the measurement.
The table shows the optimized DFG parameter values and their uncertainties for each case.
try and interferometer technology aw well as second order effects, error propagation
and safety considerations are necessary and are already on their way.
The authors are grateful for the support of the United States National Science
Foundation (PHY-04-57528) and Columbia University. We are indebted to many
of our colleagues, in particular to G.Giordano, R.Adhikari, V.Sandberg, M.Landry,
P.Sutton, P.Shawlian, D. Sigg, R.DeSalvo, H.Yamamoto, Y. Aso.
References
1. http://www.ligo.caltcch.edu/advLTGO.
2. H. Hirakawa, K. Taubono, and K. Okie. Nature, 283:184. 1980; K. Kuroda and H. Hi-
rakawa. Phys. Rev. D., 32:342, July 1985;
3. F. Acernese et al. Glass. Quantum Grav., 23:S63, 2006.
4. R. G. Adelberger, B. R. Heckel, A. E. Nelson. Ann. Rev. Nncl. Part. Set., 53:77, 2003.
5. P. E. Boyoton et al. gr-qc/0609095, 2006.
6. P. Astone et al. Z. Phys. C, 50:21, 1991 and Eur. Phys. J. C, 5:651, 1998.
7. J. K. Hoskins, R. D. Newman, R. Spero, and J. Schultz. Phys. Rev. I)., 32:3084, 1985.
8. L. Matone et al. Class. Quantum Grav., accepted for publication, gr-qc/070HBJt.
9. M. V. Moody and H. J. Paik. Phys. Rev. Lett., 70:1195, 1993.
10. P. Ratfai, L. Matone, S. Marka, I. Bartos, Z. Marka. Phys. Rev. D., to be submitted.
J I. D. Sigg and the LIGO Science Collaboration. Class. Quantum Grav., 23:51, 2006.
J 2. J. Sinsky and J. Weber. Phys. Rev. Lett, 18:795, 1967 and Phys. Rev., 167:1145, L968.
ASTROPHYSICAL SOURCES OF GRAVITATIONAL WAVES
V. M. LIPUNOV
Sternberg Astronomical Institute,
Moscow, 119992, Russia
lipunov@xray.sai.msu.ru
The most realistic sources for LIGO-type detectors are discussed.
Keywords: Gravitational waves; Evolution of stars; Neutron stars; Close binaries.
1. Introduction
Relativistic stars (neutron stars and black holes) merging can be discussed like
"astrophysics" reaction of "elementary particle" interaction. This merging is analogous
to elementary physics processes in the world of elementary particles.1 There is no
doubt, that there are the following processes in the Universe:
NS + NS =^ NS + GWB;
NS + NS =^ BH + GWB;
The result depends on the mass of neutron star and Oppenheimer-Volkoff limit.
NS + BH =^ BH + GWB;
BH + BH =^ BH + GWB;
where GWB is the Gravitational Wave Burst.
The "cross-section" or probability calculation of these processes in the
Universe is of principal importance not only for astrophysics, but, first, for
fundamental physics, so as exactly these processes are accompanied by the most powerful
gravitational-wave emission. This emission has an impulse character, which can be
detected by gravitational wave antenna like LIGO.
The powerful gravitation wave emission in these processes mount to the
maximum possible value in nature (even if we take into account the future theory of
quantum gravitation1):
Lmax = M2JRgc = Epl/tpl = c5/G « 4 ■ 1059erg/s
(The detection of such processes possibility has been done by only 2
ways last 20 years. — Eto predlozhenije nikto ne pojmet. Ya tozhe.) First one is
to use our understanding of binary stars evolution processes and to use observed
astronomical data in all wave lengths. Second way is based upon radio astronomical
data about radio pulsars. Let's consider them.
2. Two methods of merging rate estimations
Both methods are based upon the observed data of our Galaxy with the following
generalization to the whole Universe.
But historically first one is called by "theoretical", and the second one is called
by "observed". It's not right as a matter of fact, but let's use this terminology.
The possibility of the processes or the cross-section can be characterized in
2385
2386
the terms of "merging rate", normalized to the galaxy like our one. Practically,
normalization on 1011 Mq of luminous barion matter is suggested.
"Theoretical" estimation is always attached to the following chain:
• - Merging Rate is equal to the Star Formation Rate in the Galaxy (Salpeter
Function);
• - the part of binary stars, that can form the relativistic star (the distribution
function by the relation of masses of binary components);
• - the part of the stars, which survive after the first supernova explosion (it
strongly depends on the anisotropy of the collapse or so called kick velocity);
• - the part of neutron stars after the second explosion and
• - the part of double relativistic stars, which can merge in Hubble time.
The most weak link in this chain is our lack of knowledge of possibility of collapse
anisotropy. But the "theoretical" method, that was realized in the most completely
realization (see monograph "Scenario Machine",2 and3) suggests the obligatory
calibration of unknown parameters by the observed data in all wavelength (from radio
to X-rays). So, if the mean output velocity is too large, all massive X-ray stars like
X-ray pulsars must be disappear from the sky (Gen X-3, Vela X-l, etc., the total
number is about 50), so as in case of large anisotropy of the collapse the binary
stars will be too quickly broken. On the contrary there will be too much of such
systems at small anisotropy, and there will be contradiction with observed number
of binary radio-pulsars. The first method4 gave the estimation 10~4 in the galaxy
like our own (see Fig. 1).
Fig. 1. Gravitational Wave Spectra from astrophysical sources.4 NS merging rate (year~1) for
distances less than 20 Mpc (line e). It corresponds to Merging Rate in 1/104 years per 1011 solar
Mass.
Second "observational" method was first used by Phinney.5 It was based on
observed parameters of binary radio pulsars, which can merge in Hubble time. In
2387
1991 there was only one such pulsar, and the estimation was 10 6year J in our
Galaxy (10nMO).
This wrong (in my opinion) estimation served to begin the building the gravity
interferometer LIGO.
The main problem of observational method is not in the fact, that there was
used only one observed example for statistical estimation, and is not in the fact,
that interpretation of observations always was difficult from the selection effects
(uncertainty of the distance, collimation angle, life-time, horizon of sensitivity that
is much more smaller, than Galaxy; we see less than 1% of all radio-pulsars). The
main problem is in the interest to the process of neutron stars merging, no radio
pulsars. Simple analysis shows, that neutron star passes not less than 6 physically
different states during evolution of its rotation. The phenomena of radio pulsar is
very specific among these states, and the neutron star can be invisible in radio
waves.6
There are the change of the probability estimation of the process of neutron stars
merging during the last 25 years (see in the Tables 1 and in the Fig. 2). One can
compare them. I assert, that most adequate to modern standard of interpretation
of binary and relativistic stars evolution "theoretical" estimation didn't change
during the last 17 years and beginning from the 1987 year always gave the value
10_4±0'5/year in the levels of reduced precision.
I
I
■u
|
«5
.
1E-4-
:
-
1E-5-
1E-6-
Clark et al
i <
LPP
■
Mils et al
■ ■ ■
LPP Tutukov,
Yungelson
1
* {
Phirtney * * Natayart et
1 ' I r— 1
LPP
Bethe, Brown
■ *
Q
Burgs
Portegtes et ai.
Portsgies et al.
+ Bailes
Van ttert Hsuvel, Lorimer
Jurran. Lorimer
al
, ( _, 1
1975
1980
1985
1990
1995
2000
2005
Fig. 2. Merging Rate estimation by different authors. Squares are the "theoretical" method,
"stars" are the observational one. if bh is the part of pre-supernova mass which collapsed into the
Black Hole.
This estimation corresponds to one merge per minute in whole Universe and to
2388
Table 1. "Theoretical" estimations of Neutron Stars Merging normalized to 1O11M0 (left) and
"observational" estimations of Neutron Stars Merging Rate (right).
Reference
10
4
12
3
13
19
2
20
8
Type
1/104 - 1/106
1/104
1/104
1/104
< 3/104
3/105
3/104 - 3/105
1/104 - 3/105
1/104
Reference
5
18
11
21
7
9
Type
1/106
1/106
3/106
8/106
< 1/105
1/104
1 event per year at the gravitational wave detector with 10~21 sensitivity.
More difficult problem is to estimate the frequency of the reaction with black
hole participation. Our understanding of the evolution is essentially worse here.
Nevertheless,15,16 could get round the theoretical uncertainty, using simple observed
limits. They are the following: there is no any radio pulsar with black hole on the
sky (this is upper limit) and there is at least several black holes in the binary with
massive optical stars (for example, Cyg X-l) in the Galaxy. As it was shown in,15,16
it is more possible to register the gravitational wave impulse from the black holes
merging:
BH + NS => BH + GWB
BH + BH => BH + GWB
and the frequency is 10_5/year/galaxy.
So as the mean black hole mass can be in 8-10 times more than the mass of
the neutron star, the frequency of these processes at the detector can be essentially
more, than from the neutron stars merging (see Fig.3).
Recently,17 proved that preliminary possibility of last two processes can be
increased up to 5-7 times.
3. Conclusions
(1) So called "theoretical" estimations give us the merging rate io_4±0-5 from.4 One
must accentuate, that the most full and correct model of binary stars evolution
is the "Scenario Machine", that takes into account the evolution of magnetized
neutron stars (see for details2).
(2) The "observed" estimations, which use radio-pulsars data, were always
burdened by selection effects.
(3) The gravitation impulses from the merging with black holes participation must
be the first events on the interferometers like LIGO.15'16
2389
,0M,r.S/N=l
:f=100Hz
'2 l't
Tntxl
\ 'NS+NS
0.1
0.0 0.2 0,4 O.B 0.8 1,0
kb)1
Fig. 3. Predicted Detector rates for Neutron Stars (horizontal branch) and Neutron Stars
- Black Holes and BH + BH - dark area.15
10 -
■t
t
10 ,
r
I
1 bM bh
bh+ns
re»+ns
BH+BH
BH+NS
NS+NS
wl
10
10°
10"
10"
10"
-M- ^ i
6.2 ' ' 0 4
1 ft
'1
0.6
08
Fig. 4. ZDES' NADO VSTAVIT' ZAGOLOVOK (oeobyaztel'no zhiriiij). I dobavit' paru
predlozhenij v tekst.
2390
References
1. V. Lipunov, 1993, in Volcano workshop 1992 Conference Proceedings 40, 499 (ed. F.
Giovannelli h G. Mannocchi, Bologna, 1993)
2. V. Lipunov, K. Postnov, M. Prokhorov, Atrophysics and Space Physics Reviews 9,
part 4, 1 (1996)
3. A. Tutukov, L. Yungelson, Astronomy Reports 37, 411 (1993)
4. V. Lipunov, K. Postnov, M. Prokhorov, AhA 176, LI (1987)
5. E. Phinney, ApJ 380, L17 (1991)
6. V. Lipunov, Astrophysics of Neutron Stars (Springer Verlag, 1992).
7. M. Bailes, in Compact Stars in Binaries: Proc. of the 165th Symp. of Inst. Astron.
Union, The Netherlands, 1994 213 (ed. J. Van Paradijs, E. P. J. Van den Huevel, E.
Kuulkers, Dordrecht: Kluwer Acad. Publ., 1996)
8. H. Bethe, G. Brown, ApJ 517, 318 (1999)
9. M. Burgay, N. D'Amico, A. Posseti, R. Manchester, A. Lyne, B. Joshi, M.
McLaughlin, M. Kramer, J. Sarkisian, F. Camilo, V. Kalogera, C. Kim, D. Lorimer, astro-
ph/0312071 (2003)
10. J. Clark, E. P. J. van den Huevel, W. Sutantyo, Ah A 72, 120 (1979)
11. S. Curran, D. Lorimer, MNRAS 276, 347 (1995)
12. D. Hills, P. Bender, R. Webbink, ApJ 360, 75 (1990)
13. V. Lipunov, K. Postnov, M. Prokhorov, AhA 298, 677 (1995)
14. V. Lipunov, K. Postnov, M. Prokhorov, MNRAS 288, 245 (1997a)
15. V. Lipunov, K. Postnov, M. Prokhorov, Astromy Letters 23, 492 (1997b)
16. V. Lipunov, K. Postnov, M. Prokhorov, New Astronomy 2, 43 (1997c)
17. V. Lipunov, E. Panchenko, AFP Conference Proceedings 686 (ed. J. M. Centrella,
2003)
18. R. Narayan, T. Piran, A. Shemi, ApJ 379, L17 (1991)
19. S. Portegies Zwart, R. Spreeuw, AhA 312, 670 (1996)
20. S. Portegies Zwart, L. Yungelson, AhA 332, 173 (1998)
21. E. P. J. van den Heuvel, D. Lorimer, MNRAS 283, L37 (1996)
Space and Third Generation
GW Detectors
This page intentionally left blank
DECIGO: THE JAPANESE SPACE GRAVITATIONAL WAVE
ANTENNA
MASAKI ANDO1'*, SEIJI KAWAMURA2, TAKASHI NAKAMURA3, NAOKI SETO4, KIMIO
TSUBONO1, KENJI NUMATA5, RYUICHI TAKAHASHI6, MITSURU MUSHA7, KEN-ICHI
UEDA7, IKKOH FUNAKI8, SHIGENORI MORIWAKI9, TAKESHI TAKASHIMA8,
SHIN-ICHIRO SAKAI8, TAKASHI SATO10, NOBUYUKI KANDA", SHIGEO NAGANO12,
MIZUHIKO HOSOKAWA12, TAKEHIKO ISHIKAWA13, SHUICHI SATO2, YOICHI ASO1,
MUTSUKO Y. MORIMOTO2, KAZUHIRO AGATSUMA14, TOMOMI AKUTSU14,
TOMOTADA AKUTSU56, KOH-SUKE AOYANAGI15, KOJI ARAI2, YUTA ARASE14,
AKITO ARAYA16, HIDEKI ASADA17, TAKESHI CHIBA18, TOSHIKAZU EBISUZAKI19,
MOTOHIRO ENOKI20, YOSHIHARU ERIGUCHI21, FENG-LEI HONG30, MASA-KATSU
FUJIMOTO2, MITSUHIRO FUKUSHIMA22, TOSHIFUMI FUTAMASE23, KATSUHIKO
GANZU3, TOMOHIRO HARADA24, TATSUAKI HASHIMOTO8, KAZUHIRO HAYAMA25,
WATARU HIKIDA26, YOSHIAKI HIMEMOTO27, HISASHI HIRABAYASHI8, TAKASHI
HIRAMATSU27, HIDEYUKI HORISAWA28, KIYOTOMO ICHIKI6, TAKESHI IKEGAMI30,
KAIKI T. INOUE31, KUNIHITO IOKA3, KOJI ISHIDOSHIRO1, HIROYUKI ITO12,
YOUSUKE ITOH32, SHOGO KAMAGASAKO14, NOBUKI KAWASHIMA31, FUMIKO
KAWAZOE33, HIROYUKI KIRIHARA14, NAOKO KISHIMOTO8, KENTA KIUCHI15,
WERNER KLAUS12, SHIHO KOBAYASHI34, KAZUNORI KOHRI35, HIROYUKI
KOIZUMI36, YASUFUMI KOJIMA37, KEIKO KOKEYAMA33, WATARU KOKUYAMA1, KEI
KOTAKE15, YOSHIHIDE KOZAI38, HIDEAKI KUDOH27, HIROO KUNIMORI12, HITOSHI
KUNINAKA8, KAZUAKI KURODA14, KEI-ICHI MAEDA15, HIDEO MATSUHARA13,
YASUSHI MINO39, JUN-ICHI MIURA7, OSAMU MIYAKAWA40, SHINJI MIYOKI14,
TOMOKO MORIOKA33, TOSHIYUKI MORISAWA26, SHINJI MUKOHYAMA27, ISAO
NAITO41, NORIYASU NAKAGAWA14, KOUJI NAKAMURA6, HIROYUKI NAKANO1,
KENICHI NAKAO", SHINICHI NAKASUKA36, YOSHINORI NAKAYAMA42, ERINA
NISHIDA33, KAZUTAKA NISHIYAMA8, ATSUSHI NISHIZAWA43, YOSHITO NIWA43,
MASATAKE OHASHI14, NAOKO OHISHI44, MASASHI OHKAWA45, AKIRA OKUTOMI14,
KOUJI ONOZATO1, KENICHI OOHARA45, NORICHIKA SAGO46, MOTOYUKI SAIJO47,
MASAAKI SAKAGAMI43, SHIHORI SAKATA33, MISAO SASAKI26, MASARU SHIBATA21,
HISAAKI SHINKAI48, KENTARO SOMIYA49, HAJIME SOTANI50, NAOSHI SUGIYAMA6,
HIDEYUKI TAGOSHI46, TADAYUKI TAKAHASHI8, RYUTARO TAKAHASHI2, KAKERU
TAKAHASHI1, HIROTAKA TAKAHASHI49, TAKAMORI AKITERU16, TADASHI
TAKANO8, TAKAHIRO TANAKA3, KEISUKE TANIGUCHI51, ATSUSHI TARUYA27,
HIROYUKI TASHIRO3, MITSURU TOKUDA11, MASAO TOKUNARI14, MORIO
TOYOSHIMA12, SHINJI TSUJIKAWA38, YOSHIKI TSUNESADA52, MASAYOSHI
UTASHIMA8, HIROSHI YAMAKAWA53, KAZUHIRO YAMAMOTO14, TOSHITAKA
YAMAZAKI2, JUN'ICHI YOKOYAMA29, CHUL-MOON YOO11, SHIJUN YOSHIDA54,
TAIZOH YOSHINO55
1 Department of Physics, The University of Tokyo, Bunkyo, Tokyo 113-0033, Japan,
* E-mail: ando@granite.phys-s.u-tokyo.ac.jp
2 TAMA Project, National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan,
3 Department of Physics, Kyoto University, Kyoto 606-8502, Japan,
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2394
4Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575,
U.S.A.,
5 NASA Goddard Space Flight Center, Code 663, 8800 Greenbelt Rd., Greenbelt, MD20771,
U.S.A.,
6Division of Theoretical Astronomy, National Astronomical Observatory of Japan, Mitaka,
Tokyo 181-8588, Japan,
7Institute for Laser Science, The University of Electro-Communications, Chofu, Tokyo
182-8585, Japan,
8Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency,
Sagamihara, Kanagawa 229-8510, Japan,
9 Department of Advanced Materials Science, The University of Tokyo, 5-1-5 Kashiwanoha,
Kashiwa, Chiba 277-8561, Japan,
10 Department of Electrical and Electronic Engineering, Faculty of Engineering, Niigata
University, Niigata, Niigata 950-2181, Japan,
11 Department of Physics, Osaka City University, Osaka, Osaka 558-8585, Japan,
12 National Institute of Information and Communications Technology (NICT), Koganei, Tokyo
184-8795, Japan,
13Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Tsukuba,
Ibaraki, 305-8505, Japan,
14 Institute for Cosmic Ray Research, The University of Tokyo, Kashiwa, Chiba 277-8582,
Japan,
15 Department of Physics, Science and Engineering, Waseda University, Shinjuku, Tokyo,
169-8555, Japan,
16Earthquake Research Institute, The University of Tokyo, Bunkyo, Tokyo 113-0032, Japan,
17Department of Earth and Environmental Sciences, Hirosaki University, Hirosaki, Aomori
036-8560, Japan,
18 Department of Physics, College of Humanities and Sciences, Nihon University, Setagaya,
Tokyo 156-8550, Japan,
l9RIKEN, 2-1 Hirosawa Wako 351-0198, Japan,
20 Astronomical Data Center, National Astronomical Observatory of Japan 2-21-1, Osawa,
Mitaka, Tokyo 181-8588, Japan,
21 Department of Earth Science and Astronomy, The University of Tokyo, Komaba, Meguro,
Tokyo 153-8902, Japan,
2 Advanced Technology Center, National Astronomical Observatory of Japan, Mitaka, Tokyo
181-8588, Japan,
23 Astronomical Institute, Tohoku University, Sendai 980-8578, Japan,
24Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan,
25 SO Fort Brown, Brownsville 78520, Texas, U.S.A.,
26 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Kyoto 606-8502, Japan,
27 Theoretical Astrophysics Group, Department of Physics, The University of Tokyo, Bunkyo-ku,
113-0033, Japan,
28 Department of Aeronautics and Astronautics, School of Engineering, Tokai University,
29 Research Center for the Early Universe, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku,
Tokyo, 113-0033, Japan,
30 National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaragi
305-8563, Japan,
31 Kinki University School of Science and Engineering, Higashi-Osaka, Osaka 577-8502, Japan,
32Physics Department, University of Wisconsin - Milwaukee, P.O. Box 413, 2200 E. Kenwood
Blvd., Milwaukee, WI 53201-0413, U.S.A.,
33 Ochanomizu University Graduate School of Humanities and Sciences, Bunkyo, Tokyo,
112-8610 Japan,
34 Astrophysics Research Institute, Liverpool John Moores University, Twelve Quays House,
Egerton Wharf, Birkenhead L41 1LD, UK,
35Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, U.S.A.,
36 Department of Aeronautics and Astronautics, University of Tokyo, Hongo 7-3-1, Bunkyo-ku,
2395
Tokyo 113-8656, Japan,
37Hiroshima University, Graduate School of Science, Higashi-hiroshima, Hiroshima 739-8526,
Japan,
38 Gunma Astronomical Observatory, Agatsuma-gun, Gunma 377-0702, Japan,
39Theor. Astrophysics, California Institute of Technology, Pasadena, CA 91125, U.S.A.,
40 LIGO Laboratory, California Institute of Technology, M/C 18-34, Pasadena, CA 91125,
U.S.A.,
41 Numakage, Saitama-shi, Saitama 336-0027 Japan,
42 Department of Aerospace Engineering, National Defense Academy, 1-10-20, Hashirimizu,
Yokosuka 239-8686, Japan,
43Faculty of Intergrated Human Studies, Kyoto University, Kyoto 606-8501, Japan,
44MIRA Project,, National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan,
45 Department of Biocybernetics, Faculty of Engineering, Niigata University, Niigata, Niigata
950-2181, Japan,
46Department of Earth and Space Science, Osaka University, Toyonaka, Osaka 560-0043, Japan,
47Highfield, Southampton S017 1BJ, United Kingdom,
48Department of Information Science, Osaka Institute of Technology, Kitayama 1-79-1,
Hirakata, Osaka 573-0196, Japan,
49 Max-Planck-Institut fur Gravitationsphysik, Albert-Einstein-Institut, Am Muhlenberg 1,
D-14476 Golm bei Potsdam, Germany,
50 Department of Physics, Section Astrophysics, Astronomy and Mechanics, Aristotle University
of Thessaloniki, Thessaloniki 54124, GREECE,
51 Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street,
Loomis Laboratory of Physics, Urbana, IL 61801, U.S.A.,
52 Graduate School of Science and Engineering / Physics, Tokyo Institute of Technology,
Ookayama, Meguro-ku, Tokyo, 152-8550, Japan,
53 Research Institute for Sustainable Humanosphere, Kyoto University, Gokasho, Uji, Kyoto
611-0011, Japan,
54 Pure and Applied Physics, Science and Engineering, Waseda University, Shinjuku, Tokyo
169-8555, Japan,
55Nakamura-minami, Nerima, Tokyo 176-0025, Japan,
56Department of Astronomy, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo
113-0033, Japan
DECIGO (DECI-hertz interferometer Gravitational wave Observatory) is the future
Japanese space gravitational wave antenna with observation band around 0.1 Hz. It aims
at detecting gravitational waves from various kinds of sources, with sufficient sensitivity
to establish the gravitational wave astronomy. In the pre-conceptual design, DECIGO
is formed by three drag-free spacecraft, 1000 km apart from one another. The relative
displacements between proof masses housed in these spacecraft are measured by Fabry-
Perot interferometers. We plan to launch DECIGO in 2024 after research and
development phase, including two milestone missions (DECIGO pathfinder and Pre-DECIGO)
for verification of required technologies.
Keywords: DECIGO, Gravitational waves, Astronomy, Space Mission
1. DECIGO
DECIGO (DECI-hertz interferometer Gravitational wave Observatory) is the future
Japanese space gravitational wave (GW) antenna,1 with observation frequency band
of around 0.1 Hz (Fig. 1). This frequency band is the gap region between LISA (Laser
Interferometer Space Antenna)2 and terrestrial detectors such as Advanced LIGO3
and LCGT (Large-scale Cryogenic Gravitational-wave Telescope).4 In addition, this
2396
Brag-Frea
Spacecraft 3
^
10"
10"
do, )«<?:'
/ •'•!C1
I 10-
J vr»\
10'
V
LZ.W*
6iStt**y
10 lO-^ 10" 10'
Frequency [Hz]
10'
Fig. 1. Pre-conceptual design of DECIGO (left) and its design sensitivity (right).
band opens the possibility to observe GWs from cosmological distance, because it is
free from the confusion noises, irresolvable GW signals, from too many white dwarf
binaries in our Galaxy.
Main targets of DECIGO are GWs from binary inspirals of compact binaries,
and from the early universe. DECIGO will have sufficient sensitivity to observe GWs
from distant (redshift of z ~ 1) neutron-star binaries which are a few months to 5
years before merger. By resolving GW signals emitted from many (about 3 x 105)
binaries in this range, we will obtain information of mass distribution of neutron-
stars, and thus, on the theory of the evolution of massive stars and on the equation
of state of high-density matters. Moreover, observing distant binaries, which play
as precise clocks, it will be possible to measure the acceleration of the expansion of
the universe from their redshift change.1 As for black-hole binaries, DECIGO will
observe GWs from coalescences of intermediate-mass (1O3M0) black hole binaries,
which could reveal the mechanism of the formation of super-massive black holes
in the center of galaxies. The extremely good sensitivity of DECIGO would enable
us to detect GWs from the very early universe, which could provide important
information to understand the beginning of the universe.
2. Pre-conceptual design of DECIGO
In the pre-conceptual design, DECIGO is formed by three drag-free spacecraft,
1000 km apart from one another. Relative displacements of the proof masses
(mirrors) inside the spacecraft are measured by Fabry-Perot interferometers (See Fig. 1).
We adopted the Fabry-Perot configuration because it provides a better best
sensitivity at 0.1 Hz band than an optical transponder configuration which is adopted
by LISA. Although the Fabry-Perot configuration with shorter arm length has the
larger acceleration noises by laser radiation-pressure noise and practical force
fluctuations than transponder configuration with long arm length does, these noises
would be still slightly lower than the confusion noise by Galactic binaries.
The distance between spacecraft (Fabry-Perot cavity arm length) was chosen
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to be 1000 km. This arm length was chosen so as to be short enough to avoid
refraction losses of laser power, and to form Fabry-Perot cavities, and yet so as to
be long enough to ensure the high sensitivity for GW signals. The mirrors forming
the cavities, which works as proof masses in spacecraft, have a diameter of 1 m, with
moderate reflectivity to realize the cavity finesse of 10. The mass of mirror (about
100 kg) was simply chosen to be the largest we could fabricate and handle. The
laser source of DECIGO will have an effective power of 10 W with a wavelength of
532 nm. The orbit and constellation of DECIGO is to be determined, considering
the gravity disturbances by the sun and planets, durability of the thruster fuels,
solar power supply, and the required angle resolution for the GW source, and so on.
3. Milestone missions for DECIGO
Long and intensive development phase will be required in order to realize DECIGO.
We plan to launch DECIGO in 2024 after design (a pre-conceptual design, a
conceptual design, a preliminary design, and finally a final design) and prototype tests
with the help of research and development with table top experiments. We also
have two milestone missions, DECIGO pathfinder (DPF) and Pre-DECIGO, before
the launch of DECIGO. DPF will be one small satellite consists of two proof mass
mirrors, which form a short Fabry-Perot cavity. The cavity length is measured by
a stabilized laser source, and the mirrors are kept in the satellite with a drag-free
control. The target of DPF will the technical demonstrations: a drag free control,
laser stabilization in space, precise measurement with Fabry-Perot cavity, and
mirror clump system used at the launch of the satellite. In addition, since DPF will have
a modest sensitivity for GW events, we expect some scientific results with
continuous observation at the DECIGO frequency band. The objectives and a conceptual
design of Pre-DECIGO will be determined during the research and development
phase of DECIGO.
4. Conclusions
We have started a serious investigation to realize DECIGO by determining the pre-
conceptual design. Although hard efforts will be required before its launch, DECIGO
will provide fruitful scientific results by opening a new astronomy with gravitational
waves.
References
1. N. Seto, S. Kawamura, and T. Nakamura, Phys. Rev. Lett, 87 (2001) 221103,
S. Kawamura et al., Class. Quantum Grav. 23 (2006) S125.
2. LISA: System and Technology Study Report, ESA document ESA-SCI (2000).
3. "LIGO II Conceptual Project BooK\ LIGO M990288-A-M (1999).
4. K. Kuroda, et al, Class. Quantum Grav. 19 (2002) 1237.
DESIGN AND CONSTRUCTION OF THE LISA TECHNOLOGY
PACKAGE OPTICAL BENCH INTERFEROMETER
CHRISTIAN J KILLOW, JOHANNA BOGENSTAHL, MICHAEL PERREUR-LLOYD,
DAVID I ROBERTSON and HENRY WARD
Institute for Gravitational Research,
Department of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ
c.killow@physics.gla. ac. uk
FELIPE GUZMAN CERVANTES and FRANK STEIER
Max Planck Institute for Gravitational Physics (Albert Einstein Institute),
Callinstrafie 38, D-30167 Hannover, Germany
The LISA Technology Package (LTP) is an experiment that will fly on board the space
based gravitational wave demonstrator mission, LISA Pathfinder. The LTP optical bench
interferometer will be used to monitor the changes in separation between two test masses
with a sensitivity of lOpm/v'Hz in the measurement band of 3mHz to 30mHz. The
precision alignment processes required to manufacture this ultra-stable, space-worthy
optical bench are described and the design and construction status presented.
1. Introduction
The Laser Interferometer Space Antenna (LISA) is a planned spaceborne
gravitational wave detector.1 The mission is being undertaken jointly by the European
Space Agency and NASA. In order to demonstrate some of the required technologies
for LISA that cannot be adequately tested in the lg earth environment a
demonstrator mission called LISA Pathfinder (LPF) is being constructed. LISA Pathfinder will
house an experiment called the LISA Technology Package (LTP).2 In this package
the relative motion of two inertial test masses will be interferometrically monitored
to reveal the level of residual differential acceleration noise to within an order of
magnitude of the level required for LISA. This requires positioning monitoring at
a sensitivity of ~ 10pm/vTIz. The acceleration noise sources and couplings will be
characterised to give confidence that the LISA goals can be met.
Central to LTP is the Optical Bench Interferometer (OBI). The approach is
to use Mach-Zehnder interferometers with beams separated in frequency by ~kHz.
Separate interferometers monitor the spacing between the two test masses and the
distance between one test mass and the interferometer structure. Readout of each
interferometer is by comparing the phase of the output signal with that, of a reference
interference point on the optical bench.
The optical bench itself is a 212 x 200 x 45 mm block of Zerodur© with fused
silica mirrors and beamsplitters of dimensions ~ 20 x 15 x 7mm jointed to it to
form the multiple interferometers. The components are attached using a specifically
developed technique that utilises a process called hydroxide-catalysis bonding. This
has many advantages3,4 over gluing and optical contacting that are particularly
suited to this application. A CAD model of the flight model optical bench currently
under construction in Glasgow is shown in Figure 1.
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Fig. 1. CAD model of the flight model Optical Bench Interferometer. The light is introduced
onto the bench using two fibre injectors designed specifically for this mission (top left). Twenty-two
mirror and beamsplitter components form three Mach-Zehnder interferometers to monitor the test
mass positions and angles and laser frequency noise. Eight photodiodes readout the interferometers,
and there are two further power monitor photodiodes. All photodiodes can be seem towards the
periphery of the Zerodur® baseplate.
2. Precision positioning and measurement
The mirror and beamsplitter components have to be positioned with resolution of
order a micron in order to obtain sufficient, contrast at the interferometer outputs
and to hit the nominal test mass reflection points to within the required ±25/im*.
This raises many practical issues, not least the need to align the components in
a maimer compatible with the bonding technique. During the construction of a
prototype optical bench5 and the LTP OBI engineering model6 a 'floating alignment'
method was used in which the component to be bonded was initially floated on a
layer of buffer fluid while its angular alignment was optimised. Stops were then used
to define the required position during the bonding process. For the flight model
construction the tighter alignment tolerances have led to the development of ail
additional step known as 'hovering alignment'.
Iii order to gain knowledge of the physical component positions, a coordinate
measuring machine with micron level measurement precision is used. This will be
used to measure the position of pre-calibrated quadrant photodiodes (CQPDs) to
provide a target for alignment. This idea is a new development for the flight model
OBI. In some instances the CQPDs must also be positioned to a few microns in
as many as five degrees of freedom. To cater for this need two six-axis parallel
kinematics translation stages are used (Physik Instrument*'s Hexapod model M-
8247), having a repeatability of ±0.5 /rm. These are also used during the hovering
alignment stage.
"This is the OBI apportionment of the total budget of ±50 fan.
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Fig. 2. CAD drawings of a hydroxide-catalysis bonded Fibre Injector Optical Subassembly. The
long side of the rectangular baseplate has length ~ 3 cm. The fibre strain relief can be seen to
the left of the pictures. The fibre is glued into a custom drawn capillary tube and then into the
cylinder, which lias a hole 1 mm in diameter. The light passes from the fibre, through a precision
fused silica spacer and a spherical lens. The beam then leaves silica and travels in free space to
the second, aspheric, lens which collimates the beam. A polariser then completes the FIOS.
3. Fibre Injector Optical Subassemblies
The practicalities of coupling light onto the bench in a stable, non-magnetic and
robust way are considerable, especially when coupled with the demanding beam
quality required. The approach taken has been to construct quasi-monolithic
subassemblies, the Fibre Injector Optical Subassemblies (FIOS), taking the single mode
polarisation maintaining fibre and constructing a precision aligned, hydroxide-catalysis
bonded structure. CAD drawings of the FIOS can be seen in Figure 2.
Acknowledgments
We wish to acknowledge the support of PPARC and the University of Glasgow.
F. Steier and F. Guzman Cervantes wish to thank the European Graduate College.
References
1. Laser Interferometer Space Antenna: A Cornerstone Mission for the Observation of
Gravitational Waves. System and Technology Report,. (ESA SCI 11, 2000).
2. S. Anza and the LTP Team, Class. Quantum Grav. 22, S125-S138 (2005).
3. E. J. Ellife, J. Bogenstahl, A. Deshpande, J. Hough, C. Kiilow, S. Reid, D. Robertson,
S. Rowan, H. Ward and G. Cagnoli, Class. Quantum Grav. 22, S257-S267 (2005).
4. S. Reid, G. Cagnoli, E. Elliffe, J. Faller, J. Hough, I.Martin and S. Rowan Submitted
to Physics Letters A, doi:10.1()16/j.physlet,a.2006.11.068 (2006).
5. D. Robertson, C. Kiilow, H. Ward, J. Hough, G. Heiuzel, A. Garcia, V. Wand, U.
Johann and C. Braxmaier, Class. Quantum Grav. 22, S155-S163 (2005).
6. G. Heinzel, C. Braxmaier, M. Caldwell, K. Danzmann, F. Draaisma, A. Garcia, J.
Hough, O. Jennrich, U. Johann, C. Kiilow, K. Middleton, M. te Plate, D. Robertson,
A. Riidiger, R. Schilling, F. Steier, V. Wand and H. Ward, Class. Quantum Grav. 22,
S149-S154 (2005).
7. Physik lnstrumente GmbH, http://www.physikinstrumente.com/
COMPACT BINARY INSPIRAL AND THE SCIENCE POTENTIAL
OP THIRD-GENERATION GROUND-BASED GRAVITATIONAL
WAVE DETECTORS*
CHRIS VAN DEN BROECK and ANAND S. SENGUPTA
School of Physics and Astronomy, Cardiff University,
Queen's Buildings, The Parade, Cardiff CF24 3AA, United Kingdom
Chris.van-den-Broeck@astro.cf.ac.uk, Anand.Sengupta@astro.cf.ac.uk
We consider EGO as a possible third-generation ground-based gravitational wave
detector and evaluate its capabilities for the detection and interpretation of compact binary
inspiral signals. We identify areas of astrophysics and cosmology where EGO would have
qualitative advantages, using Advanced LIGO as a benchmark for comparison.
Compact binary inspiral. Inspirals of compact binary objects (black holes and/or
neutron stars) are among the most promising sources for ground-based gravitational
wave detectors,1 and as such they are eminently suitable to evaluate the science
potential of future observatories. In the quasi-circular, adiabatic regime, where the
periods of orbits are much smaller than the inspiral timescale, gravitational
waveforms have been computed in the post-Newtonian (PN) approximation,2 where the
signal is a superposition of harmonics in the orbital phase. Recently the full
waveforms, with inclusion of PN amplitude corrections, were used to accurately assess
the potential of Advanced LIGO and EGO in terms of redshift reach, detection
rates, and parameter estimation.3 Here we briefly discuss possible implications for
astrophysics and cosmology; for the theoretical underpinnings as well as complete
references we refer to these more technical papers.
EGO as a third-generation detector. EGO is not yet on the drawing boards;
rather, its strain sensitivity as plotted in Fig. 1 should be viewed as a summary of
what is believed to be possible with steadfast advances in interferometer technology
over the next decade or so. In most of the frequency interval shown, the difference in
sensitivity between EGO and Advanced LIGO is a factor of a few; at low frequencies,
which are of interest for compact binary inspiral, the difference is about an order
of magnitude.
Redshift and mass reach. The right hand panel of Fig. 1 shows how these
sensitivities translate into redshift reach as a function of total mass M for a fixed ratio
of the component masses mi, mi. The mass reach of Advanced LIGO is slightly
over 400 M0 while EGO can see systems that are three times heavier. It is useful
to make a distinction between stellar mass systems with M < 100 M0 and the
heavier intermediate mass binaries with M up to (a few) x 1000Mq. The latter
systems may form in the centers of galaxies and in globular clusters, and they are
expected to be rather asymmetric; hence our choice vaijra^ = 0.1. (Note that the
redshift reach would be larger in the equal mass case, and for a more convenient sky
"This research was supported in part by PPARC grant PP/B500731/1.
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10-22
CJ
110"23
1024
in25
SS::
hS^ :::
"
mm.
i |:
::;.:::. :i:|::
^
— Advanced LIGCJ
---EGO J
;:;::;:;::- rh^M^iAk
10
10
10'
f(Hz)
10'
... I
mm :
— Advanced LIGC
---EGO
^=---..: ::: =
=ilti|i= miimmmi
200
400 600
M/M„„
800 1000 1200
Fig. 1. Plots of the stain sensitivities of EGO and Advanced LIGO (left) and their redshift reach
for a fixed SNR of 10 (right). On the right hand side we have fixed m\/m2 =0.1 and angles
6 = 4> = tt/6, V = w/4, i = tt/3.
position and orientation.) We see that EGO would be able to detect stellar mass
inspirals through much of the visible Universe. Detection rates in EGO have been
conservatively estimated to be at least 700 times higher than in Advanced LIGO.3
Measuring component masses. How well can parameters be extracted from a
signal in EGO? The individual component masses m-i, m-2 enter the waveforms
through particular combinations, the chirp mass M. and the symmetric mass ratio
7]. As a consequence, the latter tend to be measurable with good accuracy, while
mi, m-2, which are of direct astrophysical interest, generally are not well-determined
in initial detectors. In the left panel of Fig. 2 we see that for a distance of 100 Mpc,
— Advanced LIGC
---EGO
200 300
M/M„„,
400
500
— Advanced LIGC
---EGO
200 300
M/M ,
sol
400
500
Fig. 2. Relative error on component mass (left) and error on the spin-orbit parameter (right) at
a distance of 100 Mpc, again setting mi/m,2 = 0.1.
in Advanced LIGO the relative error on component mass vax never goes below 5%,
while in EGO it is only a few percent in a very large mass range. EGO would enable
us to "map" the mass distribution of black holes. It would give us a direct view on
the way intermediate mass black holes grow through successive coalescences with
2403
smaller compact objects. Fig. 1 indicates that for stellar mass systems, parameter
estimation in EGO up to redshift z ~ 2 (corresponding to a luminosity distance ~
16 Gpc) would be as good as in Advanced LIGO up to only z ~ 0.2 (or ~ 1 Gpc).
With a network of detectors one could also measure distance. This opens up the
possibility of studying the population evolution of black holes (and indirectly of the
stars that produce them) over cosmological distances.
Restricting component spins. In the right panel of Fig. 2 we have plotted the
error on the parameter (3. which encodes the interaction between the components'
spins and orbital angular momentum; its precise definition can be found in Ref. 3.
To a first approximation one can neglect spin-induced precession of the orbital plane
and take (3 to be a constant. An important point is that if \(3\ > 113/12 then the
spin of at least one component of the binary violates the Kerr bound, indicating a
naked singularity, a boson star, or a still more exotic object. As seen in the right
panel of Fig. 2, EGO could measure j/3j to within 5% of its abovementioned bound
for masses up to almost 500 Mq, in stark contrast with Advanced LIGO. A more
in-depth analysis has appeared elsewhere.3
Other possible applications. The large redshift reach of EGO would make it an
ideal tool for cosmology; we confine ourselves to two more examples which were
already foreseen by Schutz4 in the context of LIGO and deserve to be revisited
with a view on third-generation detectors, (i) With multiple detectors one can
determine sky position and it becomes possible to identify the host galaxy (or cluster
of galaxies), which will have some redshift z. From the gravitational wave signal
the luminosity distance D can be extracted. In a flat Universe there is a definite
relationship D(z) which depends on the Hubble constant Ho as well as parameters
f2o and !"2a set by the mass density of the Universe and a possible cosmological
constant, respectively. Given a sufficient number of events at different distances one
could fit the function D(z), which would amount to measuring Hq, Qq, and Q\.
(ii) At the largest scales, galaxy clusters tend to be on the surfaces of "bubbles''
surrounding relative "voids". It is natural to ask whether black hole binaries are
similarly distributed, which may be relevant to dark matter studies.
References
1. L.P. Grishchuk, in Astrophysics Update, ed. J.W. Mason (Springer-Praxis, Berlin,
2004).
2. L. Blanchet, Liv. Rev. Rel. 5, 3 (2002).
3. C. Van Den Broeck and A.S. Sengupta, Class. Quantum Grav. 24, 155-176 (2007);
C. Van Den Broeck and A.S. Sengupta, gr-qc/0610126.
4. B.F. Schutz, Class. Quantum Grav. 6, 1761-1780 (1989).
DISCRETE SAMPLING VARIATION MEASUREMENT
TECHNIQUE FOR SUB-SQL SENSITIVITY DETECTION OF
GRAVITATIONAL WAVES*
S. L. DANILISHIN* and F. YA. KHALILI
Dept. of Physics of Oscillations, Faculty of Physics, Moscow State University,
Moscow 119992, GSP-2, Russia, Leninskie Gory, 1, bid. 2,
* stefan@hbar.phys.msu.ru
We propose a new method of discrete sampling variation measurement (DSVM) which
allow to overcome the quantum limitations of sensitivity imposed by uncertainty relation
and, therefore, increase the sensitivity of advanced gravitational wave detectors.
1. Introduction
Recent progress in experimental gravitational wave (GW) astronomy allows to hope
that first detection of GWs will take place in near future. Gravitational wave
observatories are built all over the world and have been already comissioned to operation
and started the scientific search (LIGO1 in USA, GEO 6002 in Germany, TAMA
3003 in Japan, VIRGO4 in Italy) for GWs. Sensitivity of operating observatories
has reached extra-galactic distances of ~ 10 Mpc.5 However, contemporary
theoretical predictions of astrophysicists concerning the rate of detectable events (see
the review6) imply that sensitivity of modern detectors should be increased
drastically. Therefore the new generation of detectors with sensitivity close or equal to
the ultimate standard quantum limit (SQL)7-9 is being developed and planned to
be built within the next decade (Advanced LIGO, LCGT, AIGO). Nethertheless,
in order to reach cosmological distances the sensitivity of GW antennae should be
better than SQL. It should be noted that SQL arises due to perturbation of
measured quantity by the meter during the process of measurement. This perturbation
is inevitable consequence of fundamental laws of quantum mechanics and arises
due to non-commutativity of measured observable with itself at different times9'10
which is a common issue for all kinds of displacement measurements. However,
this perturbation can be overcome if one choose to measure such obsevable that is
not influenced by this perturbation. This idea lies in the foundation of all SQL-free
methods of measurement. For example, in variation measurement11-14 it is proposed
to measure such quadrature of the optical field reflected from the probe body that
contains information about the optimal linear combination of body displacement
and momentum that have minimal uncertainty at the moment. This measurement
is equivalent to introduction of such cross-correlation between the displacement
measurement noise and back-action noise that cancels perturbation of measured
quantity by back-action and eliminates it from the output signal. The main idea
of variation measurement forms the basis of proposed method of discrete sampling
*This research has been partially supported by Russian President grant MK-6859.2006.2 and
Marcel Grossmann Foundation grant.
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variation measurement (DSVM) which will be described below. Those readers who
are interested in more details can find additional information in papers.15'16
2. Discrete sampling variation measurement
In GW interferometers laser light is used to detect small variations of interferometer
arms lengths caused by GWs. Quantum fluctuations of laser light phase determine
the measurement uncertainty called shot noise (SN). Fluctuating amplitude of laser
light causes stochastic radiation pressure force acting on the interferometer mirrors
and perturbs their positions. This noise is known as radiation pressure noise (RPN).
If we represent the light wave coming out the interferometer in terms of amlitude
and phase quadratures, then one can readily show that GW signal will be in the
second (phase) quadrature.17 But if one measures this quadrature of the output
light he will confront with SQL, because SN and RPN are not independent and
satisfy Heisenberg's relation. In order to overcome the SQL, measurement of mixed
quadrature chosen by adjusting local oscillator (LO) phase £ in homodyne readout
scheme was proposed (see Sec. II C in13) which is equivalent to introduction of
certain cross correlation between SN and RPN. But to achieve sub-SQL sensitivity
in wide frequency band it is necessary either to have frequency dependent LO phase
C(f2) which requires kilometer scale additional cavities, or allow £ to vary in time.
The last case can be implemented relatively easy, but optimal £(£) occurs to depend
on measured signal.18 We propose to overcome this difficulty using the following
measurement strategy which we call discrete sampling variation measurement:
(1) Upper frequency ilmax of expected signal should be known before the
measurement;
(2) All the measurement time is divided into short time intervals with duration
t ^ ir/£lmax. Obviously, optimal £(£) is a periodic function: £(£ + nr) = £(£).
(3) During the measurement LO phase is modulated according to £(£) and data
record with additive noise is obtained: s(t) = ssignai(t) + snoise(t).
(4) Using optimal digital filter v(t) fitted to C,(t) the experimentalist gets the
sequence of data samples
/>oo
dtv(t — riT)s(t).
(5) Using the restore function r(t) the reconstruction sr(t) of the signal is obtained:
r(*)=ry -^ => sr(t)= jrSnr(t-nr),
/-co 2^ v(Q)
where ii(il) is the Fourier transform of v(t).
n— — oo
It can be shown that optimal v(t) and ((t) should minimize the functional of
signal-to-noise-ratio (SNR) with simplest signal spectrum template (e.g. h(Cl) oc
Q~e for inspiral phase of compact binary coalescense6). It should be noted that to
2406
obtain a wide-band sensitivity gain in GW interferometers using DSVM one should
include mirrors thermal noise term along with quantum noise into the SNR before
optimizing it. If all the above requirements are satisfied it can be shown that SNR of
DSVM method is higher than one for conventional SQL-limited GW interferometer.
In article16 we demonstrated the possibility to beat the SQL threefold in wide
frequency band using DSVM together with intracavity design of detector.
3. Conclusion
We have shown the possibility to use DSVM method for detection of GWs with
sub-SQL sensitivity in advanced GW interferometers. This possibility is due to the
introduction of proper cross-correlation between the displacement and back-action
noise terms which minimize or completely eliminates back-action from the output
signal, thus increasing the output SNR of the detector. We believe that method of
DSVM is a good candidate to be implemented in future GW detectors.
Acknowledgments
Authors are pleased to express their deep gratitude to organizers of MG 11
conference and especially to A. Kleinert, H. Kleinert and R. Jantzen, for their hospitality,
outstanding organizational efforts and eagerness to help. Special thanks to Y. Chen
for fruitful discussions and numerous useful advice.
This work is supported in part by Russian President Grant for young researchers
No. MK-6859.2006.2 and Marcel Grossmann Foundation grant.
References
1. A.Abramovici et. al., Science 256, 325 (1992).
2. H. Luck et al., Class. Quantum Grav. 14, p. 1471 (1997).
3. M. Ando et. al., Phys. Rev. Lett. 86, p. 3950 (2001).
4. B. Caron et al., Class. Quantum Grav. 14, p. 1461 (1997).
5. D. Sigg, Class. Quantum Grav. 23, S51 (2006).
6. L. R. Y. K. A. Postnov, Living Reviews in Relativity 9 (2006).
7. V. B. Braginsky, Sov. Phys. JETP 26, p. 831 (1968).
8. V. B. Braginsky, M. L. Gorodetsky, F. Ya. Khalili, A. B. Matsko, K. S. Thorne and
S. P. Vyatchanin, Phys. Rev. D 67, p. 082001 (2003).
9. V. B. Braginsky, F. Ya. Khalili, Quantum Measurement (Cambridge University Press,
1992).
10. A.Buonanno, Y.Chen, Phys. Rev. D 65, p. 042001 (2002).
11. S. P. Vyatchanin and E. A. Zubova, Phys. Lett. A 201, 269 (1995).
12. S. P. Vyatchanin and A. B. Matsko, JETP 109, 1873 (1996).
13. H.J.Kimble, Yu.Levin, A.B.Matsko, K.S.Thorne and S.P.Vyatchanin, Phys. Rev. D
65, p. 022002 (2002).
14. A. Buonanno and Y. Chen, Phys. Rev. D 69, p. 102004 (2004).
15. S. L. Danilishin, F. Ya. Khalili and S. P. Vyatchanin, Phys. Lett. A 278, 123 (2000).
16. S. L. Danilishin, F. Ya. Khalili, Phys. Rev. D 73, p. 022002 (2006).
17. C. M. Caves, Phys. Rev. D 23, 1693 (1981).
18. S. P. Vyatchanin, Phys. Lett. A 239, 201 (1998).
THE DETECTION OF GRAVITATIONAL WAVES WITH MATTER
WAVE INTERFEROMETERS
P. DELVA*, M.-C. ANGONIN and Ph. TOURRENC
ERGA, Universite P. et M. Curie,
F-75252, Pans Cedex 05, France
* E-mail: pacome.delva@obspm.fr
aramis. obspm.fr/~erga/
We present the phase differences of fixed and free interferometers for different
configurations, and the required main characteristics of a matter wave interferometer to detect
gravitational waves.
Keywords: Gravitational waves detection; Matter wave; Interferometry.
1. Introduction
The first demonstration of the wave behavior of a massive particle, as predicted in
1925 by Louis de Broglie,1 was an electron diffraction experiment2 in 1927. Then
matter wave interferometry was only a matter of time. Electron interferometry
began in 19533 and neutron interferometry in 1962.4 In 1991, four atom interferometers
gave their first signals.5~8 Finally molecule interferometry was observed for the first
time in 1994.9
The possibility to detect Gravitational Waves (GWs) with Matter Wave
Interferometers (MWIs) has been explored since 1976 with different approaches.10-17
Recently a controversy has begun on the calculation of the phase difference. 18~20
We think that it comes from a wrong description of the experiment.21
In this proceeding we recall the main results of our article,22 where we compute
the phase difference for two kinds of experiments: fixed and free interferometers, in
different configurations.
2. Different configurations
We computed the phase difference for a Michelson Morley free configuration. The
method10 gives the same formal result for a photon or a massive particle, depending
on the wavelenght A, where A is the de Broglie wavelength for a massive particle. If
L <C V/c a the phase difference amplitude reads
A0 = 4^+|.^ (1)
where h+ is the GW amplitude of the + polarization. This result is well-known
for Light Wave Interferometers23 (LWIs).
L is the arm length, V = c for photons and V = vq, the initial velocity, for atoms.
2407
2408
For a rigid MWI in a Michelson Morley configuration, there are two regimes.20,22
In the first one (L < v0/c) Acfi ~ 0. In the second one (v0/c < L < A)b one finds
again the free phase difference (1). Unfortunately the low atom velocities permitted
in this regime limits the sensitivity device.
A rigid MWI with a Ramsey-Borde geometry is sensitive in the first regime17,22
to the cross polarization. The phase difference is of the same order of (1) if h+ —> hx
and if the angle of separation of the two matter beams is of order ir/4. The second
regime is sensitive to the two polarizations, in a ratio that depends on the atoms
velocity.
3. The MWI main characteristics
From Equ.(l) we compared the sensitivities of MWIs and LWIs when they are
limited only by the shot noise, for the same integration time. Figs. 1 & 2 represent
the required characteristics0 of a MWI to reach the sensitivity of Virgo and LISA.
The cross on each figure corresponds to the MWI described by Gustavson et al.24
vn(ra. s "■)
S-Js 'I
^-..(s1)
Fig. 1. Required characteristics of a MWI
necessary to reach the sensitivity of Virgo.
Fig. 2. Required characteristics of a MWI
necessary to reach the sensitivity of LISA.
One sees that relativistic velocities are required to reach the sensitivity of Virgo.
To reach LISA sensitivity, one would need a 1 km interferometer with a thermal
velocity. The development of atomic cavities25,26 could reduce this length. A one
meter MWI with ~ 1000 round-trips in each arm would reach LISA sensitivity.
However an atomic cavity has never been coupled to a MWI.
bA is the GW wavelength.
cThe curves are drawn for the caesium mass, vg, Lmw and NmW are respectively the initial atom
wave group velocity, the MWI arm length and the atom flux.
2409
Major improvements remain to be done, one of the most challenging could be
the beam separation.
The sensitivity that we have estimated is not the only crucial parameter for a
detector. We have already estimated the thermal noise22 but a complete study is
still missing.
To conclude, we think that MWIs will not compete in the future with high
frequency earth-based interferometer, but could reach the sensitivity of low frequency
space-based interferometers in a much more compact way.
References
1. L. de Broglie, Ann. Phys. Ill, 22 (1925).
2. C. Davisson and L. H. Germer, Phys. Rev. 30, 705(1927).
3. L. Marton, J. A. Simpson and J. A. Suddeth, Phys. Rev. 90, 490(1953).
4. H. Maier-Leibnitz and T. Springer, Z. Phys. 167, 386 (1962).
5. O. Carnal and J. Mlynek, Phys. Rev. Lett. 66, 2689(1991).
6. D. W. Keith, C. R. Ekstrom, Q. A. Turchette and D. E. Pritchard, Phys. Rev. Lett.
66, 2693(1991).
7. M. Kasevich and S. Chu, Phys. Rev. Lett. 67, 181(1991).
8. F. Riehle, T. Kisters, A. Witte, J. Helmcke and C. J. Borde, Phys. Rev. Lett. 67,
177(1991).
9. C. J. Borde, N. Courtier, F. Du Burck, A. N. Goncharov and M. Gorlicki, Phys. Lett.
A 188, 187(1994).
10. B. Linet and P. Tourrenc, Can. J. Phys. 54, p. 1129(1976).
11. L. Stodolsky, Gen. Relativ. Gravitation 11, 391(1979).
12. Y. Q. Cai and G. Papini, Classical Quantum Gravity 6, 407(1989).
13. C. J. Borde, A. Karasiewicz and P. Tourrenc, Int. J. Mod. Phys. D 3, 157 (1994).
14. C. J. Borde, Atom Interjerometry (Academic Press, 1997), ch. 7, pp. 257-292.
15. C. J. Borde, C R. Acad. Sci. Paris, Phys. 2, 509(2001).
16. C. J. Borde, Gen. Relativ. Gravitation 36, 475(2004).
17. F. Vetrano, G. Tino and C. J. Borde, Can we use atom interferometers in searching
for gravitational waves?, in Aspen Winter Conference on Gravitational Waves, 2004.
18. R. Y. Chiao and A. D. Speliotopoulos, J. Mod. Opt. 51, 861(2004).
19. S. Foffa, A. Gasparini, M. Papucci and R. Sturani, Phys. Rev. D 73, 022001 (2006).
20. A. Roura, D. R. Brill, B. L. Hu, C. W. Misner and W. D. Phillips, Phys. Rev. D 73,
084018(2006).
21. P. Delva, M.-C. Angonin and P. Tourrenc, Matter waves and the detection of
gravitational waves, in to be published in Journal of Physics: Conference Series, 2007.
22. P. Delva, M.-C. Angonin and P. Tourrenc, Phys. Lett. A 357, 249(2006).
23. P. Tourrenc, General relativity and gravitational waves, in Experimental Physics of
Gravitational Waves, eds. M. Barone, G. Calamai, M. Mazzoni, R. Stanga and F.
Vetrano (World Scientific Publishing Co. Pte. Ltd., 1999).
24. T. L. Gustavson, A. Landragin and M. A. Kasevich, Classical Quantum Gravity 17,
2385(2000).
25. V. I. Balykin, V. G. Minogin and V. S. Letokhov, Rep. Prog. Phys. 63, 1429(2000).
26. F. Impens, P. Bouyer and C. J. Borde, Applied Physics B: Lasers and Optics 84,
603(2006).
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GW Data Analysis
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DETECTING LISA SOURCES USING
TIME-FREQUENCY TECHNIQUES
JONATHAN R GAIR
Institute of Astronomy, University of Cambridge, Cambridge, CBS OH A, UK
jgair@ast. cam. ac. uk
GARETH JONES
Cardiff School of Physics and Astronomy, Cardiff University, Queens Buildings, The Parade,
Cardiff, CF24 3AA, UK
Gareth. Jones@astro. cf ac.uk
1. Introduction
The LISA data stream will contain many gravitational wave (GW) signals from
different types of source, overlapping in time and frequency. We expect to detect
signals from compact binaries (composed of white dwarfs (WDs) or neutron stars
(NSs)), in the nearby Universe. At low frequencies these will form a confusion
foreground, but we also hope to individually resolve ~ 10, 000 of these sources1 at high
frequencies. LISA will also detect 1-10 signals per year2 from the merger of super-
massive black holes (SMBHs) of appropriate mass (~ 105MQ - 107MQ). Thirdly,
LISA should detect GWs from extreme mass ratio inspirals (EMRIs) — the inspiral
of a compact object (a WD, NS or BH) into a SMBH in the centre of a galaxy.
The astrophysical rate is very uncertain, but LISA could resolve as many as several
hundred EMRIs3 and may also see a confusion background from distant events.4
The development of techniques to analyze LISA data is the subject of much
current research. One promising approach is to use Markov Chain Monte Carlo
(MCMC) methods. These have proven effective for detecting compact binaries,5
SMBH mergers6 and for the detection of a single simplified EMRI signal.7
Although MCMC techniques can be used to fit simultaneously for many signals of
several types, it is not yet clear whether this will be practical for the EMRI search.
This is because of the high computational cost associated with constructing
sufficiently accurate EMRI waveform templates, even when using kludge models.3 It
may therefore by impractical to use MCMC for the EMRI search unless some
advance estimate has been made of the source parameters. One alternative approach
to LISA data analysis is to use time-frequency (t-f) techniques. These could be used
to estimate the parameters of the loudest EMRIs in the LISA data stream and for
the detection of unexpected GW events. A t-f analysis will consist of two stages —
detection of a source in the data and parameter estimation for that source.
2. Source Detection
We consider a simplified model of the LISA data stream in which there is a single
source embedded in instrumental noise. We divide the data stream into M segments
of length T, carry out a Fourier transform on each segment and hence construct
2413
2414
a spectrogram of the data, S°, with power P^0- in pixel (i,j). We then search this
spectrogram for features. The simplest technique is to look for individual pixels that
are unusually bright, i.e., with p9. > 77, for some suitably chosen threshold rj. To
improve the performance, we generate and search a sequence of binned spectrograms,
Sk, in which the power in pixel (i,j) is defined to be
nk — Ilk— 1
a=0 6=0
Using bins of the form nk = 2P, Ik = 2q, for all possible p and q, a segment length
T = 220s, and assuming a 3 year LISA mission, this simple excess power search has
a reach of ~ 2.5Gpc for a typical EMRI event (we take the reach to be the distance
at which the detection rate is only 20% for a search false alarm probability of 10%).
The range is somewhat higher for EMRIs on nearly circular orbits. This method
and these results are described in Wen & Gair 20058 and Gair & Wen 2005.9
A more sophisticated technique is to look for clusters of bright pixels. One
algorithm is the Hierarchical Algorithm for Clusters and Ridges (HACR). This involves
identifying black pixels with Pitj > rjup, and then counting the number of grey pixels
with Pij > rjiow (< rjup) that are connected to the black pixel. If the number of
pixels in the cluster, Np, exceeds a threshold, Nc, then the cluster constitutes a
detection. The three thresholds can be tuned to make the search sensitive to a
particular source or chosen to make the search generally sensitive to a variety of source
types. After tuning, HACR has a detection rate 10 — 15% higher than the simple
excess power search at fixed overall false alarm probability for a typical EMRI.
This represents a significant improvement in LISA event rate. The HACR search
is described in more detail in Gair & Jones 2006.10 HACR can also detect SMBH
mergers at redshift up to ~ 3.5 and compact binaries at up to ~ 12kpc.10
3. Parameter Extraction
Once a source has been identified in the data, we would like to estimate its
parameters to allow a targeted follow up with matched filtering. The time-frequency
structure of an event tells us about the type of signal — a WD-WD binary is
almost monochromatic (the track is therefore long in time but narrow in frequency),
while EMRI and SMBH merger signals "chirp" over time. EMRIs chirp slowly and
are likely to be on eccentric orbits, indicated by the presence of several tracks at
different frequencies that evolve in a similar fashion. By contrast, SMBH mergers
are likely to be circular and evolve much more rapidly. The time, central frequency,
frequency derivatives and power profile of an event can all be extracted from a t-f
map and provide information on the system, as does the bin size used to generate
the spectrogram in which the detection is made. If multiple tracks can be
associated with the same event we get this information for each track. The shape of the
boundary of a track provides a way to distinguish a single event from two crossing
tracks or a noise burst. The shape parameters (curvature, area, perimeter), skeleton
2415
and convex hull of a cluster provide further information.11 This information can be
extracted directly from clusters identified by HACR (see discussion in Gair & Jones
200610 and Gair & Jones 2007 in prep.). The excess power search identifies
individual pixels only, so this search must be followed by a second track identification
search before information can be extracted.12
4. Application to LISA Data Analysis
Time-frequency searches of the nature described here could play a useful role in the
LISA data analysis pipeline. These methods should be able to detect the loudest
events in the LISA data stream at much lower computational cost than matched
filtering searches. They also provide a method to find unexpected sources in the LISA
data, since they do not rely on the observer having a model of the source. The main
issue that will limit the sensitivity of time-frequency techniques is source confusion.
The analyses described here have considered the detection of single isolated events,
which is not the situation we expect for LISA. To deal with confusion, we could
apply t-f techniques only to analyze a "cleaned" spectrogram, i.e., with the loudest
recognizable events extracted as well as possible by other techniques. This could
find events missed at the first stage of the analysis, but the effect of cleaning must
be carefully explored. Alternatively, we can use percolation techniques — set a high
threshold and gradually reduce it until a track appears. We can then extract this
loudest event before lowering the threshold further to find the next event etc. This
approach will be examined further in the future. Although our focus has been on
LISA, the methods discussed here could also be applied to searches of Advanced
LIGO data, e.g., for detection of intermediate mass ratio inspiral sources.
Acknowledgments
This work was supported by St. Catharine's College, Cambridge (JG) and by the
School of Physics and Astronomy, Cardiff University (GJ).
References
1. Farmer A J and Phinney E S Mon. Not. Roy. Astron. Soc. 346, 1197 (2003).
2. Sesana A, Haardt F, Madau P and Volonteri M Astrophys. J. 623, 23 (2005).
3. Gair J R, Barack L, Creighton T, Cutler C, Larson S L, Phinney E S and Vallisneri
M Class. Quantum Grav. 21, S1595 (2004).
4. Barack L and Cutler C Phys. Rev. D 70, 122002 (2004).
5. Cornish N J and Crowder J Phys. Rev. D 72, 043005 (2005).
6. Cornish N J and Porter E K preprint gr-qc/0612091 (2006).
7. Stroeer A, Gair J R and Vecchio A preprint gr-qc/0605227 (2006).
8. Wen L and Gair J R, Class. Quantum Grav. 22, S445 (2005).
9. Gair J R and Wen L, Class. Quantum Grav. 22, S1359 (2005).
10. Gair J R and Jones G, Class. Quantum Grav. submitted, preprint gr-qc/0610046.
11. Russ J C, Image Processing Handbook (Boca Raton: CRC Press) (2002).
12. Wen L, Chen, Y and Gair J R, preprint gr-qc/0612037.
DETERMINING THE NEUTRON STAR EQUATION OF STATE
USING THE NARROW-BAND GRAVITATIONAL WAVE
DETECTOR SCHENBERG
J. C. N. de ARAUJO
Divisao de Astrofisica - Instituto Nacional de Pesquisas Espaciais
Av. dos Astronautas 1758, Sao Jose dos Campos, 12227-010 SP, Brazil
jcarlos@das.inpe. br
G. F. MARRANGHELLO
UniPampa/Bage - Universidade Federal de Pelotas
Av. Carlos Barbosa, s/n, 96400-970 Bage/RS, Brazil
gfrederico.unipampa@ufpel. edit, br
A new window to the Universe is about to be opened. With the detection of
gravitational waves, astrophysicists expect to have the answer to many questions, as
well as new ones. There are many detectors all around the world and, soon, even
above us. With interferometric or resonant mass detectors, at low and high
frequencies, we shall be able to see waves coming from coalescing binary systems, from a
cosmological background, from catastrophic events etc.
We take a special attention to a small region of this spectrum, localized at 3.0-
3.4 kHz, which is the region of operation of the resonant spherical antennas Mario
Schenberg (Brazilian)1 and mini-Grail (Dutch).2
When we take such a small window (3.0-3.4 kHz) to such a giant garden named
the Universe, we are sure that we are losing a great amount of information. However,
if we are able to focus with good accuracy to this small region, we can also expect
to see all the magnificence of it. This is the case we are proposing in this work.
In Benhar et al3 the authors have calculated the properties of the neutron star
(NS) oscillating modes using a wide sample of equations of state. As the main
results, they have obtained empirical formulae for, among others, the frequency of
the f- and first p-mode
[W
vf = 0.79(±0.09) + 33(±2) \ — , (1)
1
M
M
-1.5(±0.8) + 79(±4) —
H
(2)
where the masses and radii are in units of km.
The aim of this work is to invert those relations and obtain general information
about the mass and radius of NSs. Once this work is done, the next step is to
investigate nuclear matter models for NSs and, finally, obtain the information about the
equation of state based on gravitational wave observations. The figure 1 shows some
mass-radius relations for different nuclear models and the Schenberg constraints for
a possible detection of neutron stars f- and p-modes.
2416
2417
Fig. 1. Mass-radius relation for the Benhar et al3 empirical relation and for different EOSs. The
light shaded regions represent the empirical relations for the f- (lower) and first p-mode (upper)
for the Schenberg bandwidth. The results are compared, for the NL model, to the normal static
(solid line) and (maximum) rotating NS (dotted line) and the Taurines model with K=220MeV
(dashed line). Static strange quark stars for the MIT bag model are plotted with B = 60MeVffm3
(dot-double dashed line) and B = WOMeV/fm3 (dash-double dotted line) and Chromo-dieletric
model (dot-dashed line). We refer the reader to the paper by Marranghello and de Araujo4 for
further details.
A relevant question is how to know, in a putative detection by Schenberg, if the
source is a NS or a Black hole (BH). The basic way to distinguish them can be
through the damping times of their oscillating modes. The damping time for the
quasi-normal modes of BHs is orders of magnitude shorter than the f- and p- modes
of NSs.
There is a simple relation correlating a Schwarzschild BH mass to the frequency
of its fundamental (quadrupole) quasi-normal mode. Such a relation implies that
the Schenberg antenna will only see BHs if their masses are about 3.5 — 4.0Mq. The
corresponding damping time is ~ 0.2 ms.
The same procedure is now applied to identify f- or p-modes in the Schenberg
antenna. A NS f-mode with frequency about 3.0 kHz presents a damping time much
larger then those presented by a BH, being of the order of 100 ms. This is also the
case for the first p-mode, which would have damping times greater then a few
seconds. So, the differences of a BH ringdown and a, NS f- and p-modes are easily
identified by their corresponding damping times.
Even though the Schenberg spherical antenna cannot determine the properties
of the damping time with low errors, we applied the empirical relations obtained
by Benhar et al, as we have done before, to describe its properties, considering the
NSs in which the f-mode frequency lies between 3.0-3.4 kHz.
In Benhar et al3 the authors also found an empirical relation for the f-mode
damping time described by
-i
T/ =
i?4
cM3
3.7 ± 0.2) • If)-2 + (-0.271 ± 0.009)
M
(3)
2418
where R is the radius and M is the gravitational mass, both in units of kin and c
is the speed of light. In addition to the frequency equation, Eq.l, that gives rise to
the detectable region in the mass-radius diagram, we obtained the diagram drawn
in Fig.2 using the above equation.
12
10
1 ■
[g 0.06 -0.07 s
#t 0.07-0.08 s
, 0.08- 0.10 s
0.10-0.20 s
> 0.20 s
„! 1 1 L 1 L 1 , 1
0 0.5 1 1.5 2
M (Ms_)
Fig. 2. Mass-radius relation for the Benhar etal empirical relation. The shaded regions represent
the empirical relations for the f-niode with different damping times.
Assuming a detection, we were able to find, for example, a very important
constraint for the compression modulus, restricting its value around 220 MeV. The
same analysis could be done to some other physical properties as the bag constant
or the effective hadron masses. A deeper discussion of this work can be found in
Marranghello and de Araujo.4
Acknowledgments
GFM would like to thank CNPq for financial support. JCNA would like to thank
FAPESP and CNPq for financial support.
References
1. O. D. Aguiar et. al. 2005 Class. Quant. Grav. 22 S209
2. A. de Waard et. al. 2005 Class. Quant. Grav. 22 S215
3. O. Benhar, V. Ferrari, L. Gualtieri 2004 Phys. Rev. D70 124015
4. G. F. Marranghello, J. C. N. de Araujo, Class. Quant. Grav. 23 6345.
APPROXIMATE WAVEFORM TEMPLATES FOR DETECTION OF
EXTREME MASS RATIO INSPIRALS WITH LISA
JONATHAN R GAIR
Institute of Astronomy, University of Cambridge, Cambridge, CB3 OHA, UK
jgair@ast.cam.ac.uk
1. Introduction
One of the most interesting potential sources of low frequency gravitational waves
(GWs) for LISA are the inspirals of stellar mass compact objects (white dwarfs,
neutron stars or black holes) into supermassive black holes (SMBHs) in the centers
of galaxies. The mass ratio is typically 10 : 106, so these events are termed extreme
mass ratio inspirals (EMRIs). Detection and parameter estimation for these events
is likely to involve matched filtering. LISA will observe EMRIs for the last several
years of inspiral prior to plunge, so the search templates will need to match the phase
of the signal over a fewxlO5 cycles. The extreme mass ratio ensures that templates
of sufficient accuracy can be computed using black hole perturbation theory —
the "self-force" formalism.1 Such templates are not yet available, however, and will
be very computationally expensive when they are ready. The number of templates
required to cover the parameter space of possible EMRI signals is very large,2 so
there is a need for approximate models that are quick to generate while also being
able to estimate the parameters of the source with sufficient accuracy that follow-up
with more accurate waveforms is possible.
One family of models are "adiabatic" templates, which are based on accurate
evaluation of the dissipative part of the self-force, combined with the assumption
that the orbital inspiral occurs slowly compared to the orbital period.3,4 Adiabatic
waveforms are likely to play a role at some stage of the LISA data analysis pipeline,
but they are still computationally expensive. For scoping out LISA data analysis,
waveforms must be generated in large numbers, so two families of approximate,
quick-to-compute, "kludge" waveforms have also been developed. The "analytic
kludge" is a phenomenological model based on Keplerian waveforms with relativistic
inspiral and precession imposed.5 The "numerical kludge" (NK) will be described
here. These NK waveforms are sufficiently faithful that they may play a role in
source detection for LISA and perhaps in source characterization.
2. Numerical Kludge Waveform Model
The NK family of waveforms are designed to be faithful models of true EMRI GW
signals. The waveform parameters are the same as for true EMRI signals — using
the NK model does not reduce the size of the parameter space or the number of
waveforms required to cover it. For a given set of parameters, however, the model
is much simpler to evaluate than a perturbative waveform and that is where the
2419
2420
computational savings for data analysis arise. Construction of an NK waveform
is done in two stages — (i) generation of the trajectory for an object inspiralling
through a sequence of quasi-geodesic orbits; (ii) construction of an approximate
waveform for an object moving on this trajectory.
2.1. Inspiral Trajectory Generation
To construct the inspiral trajectory, we first compute the phase space evolution
of the object, i.e., how the energy (E), angular momentum (Lz) and Carter
constant (Q) of the orbit evolve with time. This is done by deriving suitable
expressions for dE/dt etc. as a function of the orbital parameters, and then integrating
them through phase space. The expressions we use are built on second order post-
Newtonian (PN) results.6,7 Using 2PN results directly leads to pathological behavior
for nearly circular orbits, but this can be corrected by amending the circular pieces
of the fluxes.7 The trajectories can be further improved by using fits to data derived
from perturbation theory, i.e., based on solution of the Teukolsky equation. We have
done this for circular orbits of arbitrary inclination, but not yet for generic orbits
since perturbative data for such situations is only now becoming available.4 The
resulting phase space evolution equations are given in detail in Gair & Glampedakis
2006.7 For circular inclined inspirals, these fluxes agree with perturbative results to
an accuracy of 1% for orbits with periapse greater than ~ 5M, and to an accuracy
of < 5% for orbits with periapse greater than ~ 2M. For eccentric orbits, the fluxes
agree to ~ 5% for orbits with periapse greater than 5M, but this increases to a few
tens of percent for orbits that come very close to the central black hole.
Once the phase space trajectory (E(t), Lz(t), Q(t)) has been obtained, the
inspiral trajectory is derived by integrating the Kerr geodesic equations dr/dt =
R(r,B,E,Lz,Q), dO/dt = G(r,0,E,Lz,Q), d0/di = $(r,6,E,Lz,Q), with the
time-dependent E, Lz and Q inserted on the right hand side. We thus obtain the
particle trajectory in Boyer-Lindquist coordinates, (r(£),#(£), </>(£)).
2.2. Waveform Construction
After computing the particle trajectory in Boyer-Lindquist coordinates, we may
construct a corresponding trajectory in a pseudo-fiat space by identifying these
coordinates with spherical polar coordinates. A waveform can then be obtained by
supposing that there was a particle moving on such a trajectory in fiat space, and
using a weak-field GW emission formula. This approach is inconsistent in the sense
that it neglects the stress-energy that is causing the particle to move on the
trajectory, but it appears post facto to work well. We have constructed waveforms
using the Press formula8 (valid for weak-field, fast motion sources) and also
using the quadrupole and quadrupole-octupole formulae obtained by expanding the
Press expression in v/c. Based on a balance between ease of computation and
accuracy, it appears that the quadrupole-octupole formula is optimal. This waveform
construction is described in more detail in Babak et al. 2006.9
2421
3. Application to LISA
For both generic geodesic orbits, and for circular inclined inspiral orbits, the overlap
between the NK waveforms and more accurate adiabatic waveforms is very high.
For orbits with periapse greater than ~ 5Af the overlaps are typically greater than
95%, but this degrades for orbits that come deep into the strong field near the black
hole.9 The waveforms are sufficiently cheap to be generated in the large numbers
required for LISA data analysis, while their high faithfulness suggests that they
will also be able to constrain the source parameters quite well. NK waveforms are
already being used for scoping out LISA data analysis,2 and their high accuracy
indicates that they could play an important role in source detection for LISA, and
quite possibly for parameter estimation as the first stage of a hierarchical search.
The NK waveforms can be further improved in several ways — (i) inclusion of
PN conservative self-force corrections, i.e., the piece of the self-force that does not
dissipate. We have already demonstrated how this can be done to lowest order for
the simple case of circular inspirals in the Schwarzschild spacetime.9 Inclusion of this
effect will provide information on the relative influence of conservative corrections
on the phasing of generic EMRI waveforms, currently a matter of some debate,
(ii) Addition of "tail terms", i.e., the effect of radiation back-scattering off the
background geometry. This can be done by expanding the Teukolsky function and
should help to improve the accuracy of the NK waveforms for strong-field orbits,
(iii) Improvement of the flux expressions, i.e., dE/dt etc., for eccentric orbits by
using fits to perturbative data. This will ensure the NK waveforms can match true
EMRI signals for longer segments of the inspiral. These three improvements will be
implemented in the future to further develop this model as a tool for data analysis.
Acknowledgments
The work described in this paper was done in collaboration with Stanislav Babak,
Hua Fang, Kostas Glampedakis and Scott Hughes.7'9 JG's work was supported by
St.Catharine's College, Cambridge.
References
1. Poisson, E., "The Motion of Point Particles in Curved Spacetime", Living Rev.
Relativity 7, 6 (2004). [Online article]: cited on 26/12/2006.
2. Gair J R, Barack L, Creighton T, Cutler C, Larson S L, Phinney E S and Vallisneri
M, Class. Quantum Grav. 21, S1595 (2004).
3. Hughes S A, Drasco S, Flanagan E E and Franklin J,Phys. Rev. Lett. 94, 221101
(2005).
4. Drasco S and Hughes S A, Phys. Rev. D73, 024027 (2006).
5. Barack L and Cutler C, Phys. Rev. D69, 082005 (2004).
6. Glampedakis K, Hughes S A and Kennefick D, Phys. Rev. D66, 064005 (2002).
7. Gair J R and Glampedakis K, Phys. Rev. D73, 064037 (2006).
8. Press W H, Phys. Rev. D15, 965 (1977).
9. Babak S V, Fang H, Gair J R, Glampedakis K and Hughes S A, Phys. Rev. D accepted,
preprint gr-qc/0607007 (2006).
GW - DETECTOR'S OUTPUT PROCESSING
AT THE NON-GAUSSIAN NOISE BACKGROUND
A.V.Gusev, S.M.Popov, V.N.Rudenko
SAL MSU, Moscow
Russia
Optimal data processing algorithms for output realization of GW detectors are considered
under a presence of non-Gaussian component of noises. It is shown that a shielding of
non-Gaussian hindrances might be carried out through an additional filter so called
"non linear non inertial transformer" (NNT). Ways of composing of such transformer are
discussed in a half empirical manner.
1. Introduction
The cryogenic resonance bar detectors Explorer and Nautilus are gravitational wave
antennae which already have accumulated simultaneous data during of several years
observational time. These data are available for an off line coincidence analysis in a
searching for weak gravitational signals associated with transient relativistic sources
in the Galaxy and its close environment (see for example recent papers [1-3] ). The
data might be processed by special algorithms adapted to specific model of sources
depending on hypothesis closed to be tested. However a preliminary processing
presents some common procedure (so called a prefiltering of the bar's output
realization) described in the paper [4]. It consists in "whitening" (WF), "matching"
(MF) filters and Winer-Kolmogorov filter to cut off an additive read out noise.
Such filtering is considered as an optimal in the case of Gaussian noise background
and "S-pulse" signals. But in reality the bar's stochastic output realization has a
significant non-Gaussian noise and due to this the "prefiltering" procedure must
be changed. In this short note we discuss some adaptive quasi optimal algorithms
which could help to suppress non-Gaussian components of the bar's output noises.
Below we will describe the bar's output process by the sum of signal and noise
components
x{t) = \s{t) + n(t)
where A = (0,1) is "detection parameter".
After a proper ADC it is presented as a discreet stretch of counts x =
(ii,...,im), £k = x(kAt), At is a sampling time. A joint probability density
of the values x is defined as
■Wx(x\\) = Wn(x-\a),
where Wn(n) is a joint n-dimensional probability density of the stochastic counts
n = (rc,...., nM), nk = n{k,At); s = (s1; sju), sk = s(kAt).
2422
2423
2. Local optimal algorithm for a detection of weak gravitational
pulses
In general a sufficient statistics has to be proportional to the likely hood ratio or
its logarithm. The conditional likely hood (LHR) ratio read as
Abdsl = ^(X'A = 1) = ^(x's)
1 ' J iy,(x|A = 0) W„(x)
The unconditional LHR A[x] one gets by averaging this expression over the
stochastic signal's parameters:
A[x] = (A[x|s])s.
For the small signals at the arbitrary noise background one can use the following
expansion of the a posterior density probability [5, 6, 7]
M „TI. , N -MM
^(x-s)«^x)-£^Sfc + i££
Wn(x) , 1^^92iy„(x)
, , dxk 2 f-^ ^ dxkdxl
a—l fc=ii=i
SkSi- (1)
The last term in this formula might be omitted if small signals are considered as
unknown but not stochastic. So it reduces to
M
taHx]hA[x]»£rH«.rH = =^_gGE>. (2)
k—1
Thus for a forming the sufficient statistics at practice one needs to know the n-
dimensional density probability of the output noises Wn(n). The one dimensional
density probability can be estimated in the class of so called " e-contaminated
distributions" [5,6]. In particular one can seek the output fluctuation of GW bar detector
as an additive mixture of two Gaussian components with different variances <7q and
a\ empirically estimated together with a parameter of mixture 0 ^ e ^ 1. Then the
unknown one dimensional density probability W\{n) (A = 0) read
W1(n) = (1 - e)W*(x,al) + eW*(n,a^), -oo < n < oo,
where W^{n,a2) is the one dimensional Gaussian probability density with
parameters (0, a2). If e <$; 1 and a2 3> <7q the output density probability is almost Gaussian
one but with abnormally heavy tails.
In a general case an estimate of the n-dimeusional density probability is a serious
problem. However at practice one can use the approximation of the "non-Gaussian
white noise". Then the n-dimensional density probability is factorized and can be
presented as [5, 6, 7]:
M
Wn(*) = \{Wl{xk).
k=\
Now a come back to the formula of sufficient statistics (2) results in
M W'(r\
lnA[x]«£/[sfc]s*, /[s] = ^. (3)
k=i 1^")
2424
According to this expression the optimal receiver at such quasi Gaussian background
has in its composition: a) a non inertial nonlinear transformer (NNT) with specific
characteristics defined by the f(x) , b) a discreet matched filter (MF) (3). Our
approximation of the probability density as a "white non-Gaussian noise" provides
a low limit estimation of SNR as well as detection characteristics [5, 6, 7].
3. Maximum SNR criteria for depressing of non-Gaussian
hindrances
At the correlated Gaussian noise background a structure of the receiver optimized
on the maximum SNR criteria contains two principal links:
OF -► WF — MF.
This scheme must be changed in a presence of a correlated non-Gaussian fluctuation.
In general a solution of this problem is enough complex [8], but a quasi optimal
receiver might be constructed by incorporating in the " Gaussian structure" a new
link, which is just the NNT filter discussed above. One can show [5] that NNT filter
with the characteristics optimized on the SNR criteria , provides some shielding of
non-Gaussian hindrances. Depending on the order of filtering links the two variants
of data processing could be recommended:
NNT - WF — MF and WF — NNT — MF .
* v, ' ^ v '
I II
In the scheme I the "input NNT" produces an "amplitude suppressing" of big
non-Gaussian hindrances; then the linear part "WF — MF" provides an additional
improvement of SNR due to some "frequency compressing" of the non-Gaussian
correlated noises after NNT. The optimal characteristics of NNT ^[x] is proportional
to f[x] which is the cliaracteristics of NNT at the white non-Gaussian background
(3). A coefficient of the amplitude suppressing of a big additive noise n(t) for the
scheme I is defined by the formulae [5, 6, 7]
oo
p = a\ls 2 1, If = J
— oo
where If is a so called "Fisher Information" [5, 6. 7].
In the second variant of the "modified filtering scheme" II NNT is placed
between the WF and MF. This variant provides an "frequency - amplitude" depressing
of non-Gaussian correlated hindrances. A relative efficiency of the both scheme was
discussed in the monograph [6] at the qualitative level. In particular under strongly
correlated noise background a preliminary whitening of noises is considered as a
preferable step. In fact a general treatment of information in the discreet time [5, 7]
deals with the following local optimal algorithm of deterministic signals detection
at a correlated non-Gaussian background:
W[(x)
WJx)
z
W\{x)dx
NIDC - NNT - MF,
(4)
2425
where NIDC is a so called " nonlinear inertial decorrelator" .
The scheme II in our consideration above has the only difference with (4) : the
linear WF filter substitutes the nonlinear NIDC link. If the sampling time A is much
less the correlation time of additive noises at the WF output the both schemes are
statistically equivalent.
4. Conclusions
A. For a shielding of the traditional scheme of gravitational data processing (at
the output of gravitational wave detectors ) from non- Gaussian hindrances it is
useful to introduce the additional filtering link called as NNT. Characteristic of the
optimal NNT on the "criteria SNR" is depends on the one dimension probability
density of input noises. The last one is estimated through an empirical investigation
of noises n(t) at the GW detector output (a non parametrical estimate).
B. In is possible to perform a parametrical estimate of the one dimension
probability density of input noises in the class of "e-contaminated" distributions. Then
unknown parameters - variances (Jq , a\ and a " part of mixture e", are found though
empirical stretches of samples using the " method of initial moments" [5].
C. Modification of the "Gaussian" data processing algorithm associated with
introduction of the new nonlinear filter NNT. It might be used before Gaussian
algorithm for a strongly correlated noise background (the method mostly adapted for
bar detectors), or might be inserted between the whitening and matched filters (the
method recommended mostly for interferometers).
References
1. Astone P., Bassan M., Bonifazi P. et al // Phys. Rev. D66, 102002 (2002).
2. Astone P., Babusci D., Bassan et al// Phys. Rev. D71, 042001 (2005).
3. Babusci D, Giordano G., Murtas G.P., Pizzella G.
Astronomy & Astrophysics, 421, p. 811-813 (2004).
4. Astone P., Buttiglione S., Frasca S., Pallottino G. V., Pizzella G.// IL Nuovo Cimento.
Vol.20C,Nl.P.9.(1997).
5. Sheluhin O.I. Non-Gaussian processes in radiotechnics (in Russian). Moscow, "Radio
fesvyaz", (1999).
6. Kassam S.A. SignalDetection in Non-Gaussian Noise. Dowden & Culler, Inc.,Springer-
Verlag, New York (1988).
7. Levin B.R Theoretical annals of statistical radiotechnics (in Russian).Moscow, "Radio
fesvyaz", (1988). '
8. Sosulin Y.G. Theoretical annals of radio ranging and radio navigation (in Russian),
Moscow, "Radio & svyaz", (1992).
DETECTING A STOCHASTIC BACKGROUND OF
GRAVITATIONAL WAVES IN THE PRESENCE OF
NON-GAUSSIAN NOISE
YOSHIAKI HIMEMOTO
Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan
himemoto @utap .phys. s.u-tokyo .ac.jp
We discuss a robust data analysis method to detect a stochastic background of
gravitational waves in the presence of non-Gaussian noise. In contrast to the standard cross-
correlation (SCC) statistic frequently used in the stochastic background searches, we
consider a generalized cross-correlation (GCC) statistic, which is nearly optimal even
in the presence of non-Gaussian noise. The detection efficiency of the GCC statistic
is investigated analytically, particularly focusing on the statistical relation between the
false-alarm and the false-dismissal probabilities, and the minimum detectable amplitude
of gravitational-wave signals.
1. Introduction
The stochastic gravitational waves are the random superposition of plane waves,
whose statistical nature basically follows from the cosmological population of astro-
physical compact sources and/or diffusive high-energy sources in the early universe.
Especially, if we could detect a stochastic background of cosmological gravitational
waves, we may observe the very early universe directly. Therefore it is very
important and interesting to develop the detection methods for a stochastic background
of gravitational waves.
A stochastic background of gravitational waves has very tiny signal. This means
that we have no practical way to discriminate between detector noise and a
gravitational signal using a single gravitational detector. Then in order to search for a
stochastic background we use the cross-correlation statistic of the outputs at the
different detector.1
Most of the data analysis of a stochastic background of gravitational waves have
been studied under the assumption that the detector noise is Gaussian. However
the almost gravitational wave detectors do not have the pure Gaussian noise. In the
previous work,2 the standard cross-correlation (SCC) method has been extended
to deal with the more realistic detector noise efficiently. This modified statistic is
called by the generalized cross-correlation statistic (GCC). In this paper considering
the output data including the non-Gaussian noise, we analytically and numerically
discuss the detection efficiency of the GCC statistic compared to the SCC one.
2. Optimal Detection Statistics in the presence of non-Gaussian
Noise
In a single detector, we can not extract a stochastic background of gravitational
waves from the observational data including detector noise. Therefore we consider
two gravitational wave detectors to use the cross-correlation analysis and to search
for a common signal between their detectors. We denote the output of each detector
2426
2427
by s,f, with
s?=ft?+n?, (1)
where i = 1,2 labels the detector, and A; = l,...,iV is time index. Here h\ is a
gravitational signal and n\ is the detector noise. Considering that the gravitational
signals originating from a stochastic background are very weak, we assume that the
signal amplitude \h\\ = e is very small in this paper.
The GCC statistic is given by2
1 N
AGcc =-£/{(*{)/£(*£). (2)
fc=l
Here we introduced an arbitrary function /j(n*) to express non -Gaussian
distribution. If we set that this function has the quadratic form, namely Gaussian noise,
this statistic (2) reduces to the standard cross-correlation (SCC) statistic:
N
i
Ascc
^E'f* (3)
k=i
Hence we call Eq.(2) the generalized cross-correlation (GCC) statistic.
3. Performance Comparison between the GCC and the SCC
statistic
In this section we compare the performances of the GCC and the SCC for Gaussian
signal in the presence of non-Gaussian noise. Here we apply the two-component
Gaussian noise model given by
PnAx) = e-*(*> = [^ l) e-*2/2^.* + —^e-x2'2<> , (i = 1,2). (4)
n,i v27rcrt.i
(l-Pi) ^_x2/2ali t P{
to non-Gaussian noise model.2 This model can be characterized by the two
parameters, i.e., the ratio of variances, (<7t,i/o~m,i)2 and the fraction of non-Gaussian
tail, Pi. Here, Pi means the total probability of the non-Gaussian tail. Under this
noise model, we analytically calculate the probability of false alarm (-Pfa) versus
the probability of false dismissal (Pfd) curves (for the detail see Ref.[3]).
Furthermore, from the Pfa~Pfa relation, we obtain the minimum detectable amplitude of
gravitational waves for the threshold value of two error probabilities (Pp^, Pfd)-
In the left panel of Fig.l, we plot the analytic Pfa-Pfd curves for various signal
amplitudes. Here, the parameters P, at/am and N are specifically chosen to P =
0.01, <7t/o"m = 4 and N = 104. The solid and dotted lines represent the Pfa-Pfd
curves for the GCC and the SCC statistics, respectively. In each signal amplitude
e, the false dismissal probability Pfd of the GCC statistic is always smaller than
that of the SCC statistic for any Pfa- As expected, the performance of the GCC
statistic improves as the parameter e increases.
SCC =
GCC •
o
^ o ° P = 0.1
., » ° N= 10000
"0 0.2 0.4 0.6 0.8 1 2 4 6 8 10
PFA a\./°m
Fig. 1. Analytic Pfa — Pfd curves and minimum detectable amplitude of the gravitational-wave
signals for the SCC and GCC statistics in the non-Gaussian model (4).
We plot in the right panel of Fig.l the dependence of the amplitude edetect on
the ratio of variance at/am. In this plot, we specifically set the detection point to
(PpA, -Pfd) = (0.1, 0.1). The solid and dotted lines represent the analytic estimates
of the minimum amplitude for GCC and SCC statistics, respectively. Filled (GCC)
and open (SCC) circles represent the simulation results. This Figure shows that
the minimum detectable signal amplitude for the GCC statistic is insensitive to the
value of the variance ratio. Therefore we find that the GCC statistic performs much
better than the SCC one as the tail variance becomes large.
To summarize, using the analytical and numerical approach for Pfa-Pfd
relation, we confirmed that the GCC statistic performs better than the SCC one in the
presence of non-Gaussian noise. We believe that this strategy is useful for the future
plan of the search for a stochastic background of gravitational wave.
References
1. B. Allen and J.D. Romano Phys. Rev. D 59, 102001 (1999)
2. B. Allen, J.D.E. Creighton, E.E. Flanagan and J.D. Romano Phys. Rev. D 65, 122002
(2002)
3. Y. Himemoto, A. Taruya, H.Kudoh and T. Hiramatsu gr-qc/0607015
2428
0.6
0.4
02
COINCIDENCES BETWEEN THE GRAVITATIONAL WAVE
DETECTORS EXPLORER AND NAUTILUS IN THE YEARS 1998,
2001, 2003 AND 2004
G.PIZZELLA
Dipartimento di Fisica, Universitd di Roma "Tor Vergata"
Via Ricerca Scientifica 1, 00133 Roma, Italy and
INFN Laboratori Nazionali di Frascati
guido .pizzella@lnf-infn.it
We report here the results of the search for gravitational waves with the EXPLORER-
NAUTILUS experiment during the years 1998, 2001, 2003 and 2004. We find that in all
years a small consistent coincidence excess occurs at the sidereal time when the two bars
are oriented perpendicularly to the galactic plane. No physical interpretation is given,
although the statistical evidence appears robust.
PACS numbers: 0480, 0430
1. Introduction
In 2001 and 2002 the ROG collaboration presented the results of searches for GW
bursts with the EXPLORER and NAUTILUS cryogenic bar detectors operating in
1998 for six months1 and in the year 2001 for nine months2'3 . In those papers a
sidereal time analysis was performed in order to look for specific galactic signatures.
A small excess of events with respect to the expected background was found*,
concentrated around sidereal hour four. At this sidereal hour the two bars, which
are oriented parallel to each other, are perpendicular to the galactic plane, and
therefore their sensitivity for galactic sources of GW is maximal4 .
After an upgrade of the detectors, other data of EXPLORER and NAUTILUS
from the 2003 run were analyzed and the results reported in a recent paper5 , where
again a small coincidence excess, not significant by itself, was found. New analysis
with the new 2004 data will be reported here, again showing a small coincidence
excess.
The purpose of this presentation is to try to give a statistical assessment to the
coincidence excess which we have consistently found from 1998 to 2004, without
grabbing the difficult task to discuss the physical mechanisms involved. The total
time period considered (549.7 days from 1998 to 2004) corresponds to the longest
coincidence study of GW detectors ever. In the considered period Explorer and
Nautilus were the only detectors in continuous data taking, and with a good working
stability.
*In the conclusions of the 1998 paper that was discussed within the IGEC collaboration in 1999
and 2000-.. . . we find an excess of coincidences at zero time delay in the direction of the galactic
centre. We report this conclusion because it sets the line for the following data analyses.
2429
2430
2. Experimental data in 2004
The data, sampled at intervals of 3.2 ms, are filtered with an adaptive filter matched
to delta-like signals for the detection of short bursts6 . This search for bursts is
suitable for any transient GW which shows a nearly flat Fourier spectrum at the
two resonant frequencies of each detector. The metric perturbation h(t) can either
be a millisecond pulse, a signal made by a few millisecond cycles, or a signal sweeping
in frequency through the detector resonances. This search is therefore sensitive to
different kinds of GW sources, such as a stellar gravitational collapse, the last stable
orbits of an inspiraling neutron star or black hole binary, its merging and its final
ringdown.
Let x(t) be the filtered output of the detector. This quantity is normalized, using
the detector calibration, such that its square gives the energy innovation E of the
oscillation for each sample, expressed in kelvin units.
For well behaved noise due only to the thermal motion of the oscillator and to the
electronic noise of the amplifier, the distribution of x{t) is normal with zero mean.
Its variance (average value of the square of x(t)) is called effective temperature
and is indicated with Teff. The distribution of x(t) is
1
V27rTe//
In order to extract from the filtered data sequence events to be analyzed we set
a threshold for x2. The threshold is set at Et = 19.5 Teff in order to obtain, in
the presence of thermal and electronic noise alone, a reasonable low number of
events per day (see Ref. 7). When x2 goes above the threshold, its time behaviour
is considered until it falls back below the threshold for longer than one second. The
maximum amplitude Es and its occurrence time define the event.
Computation of the GW dimensionless amplitude h from the energy signal Es
requires a model for the signal shape. A conventionally chosen shape is a short pulse
lasting a time of rg, resulting (for optimal orientation, see later) in the relationship
L 1 kEs
where vs is the sound velocity in the bar, L and M the length and the mass of the
bar and rg is conventionally assumed equal to 1 ms (for instance, for E8 = 1 mK
we have h = 2.5 10"19, for both EXPLORER and NAUTILUS).
3. Experimental results
Before searching for coincidences we must make two important choices: a) the
threshold SNRt used for the event definition, b) the coincidence window. We must
also consider whether to apply the energy filter that we adopted in Refs. 1,2 (not
applied to the 2003 data published in Ref. 5). This filter eliminates the
coincidences between events whose energies are not compatible, taking into account the
uncertainty due to the noise contribution to the measured event energy.
2431
We have calibrated the apparatuses by means of known small forces applied to
piezoelectric ceramics. In our case we can test if the calibration was properly done
by making use of the cosmic ray showers (CRS)t, as follows8 . We use the data
obtained in coincidence with 2508 showers for EXPLORER and 1189 showers for
NAUTILUS, with multiplicity (number of secondaries measured at the bottom of
the bar) below 1500aecoffi"ea, and we take the averages of the responses of the
two apparatuses referred to the time of each CRS. The result is shown in fig.l. We
0.5
0.4
0.2 -
0 1
-0.2
-0.1 -0.05 0 0.05 0.1 -0.1 -0.05 0 0.05 0.
seconds seconds
-
- nautilus 2004
: 1
{ f \
[ I M
: I
_
f 1
t
11
In
i1 i
r
1
lit It A
|! i i hi if'
J 1 1
Fig. 1. Cumulative response to CRS for the multiplicity interval 400 < A < 1500
The background has been subtracted (this accounts for the small negative values).
secondaries
note that both EXPLORER and NAUTILUS respond in the same way to CRS, so
for each coincidence we can compare the event energy of EXPLORER with that of
NAUTILUS.
It is very important to remark that the use of the CRS ensure that the detectors
are indeed able to observe very tiny vibrations. EXPLORER and NAUTILUS are
the only GW detectors equipped with cosmic rays apparatuses.
Since we want to compare the new results obtained with the 2004 data with the
tBoth EXPLORER and NAUTILUS are equipped with cosmic ray apparatuses.
2432
previous results, we must use the same choices, whenever possible. Thus we take the
energy threshold for the events set for our past analyses at SNRt — 19.5. For the
coincidence window, because of the larger bandwidth of the detectors after 2001,
we use the same window applied for the 2003 data5 , that is w = ±30 ms, after a
carefull examination of the detector response to cosmic rays.
Searching for coincidences we have also determined the accidental ones by time
shifting one of the two event list with respect to the other one by time steps of
2 seconds from -100 s to +100 s. The number of coincidences nc at zero delay is
then compared with the average number n of the accidentals obtained with the one
hundred time-shifts. For a Poissonian distribution we expect:
NexpiNnaut2w
nth = , , ... 3
totaltime
We search for coincidences with and without the energy filter, and the summary
result is given in the table 1. During 5196 hours of common operation we have 50464
EXPLORER events and 66756 NAUTILUS events, with average values < T*ff >=
3.09 mK for EXPLORER and < T™. >= 1.79 mK for NAUTILUS. Applying Eq.3
Table 1. Results of the
coincidence search.
filter nth nc
no filter 10.7 13
filter at 68% 8.1 12
we calculate the no-filter accidental coincidences nth = 10.8, very compatible with
the experimental n = 10.7. We notice form the Table 1 a small coincidence excess,
statistically non very significant.
In the papers Refs. 2,3,5 we analysed the data taking into consideration that, as
the Earth rotates around its axis during the day, the detectors happen to be variably
oriented with respect to a given source at an unknown location. Thus we expect the
detector sensitivity to that hypothetical source (therefore, the coincidence excess
rate) to be modulated during the day; more precisely the modulation is expected to
have a period of one or half sidereal day, since the GW sources, if any, are certainly
located far outside our Solar System.
Applying the same method to the 2004 data we obtain the result shown in
fig.2. We notice a small coincidence excess at sidereal hour 4-5, consistent with our
previous findings.
As discussed in Refs. 5,9 , we should investigate how the coincidence excess
although small, is distributed during the year 2004. We search for coincidences
during moving five days periods and obtain the result shown in fig. 3. It is intriguing
that most of the coincidence excess is concentrated in the time interval day 205-
215, in particular at days 212-214. More intriguing is the fact that on day 213.76
(31 July 2004) the supernova SN2004dj was observed10 , the brightest supernova
2433
5 10 15 20
5 10 15 20
sidereo nour
-1
-2
IAIVW
-»
T-
V
5 10 15
solar hour
20
Fig. 2. The upper graphs show the hourly number nc of coincidences (continuos line) and the
average number n of accidentals (dashed line) versus the sidereal and solar hour. The lower graphs
show the corresponding Poisson probability to obtain a number of coincidences nc greater than or
equal to nc. The energy filter has been applied.
detected for several years11 , in the nearby spiral galaxy NGC 2403. However, this
is the day the supernova was firstly visually observed, and we do not know the
very time of the supernova explosion, therefore it is difficult to infer a correlation
of this SN occurrence with the coincidence excess observed with the EXPLORER
and NAUTILUS gravitational wave detectors.
4. Comparing the 2004 results with the published 1998, 2001 and
2003 results
During the years 1998, 2001, 2003 and 2004 only EXPLORER and NAUTILUS were
in continuos operation. We think it is important to compare the small coincidence
excess observed during 2004 with our previous results1-3'5 obtained during 1998,
2001 and 2003. We show in the Table 2 the main characteristics of the apparatuses
during the time of the coincidence search.
During 2001 we first2 used a variable coincidence window as suggested by
simulations, later the use of the cosmic ray apparatuses has shown that a fixed coincidence
2434
F
2.5 E-
I
2 -
1.5 F-
0.3
0
k±
A
30
'50
200
250
300
350
10 -
TTi r
i
'50 200 250
dcy o~~ year 2004
350
Fig. 3. In the upper graph the coincidences (continuos line) and the average accidentals (dashed
line) for five-day periods. In the lower graph the corresponding Poisson probabilities.
window has to be preferred (see Ref. 5). In Ref. 3 we have used w = ±0.5 s, and so
we do in the present paper.
For the 2003 data we found8 that, by using the cosmic rays, the EXPLORER
event energy needs to be multiplied by a factor 3.3 (because of a mis-calibration of
the SQUID apparatus during 2003) and so we apply this factor for the energy filter
in the present paper.
In the process of combining experimental data obtained in different situations,
as in our case because of the continuos upgrades of the apparatuses, we are faced
with the danger to make, perhaps unwilling, choices which would affect the final
statistical significance. Being aware of this, we have been careful to apply to the
coincidence searches the same procedure whenever possible, in order to verify the
initial result obtained in 1998 (see footnote).
The only change applied here to the 1998 and 2001 data analysis has been to
present the results in terms of the sidereal time at the Greenwich longitude, as
already done in Ref. 5 . We show the results for all years in the fig.4.
In the Table 3 we give the number of coincidences and average number of
accidentals for the four years. We also give the same information for the side-
2435
Table 2. Main characteristics of the detectors for the coincidence search in the
four years, time refer to the common time of operation. The coincidence window
has been determined with the cosmic ray apparatuses when available.
year
detector
time
1998 EXPLORER 94.5 days
NAUTILUS
2001 EXPLORER 90 days
NAUTILUS
2003 EXPLORER 148.7 days
NAUTILUS
2004 EXPLORER 216.5 days
NAUTILUS
frequencies
904.7, 921.3 Hz
907.0, 922.5 Hz
904.7, 921.3 Hz
907.0, 922.5 Hz
904.7, 921.3 Hz
926.3, 941.5 Hz
904.7, 921.3 Hz
926.3, 941.5 Hz
bandwidth
~0.4 Hz
~0.4 Hz
~9 Hz
~0.4 Hz
8.7 Hz
9.6 Hz
8.7 Hz
9.6 Hz
window
±1 s
3ct ~0.5 s
±30 ms
±30 ms
\
1998
fa ;
rj \ A f, a f
0 5
15 20
0 15 20
r !\
5 20
:-R7yv-x=
-2t /
c r
:0G3
, , \ , ,
0 5 1C 15 20
4 Z
3 1 II
2 " A „ A
u
0 5 10 15 20
side roc nour
10
-2F
i / v
10 f I 2004
(3 5
10 15 20
'oereal nour
Fig. 4. Coincidences and Poisson probabilities for 1998, 2001, 2003 and 2004. See text for
explanation.
real hour range when we expect signals due to sources in the galactic disk. This
range, for a two-detector coincidence search has been calculated in Ref. 4 and is
~ 3.5 ± 1.5 sidereal hours.
We now must attempt to combine all data in a single result. We do this by
2436
Table 3. In the second column we give the threshold, expressed
in terms of the adimentional perturbation h, corresponding to
SNRt = 19.5. In the third, fourth and fifth columns the total number
nc of coincidences, average accidentals n and the Poisson probability.
In the remaining columns the number of coincidences in the sidereal
hour range 2-5 (see ref.4) and relative Poisson probability. In total
we have 132 coincidences and 103.7 average accidental for a Poisson
probability of 1.2 10—2. In the 2-5 sidereal hour range we have 29
coincidences and 12.4 average accidental for a Poisson probability of
1.9 10~4.
Tic
12
8
6
3
n
6.26
3.06
1.60
1.23
poisson
2.7 10~2
1.3 10"2
6.0 10-3
13 10~2
applying the following formula (see Ref. 12)
3 1
P =P\P2P3P4^2^\log(pip2P3P4)\J (4)
3=0 3'
where pi, i = 1,2,3,4 are the Poisson probabilities obtained for the four years 1998,
2001, 2003 and 2004. We get the result shown in the fig.5.
We must conclude that in each year a small coincidence excess, a small excess
during each year, is present at sidereal hours compatible with gravitational wave
sources in the galactic disk.
The physical interpretation appears difficult with our present knowledge, also in
consideration of the fact that the sensitivity of our apparatuses has changed during
the years. Gravitational waves would require a cross-section larger by at least two
orders of magnitude for producing the signals. But, one should not rule out, in
addition, the possibility that dark matter be the cause of the observed coincidence
excess.
Acknowledgments
I thank the ROG Collaboration for making available the experimental data, and
Gianfranco Giordano and David Blair for useful discussions.
References
1. P.Astone et al.: Class. Quantum Grav. 18 , 243 (2001)
2. P.Astone et al.:, Class. Quantum Grav. 19, 5449 (2002)
3. Pizzella, G. : Tenth Marcel Grossmann Meeting on General Relativity, (M. Novello, S.
Perez-Bergliaffa, R. Ruffini, Eds.) (2003)
4. D.Babusci et al.: Astron.Astrophys. 421, 811 (2005)
5. P.Astone et al.: Class.Quant.Grav .23, S169 (2006)
6. P.Astone, C.Buttiglione, S.Frasca, G.V.Pallottino and G.Pizzella, II Nuovo Cimento
20, 9 (1997)
year
1998
2001
2003
2004
threshold
4.3 10~18
1.6 10"18
1.9 10"18
1.2 10~18
nc
64
37
19
12
n
52.1
31.4
12.1
8.1
poisson
6.1 10"2
18 10~2
4.1 10~2
12 10"2
2437
10 12.5 15 17.:
solar hour
Fig. 5. Combining the probabilities for the four years 1998, 2001, 2003 and 2004, according to
Eq. 4.
7. P.Astone et al.: Phys.Rev. D59, 122001 (1999)
8. G.Modestino,G.Pizzeria,F.Ronga,LNF-05/27(IR)(2005)
http://www.lnf.infn.it/sis/preprint/pdf/LNF-05-27(IR).pdf
9. I.Modena and G.Pizzella, 2006, Int. J. of Modern Phys. D 15, 485 (2006)
10. S.Nakano et al.: IAU Circ. 8377
11. R.J.Beswick et al.: The Astrophysical Journal623 :L21-L24 (2005)
12. B.P.Roe, Probability and Statistics in Experimental Physics, pag.164 (Springer, 2001)
INCOHERENT STRATEGIES FOR THE NETWORK DETECTION
OF PERIODIC GRAVITATIONAL WAVES
P. ASTONE, S. FRASCA and C. PALOMBA
INFN, Sezione di Roma and Universita "La Sapienza", Roma, Italia.
cristiano.palomba@romal. infn. it
In the Virgo Collaboration, a hierarchical procedure for the blind search of continuous
gravitational signals has been developed. A brief description of the method with some
bibliographic references and of the preliminary results obtained on the data of C6 and
C7 Comissioning Runs can be found elsewhere in these Proceedings.1 In this paper we
focus attention on an important part of the analysis, consisting in doing coincidences
among the candidates found in two or more data sets, which strongly reduces the false
alarm probability. Data sets can indifferently belong to a single or more detectors.
Keywords: Gravitational waves; Continuous sources; Virgo detector.
1. Need for coincidences
In the hierarchical procedure developed in Virgo for the search of continuous
gravitational signals we select candidates in a given data set putting a threshold on the
critical ratio (CR) of the Hough sky histograms, defined as CR = ^^ where n
is the number count in a given cell of the histogram, /i is the mean number count
and a the standard deviation. The value of the threshold is chosen as a compromise
between the need to minimize the sensitivity loss and to have a manageable number
of candidates. By doing coincidences among candidates of two or more data sets
we strongly reduce the false alarm probability P/a. This is a very important point
because to claim a detection we need to reduce it to values such that Pja <c iV"1,
where Np is the total number of points in the source parameter space. Making
coincidences means to check if the parameters of a pair of candidates are within a given
coincidence window. To perform coincidences we need at least two data sets, from
one or more detectors. We can choose them in different ways and, as we will see,
not all the choices are equivalent. Here, three different choices are presented.
• Distinct data sets
Each data set can correspond to a detector run. For a given minimum spin-down age
we have that the number of spin-down values to be analyzed is minimum. From one
hand this reduces the computational load, on the other reduces also the resolution
in spin-down. If the data sets cover a short time interval1 'spurious' candidates can
appear in each data set, and then survive in the coincidences. This not only affects
the false alarm probability but also the accuracy with which the parameters of a
source, especially the position, can be determined.
• Twofold 'mixed' data sets
We can take the two original data sets (call them ag and £>0) and suitably mix them
creating two new sets (a1 and bi). A simple choice would consist, for instance, in
2438
2439
taking aj as the first half of a0 plus the first half of 60 and b\ as the second half of
ao plus the second half of bo. In this way the time interval covered by each of them
is larger, thus increasing the resolution in spin-down and reducing the number of
spurious coincidences, if each original data set was short.
• N-fold 'mixed' data sets
We can generalize the previous choice by mixing more pieces of the original data sets.
A particularly convenient choice is to produce new sets with approximately the same
sensitivity. If we call a,i and bi, with i = 1, ..n; n > 2, the pieces, one new set could
be done, e.g., as a: +a3 + ... + bi +&3 + ... and the other one as 0-2 + 04 + ... + b2 + b4 +....
In this situation, as will be shown in the following, the sensitivity of the analysis
may be larger, at least if disturbances, as expected, are present in the data. Let us
now show that, if coincidences are done, it is better to use data sets with the same
sensitivity. Let us assume to have two data sets with corresponding linear signal to
noise ratio SNRi and SNR2, for a unitary amplitude signal in arbitrary units. By
re-organizing them in two new data sets with equal sensitivity, the resulting SNR for
both is SNR = y ^ - assuming the incoherent step of the hieararchical
procedure (the Hough transform) is done adaptively.2 The critical ratios for the
original data sets are
CR, = G(0; 1) + SNR\ ■ h2gw, CR2 = G(0; 1) + SNR22 ■ h2gw
where G(0; 1) is a value taken from the CR distribution in absence of signals, which
follows a standard gaussian with heavier tails due to disturbances, and hgw is the
amplitude of the gravitational signal. In the case of data sets with the same
sensitivity we have
CR, = G(0; 1) + SNR2 ■ h2gw, CR2 = G(0; 1) + SNR2 ■ h2gw
The CR of a coincidence is CRCOin = min(CRi, CR2) where CR\ and CR2 refers to
the two coincident candidates; then, given a threshold z, we can take the probability
P(CRcoin > z) as a measure of 'effectiveness' which allows us to compare the two
cases, see Fig.(l), obtained with a Monte Carlo simulation. For equal sensitivity
data sets the probability is larger, that is we could choose a lower threshold for
candidate selection with a lower sensitivity loss at fixed false alarm probability. Or,
viceversa, taking fixed the threshold we have a lower false alarm probability. Let
us now indicate with Pfa(z) the false alarm probability (f.a.p.) for a given data
set, depending on z, the threshold on the critical ratio of the Hough map used to
select candidates. If a gravitational signal of amplitude ho is present in the data,
the corresponding detection probability can be expressed as Pd(z; A) = Pfa{z — A),
where for small signals A ~ 0.830 • VN ■ °2gFT, being N the number of spectra,
each with length Tfft, and Sn is the unilateral noise spectral power densitiy of the
detector. Let us now compare this case with the coincidences among M data sets
with the same sensitivity obtained from that. In each subset we have a sensitivity
loss \f~M because the sensitivity of the incoherent step scales as \jTobsjTFFT- Given
2440
Fig. 1. 'Effectiveness' of the coincidence method (see text for the definition), as a function of the
CR threshold z, for the two original data sets (solid line, with SNR2 = SNRi/2) and two sets
with equal sensitivity obtained from them (dashed).
a threshold for candidate selection, making coincidences among the candidates of the
M subsets the f.a.p. reduces by the Mth power: P^] (z) = P™(z). The detection
P¥(z — A/M). We make the comparison by computing
probability is Pj (z; A)
the ROC curves at fixed signal amplitude and the detection probability curves as a
function of signal amplitude at a f.a.p., see Fig.(2). The computation is done using
a CR distribution which approximates that found in C7 Virgo data. Similar results
have been found for C6 data. From Fig.(2a) we see that coincidences give an higher
False alarm probability
10 12 14 16
(a)
(b)
Fig. 2. (a) ROC curves for signal amplitude A = 6 and (b) Detection probability vs. signal
amplitude at false alarm probability 10~ls for N = 1 (solid line), N = 2 (dashed), N = 3 (dot-
dashed), N = 4 (dot bold) and N = 5 (dot) detectors.
detection probability as soon as we choose a f.a.p lower than ~ 10 . For a f.a.p. of
~ 10_ 15 the detection probailities for M > 3 are similar, with a slightly larger value
for M = 3. For still smaller f.a.p. larger values of M are favoured. From Fig.(2b) we
have that the detection probabilities at fixed f.a.p (10-15) as a function of signal
amplitude are very similar for M > 3 and much larger than for M < 2.
References
1. F. Acernese et al., First coincidence search among gravitational wave periodic source
candidates using Virgo data, these Proceedings.
2. C. Palomba, P. Astone, S. Frasca, Classical & Quantum Gravity 22, S1255 (2005).
SEARCH FOR CONTINUOUS GRAVITATIONAL WAVES: SIMPLE
CRITERION FOR OPTIMAL DETECTOR NETWORKS
REINHARD PRIX
Max-Planck-Institut fur Gravitationsphysik, Albert-Einstein-Institut,
D-14476 Golm, Germany
We derive a simple algebraic criterion to select the optimal detector network for a
coherent wide parameter-space (all-sky) search for continuous gravitational waves. Optimality
in this context is denned as providing the highest (average) sensitivity per computing
cost. This criterion is a direct consequence of the properties of the multi-detector T-
statistic metric, which has been derived recently. Interestingly, the choice of the optimal
network only depends on the noise-levels and duty-cycles of the respective detectors, and
not on the available computing power.
1. Multi-detector matched filtering
The ^"-statistic2 is a coherent matched-filtering detection statistic for continuous
gravitational waves (GWs). We follow the expressions and notation of our previous
work1 (Paper I) on the multi-detector ^"-statistic metric. We consider a set of N
detectors with (uncorrelated) noise power-spectra Sx, where X is the detector index,
X = 1,..., N. Let T be the total observation time spanned by the data to be
analyzed. The corresponding multi-detector scalar product for narrow-band continuous
waves can be written as
(x\y)=TS-l(xy)s, (1)
where boldface notation denotes multi-detector vectors, i.e. {x{t)} = xx(t). We
can allow for the fact that each detector will be in lock only for a duration Tx < T,
so each detector can be characterized by a "duty cycle", dx = Tx/T < 1. This
is a slight, but straightforward generalization with respect to Paper I, and the
corresponding noise-weighted time average (.)s is defined as
T
(Q)s=^J2wxf QX(t)dt, (2)
where the weights u>x and the total inverse noise-power <S_1 are defined as
s-i N
wx=dx-~, where <S_1 = ^ dx S^1. (3)
x=i
The importance of Eq. (1) is that it separates out the scaling with the total
observation time T and the set of detectors (via <S_1), from the averaged contribution
(xy)s, which does not scale with T or the number of detectors. In terms of this
scalar product (1), the optimal signal-to-noise ratio (SNR) for a perfectly-matched
signal s(t) can be obtained as
p(0) = ^W) = VTS^^%. (4)
2441
2442
It is obvious from this expression that the SNR increases when increasing the
observation time T or the number of detectors N. However, here we are interested
in the case of wide parameter-space searches, in which the highest achievable SNR
is computationally limited. We therefore need to find the optimal sensitivity per
computing cost.
2. Optimizing sensitivity per computing cost
For simplicity we only consider the sensitivity to an "average"' sky-position, so we
disregard the dependence of (s2)s to both the sky-position as well as the relative
orientation of the different detectors. Both should be small effects on average. In
addition to Eq. (4) for the SNR, the second ingredient for the optimal network is
the computing cost of a wide-parameter search. For the sake of example we consider
a search for GWs from unknown isolated neutron stars, with unknown intrinsic GW
frequency /, sky-position a, 8 and one spindown-parameter /. One can show1 that
in this case the number of required templates Np scales (at least) as J\fp oc T6,
which severely limits the computationally affordable observation time Tmax. Most
importantly, however, the number of templates does not scale with the number N
of detectors.1 The corresponding computing cost Cp required to search these Np
templates can be estimated as Cp oc NT7 for a "straightforward" computation,
while it could be reduced down to about Cp oc N T6 if the FFT-algorithm is used.2
Generally, we can write
CptxNT", (5)
where typically k ~ 6 — 7 for isolated neutron-star searches. The linear scaling with
N comes from the fact that we need to compute the correlation of each template
with each of the N detector time-series xx(t).
The question we are trying to answer is the following: for given computing power
Cp and a set of N detectors, which (sub)-set of N < N detectors {X} C {X} yields
the highest SNR? Using (5), we can express Tmax oc (Cp/N)1^, and inserting this
into (4), we find p(0) oc Cp/{2k) ^({X}), where the " gain function" 7 is defined as
N
7({X})^iV-1/^dx5-1. (6)
x=i
This simple algebraic function provides the sought-for criterion for the optimal
detector-network {X}, depending only on the respective noise-floors Sx and duty-
cycles dx- The optimal detector network is simply the subset {X} of detectors that
maximizes the gain-function 7({X}).
This optimal subset can be found in the following simple way: we label the
detectors X in order of decreasing dxS^1, and include exactly the first X = 1,..., N
detectors in (6) where 7 reaches a maximum. It is easy to see that this arrangement
is optimal, as either adding further detectors, or replacing any term d-^S*"1 in the
sum by another detector X' > N reduces 7.
2443
In the special case of identical detectors, the gain function 7 is strictly monotonic
with N, and so the optimal network simply consists of using as many detectors as
possible, reducing the observation time T.
3. Example application
As an example, consider a set of "typical" detectors as given in Table 1. The assumed
parameters are: LIGO (HI, H2, LI) at design sensitivity, with S5 duty-cycles, GEO
(Gl) at S5 sensitivity, and S4 duty-cycle, Virgo (V2) at design sensitivity, assuming
a "typical" LIGO duty-cycle. We see that our simple criterion tells us that for a
Table 1. Example set of detectors with "typical" sensitivities and duty-
cycles.
dx
V^x" [10-23/>/Hi]
V'Sx" [10-23/VBi)
Frequency
/ = 200 Hz
/ = 600 Hz
HI
0.71
2.9
7.5
LI
0.59
2.9
7.5
H2
0.78
5.8
15
Gl
0.97
73
39
V2
0.7
4.4
5.5
-1 1.35 | , 1 , , ,
1.3 - r_-_- ----- " ,
/'
1.25 - /
/
1.2 - /
/
- l^_ 1.15 -
1-1 "
1.05 -
0.95 I ' ' ' ' '
V2 +H1 +L1 +H2 +G1
Fig. 1. SNR gain \J"t{{X}) (assuming k = 6) as a function of the detector network, normalized
to the single-detector case Left figure: at f = 200 Hz. Right figure: at f = 600 Hz.
search at / = 200 Hz we should include HI, LI, V2, and H2 for the best all-sky
sensitivity per computing cost, gaining on average a total of about 40% in SNR over
HI alone. Similarly, at / = 600 Hz, we find the same set of detectors to be optimal,
with H2 providing a smaller marginal improvement.
References
1. R. Prix, Phys. Rev. D. 75, p. 023004 (2007), (preprint gr-qc/0606088).
2. P. Jaranowski, A. Krolak and B. F. Schutz, Phys. Rev. D. 58, p. 063001 (1998).
^
1.45
1.4
1.35
1.3
1.25
1.2
1.15
1.1
1.05
1
0.95
FIRST COINCIDENCE SEARCH AMONG PERIODIC
GRAVITATIONAL WAVE SOURCE CANDIDATES
USING VIRGO DATA
F. ACERNESE6, P. AMICO10, M. ALSHOURBAGY11, F. ANTONUCCI 12, S. AOUDIA7,
P. ASTONE12, S. AVINO6, D. BABUSCI4, G. BALLARDIN2, F. BARONE6, L. BARSOTTI11,
M. BARSUGLIA8 , F. BEAUVILLE1, S. BIGOTTA11, S. BIRINDELLI11, M.A. BIZOUARD8,
C. BOCCARA9, F. BONDU7, L. BOSI10, C. BRADASCHIA11, S. BRACCINIU,A. BRILLET7,
V. BRISSON8, L. BROCCO12, D. BUSKULIC1, E. CALLONI6, E. CAMPAGNA3,
F. CARBOGNANI2, F. CAVALIER8, R. CAVALIERI2, G. CELLA11, E. CESARINI3,
E. CHASSANDE-MOTTIN7, N. CHRISTENSEN2, C. CORDA11, A. CORSI12,
F. COTTONE10, A.-C. CLAPSON8, F. CLEVA7, J.-P. COULON7, E. CUOCO2, A. DARI10,
V. DATTILO2, M. DAVIER8, M. del PRETE2, R. de ROSA6, L. di FIORE6,
A. di VIRGILIO11, B. DUJARDIN7, A. ELEUTERI6, I. FERRANTE11, F. FIDECARO11,
I. FIORI11, R. FLAMINIO1^, J.-D. FOURNIER7, O.FRANCOIS2, S. FRASCA12,
F. FRASCONI2, 11, L. GAMMAITONI10, F. GARUFI6, E. GENIN2, A. GENNAI11,
A. GIAZOTTO11, G. GIORDANO4, L. GIORDANO6, R. GOUATY1, D. GROSJEAN1,
G. GUIDI3, S. HEBRI2, H. HEITMANN7, P. HELLO8, S. KARKAR1, S. KRECKELBERGH8,
P. La PENNA2, M. LAVAL7, N. LEROY8, N. LETENDRE1, B. LOPEZ2, LORENZINI3,
V. LORIETTE9, G. LOSURDO3, J.-M. MACKOWSKI5, E. MAJORANA12, C. N. MAN7,
M. MANTOVANI11, F. MARCHESONI10, F. MARION1, J. MARQUE2, F. MARTELLI3,
A. MASSEROT1, M. MAZZONI3, L. MILANO6, F. MENZINGER2, C. MOINS2,
J. MOREAU9, N. MORGADO5, B. MOURS1, F. NOCERA2, A. PAI12, C. PALOMBA12,
F. PAOLETTI2,ll, S. PARDI6, A. PASQUALETTI2, R. PASSAQUIETI11, D. PASSUELLO11,
B. PERNIOLA3, F. PIERGIOVANNI3, L. PINARD5, R. POGGIANI11, M. PUNTURO10,
P. PUPPO12, K. QIPIANI6, P. RAPAGNANI12, V. REITA9, A. REMILLIEUX5, F. RICCI12,
I. RICCIARDI6, P. RUGGI 2, G. RUSSO6, S. SOLIMENO6, A. SPALLICCI7, R. STANGA3,
T. MARCO11, M. TONELLI11, A. TONCELLI11, E. TOURNEFIER1, F. TRAVASSO10,
C. TREMOLA11, G. VAJENTE u, D. VERKINDT1, F. VETRANO3, A. VICERE3,
J.-Y. VINET7, H. VOCCA10 and M. YVERT1
1Laboratoire d'Annecy-le-Vieux de Physique des Particules (LAPP), IN2P3/CNRS, Universite
de Savoie, Annecy-le-Vieux, France
2 European Gravitational Observatory (EGO), Cascina (Pi), Italia
INFN, Sezione di Firenze/XJrhino, Sesto Fiorentino, and/or Universita di Firenze, and/or
Universitd di Urbino, Italia
INFN, Laboratori Nazionali di Frascati, Frascati (Rm), Italia
5LMA, Villeurbanne, Lyon, France
6INFN, sezione di Napoli and/or Universita di Napoh "Federico II" Complesso Universitario di
Monte S.Angelo, and/or Universita di Salerno, Fisciano (Sa), Italia
7Departement Artemis - Observatoire de la Cote d'Azur, BP 42209 06304 Nice, Cedex 4, France
8Laboratoire de VAccelerateur Lmeaire (LAL), IN2P3/CNRS Universite de Paris-Sud, Orsay,
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France
9ESPCI, Paris, France
10INFN, Sezione di Perugia and/or Universita di Perugia, Perugia, Italia
11 INFN, Sezione di Pisa and/or Universita di Pisa, Pisa, Italia
1 INFN, Sezione di Roma and/or Universita "La Sapienza", Roma, Italia
cristiano.palomba@romal.infn.it
This paper describes the ongoing work we are doing on the blind search for continuous
gravitational waves emitted by isolated asymmetric rotating neutron stars in the data
of the interferometric detector Virgo. An optimal blind search for continuous sources
cannot be done with the presently available computing power. We have developed a
hierarchical procedure which strongly cut the computational needs, with respect to the
optimal analysis, at the cost of a small reduction in sensitivity1 We have used the data of
the two commissioning runs C6 and C7 to build two periodic source candidate data bases.
Each candidate is denned by the physical parameters of the source, namely frequency,
sky position and value of the spin-down first order parameter. We have performed an
all sky analysis, covering the frequency band 50 — 1050 Hz and spin-down in 0 — 1.52 •
10—4 Hz/day. We have done a preliminary search for coincidences between the physical
parameters of the two candidate sets. We present the full procedure and the results.
Keywords: Gravitational waves; Continuous sources; Virgo detector.
1. From the short FFT database to the Hough transform
For each data set we start from the 4 kHz h-reconstructed data and apply a data
quality procedure, which consists in the identification and removal of impulsive
disturbances. From these cleaned data the short FFT database is built. The time
duration Tfft of each FFT is chosen in such a way that the Doppler shift is less than
a frequency bin, so that the power of a periodic signal would not be spread among
more bins. This would lead to a maximum duration Tfft,max = 1-1 • 105/\/7 s
where / is the search frequency in Hertz. However in this work, for simplicity and
for saving computing power, we have decided to use the same Tfft = 1048.576 s
in the whole frequency band, with a resulting low sky resolution at low frequency.
Each FFT in the database contains also a very short periodogram, which is the
estimation of the average power, computed with an autoregressive procedure in
the frequency domain, in such a way to be not affected by narrow spectral peaks.
Then, we compute the ratio between each spectrum and its estimation, and select
local maxima above a threshold, so we build the time-frequency peak map, which
covers the frequency band [0, 2kHz] and the whole observation time for both C6
and C7, see Astone et al? for more details. The peak map is cleaned removing the
most noisy frequency intervals by setting a further threshold on the peaks frequency
distribution.
The Hough transform connects the time-frequency plane to the source parameter
space: it takes the peak map at input and produce a set of candidates at output, each
2446
defined by 4 parameters: position in the sky, frequency / and frequency derivative
/. We have carried the analysis over the frequency band [50Hz, 1050Hz], with
frequency resolution Sf = 9.5367-10~4 Hz. The sky resolution varies with frequency
from 10° at 50 Hz up to 0.5° at 1050 Hz. We searched for sources with minimum
spin-down age from 100 yr (at 50 Hz) to 2100 yr (at 1050 Hz), corresponding to
/ between 0 and 1.52 • 10~4 Hz/day; this range is covered by 40 values of / for C6
and 10 for C7, the different values being due to the different observation times. The
analysis has been partly carried on the INFN Production Grida.
2. Candidate selection and coincidences
On each Hough histogram, corresponding to a given value of / and /, we select
candidates by the use of a suitable threshold, finding nearly 5 • 108 candidates for
C6 (with false alarm probability 1.1 • 10~4) and more than 1.5 • 108 for C7 (with
false alarm probability of 1.7 • 10-4). The frequency distribution of candidates is
shown in Fig.(l). We have an excess of candidates at several frequencies, due to
Fig. 1. Candidates frequency distribution for C6 (a) and C7 (b).
disturbances in the data, even if some cleaning has been done as previoulsy said.
Moreover, there are many 'spurious' candidates due to the short observation time.
We have estimated the sensitivity of our analysis, on the basis of the data and of
the search parameters we use, see Fig.(2). With respect to the optimal analysis,
we have an effective sensitivity loss factor of 2.4 for C6 and 1.8 for C7. In a future,
work we will discuss the injection of simulated signals in the data. We have found
9.6-105 coincidences among candidates found in C6 and C7 data. The corresponding
false alarm probability is reduced at the level of 2.2 • 10-7. The coherent "follow-
up", which is not discussed here, would be done only on the coincidences with a
computational cost negligible with respect to that of the incoherent step.
We have also performed the analysis, in the frequency band [50 Hz, 550 Hz], over
two sets of data obtained by a suitable mixing of the C6 and C7 data sets, in such a
way that each of the new sets covers a larger time interval. In this way we have found,
ahttp://grid-it.cnaf.infn.it/
2447
Fig. 2. Search sensitivity for C6 analysis (red) and C7 (blue).
as expected, less 'spurious' candidates and a lower number of coincidences. A more
detailed description of the method can be found elsewhere in these Proceedings.3
References
1. S. Frasca, P. Astone, C. Palomba, Classical & Quantum Gravity 22, S1013 (2005).
2. P. Astone, S. Frasca, C. Palomba, Classical & Quantum Gravity 22, S1197 (2005).
3. P. Astone, S. Frasca, C. Palomba, this Proceedings.
PRIMORDIAL BLACK-HOLE GRAVITATIONAL WAVE
BACKGROUND NOISE IN THE LISA, DECIGO AND BBO
FREQUENCY BANDS
J. C. N. de ARAUJO*, O. D. AGUIAR§ and O. D. MIRANDAt
Divisao de Astrofisica - Institute: Nacional de Pesquisas Espaciais
Avenida dos Astronautas 1758 - Sao Jose dos Campos - 12227-010 SP - Brazil
* jcarlos@das.inpe.br § Odylio@das.inpe.br t oswaldo@das.inpe.br
According to the standard model primordial black holes (PBHs) could have been
generated during the first few moments after the big bang as consequence of density
fluctuations of matter. The Laser Interferometer Space Antenna (LISA), the DECihertz
Interferometer Gravitational wave Observatory (DECIGO), and the Big Bang Observer
(BBO) will probably detect a gravitational wave background produced by these PBHs.
Here we calculated this background as a function of the PBH population of the Galaxy.
Depending on what population is assumed the gravitational wave background produced
may give trouble for these space interferometers in their task to detect other signals. Very
large ground base interferometers such as LIGO and VIRGO can soon give information
that would put stringent constraints on this population.
1. Introduction
There is evidence from gravitational microlensing surveys of the Large Magellanic
Cloud (LMC) that ~ 20% of the Galactic halo is composed of massive compact
halo objects (MACHOs) with masses 0.15 - 0.9 Mq.1 Although the nature of
these objects is unknown, PBHs with masses of ~ 0.5 MQ have been proposed as
possible MACHO candidates.2,3 If this scenario is correct, the PBH binaries could
be a relevant source of gravitational waves (GWs) for both the ground base and
space detectors. Long baseline interferometers and resonant mass detectors could
detect their merger signals, and, on the other hand, the Laser Interferometer Space
Antenna (LISA), the DECihertz Interferometer Gravitational wave Observatory
(DECIGO), and the Big Bang Observer (BBO) could detect a background composed
by a superposition of their (almost) periodic gravitational wave signals.
This contribution addresses the question whether these background signals could
be resolvable or not for the space interferometers, in the case that PBHs exist.
2. The Background from the PBH Distribution Function
There exists in the literature a series of papers concerning the GW generated
by an ensemble of PBHs. Nakamura et al4 calculated the probability distribution
function of the PBHMACHOs binaries. They estimated the coalescence rate of
these binaries, which could be seen by the earth based interferometers. Hiscock5
and Ioka6 calculated the low-frequency GWs from these Macho binaries, de Araujo
et al7 estimated the Machos that Schenberg and Mini-Grail could see.
Since there are many papers on this issue a relevant question is: why to revisit
the PBHMACHOs? There two good reasons to do so. The first one is the paper by
Abbott et al8 (based on the LIGO second science run), in which it is estimated the
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N
]>
CO
c
cu
CO
Merger rate
in the galaxy
1 x 102/year
1 x 1Cr5/year
1 x 10-8/year
frequency (Hz)
Fig. 1. The amplitude of the spectral density for different merger rates; also plotted are the
LISA, DECIGO, and BBO curves taken from Takahashi and Nakamura.9 The dots separate the
background curve into two regions: the resolvable source region (right) and the confusion noise
(left).
rate of PBH binary coalescence in the Galactic halo. The second one is related to
the two new space projects for the detection of GWs, namely, DECIGO and BBO.
The first reason affects directly the estimates of the background predicted for
the space interferometer. Since the higher (lower) the coalescence rate, the greater
(smaller) is the number of PBH in the Galactic halo, as a result the higher (lower)
is the amplitude of the background produced by the PBHs.
We have calculated the PBH background noise starting from the distribution
function proposed by Nakamura et al4 for the PBH binaries created in the early
Universe, and evolve this distribution function to the present. We have assumed
circular orbits, that all PBHs have 0.5 M©, and a Milky Way (MW) halo radius of
50 kpc. Our results agree quite well with Hiscock5 and Ioka et al6 for / > 10~3 Hz.
Recall that they take into account the contributions of higher harmonics.
Fig. 1 shows the amplitude of the spectral density for different merger rates;
also plotted are the LISA, DECIGO, and BBO curves taken from Takahashi and
Nakamura.9
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One important question is whether this background is a resolvable one or not. In
order to answer this question, one needs to calculate the density of PBH binaries per
frequency bandwidth for each frequency of the background curve. In Fig. 1, the dots
mark the positions on the background curve where an integration time separates
it into two regions: a resolvable source region, on the right, and a confusion noise
region, on the left. Then, for each integration time used, one can define two spectral
regions of PBH binaries: one resolvable and another non resolvable.
It can be seen from Fig. 1 that LISA could face a confusion noise background
at its low end of the sensitivity band in the case of the highest estimated rate of
PBH mergers in the Galactic halo. However, Hiscock5 and loka et al6 taking into
account the eccentricity of the PBH binary systems showed that for frequencies
below 10~3 Hz the background curve for PBH binaries becomes approximately
constant. Therefore, all background curves below 10~3 Hz in Fig. 1 should consider
this correction. However, the only background curve in Fig. 1 where this correction
makes any difference is the one for the merger rate of 1 x 10_2/year. So, a horizontal
dotted segment on that curve represents the corrected confusion background noise
below 10-3 Hz.
Note that from Fig. 1 it is clear that DECIGO and BBO are free from facing a
PBH confusion noise, because all PBH could be resolvable in their frequency bands.
3. Conclusions
If PBH binaries exist they will probably be seen by the three space interferometers.
Even for the highest estimated rate of PBH mergers for the Milky Way (~ 10~2
yr_1 MWH-1), we do not expect that PBH binaries will produce a confusion noise
very much above the low end of the LISA sensitivity band.
DECIGO and BBO will, in any case, be free from facing a PBH binary confusion
noise, because all PBH signals for them could be resolvable.
Acknowledgments
JCNA and ODA would like to thank the Brazilian agencies CNPq and FAPESP for
partial support.
References
1. C. Alcock et al. (MACHO) Astrophys. J. 542, 281 (2000)
2. K. Jedamzik Phys. Rev. D 55, 5871 (1997)
3. J. Yokoyama Prog. Theor. Phys. Suppl. 136, 338 (1999)
4. T. Nakamura et al.Astrophys. J. 487, L139 (1997)
5. W. A. Hiscock, Astrophys. J. 509, L101-L104 (1998)
6. K. Ioka, T. Tanaka, and T. Nakamura, Phys. Rev. D 60
7. J. C. N. de Araujo et al., Class. Quantum Grav. 21, S521 (2004)
8. B. Abbott et al. Phys. Rev. D 72, 082002(2005)
9. R. Takahashi and T. Nakamura Prog. Theor.Phys. 113 63 (2005)
Recent Advances in the
History of General Relativity
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THE EINSTEIN-VARICAK CORRESPONDENCE ON
RELATIVISTIC RIGID ROTATION
TILMAN SAUER
Einstein Papers Project,
California Institute of Technology 20-7, Pasadena, CA 91125, USA
tilman@einstein. caltech. edu
The historical significance of the problem of relativistic rigid rotation is reviewed in light
of recently published correspondence between Einstein and the mathematician Vladimir
Varicak from the years 1909 to 1913.
1. Introduction
The rigidly rotating disk has long been recognized as a crucial 'missing link' in our
historical reconstruction of Einstein's recognition of the non-Euclidean nature of
spacetime in his path toward general relativity.1'2 Relativistic rigid rotation
combines several different but related problems: the issue of a Lorentz-covariant
definition of rigid motion, the number of degrees of freedom of a rigid body, the reality
of length contraction,3 as well as Ehrenfest's paradox4 and the introduction of non-
Euclidean geometric concepts into the theory of relativity.5
2. Relativistic rigid motion
A relativistic definition of rigid motion was first given by Max Born.6 The definition
was given in the context of a theory of the dynamics of a model of an extended,
rigid electron, and defined a rigid body as one whose infinitesimal volume elements
appear undeformed for any observer that is comoving instantaneously with the
(center of the) respective volume element. The definition and its implications were
discussed at the 81st meeting of the Gesellschaft Deutscher Naturforscher und Arzte
in Salzburg in late September 1909.
Gustav Herglotz and Fritz Noether, in papers received by the Annalen der Physik
on 7 and 27 December, respectively, further elaborated on the mathematical
consequences of Born's definition.7 Herglotz, in particular, reformulated the definition
in more geometric terms: A continuum performs rigid motion if the world lines of
all its points are equidistant curves. The analysis showed that Born's infinitesimal
condition of rigidity can only be extended to the motion of a finite continuum in
special cases. It implied that a rigid body has only three degrees of freedom. The
motion of one of its points fully determines its motion. Translation and uniform
rotation are special cases. In particular, the definition does not allow for acceleration
of a rigid disk from rest to a state of uniform rotation with finite angular velocity.
In view of these consequences, various other definitions of a rigid body were
suggested, e.g. by Born and Noether,7,8 until it became clear that special relativity
does not allow for the usual concept of a rigid body. In other words, a relativistic
rigid body necessarily has an infinite number of degrees of freedom.9
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On 22 November 1909, a short note appeared by Paul Ehrenfest pointing to a
paradox that follows from Born's relativistic definition of rigid motion of a
continuum.10 He considered a rigid cylinder rotating around its axis and contended that
its radius would have to meet two contradictory requirements. The periphery must
be Lorentz-contracted, while its diameter would show no Lorentz contraction. The
difficulty became known as the "Ehrenfest paradox." In a polemic exchange with
von Ignatowsky,11 Ehrenfest devised the following thought experiment to illustrate
the difficulty. He imagined the rotating disk to be equipped with markers along the
diameter and the periphery. If their positions were marked onto tracing paper in
the rest frame at a fixed instant, with the disk both at rest and in uniform rotation,
the two images should show the same radius but different circumferences.
3. The Einstein-Varicak correspondence
Immediately after the 1909 Salzburg meeting, Einstein wrote to Arnold Sommerfeld
that "the treatment of the uniformly rotating rigid body seems to me of great
importance because of an extension of the relativity principle to uniformly rotating
systems."12 This was a necessary step for Einstein following the heuristics of his
equivalence hypothesis, but only in spring 1912, a few weeks before he made the
crucial transition from a scalar to a tensorial theory of gravitation based on a general
spacetime metric,5 do we find another hint at the problem in his writings.1,2
The Collected Papers of Albert Einstein recently published13 nine letters by
Einstein to Vladimir Varicak (1865-1942), professor of mathematics at Agram
(now Zagreb, Croatia). Varicak had published on non-Euclidean geometry14 and
is known for representing special relativistic relations in terms of real hyperbolic
geometry.15,16 The correspondence seems to have been initiated by Varicak asking
for offprints of Einstein's papers. In his response, Einstein added a personal tone
to it with his wife Mileva Marie, a native Hungarian Serb, writing the address in
Cyrillic script in order to raise Varicak's curiosity. After exchanging publications,
Varicak soon commented on Einstein's (now) famous 1905 special relativity paper,
pointing to misprints but also raising doubts about his treatment of reflection of
light rays off moving mirrors. These were rebutted by Einstein in a response of 28
February 1910 in which he also, with reference to Ehrenfest's paradox, referred to
the rigidly rotating disk as the "most interesting problem" that the theory of
relativity would presently have to offer. In his next two letters, dated 5 and 11 April
1910 respectively, Einstein argued against the existence of rigid bodies invoking the
impossibility of superluminal signalling, and also discussed the rigidly rotating disk.
A resolution of Ehrenfest's paradox, suggested by Varicak, in terms of a distortion
of the radial lines so as to preserve the ratio of ir with the Lorentz contracted
circumference, was called interesting but not viable. The radial and tangential lines
would not be orthogonal in spite of the fact that an inertial observer comoving with
a circumferential point would only see a pure rotation of the disk's neighborhood.
About a year later, Einstein and Varicak corresponded once more. Varicak had
2455
contributed to the polemic between Ehrenfest and von Ignatowsky by suggesting a
distinction between 'real' and 'apparent' length contraction. The reality of
relativistic length contraction was discussed in terms of Ehrenfest's tracing paper
experiment, but for linear relative motion. According to Varicak, the experiment would
show that the contraction is only a psychological effect whereas Einstein argued
that the effect will be observable in the distance of the recorded marker positions.
When Varicak published his note, Einstein responded with a brief rebuttal.17
Despite their differences in opinion, the relationship remained friendly. In 1913,
Einstein and his wife thanked Varicak for sending them a gift, commented
favorably on his son who stayed in Zurich at the time, and Einstein announced sending a
copy of his recent work on a relativistic theory of gravitation. The Einstein-Varicak
correspondence thus gives us additional insights into a significant debate. It shows
Einstein's awareness of the intricacies of relativistic rigid rotation and bears
testimony to the broader context of the conceptual clarifications in the establishment of
the special and the genesis of the general theory of relativity.
References
1. J. Stachel, Einstein and the Rigidly Rotating Disk, in General Relativity and
Gravitation: One Hundred Years after the Birth of Albert Einstein. Vol. 1, ed. A. Held
(Plenum, 1980), 1-15; see also "The First Two Acts," in J. Stachel. Einstein from 'B'
to 'Z' (Birkhauser, 2002), 261-292.
2. G. Maltese and L. Orlando. Stud. Hist. Phil. Mod. Phys. 26, 263 (1995).
3. M. Klein et al. (ed.) The Collected Papers of Albert Einstein. Vol. 3. The Swiss Years:
Writings, 1909-1911. (Princeton University Press, 1993), 478-480.
4. M. Klein. Paul Ehrenfest: The Making of a Theoretical Physicist. (North-Holland,
1970), 152-154.
5. M. Janssen, J. Norton, J. Renn, T. Sauer, J. Stachel. The Genesis of General
Relativity: Einstein's Zurich Notebook. Vol. 1. Introduction and Source. Vol. 2. Commentary
and Essays. (Springer, 2007).
6. M. Born. Ann. Phys. 30, 1 (1909); Phys. Zs. 10, 814 (1909).
7. G. Herglotz, Ann. Phys. 31, 393 (1910); F. Noether, Ann Phys. 31, 919 (1910).
8. M. Born, Nachr. Konigl. Ges. d. Wiss. (Gottingen) 161 (1910).
9. A. Einstein, Jahrb. Radioaktiv. Elektr. 4, 411 (1907); M. Laue, Phys. Zs. 12, 85 (1911).
10. P. Ehrenfest, Phys. Zs. 10, 918 (1909).
11. P. Ehrenfest, Phys. Zs. 11, 1127 (1910); 12, 412 (1911); W.v.Ignatowsky, Ann. Phys.
33, 607 (1910); Phys. Zs. 12, 164, 606 (1911).
12. M. Klein et al. (ed.) The Collected Papers of Albert Einstein. Vol. 5. The Swiss Years:
Correspondence, 1902-1914- (Princeton University Press, 1993).
13. D. Buchwald et al. (ed.) The Collected Papers of Albert Einstein. Vol. 10. The
Berlin Years: Correspondence, May-December 1920 and Supplementary
Correspondence, 1909-1920. (Princeton University Press, 2006).
14. V. Varicak. Jahresber. dt. Math. Ver. 17, 70 (1908): Atti del Cong, internal del Mat.
2, 213 (1909).
15. V. Varicak. Phys. Zs. 11, 93, 287, 586 (1910); Jahresber. dt. Math. Ver. 21, 103 (1912).
16. S. Walter. The Non-Euclidean Style of Minkowskian Relativity, in The Symbolic
Universe, ed. J. Gray (Oxford University Press, 1999), 91-127.
17. V. Varicak, Phys. Zs. 12, 169 (1911); A. Einstein. Phys. Zs. 12, 509 (1911).
THE HISTORY OF THE SO-CALLED LENSE-THIRRING EFFECT
H. PFISTER
Institute for Theoretical Physics, University of Tubingen,
D-72076 Tubingen, Germany
* herbert.pfister@uni-tuebingen. de
Some historical documents, especially the Einstein—Besso manuscript from 1913, an
extensive notebook by Thirring from 1917, and a correspondence between Thirring and
Einstein from 1917 reveal that most of the credit for the so-called Lense—Thirring effect
belongs to Einstein. I also comment on the later history of the problem of a correct
centrifugal force inside a rotating mass shell which was resolved only relatively recently.
1. The history of the so-called Lense—Thirring effect
The idea that rotating bodies may exert on test particles a "dragging force"
deflecting the particles in the direction of the rotation, was first put forward in Mach's
mechanics.1 And although Mach did not provide a concrete extension of Newton's
laws of inertia and gravitation, and although he did not perform any "dragging
experiments", Mach's mechanics was a decisive stimulus for other physicists, like
Priedlaender2 and Foppl,3 to do such things. The first concrete calculation of a
Machian dragging effect was performed by Einstein4 within a preliminary, scalar,
relativistic gravity theory. The first tensorial, relativistic gravity theory was the
Entwurf-theory of Einstein and Grossmann.5 The first applications of this
theory were performed in the so-called Einstein-Besso manuscript.6 Besides the main
objective 'perihelion advance of Mercury', this manuscript contains the following
interesting results: They derive a Coriolis force inside a spherical, rotating mass shell
(mass M, radius R), and calculate the resulting "dragging" of test particles. For the
ratio d between the induced angular velocity of test particles and the angular
velocity of the mass shell they get (in units where the gravitational constant and the light
velocity have value 1) d = 2M/3R, half the value which Thirring derived in 19187
in general relativity. This result entered also Einstein's paper,8 where he remarks
that "unfortunately the expected effect is so small that we cannot hope to verify it
in terrestrial experiments or in astronomy". Einstein and Besso also calculate the
motion of the nodes of planets in the field of the rotating sun. In comparison with
the later result of Lense and Thirring,9 the effect in the Entwurf-theory is only 1/4
of the effect in general relativity.
The history of the origin and rise of the two main papers by Thirring and
Lense and Thirring9 can be disclosed quite well from Thirring's 156 pages notebook
"Wirkung rotierender Massen" ,:0 written mainly in the time April - December 1917.
The first third of the notebook contains calculations (within the weak field limit of
general relativity) for a sphere and for a mass shell rotating with angular velocity lu,
but Thirring confines himself to the diagonal metric components whose deviations
from the Minkowski metric are of order w2. With date July 17, the notebook contains
the draft of a letter to Einstein (also published as Doc. 361 in11) in which Thirring
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tells his results, begs Einstein for his advice, and asks whether Einstein could think
of an experimental confirmation of such a "centrifugal effect" on the innermost
moon of Jupiter. Einstein's answer from August 2, 1917 (Doc. 369 in11) is quite
short but it exposes the weak points in Thirring's work hitherto in an admirably
clear and concise way: "To your example of the hollow sphere it is only to be added
that, besides the centrifugal field .... also a Coriolis field results which corresponds
to the components 541,542,543 of the potential, and which is proportional to the
first power of ui. This field acts orthogonally deflecting on moving masses, and
produces e.g. a rotation of the pendulum plane in the Foucault experiment. I have
calculated this dragging for the earth; it stays far below any measurable amount.
Such a Coriolis field is produced also by the rotation of the sun and of Jupiter, and
it causes secular changes of the orbital elements of the planets which, however, stay
far below the measurement error Nevertheless, the Coriolis field seems to be
accessible to measurement more easily than your correction terms to c/44."
The first entries in Thirring's notebook after the receipt of Einstein's letter deal
with topics he has never considered before: "Calculation of gn, 524, and 534 for the
rotating spherical shell", and "Determination of the Coriolis force". Later pages
from December 1917 contain a draft of the paper,7 and the draft of §§1-2 of paper.9
In the latter, Thirring omits the very involved expressions of order oj2 from the
notebook, and confines himself to the first order in uj. Thirring's notebook contains
no hints to §§3-4 of the paper9 (transformation of the equations of motion from
Cartesian coordinates to the orbital elements used in astronomy, and application
to the planets and moons of the solar system). Herefrom, and from other sources
it is plausible that these (and only these) parts were calculated and formulated by
Lense. From the above analysis of the Thirring notebook — for a more detailed
analysis of this notebook see the preprint12 —, and of other historical documents
I come to the conclusion that most of the credit for the so-called Lense-Thirring
effect belongs to Einstein, much less to Thirring, and even less to Lense.
2. A correct centrifugal force inside a rotating mass shell
Thirring7 who assumed that the mass shell is spherical, and consists of dust
particles, got for the acceleration of test particles inside the shell
8M . AM 2 . 8M . 4M 2 .. 8M 2
X = -^W2/+15i?WX; 2/ = ^WX+15i?W2/; Z = "l5i?WZ'
where the first parts (of first order in uj) represent a Coriolis-type acceleration 2u; x r,
with "dragging factor" d = AM/3R. However, if the interior of the rotating mass
shell would realize Mach's idea of 'relativity of rotation', the w2-parts should
represent a structurally correct centrifugal acceleration, without the axial z-component.
Later research by other authors revealed the following deficiencies of Thirring's
calculations: In order ui2, the mass shell cannot consist of dust, and it has to have a
non-spherical geometry and mass distribution. Furthermore, correct inertial forces
can only be expected if the interior of the shell is flat space-time, and if the model
2458
is treated at least up to order M2. In 1966, Brill and Cohen13 treated the mass shell
exactly in M but only in first order in u>, where the shell can still be spherical, and
where the interior flatness then is trivial. Their main (Machian) result was that in
the collapse limit M —> 2R the dragging factor attains the value d = 1: complete
dragging of test particles by the rotating shell.
An extension of this work to higher orders of w, and the final solution of the
problem of a correct, gravitationally induced centrifugal force, had to wait until 1985.
In14 it was proven in all orders u>n that a rotating flat interior metric (which
automatically establishes correct Coriolis- and centrifugal forces, and no other forces)
can be connected to a series ansatz Y17=ofi (r)-Pj(C0S^) (* = 1>--4) f°r the 4
exterior metric functions of a stationary and axisymmetric rotating body, through a
mass shell with, to begin with, unknown geometrical and material properties. The
continuity conditions (in isotropic coordinates) between interior and exterior metric
then uniquely fix the (non-spherical) shape of the shell, the degree of its differential
rotation, and the exterior functions fl (r). The discontinuities of df[ (r)/dr at the
shell then produce the (non-spherical) components of the energy-momentum tensor
of the shell.
References
1. E. Mach, Die Mechanik in ihrer Entwicklung, (Brockhaus, Leipzig, 1883).
2. B. and I. Friedlaender, Absolute oder relative Bewegung?, (Simion, Berlin, 1896).
3. A. Foppl, Sitzb. Bayer. Akad. Wiss. 34, 5 (1904).
4. A. Einstein, Vierteljahrschrift gerichtl. Medizm u. offentl. Sanitatswesen 44, 37 (1912).
5. A. Einstein and M. Grossmann, Entwurf einer verallgemeinerten Relativitatstheorie
und einer Theorie der Gravitation (Teubner, Leipzig, 1913).
6. M. J. Klein et al. (eds.), The Collected Papers of Albert Einstein, Vol.4, pp. 344-473
(Princeton Univ. Press, Princeton, 1995).
7. H. Thirring, Phys. Zs. 19, 33 (1918). Errata in Phys. Zs. 22, 29 (1921).
8. A. Einstein, Phys. Zs. 14, 1249 (1913).
9. J. Lense and H. Thirring, Phys. Zs. 19, 156 (1918).
10. H. Thirring, Wirkung rotierender Massen, (Zentralbibl. f. Physik, Univ. Wien, 1917).
11. R. Schulman et al. (eds.), The Collected Papers of Albert Einstein, Vol.8, (Princeton
Univ. Press, Princeton, 1998).
12. H. Pfister, On the history of the so-called Lense-Thirring effect, preprint 2681 at
http://philsci-archive.pitt. edu/(2005).
13. D. Brill and J. Cohen, Phys. Rev. 143, 1011 (1966).
14. H. Pfister and K. H. Braun, Class. Quant. Grav. 2, 909 (1985). Class. Quant. Grav.
3, 335 (1986). H. Pfister, Class. Quant. Grav. 6, 487 (1989).
M.-A. TONNELAT'S RESEARCH CONCERNING
UNIFIED FIELD THEORY
HUBERT GOENNER
University of Gottingen
Institute for Theoretical Physics
Priedrich-Hund-Platz 1
D-37077 Gottingen
1. Introduction
The unification of all fundamental interactions by one single theory, Unified Field
Theory (UFT), is an old dream. In the history of the development of classical UFT,
I will look at one period, i.e. the late fourties and mid fifties of the 20th century,
and one research group in Paris around Mme. M.-A. Tonnelat. At the time, the
main interest in theoretical physics had shifted to quantum mechanics and its many
applications. However, possibly due to L. de Broglie's reserve toward the statistical
interpretation of quantum mechanics, classical or semi-classical approaches
seemingly were favoured by his students and coworkers in Paris.
In the early 1920s, when Einstein started to try and realize this dream, only two
fundamental interactions were known: the electromagnetic and the gravitational. In
1937 the muon became known, since 1947 also the pion. In the mid fifties, nuclear
theory had evolved, the strong and weak nuclear forces were discussed (Beta-decay,
Fermi-theory), quantum field theory had made progress. In fact, in 1945, with L.
de Broglie and L. Leprince-Ringuet, M.-A. Tonnelat contributed to a book on the
experimental and theoretical aspects of mesons (de Broglie 1945). Nevertheless,
Einstein's concept of unifying fields via geometry remained the aim of researchers like
E. Schrodinger, M.-A. Tonnelat, and V. Hlavaty. Among the geometries studied in
Paris at the Institut Henri Poincare were both inetric-affine geometry and Rieman-
nian geometry in five dimensions (Kaluza-Jordan-Thiry). We will deal only with
the first one.
2. Metric-affine geometry
Metric-affine geometry is characterized by two independent geometrical objects: an
asymmetric metric gik, (i, k = 0,1, 2, 3) and an affine connection L-kJ. The metric
may always be decomposed into its symmetric and skew-symmetric parts:
gik = hik + fik (1)
where hik = g(lk) =: l/2(gik + gik), fik = g[ik] =: l/2(gik - gik). In order to
interpret hik as the physical metric its signature is taken to be Lorentzian, i.e.,
= ±2 in space-time; also h =: det(hik) ^ 0 is assumed. In a unified field theory
of gravitation and electromagnetism, the 2-form f^v sometimes is interpreted as
representing the electromagnetic field tensor.
2459
2460
The connection may be split into
Lij = L(ij) + SV ^2)
with the torsion tensor Si;jk = Lj.jj. In general, L{ik} ^ {kj}g =: \gk\gu,j +
9ij,i ~~ 9ij,i)- Besides the affine connection Lik the Riemannian connection {kj}h ='■
^hkl(huj + hijti — hijj) always is present. In view of general covariance, metric-
affine geometry contains (16 — 4) + 64 = 76 variables taken as degrees of freedom,
as compared with the (10 — 4) + 6 = 12 variables needed for a description of the
gravitational potentials and electromagnetic fields. Thus, 64 of the field equations
might have to be used for reducing the wealth of variables.
The components of the curvature tensor are given by;
K ijk = K i[jk}'= djLki — dkLji + LjnLki — LknL^ . (3)
Due to the asymmetry of the connection, there exists another possibility:
K ijk=djLik — dkLjj + Lmj Lik — LmkLij . (4)
+ l
In the following, we will use only K ijk and omit the plus-sign. Also, two
differing traces of the curvature tensor do exist, in general: The so-called Ricci-tensor
Kij(L) =: Klin and the homothetic curvature
H _ dLj dLi
with Lj =: i;n™. A further trace called curvature scalar oder i?icci-scalar K =:
glkKik(L) can be formed from the Ricci-tensor Kij. We note that the Ricci-
H
tensor Kij(L) is not invariant with regard to (hermitian) "conjugation"a: 9ik='-
H
gki', Lij =: Lji . On the other side, Kij(L) remains invariant with respect to
the so-called A-transformations, also introduced by Einstein:
Lki^Lkl+Skdj\. (6)
3. Unified field theory a la Einstein-Straus
Before the dynamical equations of Einstein's last Unified Field Theory can be
presented, the notion of covariant derivative in General Relativity, a tensorial
derivative, must be recalled. Its geometrical interpretation derives from the parallel
transport of objects. Applied to a tangent vector Yl and to a 1-form uii, it is:
8Yi
YV=:Q^+L»Y^ (7)
du>i i
Wiiifc=: a^ -Lki^i- (8)
aThis is the real correspondence to what has been introduced as "hermitian" in a theory with
complex valued objects by Einstein (Einstein 1945, 1948).
2461
In Riemannian geometry, the metric is covariantly constant:
9ik\\j =■ -g^j- ~ 9lk{lij} - 9il{lkj} = 0. (9)
A consequence of (9) is that norms of vectors and angles between them are conserved
during parallel transport along a curve. For an arbitrary afHne connection, Einstein
defined another covariant derivative which, if applied to the asymmetric tensor gik,
leads to:
9^ =: J^ ~ 9ikL{/- guL^ =0. (10)
(note the position of the indices in the connection!). This amounts to the use of two
+ k
connections Ly =: L{ik} + Si;jk = Li;jk and £;/ = : L^ - S{jk = Ljtk. If there
exists a geometrical interpretation of (10), it has nothing to do with the conservation
of norms and angles under parallel transport13.
In his journey through mathematical landscapes with only a very few sign posts
from empirical physics, Einstein first derived the following field equation (Einstein
1925):
dqlik}
Ktj (L) = 0; -JL_ = 0; gtm = 0, (11)
where gtk =: \/—det{gni) glk.
Later, he focussed on two other sets of field equations, both leading to vanishing
vector torsion Sj = S^1. The first is called the strong field equations (Einstein
1945)c:
Ktj (L) = 0; Si =: L^ = 0; g^ = 0. (12)
(12) reduces to Einstein's vaccum field equations in Riemannian geometry
Kij (L) ^ IUj {{lnm}h) = 0.
The second set of field equations, the Einstein-Straus weak equations (Einstein
& Straus 1946), is given by:
K(ij) {L) = 0; K[ij]k + K[ki]j + Kyk]j = 0, Si = 0; g,k}n = 0. (13)
Inclusion of the cosmological constant A leads to:
K(ij) (L) = Agij; K^^ + K[kqj + K[jk^i = A fijk, Si = 0; gzkU = 0 , (14)
where fijk =: dkfij + djfkt + difjk. In some interpretation, the electrical current
density is linked to el^k(K[ij]^k + K[ki]j + K[jk]A).
One of the main difficulties of this approach to a unification of the
gravitational and the electromagnetical fields within metric-affme geometry consists in the
bM. Bray refers to F. Maurer-Tison for having given a geometrical interpretation. Cf. Bray 1960,
p. 16-17, and Maurer-Tison 1958, p. 17-37. I have not yet been able to read her thesis.
cHere, we neglect Einstein's use of hermitian variables in a complex valued version of the theory.
2462
mismatch between the multitude of geometrical quantities and the few physical
variables. In fact, the metric alone would have sufficed to house the physical vari-
ablesd; why then introduce the connection as an independent quantity? Because
Eddington had done so? Because Weyl had claimed it formed a natural extension of
Riemannian geometry? Because then more possibilities exist for the identification
of geometric objects and physical observables? E.g., Kyik-\ may be interpreted to be
the electromagnetic field tensor.
In any case, the first step taken by those who seriously tried to solve the field
equations (13) consisted in expressing the afHne connection L -kl as a functional of
the metric and its first derivatives: L -fc' = Ljk(gmn,dkgab) • This turned out to be
a difficult task; Einstein himself was unable to find the solution.
4. Progress made by Marie-Antoinette Tonnelat
Here, contact with M.-A. Tonnelat, nee Baudot (1912-1980) and her research group
in the Institut Henri Poincare (IHP) comes naturally. She studied with Louis de
Broglie and has written her PhD-thesis "Theory of the photon in a Riemannian
space" with him in 1939 e. It seems that she received her degree only in 1941.
During the German occupation of Paris she continued to work with de Broglie
and on her own in the field of (relativistic) "spin-particles", in particular particles
with spin 1 (photon) and 2 (graviton). Perhaps, her interest in the unified field
theories of Einstein and Schrodinger was kindled during an interaction with E.
Schrodinger in Dublin, in 1946. After having been Maitre de recherche, directrice de
recherche au C.N.R.S. and Maitre de Conferences at the Sorbonne (1945-1955), she
succeeded her teacher L. de Broglie as a Professeur a la Faculte des Sciences of the
University of Paris since 1956. Within IHP, a lively interaction between theoretical
physicists, mathematicians and natural philosophers seems to have occured. One of
the mathematicians who shared Mme. Tonnelat's interest in metric-afHne geometry
was Andre Lichnerowicz (1915-1998).
M.-A. Tonnelat started from a modification of Einstein's weak field equations
(13):
g+- \\i = 0; d,fil = 0; Klk(L) = 0, K[ij]ik(L) + K[kl]J(L) + K[jkU(L) = 0, (15)
where the covariant derivative and the Ricci tensor are formed with regard to a
connection without vector torsion L^k =: Lij-fc + §<SfSj with Si =: L,J. Thus
Li =: L,J, = 0. Moreover, flk =: */^gflk with flk being the matrix reciprocal to
fik- fnf-'1 = 5'i- Equation (15) follows from a variational principle with Lagrangian
C=g'kK,k{L).
dIn fact, K. Hattori had introduced the connection {k}natton ='■ hhkl(9ii,j-i-9ij > ~9ij i) (Hattori
1928).
eThe "second" part of the thesis was done under the supervision of Francis Perrin on "Artificial
radioactivity''.
2463
ik
A primary objective was to use the first equation g-\— jj; = 0 of (15) to express
the afHne connection as a functional of the asymmetric metric L° = L°(g;dkg)-
In contrast to Riemann-Cartan theory, i.e., a theory with symmetric metric and
with torsion, in which the connection cannot be determined as a functional of the
metric and its derivatives alone, now 64 equations for 64 variables obtain.
In her first approach during the early 1950s, and summed up in her monograph
(Tonnelat 1955, 1966), the solution is achieved by splitting gu- into its irreducible
parts. If ui -fc is defined by
Lij = {ijh+uijh , (16)
the decomposition of the first equation (15) leads to
hjlUik = Sij Ilk + Skj fu, (17)
hjlSik = j hjk + hj\\k ~ -jpjfij - (Uij Ilk ~ Ukj hi), (18)
wheref Skj is the torsion tensor of the connection Li fc, and /^-j. was defined after
equation (14). The main conclusion from (17), (18) is that the affine connection
may be expressed by its antisymmetric part, the metric and its first derivatives. In
a lengthy calculation, this antisymmetric part then is expressed by {fAh, hik, fik
and its derivatives. The process works if
g(a2 + b2)^0, (19)
where a =: 2 — f + ^£ ,b =: 2,/v^y [3 — f + f] and g,h,f are the
determinants of gik, hik, fik, respectively. As a functional of the metric, its first and second
derivatives, the Ricci tensor becomes a rather complicated expression. To then find
exact solutions of the remaining field equation in (15) is a difficult task. The most
promising approaches seemed to be to investigate special cases (spherical or axial
symmetry), or aproximate solutions.
In a later approach by M. A. Tonnelat (1958), the affine connection is expressed
by the metric as above but without a decomposition of g^v - in a similar but
very much more complicated way as in the case of the Levi Civita connection (the
Christoffel symbol). An improvement of the second method was given by (Dautcourt
1959).
5. Some further developments
V. Hlavaty used still another method to express the affine connection as a
functional of the metric (Hlavaty 1957). From the mathematical point of view, these
results of Tonnelat and Hlavaty greatly simplified the study of the weak field
equations. For physics, no new insight was gained. In order to make progress, topics
Equation (18) corresponds to eq. (3.18) of (Tonnelat 1955), p. 40 but differs slightly from it.
2464
like the equations of motion of test particles, conservation laws, spherically
symmetric exact solutions, and approximate solutions (linearization of field equations)
were studied by Mme. Tonnelat, her coworkers, and also by other groups. An exact
static, spherically symmetric solution of the weak field equations had been derived
by A. Papapetrou but did not coincide asymptotically, i.e., for r —> oo or, far from
the location of the point source at r = 0, with the corresponding solution of the
Einstein-Maxwell equations (Papapetrou 1948) - as had been hoped for. This opened
a debate about the relation between geometrical objects and physical observables.
Perhaps, the metric thought to describe the gravitational potential must not be
identified with hik. Let the inverse of gu- be given by
gik = lik + mik . (20)
Lichnerowicz now suggested to use the inverse Uk ^ hik of ltk as the genuine metric
(Lichnerowicz 1955, p. 288). Schrodinger had already used this; moreover, he had
identified the electromagnetic field with the antisymmetrical part K^k] of the Ricci
tensor. It was also shown that by another, if only very contrived definition of the
metric, the Schwarzschild solution could be obtained as an exact solution of unified
field theory (Wyman 1950). His definition of the metric an- included the torsion
tensor:
O-ik =■ hik + QiQk (21)
with qi being a complicated functional of hik, fik, and Sikl = LJ,.
Another cause for concern was how to properly derive equations of motion for
charged point-particles; it turned out to be non-trivial to reach the Lorentz-equation
even in weak-field approximation. A further problem investigated by Mme. Tonnelat
was the role of matter in metric-affme geometry: how to relate observables as electric
current density, or the energy-momentum tensor to the geometric objects available
(Tonnelat 1955, Chapt.VI; Tonnelat 1958, 1962). In Einstein's understanding, no
matter variables should appear explicitely in a unified field theory; matter must be
contained within geometry. For the electromagnetic field tensor four possibilities
were claimed to be priviledged: fik; flk;K[ik]',^kl fki- M.-A. Tonnelat opted for
fik, and also for the electric current density vector J1 = —j~el^kl(fuj\^ + f[ki\ j +
f[jk],i)- The field induction is defined by: Plk = -§f^- The introduction of an energy-
momentum tensor of matter T^ is a bit more complicated; first a symmetric metric
is to be picked, e.g. hik- Then in the symmetrical part of the Ricci tensor Ku^{L)
a term of the form of the Einstein tensor Gik{hmn) is searched for. If found, then
Tik ~ Gik.
W. Pauli demanded that the fundamental objects of metric-afHne geometry must
be irreducible with regard to the permutation group (Pauli 1963, Anm. 23, 273). In
this view, an admissible Lagrangian would be L = a g^^K^ +b g^K^k] instead
of the often used C = gtkKik.
2465
6. Conclusion
In M.-A. Tonnelat's understanding, Einstein's Unified Field Theory offers a number
of new perspectives: (1) the dynamics of both the electromagnetic and the
gravitational fields are modified such that there appears to be also an influence of the
gravitational field on the electromagnetic one; (2) Because a nonlinear
electrodynamics follows, new effects will appear - as e.g. "a diffusion of light by light". (3)
The relation between field strengths and inductions are similar as in nonlinear Born-
Infeld theory (Tonnelat 1955, p. 10.). She seems to have been optimistical about
the importance of the theory although aware of the fact that its area of application
was unknown, and despite the many conceptional questions remaining unanswered.
M.-A. Tonnelat's opinion possibly is the same as the one ascribed by her to two
of her heros: "One may find with Einstein and Schrodinger a mixture of a certain
discourage and of great hopes".
Einstein's Unified Field Theory makes a good example for showing that extrinsic
influences may be as important in driving research as ideas coming from physics or
mathematics themselves. It seems that most in the group of young workers busy
with Einstein's UFT after the 2nd world war became enticed by Einstein's fame and
authority - transported also through the authority of their advicers. Many of those
who wrote a doctoral thesis in the field dropped the subject quickly afterwards in
favor of work in General Relativity proper, or in some other field. A few years after
the death of Albert Einstein, research activities in UFT decreased noticeably. The
geometrical structures studied in UFT now became the playground for alternative
gravitational theories.
References
Bray, Marcel. "Quelques solutions particulieres en theorie du champ unifie." These, Paris
1960 (Faculte des Sciences de 1' Universite de Paris).
Dautcourt, Georg. "Sur la solution de l'equation d'Einstein gi+k--l = 0- " Comptes
rendus de I'academie des Sciences 249, 2159-2161 (1959).
De Broglie, Louis, (ed.) he meson. Paris : Editions de la Revue d'optique theorique et
instrumentale (1945).
Einstein, Albert (1925). "Einheitliche Feldtheorie von Gravitation und Elektrizitat."
Sitzungsberichte der Preussischen Akademie der Wissenschaften, Nr. 22, 414-419.
Einstein, Albert (1945). "A Generalization of the relativistic theory of gravitation",
Annals of Mathematics 46, 578-584.
Einstein, Albert. "A generalized Theory of Gravitation." Review of Modern Physics 20,
320-324 (1948).
Einstein, Albert und Ernst Straus (1946). " A Generalization of the relativistic theory
of gravitation". II., Annals of Mathematics 47, 731-741.
Hattori, Kanae "Uber eine formale Erweiterung der Relativitatstheorie und ihren Zusam-
menhang mit der Theorie der Elektrizitat." Physikalische Zeitschrift 29, 538-549 (1928).
Hlavaty, Vaclav. Geometry of Einstein's unified field theory. Groningen: Noordhoff
(1957).
Lichnerowicz, Andre. Les theories relativistes de la gravitation et de I'electromagnetisme.
Paris: Masson 1955.
2466
Maurer-Tison, F. "Etude du probleme de Cauchy en theorie du champ unifie d'Einstein-
Schrodinger - 3 cones characteristiques." These, Paris, 1958. (Faculte des Sciences de 1'
Universite de Paris).
A. Papapetrou. "Static spherically symmetric solutions in the unitary field theory",
Proceedings of the Royal Irish Academy 52A, no. 6, 69-96 (1948).
Pauli, Wolfgang. Relativitatstheorie. Reprint with annotations. Torino: Boringhieri 1963.
Tonnelat, Marie-Antoinette. "Schema de 1'evolution de la theorie du meson", in de
Broglie 1945.
Tonnelat, Marie-Antoinette. La theorie du champ unifie d'Einstein et quelques-uns de
ses developpements. Paris: Gauthier-Villars 1955.
Tonnelat, Marie-Antoinette. Einstein's theory of unified fields. With a pref. by Andre
Lichnerowicz ; transl. from the French by Richard Akerib. New York ; London ; Paris :
Gordon and Breach, 1966.
Tonnelat, Marie-Antoinette. "Representation de la matiere en relativite generale et en
theeorie unitaire." C'ahiers de Physique 13, 1-11 (1958).
Tonnelat, Marie-Antoinette. "Etude critique de la representation de la matiere dans
la theorie asymetrique du champ unifie." In: Les theories relativistes de la gravitation.
Colloques internationaux du Centre National de la Recherche Scientifique. Nr. 41 (Roy-
aumont 1959), pp. 199-223 (1962).
Wyman, Max. "Unified Field Theory." Canadian Journal of Mathematics 2, 427-439
(1950).
ROSENFELD, BERGMANN, DIRAC AND THE INVENTION OF
CONSTRAINED HAMILTONIAN DYNAMICS
D. C. SALISBURY*
Department of Physics, Austin College,
Sherman, TX 75090, USA
* dsalisbury@austincollege.edu
www. austincollege. edu
In a paper appearing in Annalen der Physik in 1930 Leon Rosenfeld invented the first
procedure for producing Hamiltonian constraints. He displayed and correctly distinguished
the vanishing Hamiltonian generator of time evolution, and the vanishing generator of
gauge transformations for general relativity with Dirac electron and electrodynamic field
sources. Though he did not do so, had he chosen one of his tetrad fields to be normal
to his spacetime foliation, he would have anticipated by almost thirty years the general
relativisitic Hamiltonian first published by Paul Dirac.
Keywords: history of general relativity, constrained Hamiltonian dynamics
1. Introduction and obstacles to quantizing electrodynamics
Leon Rosenfeld produced his groundbreaking constrained Hamiltonian dynamics
formalism, published in Annalen der Physik in 1930 under the title On the
Quantization of Wave Fields,1 in those heady times shortly after Dirac had achieved his
relativistic quantum theory of the electron. Heisenberg and Pauli were quantizing
the electromagnetic field while Weyl and Fock had shown how to couple Dirac's
electron field to gravity. A fundamental unification seemed imminent. The
confident young Rosenfeld, inspired by his mentor Wolfgang Pauli, proposed precisely a
quantum field theoretic unification of gravity and electromagnetism. And he came
surprisingly close! Sadly it appears that neither he nor Peter Bergmann or Paul
Dirac, both of whom began nearly twenty years later to address the problem of
converting singular Lagrangian systems into Hamiltonian models, fully appreciated
the enormous progress he made in his 1930 paper. I will sketch in this short article
only some aspects of Rosenfeld's analysis, with an effort to highlight contributions
that were independently reinvented decades later. A full translation of Rosenfeld's
work with commentary will appear elsewhere.
Emmy Noether showed in 1918 that if a dynamical model possesses a symmetry
under a transformations involving arbitrary functions then a specific linear
combination of equations of motions must vanish identically.2 Thus, for example, the
Bianchi identity in general relativity is a reflection and consequence of the general
covariance of Einstein's equations. Similarly, since classical electrodynamics is co-
variant under the gauge transformation of the electromagnetic four-potential A^,
where the transformed potential is A'^ = A^ + d^A and A is an arbitrary spacetime
function, then Noether's theorem shows that F^v must vanish identically, where
F^v is the electromagnetic field tensor. Related to this symmetry is the vanishing
of the momentum associated with the naught component of the potential. This
2467
2468
posed a problem for researchers attempting to quantize the electromagnetic field
in the late 1920's. Heisenberg and Pauli had proposed two not entirely satisfactory
methods for dealing with this embarrassment. These procedures destroyed either
manifest gauge or manifest Lorentz symmetry. Pauli is quoted having said "Ich
warne Neugierige","I forewarn the curious". Rosenfeld was in 1929 collaborating
with Pauli in Zurich, and it was Pauli who encouraged him to construct a general
manifestly symmetric formalism. Rosenfeld writes in the 1930 article (my
translation) "As I was investigating these relations in the especially instructive example
of gravitation theory, Professor Pauli helpfully indicated to me the principles of a
simpler and more natural manner of applying the Hamiltionian procedure in the
presence of identities".
Setting equal to zero coefficients of the highest time derivatives of the arbitrary
gauge functions in Noether's identities, Rosenfeld discovered three interrelated
consequences:
(1) There are as many primary constraints, i.e., identically vanishing functions of
configuration variables and momenta (conceived as functions of configuration
and velocity), as there are arbitrary gauge functions.
(2) The Legendre matrix, consisting of second partial derivatives of the Lagrangian
with respect to velocities, is singular.
(3) Rosenfeld considered only Lagrangians quadratic in velocities. Consequently all
momenta involved contractions of the singular Legendre matrix with velocities.
Therefore it was possible to add arbitrary linear combinations of null vectors to
velocities without altering the momenta. These linear combinations reflect the
arbitrariness in evolution in time of initial data.
All of these results were obtained independently by Bergmann in 1949.3
Rosenfeld then supposed that solutions had been found for all velocities in terms
of momenta, including admissible arbitrary functions, and he constructed a
Haniiltonian with the canonical expression augmented by additional linear combinations
of primary constraints. Bergmann and Brunings obtained a similar formal result
in 1949.4 Ber gmann, Schiller, and Zatkis in 1950 invented an algorithm for solving
for the velocities in terms of the momenta.5 Rosenfeld never addressed this general
question. In 1949 Dirac approached the construction of the Hamiltonian for
singular systems from an entirely different perspective.6 His work was first published in
1950. He was motivated by a desire to choose arbitrary time foliations in flat space-
time. Dirac never concerned himself with the faithful reproduction in the canonical
Hamiltonian framework of Lagrangian symmetries. This was a principle focus of
both Rosenfeld and Bergmann.
Indeed, Rosenfeld found the correct form for canonical generators of gauge
symmetries, expressed as a sum of geometric part (determined by the tensorial nature
of the variables undergoing variations, and a transport term (reflecting the fact that
active variations were contemplated at a fixed coordinate location). He proved that
his generator produced the correct variation not only of configuration but also of
2469
momentum variables. And in a culminating tour de force he proved that while the
symmetry generator contained the primary constraints multiplying the highest time
derivatives of the gauge functions, the preservation in time of the entire generator
implied that the coefficients of all lower time derivatives of the gauge functions must
themselves be constraints. In other words, Rosenfeld was the original inventor of the
what is now referred to as the " Dirac-Bergmann" algorithmn! Indeed, the
Rosenfeld analysis yielded all constraints in a single step, a perspective that conflicts with
the terms "primary", "secondary", etc. first introduced in 1951 by Anderson and
Bergmann to characterize constraints.7
2. The Hamiltonian formulation of general relativity
Rosenfeld came surprisingly close to the breakthrough published by Dirac in 1958,8
and discovered independently at about the same time by B. DeWitt (unpublished)
and Anderson.9 Dirac showed that through subtraction of an appropriate total
derivative from the Weyl gravitational Lagrangian that time derivatives of the
naught components of the metric could be eliminated, resulting in trivially vanishing
conjugate momenta. Weyl removed second derivatives of the metric by eliminating
derivatives of the Christofel tensor through the subtraction of an appropriate total
divergence.10 Rosenfeld considered a tetrad form of gravity. Similarly to Weyl, he
eliminated second derivatives of the tetrad fields by removing derivatives of the Ricci
rotation coefficients through the subtraction from the Hilbert action of an
appropriate total divergence. It turns out that If he had simply adapted his tetrad to his
spacetime foliation by taking one of the orthonormal tetrad vectors to point
perpendicular to the fixed time hypersurfaces while the remaining triads were tangent to
the foliation, he would have obtained a Lagrangian in which no time derivatives of
the orthonormal tetrad components appear. Thus he would have anticipated Dirac,
Anderson, and DeWitt by almost three decades. Had he expressed this orthonormal
tetrad in terms of the lapse and shift functions introduced by Arnowitt, Deser and
Misner he would have obtained the triad form of their ADM Hamiltonian.11
References
1. L. Rosenfeld, Ann. Phys. 5, 113-152, (1930).
2. E. Noether, Nachr. v. d. Ges. d. Wiss. zu Gottingen 1918, 235 - 257.
3. P. G. Bergmann, Phys. Rev. 75, 680 - 685 (1949)
4. P. G. Bergmann and J. H. M. Brunings, Rev. Mod. Phys. 21, 480 - 487 (1949)
5. P. G. Bergmann, R. Penfield, R. Schiller, and H. Zatkis, Phys. Rev. 30, 81 - 88 (1950)
6. P.A. M. Dirac, Can. J. Math. 2, 129 - 148 (1950)
7. J. L. Anderson and P. G. Bergmann, Phys. Rev. 83, 1018 (1951)
8. P. A. M. Dirac, Proc. Royal Soc. London A246, 333 - 343 (1958).
9. J. L. Anderson, Phys. Rev. Ill, 965 (1958)
10. H. Weyl, Raura, Zeit, Materie, (Springer, Berlin, 1918)
11. R. Arnowitt, S. Deser, and C. Misner, in Gravitation: an introduction to current
research, L. Witten, ed. (Wiley, New York, 1962)
STELLAR AND SOLAR POSITIONS IN 1701-1703 OBSERVED BY
FRANCESCO BIANCHINI AT THE CLEMENTINE MERIDIAN
LINE IN THE BASILICA OF SANTA MARIA DEGLI ANGELI IN
ROME, AND ITS CALIBRATION CURVE
COSTANTINO SIGISMONDI
ICRA & University of Rome La. Sapienza, Piazzale Aldo Moro, 5 00185 Rome, Italy
* sigismondi@icra.it www.icra.it/solar
Stellar aberration is the largest special relativistic effect discovered in astronomy (in
1727 by James Bradley), involving the speed of light when composed with Earth orbital
motion. This effect with nutation affected the measurement of latitude with Polaris
uppper and lower transits in the first week of January, 1701 made by Francesco Bianchini
(1662-1729). Equinoxes and Solstices of 1703 were measured by timing solar and stellar
transits at the Meridian Line of Pope Clement XI built in the Basilica of S. Maria
degli Angeli in Rome. Original Eastward 4' 28.8" ± 0.6" deviation of the Line affects all
measurements. The calibration curve of Clementine Line -here firstly published after 2
years of measurements- includes also local deviations of the Line, and it is used to correct
solar and lunar ephemerides at 0.3 s level of accuracy, when meridian transits are there
observed and timed.
Keywords: History of Astronomy, Astrometry, Meridian Transits, Ephemerides
1. Introduction
Upon request of Pope Clement XI the astronomer Francesco Bianchini1 built from
1701 to 1702 the Clementine Gnomon in the Basilica of Santa Maria degli Angeli
in Rome, projected by Michelangelo in Diocletian's roman baths2. The Gnomon
is a pinhole at heigth H=20344 mm, and the Line is 44899 mm long. In order to
measure accurately the tropical year and the obliquity of the ecliptic e, Bianchini
designed the Meridian Line to evaluate by interpolation both the instants when solar
longitude was exactly 0° and 180° (spring, fall equinox) and 90° and 270° (summer,
winter solstice) by timing both solar and stellar daytime transits. By studying the
position of pinhole image Bianchini obtained also the time when declination was
exactly 0° (at equinoxes) and the value of ±e° at solstices.
2. Polaris transits and the apparent celestial pole
Averaging upper and lower transits of Polaris in January 1 to 8, 1701, Bianchini
obtained the latitude of the pinhole A = 41° 54' 27". The correction for
atmospheric refraction was included. Nutation component in declination at those dates
was —4.8", while aberration for the Polaris was +20.2" in the same direction with
a net contribution of +15.4" in declination for the apparent celestial pole. Knowing
those effects, discovered later by Bradley (1727 and 1737), Bianchini would have
obtained A = 41° 54' 11.6", in perfect agreement with GPS value of A = 41° 54' 11.2",
therefore Bianchini measured the height above horizon of the apparent pole.
2470
2471
3. Equinoxes with declination estimate
The equinoctial line, perpendicular to the meridian line, is shifted Northward of
15.4", and since near the celestial equator the solar declination changes at the rate
of 59"/hour, the spring equinox evaluated by interpolating S = 0° time is 15 minutes
before the true instant, while the fall equinox is 15 minutes in delay.
4. Solar right ascensions with stellar transits
Obscuring the Church with tents, Bianchini could observe stellar transit in daytime,
as in the case of summer solstice, when the control star was Sirius, observed at noon.
Time intervals between solar and stellar transits gave the solar right ascensions with
respect to stellar ones.
5. Azimut of the Meridian Line. I. Astrometric Recognition
The delay of solar transits at winter solstice has been measured with parallel
transits3 (average of 10 transits observed and videorecorded on 10 lines parallel
to the meridian line, to avoid seeing effects) and respect to ephemerides (Observed
- Calculated at IMCCE website - [dUTl-dUT, tabulated in IERS website] -
Eastward Line's deviation/image's speed). The transit of the Sun center on the Line in
the 2005 winter solstice occurred at 217.52 times 203.44 mm (H/100), i.e. W=44252
mm from pinhole vertical. The image's speed was 3.249 mm/s and showed a net
delay of 17.864 ±0.16 s with respect to corrected IMCCE ephemerides. dUTl-
dUT=0.656 s and Line's local Eastward deviation of d = 2.72 mm = 0.67 s have
been included in the correction. R. Boscovich and C. Maire observed a delay of
17 s at winter solstice around 17502 when dUTl=<i = 0. The center of the Sun
image covered A = 58.04 mm in that time, and the Eastward azimut of the Line is
a=tan"1(A/H/) = 4' 30.5" ± 2.4".
6. Azimut of the Meridian Line. II. Topographic Recognition with
Polaris azimut
The azimut of the Northern point of the meridian line (distant 2.207 times H from
the vertical of the pinhole) very close to winter solstice point (2.1752) with respect
to the vertical of the pinhole (0) has been transported4,5 outside the Basilica, and
observing from there the polar star in upper transit (February 2, 2006) it has been
celestially referred, and it is 4' 28.8" ±0.6". Reducing to the original pinhole vertical
(identified by the center of a square box) 5" have to be subtracted.
7. Observations in 1703
The observed transits of Sirius and of the Sun occurred both later than those
predicted by ephemrides. Sirius transits were 10.9± 1.4 s (from 1703 data) with respect
to the solar transits. In summer solstice the delay of Sirius led Bianchini to consider
lower right ascensions for the Sun, and consequently, a later estimate for the Solstice
vs IMCCE ephemerides (1703 June, 22 7:56 UT vs 7:23). Opposite situation for the
2472
Deviation of Clementine Line from laser beam
E 4
OS
01
XI
_<5
it-
o
tfj ft
« U
LU
01
I -2
+
E
E -4
-J
1
-" 1
<
y-
:
T
■J
3.9
Sfc-U
Bx •
l.0'b-U)X" H ABIb-U»X
;
H^^
■4.5^t-L^:x-
■
► t >
f;;4jl
r *'
2.3( t+uj
H
T
0
/1
55
110
partes centesimae
165
220
Fig. 1. Calibration curve of Clementine Line. Laser beam begins at 0 partes centesimae and
it ends at 220.7. Such deviations are within a range of 5 mm on the 45 m Line. The systematic delay
ranges from 5.5 s [6/21] to 18.5 s [12/22] and takes into account the overall azimut of laser beam
(4' 28.8" East) and Line's local deviations from it, as well as the different speed of solar images
ranging from 3.25 mm/s at 217.5 (winter solstice) to 1.43 mm/s at 33.3 (summer solstice). With
a pinhole exactly perpendicular to 0 point, the difference [UTC of observed transit - Ephemerides
prediction - systematic delay - current dUTl]=Ai?p/l is the correction to adopted Ephemerides.
winter solstice, where all control stars are Northern than the Sun (1703 December
22, 10:54 UT vs 11:10). They were timed always in advance with respect to the Sun,
which seems to have a larger right ascension with consequent estime of the Solstice
in advance. For equinoxes control stars are both in advance or in delay with respect
to the Sun, and the estimate is much closer to true values (1703 March, 21 8:21
UT vs 8:03 and 1703 September, 23 20:06 UT vs 19:55). Aberration in declination
influences only Polaris in January and the value of latitude; Sirius and other
control stars in conjunction with the Sun have the same aberration component in right
ascension as the Sun.
Acknowledgments
Thanks to Mons. Giuseppe Blanda and Renzo Giuliano, to Rome MCM studio for
topographic measurements with LEICA TCR703, to all collaborating students.
References
1. Bianchini, F. 1703, Be Nummo et Gnomone Clementino Roma
2. Heilbron, J. L. 1999, The Sun in the Church Harvard University Press
3. Sigismondi, C. 2006, Pinhole Solar Monitor tests in the Basilica of Santa Maria degli
Angeli in Rome, Proc. of 233 IAU Symposium, Cambridge University Press
4. Ferrari, C, Monti, C. and L. Mussio, 1977, La Meridiana Solare del Duomo di Milano,
verifica e ripristino nell'anno 1976, Veneranda Fabbrica del Duomo
5. Bezoari, G., Monti, C. and A. Selvini, 2002, Topografia Generale con Elementi di Geode-
sia, UTET, Torino
Strong Gravity and Binaries
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THE EFFACING PRINCIPLE IN THE POST-NEWTONIAN
CELESTIAL MECHANICS
SERGEI KOPEIKIN
Department of Physics and Astronomy, University of Missouri, Columbia, MO 65211, USA
kopeikins@missouri. edu
IGOR VLASOV
Department of Physics, University of Guelph, Guelph, Ontario, NIG 2W1, Canada
ivlasov@physics.uoguelph.ca
First post-Newtonian (PN) approximation of the scalar-tensor theory of gravity is used
to discuss the effacing principle in N-body system, that is dependence of equations of
motion of spherically-symmetric bodies comprising the system on their internal structure.
We demonstrate that the effacing principle is violated by terms which are proportional
to the second order rotational moment of inertia of each body coupled with /3 — 1, where
/3 is the measure of non-linearity of gravitational field. In case of general relativity, where
/3 = 1, the effacing principle is violated by terms being proportional to the rotational
moment of inertia of the forth order. For systems made of neutron stars (NS) and/or
black holes (BH) these terms contribute to the orbital equations of motion at the level
of the third and fifth PN approximation respectively.
It is well-known that in the Newtonian physics as well as in general relativity the
external gravitational field of an isolated body having non-rotating, spherically-
symmetric distribution of mass, does not depend on the specific internal structure
of the body, and is completely determined by a single parameter that is mass of
the body. This property of the gravitational field is called the effacing principle.1
Effectively, the gravitational field of the spherical body is equivalent to the field of
a point-like mass located at the center of mass of the body.
When several bodies form a self-gravitating system they interact to each other
and disturb the interior distribution of matter via tidal field. In the Newtonian
physics, this disturbance induces body's ellipticity, and leads to appearance of higher
multipole moments of the gravitational field of the body describing violation of the
effacing principle in the N-body system. Violation of the effacing principle makes the
equations of motion of the bodies different from those of the point-like masses. In the
Newtonian physics and for a given precision of calculation of equations of motion of
celestial bodies one can postpone the violation of the effacing principle by making
the characteristic distance between the bodies large enough, thus, reducing the
tidally-induced multipole moments to negligible order.2 Indeed, the tidally-induced
orbital force3
^tide ^ «tide f ^ J f^J ?N , (1)
where FN = GM2/R2 is the Newtonian gravity force for a point-like mass, M and L
are characteristic mass and size of the bodies, R is the average distance between the
bodies, G is the universal gravitational constant, ve is the body's escape velocity,
vs is the speed of sound inside the body's interior, and retide is a numerical factor
2475
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depending on the internal distribution of density. Decreasing the ratio L/R can
make Ftide < FN.
The problem of generation of gravitational waves by coalescing NS/BH binaries
makes it important to study the problem of violation of the effacing principle in
general relativity and alternative theories of gravity. We have used the scalar-tensor
theory of gravity to explore this problem in the first PN approximation.4 We assume
that each body of the N-body system has the center of spherical symmetry located
at the center of mass of the body that coincides with the origin of the local
coordinates associated with this body. It means that all functions characterizing internal
structure of the body have spherically-symmetric distribution in the local
coordinates. We also assume that each body rotates rigidly around its center-of-mass. The
rotational deformation leads to the orbital force
^^-(SHi)5^' (2)
where vr ~ uiL is the linear velocity of the body's rotation, w is the angular
rotational frequency, and rerot is a numerical factor depending on the internal
distribution of density. Making L/R sufficiently small one can neglect Frot.
In the first PN approximation of the scalar-tensor theory the orbital equations
of motion are4
MBaB =FN+ -Ftide + Frot + e2 (Feih + Fso + Fss + FB + Fi + AFB) , (3)
where e is a PN book-keeping parameter, Mb and as are the relativistic mass and
the orbital acceleration of the body B, FN is the Newtonian force, -Ftide and Frot
are perturbing forces caused by the tidal and rotational defomrmations of the body,
.Feih is the Einstein-Infeld-Hoffmann force , Fgo and F$s are the PN forces due to
the spin-orbit and spin-spin coupling, Fb is the PN force due to the second moment
of inertia of the body, F\ is the force due to the forth and higher-order moments of
inertia, and AFb is the PN force due to the second and higher order moments of
inertia that exists only in the scalar-tensor theory of gravity.
The PN forces are approximated as follows:4
FEm ^ {^f FN , (4)
*-©(t)(§)f- «5»
Wr\2 /i"2
Faa^\i) U1 Fn> (6)
2
»""®'&F"- <7)
2477
2 (L
R
2
AFb~k(/3-1) - d Fw' (8)
4
Fj-AH'm FN, (9)
where k and A are numerical factors depending on the internal distribution of density
inside the stars, and (3 is the non-linear gravity-coupling parameter of the PPN
formalism.5
One can see that if stars have finite radius, there are the PN corrections to the
EIH force that describe motion of point-like masses. The finite size PN forces (7)-
(9) are governed by the rotational moments of inertia which crucially depend on the
internal structure of the stars even if they are spherically-symmetric. This property
of the PN mechanics differs from the Newtonian mechanics of N-body problem.
PN forces Fso and Fss do not violate the effacing principle. We have proved4
that the force Fb can be completely eliminated from the equations of motion by
choosing relativistic definition of the center of mass.4 Hence, this force is not
physical, and must be excluded from the theoretical analysis of the equations of motion.
However, Fb is to be retained for proper analysis of observations as the center of
mass of each star is not known before the observations have been done, and must
be considered as a fitting parameter.6
It is instructive to evaluate the limiting case of condensed astrophysical bodies
like NS and BH. In this case, radius L of the star is close to the Schwarzshild radius
Rg ~ 2GM/c2. We assume that BH is rotating with a limiting speed approaching
c. Then, the forces (l)-(9) are reduced to the following expressions
Ftide
Feih
Fb
One can see that for the condensed astrophysical objects the effacing principle is
violated in general relativity only in terms of the 5-th PN order. In scalar-tensor
theory of gravity ((3 =£ 1) this violation is of the 3-d PN order.
References
1. Damour, T. 1983, in: Gravitational Radiation (Amsterdam: North-Holland), pp. 59-
144
2. Kopeikin, S. M. 1985, Sov. Astron., 29, 516
3. Alexander, M. E. 1973, Astrophys. Space Set., 23, 459
4. Kopeikin, S. & Vlasov, I. 2004, Phys. Rep., 400, 209
5. Will, C. M. & Nordtvedt, K. J. 1972, Astrophys. J., 177, 757
6. Kopeikin, S. & Makarov, V. 2006, arXiv. astro-ph/0611358
«tide ( — J F/v , Frot — Krot
(^) FN , Fso ^ Q Fv ,
«Q6Fv, AFb ~ (J3 - 1)FB ,
(!)"*■ 4
FSS^(|)V„.
Fl ~ A (-J Fv .
(10)
(11)
(12)
GRAVITATIONAL WAVES OF A LENSE-THIRRING SYSTEM
MATYAS VASUTH and JANOS MAJAR
KFKI Research Institute for Particle and Nuclear Physics
Budapest 114, P.O.Box 49, H-1525 Hungary
E-mail: vasuth@rmki.kfki.hu, majar@rmki.kfki.hu
We evaluate the gravitational wave polarizations for inspiralling compact binaries in the
extreme mass ratio limit and discuss the effects caused by the rotation of the central,
massive body. The formal expressions of the polarization states are given for eccentric
orbits up to 1.5 post-Newtonian order beyond the quadrupole approximation.
1. Introduction
The detection of gravitational radiation is expected by the gravitational wave
observatories in the near future. Compact binary systems are among the relevant sources
of gravitational waves since they generate well defined chirp signals. Depending on
the complexity of the system these signals can be characterized by many
parameters. Operating at low frequencies the inspiral of stellar mass compact objects into
supermassive black holes is one of the most important sources for LISA.1
As a first approximation we are considering compact binaries in the extreme mass
ratio limit. The motion of the binary system is described by the Lense-Thirring
approximation,2 i.e. with the geodesic motion around a spinning body. To analyze the
effects of the rotation up to 1.5 post-Newtonian (PN) order, we focus on terms linear
in the spin of the central, massive body. The explicit form of the vectors describing
the relative position of the binary and the detector is given which are necessary to
express the polarization states h+ and hx. Having in hand the description of the
classical motion we calculate the analytic expressions of h+ and hx of the emitted
gravitational waves for eccentric orbits including higher order corrections beyond
the quadruple term. The description of the method is completed by giving the
explicit contributions to the gravitational wave signal which belong to different PN
orders, polarizations and spin effects.
The classical motion of the binary is described by a test particle with mass
m orbiting around a massive, M 3> m, rotating body. The mass ratio m/M is
negligible and the Lagrangian of the system is
uf2 GiiM 2Gu „ ,.
£=V+^+^s-(rxr)< (1)
where /i = mM/(m + M) & m is the reduced mass of the system and S denotes the
spin vector of the central mass.
When the two bodies have comparable masses the dynamics is determined by the
equations of motion and the spin precession equations.3 Since the Lense-Thirring
dynamics describes geodesic motion the spin vector is considered constant.
Moreover, according to an order of magnitude estimate of the precession equations the
change of the spin is S ~ ^e which can be neglected in our approximation.
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To describe the dynamics of the orbiting bodies we use the results of [4], where
the complete radial and angular dynamics are given in the Lense-Thirring
approximation. Moreover, an appropriate parametrization of the orbit is developed5 for
the integration of the dynamics. We chose comoving coordinates and perform Eu-
ler rotations, r = Rz(§)Rx(i)Rz(,$)ro, to place the system in a general
orientation. This orientation is determined by the condition that in this invariant system
the z axis is aligned with the constant spin vector. In the comoving system the
components of the relative separation, velocity and spin vectors are ro = (r, 0,0),
v = («|| = r,v±,0) and S = S'(sin4'sin/,, cos'J sin/,, cos/,), respectively, t is the
angle between the Newtonian orbital angular momentum L^r and the spin and $ and
^ denotes the orientation of the separation vector and the x axis of the invariant
system with respect to the node line.
The components of the orthonormal triad (N,p, q), which vectors are used to
express the polarization states h+ and hx, is expressed in terms of the angles $.
'J, i and 7. N is the direction of the line of sight, p is chosen to be perpendicular
to the Newtonian angular momentum and q = N x p. Since N and S are constant
vectors we set the y axis of the invariant system that Ny vanishes. In this case
N = (sin7, 0,cos7), where 7 is the angle between N and S. We will use these
vectors and the comoving system to express the polarization states.
2. Polarization states
The signal of a laser interferometric gravitational wave detector can be decomposed
into two polarization states,
h(t) = F+(a,0,Z)h+(t)+Fx(a,l3,Ohx(t) , (2)
where the angles a, (3 and £ in the beam-pattern functions F+ and Fx describe the
relative orientation of the detector and the source. The independent polarization
states h+ and hx can be projected out from the metric perturbation /i^T,
1 •• 1
h+ = TfiPiPj ~ qiq3)h%TT > h-x = 2^Piqj + qiP^hTT ■ (3)
In the post-Newtonian approximation hl^T can be written as
'qH + po.5Qij + pQtJ + pQi*o + piSQij + pi-5gyQ] (4)
up to 1.5 PN order, where D is the distance between the source and the observer. The
explicit form of the different terms given in [6,7]. These terms are the quadrupole,
higher order PN corrections and spin terms. They can be given as functions of the
dynamical elements, namely r and v.
To evaluate the polarization states we substitute the components of the vectors
N, p, q and S into the transverse-traceless tensor, given by Kidder.6 Similarly to
h13
77
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hjiT the polarization states can be decomposed into different contributions,8
fr+ = (r2 ~ —) (Pi ~ ql) + 2v±f(PxPv - qxqy) + v2±{p2y - q2y) ,
hlS° = ^ [(qSK + (pS)qx] , (5)
/li5S° = -J [3v±Sz(pl - ql) + r[S X (Pxp - qxq)}x - 2wx[S X (pxp - qxq)]y] ,
h^ = 2 ( r2 J pxqx + 2v±r(pxqy + qxpy) + 2v2±pyqy ,
r
,1.5SO
2
^x = -£ [6^^^^ + r[S x (P:cq + gxp)]^ - 2w±[S x (pxq + ^p)]^] .
For the sake of simplicity the components of the vectors are substituted formally
and we have listed here the lowest order Newtonian terms and all the contributions
which are linear in spin.
3. Conclusions
We have presented a method for the calculation of the polarization states of
gravitational waves emitted by spinning compact binaries. We have considered eccentric
orbits and focused on the effects of the rotation of the central, massive body. The
results are given in terms of the components of the separation, the velocity and
spin vectors and the (N, p, q) triad. These results can be extended to more general
systems, i.e. binaries with comparable masses and spins.9
For circular orbits we have integrated the relation between the true anomaly
parameter5 and time t. In this case the lowest order, Newtonian expressions have
their frequency twice the orbital frequency.
This work was supported by OTKA grants no. TS044665, T046939 and F049429.
References
1. K. Danzmann et al., LISA Pre-Phase A Report, Report MPQ 233 (1998).
2. H. Thirring and S. Lense, Phys. Zeitschr. 19, 156 (1918), English translation: Gen.
Relativ. Gravit. 16, 727 (1984).
3. B. M. Barker and R.F. O'Connell, Gen. Relativ. Gravit. 11, 149 (1979).
4. L. A. Gergely , Z. Perjes, and M. Vasuth, Phys. Rev. D57, 876 (1998).
5. L. A. Gergely, Z. Perjes, and M. Vasuth, Astrophys. J. Suppl. Ser. 126, 79 (2000).
6. L. E. Kidder, Phys. Rev. D52, 821 (1995).
7. C. M. Will and A. G. Wiseman, Phys. Rev. D54, 4813 (1996).
8. J. Majar and M. Vasuth, Phys. Rev. D74, 124007 (2006).
9. L. A. Gergely, Phys. Rev. D62, 024007 (2000).
YORK MAP, NON-INERTIAL FRAMES
AND THE PHYSICAL INTERPRETATION
OF THE GAUGE VARIABLES OF THE
GRAVITATIONAL FIELD
LUCA LUSANNA
Sezione INFN di Firenze, Polo Scientifico,
Via Sansone 1, 50019 Sesto Fiorentino (FI), Italy
lusanna@fi.infn.it
While in Newtonian physics space and time are absolute notions, in special
relativity (SR) only space-time (with its conformal structure identified by incoming
and outgoing rays of light) is absolute. Any notion of instantaneous 3-space and
of spatial distance is observer- and frame-dependent, since it is determined by the
arbitrary choice of a convention for the synchronization of distant clocks done by a
time-like observer. Given the observer and the convention, a M0ller-admissible 3+1
splitting of Minkowski space-time (and therefore a (in general) non-inertial frame
centered on the observer) is obtained.1 It is convenient to use radar 4-coordinates
(r, o~r) adapted to the 3+1 splitting: r is observer proper time and ar are curvilinear
3-coordinates on each equal-time 3-surface T,T with origin on observer's world-line.
In the framework of parametrized Minkowski theories,2 the dynamics of every
isolated system admitting a Lagrangian formulation is formulated in such a way that
the change of the clock synchronization convention is a gauge transformation. The
rest-frame instant form of dynamics is associated with the inertial 3+1 splitting
whose instantaneous 3-spaces are orthogonal to the conserved 4-momentum of the
isolated system.
In general relativity (GR), in globally hyperbolic and topologically trivial space-
times only global non-inertial frames, associated with M0ller-admissible 3+1
splittings, centered on time-like observers, are allowed by the equivalence principle. If
the space-time is spatially non-compact and asymptotically flat, the requirement
of absence of super-translations reduces the asymptotic symmetries to the ADM
Poincare' group. Therefore, the turning off of Newton constant G allows to de-
parametrize these models of GR to Minkowski space-time with the ADM Poincare'
generators tending to the SR generators of the matter present in the space-time
(for instance the standard model of elementary particles). The requirement of no
super-translations restricts the admissible 3+1 splittings to those having the
instantaneous 3-spaces tending to Minkowski hyper-planes orthogonal to the (weak) ADM
4-momentum at spatial infinity. In this way we get a non-inertial rest-frame instant
form of canonical gravity. Each equal-time 3-space is a rest frame of the 3-universe
and there are asymptotic inertial observers to be identified with the fixed stars.
The asymptotic Minkowski metric at spatial infinity is an asymptotic background,
allowing the avoidance to define a background in the bulk in the weak field regime.
As a consequence this class of space-times is suitable for the description of
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the solar system, of our galaxy and of the universe after the recombination era
(for cosmology probably extra asymptotic terms are needed). In absence of matter
Christodoulou-Klainermann space-times fulfil these requirements.
If the 4-metric inside the ADM action for GR is expressed in terms of tetrads
(needed for the coupling to fermions and simulating the 4-velocity and three spatial
axes (gyroscopes) to be associated with any time-like observers in each point of the
world-line), we arrive at the rest-frame instant form of canonical tetrad gravity. By
relating the arbitrary tetrads to those adapted to an admissible 3+1 splitting of the
space-time with a Wigner standard boost for time-like orbits depending on three
parameters P(a), a = 1;2,3, we arrive at a canonical basis (on each instantaneous
3-space Er) containing the lapse (N = 1 + n) and shift (N(a)) functions, the boost
parameters ip(a^ and 9 fields associated with cotriad fields e^ay on Er, plus the
conjugate momenta. There are 14 first class constraints (10 primary and 4 secondary):
a) 7 are given by the vanishing of the canonical momenta conjugate to N, N^ and
tp(a)', b) 3 are rotation constraints (e^ay i—► R(a)(b)(&(c)) e(b)r)'i c) 4 (the secondary
ones) are the ordinary super-hamiltonian and super-momentum constraint. They
are generators of Hamiltonian gauge transformations.
In particular, the gauge transformations generated by the super-hamiltonian
constraint connect different admissible 3+1 splittings of space-time: instead of the
Wheeler-DeWitt interpretation (local time evolution) they imply the gauge
equivalence of the clock synchronization conventions like in SR. Instead the time evolution
is governed by the (weak) ADM energy (a consequence of the DeWitt surface term,
which has to be added to the ADM action) plus a linear combination of the first
class constraints. As a consequence there is no "frozen picture" like in spatially
compact models of GR.
Since the 3-metric can be diagonalized with a point-dependent rotation matrix,
there is a point Shanmugadhasan canonical transformation, adapted to the 10 first
class constraints a) and b), allowing to find a canonical basis implementing the York
map.3 The new configuration variables are: a) tp(a) and 3 angles a(„) (the gauge
freedom of the tetrads, i.e. the freedom in the choice of the gyroscopes and of their
transport law); j3) the lapse and shift functions (the gauge freedom in the choice of
the local unit of proper time and of the conventions about gravito-magnetism); 7)
the conformal factor <f> = (det3g)1/6 of the 3-metric (it is the 3-volume element to
be determined by the super-hamiltonian constraint); S) 3 Euler angles 8r (the gauge
freedom in the choice of the 3-coordinates on Sr); e) two functions Ra, a = 1,2,
determining the eigenvalues of a 3-metric with determinant 1. The non-vanishing
conjugate momenta are 7r#r (to be determined by the super-momentum constraints),
TTfj, (being proportional to the trace 3K of the extrinsic curvature of Er, this gauge
variable describes the gauge freedom in in the clock synchronization convention, i.e.
in the choice of the instantaneous 3-spaces Er) and IIS (conjugate to Ra.)-
While Ra and na describe the independent degrees of freedom of the
gravitational field (the tidal effects, becoming the "graviton" in the weak field regime), the
14 gauge variables (f(a), <X(a)i N, N(a)i 9r and the momentum ir^) can be interpreted
2483
as generalized relativistic inertial effects in the chosen non-inertial frame associated
to an admissible 3+1 splitting of space-time. The Dirac-Hamiltonian density, i.e
the (weak) ADM energy density plus a linear combination of the primary first class
constraints, depends on all the gauge variables, namely on the incrtial potentials of
these inertial forces (since some of them are 3-coordinate- dependent, we are facing
the interpretational problem of the energy in GR).
By fixing the 14 gauge variables, we identify a global non-inertial frame, centered
on some time-like observer, in which there are deterministic hyperbolic Hamilton
equations for R&) Ii& and matter (if present). If we solve them with admissible
Cauchy data on an instantaneous 3-space of the non-inertial frame (Cauchy
surface), we can reconstruct the 4-metric of an Einstein space-time in the chosen 4-
coordinates adapted to the 3+1 splitting. From this dynamical 4-metric in these 4-
coordinates, we can evaluate the associated dynamical lapse and shift functions and
then the dynamical extrinsic curvature tensor. By solving an inverse problem, we can
find the dynamical 3+1 splitting of this Einstein space-time, one of whose leaves is
the Cauchy surface. Therefore, there is a dynamical emergence of the instantaneous
3-spaces (i.e. a dynamical convention for clock synchronization) in accord with the
fact that the whole chrono-geometrical structure of GR (ds2 = 4gflv(x)dxfJ' dxv) is
dynamical.
References
1. D. Alba and L. Lusanna, Generalized Radar 4-Coordinates and Equal-Time Cauchy
Surfaces for Arbitrary Accelerated Observers (2005), submitted to Int. J. Mod. Phys.
D (gr-qc/0501090).
2. L. Lusanna, The Chrono-geometrical Structure of Special and General Relativity: a
Re-Visitation of Canonical Geometrodynamics, Lectures given at the 42nd Karpacz
Winter School of Theoretical Physics, "Current Mathematical Topics in Gravitation
and Cosmology," Ladek, Poland, 6-11 February 2006 (gr-qc/0604120).
3. D. Alba and L. Lusanna, The York Map as a Shanmugadhasan Canonical
Transformation in Tetrad Gravity and the Role of Non-inertial Frames in the Geometrical View of
the Gravitational Field (2006), submitted to Gen. Rel. Grav. (gr-qc/0604086).
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Post-Newtonian Dynamics in
Binary Objects
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ACCURATE AND EFFICIENT GRAVITATIONAL WAVEFORMS
FOR CERTAIN GALACTIC COMPACT BINARIES
MANUEL TESSMER and ACHAMVEEDU GOPAKUMAR
Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universitat, Max-Wien-Platz 1,
07743 Jena, Germany
in. tessmer@uni-jena. de
Stellar-mass compact binaries in eccentric orbits are almost guaranteed sources of
gravitational waves for Laser Interferometer Space Antenna. We present a prescription to
compute accurate and efficient gravitational-wave polarizations associated with bound
compact binaries of arbitrary eccentricity and mass ratio moving in slowly precessing
orbits.
keywords: gravitational waves - methods, compact binaries
1. Introduction
It is expected that the Laser Interferometer Space Antenna (LISA) will usher in a
new era for gravitational-wave (GW) astronomy. The galactic stellar-mass compact
binaries are highly promising sources for LISA. An important feature of those
binaries, consisting of neutron stars, stellar-mass black holes or a mixture of both may
be that they will have non-negligible eccentricities [sec 4 for details].
It is therefore desirable to have accurate and efficient GW templates for stellar-
mass compact binaries in eccentric orbits. We provide accurate and efficient GW
polarizations which are restricted to the quadrupolar order, /i+|q and /ix|q,
associated with compact binaries, modeled to consist of non-spinning point masses,
moving in precessing eccentric orbits. These templates, which should be useful for
LISA, are Newtonian accurate in the amplitude and 1PN accurate accurate in the
orbital motion.
2. PN accurate inputs for templates
As an example, we display only the "cross" polarization for non-spinning compact
binaries moving in non-circular orbits, up to the quadrupolar order,2
/),x|Q(n^,r,^) = -2^^|^ + r2^-f2)sm2^-2r^cos2^}. (1)
In this expression, the symmetric mass ratio reads 77 = rriirv^/m2, mi and n%2 are
the individual masses with m = m,i + m2, R' is the radial distance to the binary
and S and C stand for cos?; and sini, respectively, i being the orbital inclination.
The dynamic variables r and 4> denote the relative separation and the orbital phase
2487
2488
of the binary in a suitably defined center of mass frame, with r = j^ and <f> = ^f ?
In order to obtain a prescription that models the temporal evolutions for hx\Q,
namely the GW phasing, we invoke the following parametric descriptions, involving
the eccentric anomaly u, for r,r,<p and <j>. For simplicity, we structurally show the
parametric description for r and r [again, see 4 for further details].
r
/ /~* \ 1/3 r i
et(Gmn)l/z .
r = smw
(1 — et cosu)
1 + 0
(?)
(2a)
(2b)
where n is the 1PN accurate mean motion, defined by n = 2ir/P, P being the
orbital period, and et is the eccentricity associated with the 1PN accurate Kepler
equation (KE) displayed below.
The explicit time evolution for H+\q and /ix|q is achieved by solving the 1PN
accurate KE, present in the 1PN accurate quasi-Keplerian parameterization, which
reads
/ = n (t — to) = u — et sin u , (3)
where / is the mean anomaly. Note that Eq. (3) is structurally identical to the
classical (Newtonian accurate) KE, only if we express the PN accurate dynamics in
terms of et, one of the three eccentricities that appear in the 1PN accurate quasi-
Keplerian parameterization. This allows us to adapt the most efficient and accurate
(numerical) way of solving the classical KE, provided by Seppo Mikkola in 1987.3
3. Time & frequency domain versions for hx\Q
Without giving any technical detail, we employ Mikkola's method to create both
time & frequency domain waveforms for the case of our special binary systems.
As a result, we present scaled /ix|q(0 for stellar-mass compact binaries for
et = 0.1 and et = 0.7 for the special set of the system's parameters listed below in
Fig. 1. The associated normalized power spectrum is also displayed here. We clearly
see, as expected, as we increase the value of et, higher harmonics with appreciable
strengths appear and the total power gets distributed among several frequencies.
2489
#
2e-05
0
-2e-05
l/ , 1
e, = 0-7 l\
1,1,1.1,1
mean anomaly, 1
frequency in units of f
Fig. 1. Time & frequency domain plots of scaled hx\q{l) f°r various eccentricities. The other
orbital parameters are mi = r«2 = 1.4M© and n = 6.28 X 10~3Hz.
Acknowledgments
We are grateful to Gerhard Schafer and Seppo Mikkola for discussions and
encouragements. This work is supported by the Deutsche Forschungsgemeinschaft (DFG)
through SFB/TR7 "Gravitationswellenastronomie".
References
1. P. Colwell, Solving Kepler's Equation Over Three Centuries (Willman-Bell, Richmond,
1993)
2. T. Damour, A. Gopakumar, B. R. Iyer, Phys. Rev. D 70, 064028 (2004)
3. S. Mikkola, Celestial Mechanics 40, 329 - 334, (1987)
4. M. Tessmer, A. Gopakumar, MNRAS 374, 721 (2007)
DIMENSIONAL REGULARIZATION OF THE GRAVITATIONAL
INTERACTION OF POINT MASSES IN THE ADM FORMALISM*
THIBAULT DAMOUR
Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France
damour@ihes.fr
PIOTR JARANOWSKI
Institute of Theoretical Physics, University of Biah/stok, Biah/stok, Poland
pio ©alpha, uwb. edu.pl
GERHARD SCHAFER
Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universitat, Jena, Germany
Gerhard. Schaefer@uni-jena. de
The ADM formalism for two-point-mass systems in d space dimensions is sketched.
It is pointed out that the regularization ambiguities of the 3rd post-Newtonian ADM
Hamiltonian considered directly in d = 3 space dimensions can be cured by dimensional
continuation (to complex d's), which leads to a finite and unique Hamiltonian as d —> 3.
Some so far unpublished details of the dimensional-continuation computation of the 3rd
post-Newtonian two-point-mass ADM Hamiltonian are presented.
Keywords: binary systems, equations of motion, point masses, dimensional regularization
1. Introduction
The problem of finding the equations of motion (EOM) of a two-body system within
the post-Newtonian (PN) approximation of general relativity is solved up to the
3.5PN order of approximation for the case of compact and nonrotating bodies [by
raPN approximation we mean corrections of order (v/c)2n ~ (Gm/(rc2))n to
Newtonian gravity]. The 3PN level of accuracy was achieved only recently. There exist
two independent derivations of the 3PN EOM using distributional (Dirac delta's)
sources; either ADM-Hamiltonian-based,1'2 or harmonic-coordinate-based.3'4 There
also exists a third independent derivation of the 3PN EOM in harmonic coordinates
using a surface-integral approach.5
To cure the self-field divergencies of point particles it is necessary to use some
regularization method. It turned out that different such methods applied in d = 3
space dimensions were not able to give unique EOM at the 3PN order. Only by
employing dimensional continuation was it possible to obtain unambiguous results.2'4
In this note we review the dimensional-continuation-based derivation of the 3PN
two-point-mass ADM Hamiltonian.
*The research of RJ. has been partially supported by the KBN Grant no 1 P03B 029 27.
2490
2491
2. ADM formalism for 2-point-mass systems in d space dimensions
We use units such that c=167rGd+i=l. We work in an asymptotically flat (d + 1)-
dimensional spacetime with Minkowskian coordinates x°, x=(x1,... ,xd). Particles
are labeled by the index a £ {1, 2}; masses, positions, and momenta of the particles
are denoted by ma, xa=(^,..., xda), and pa=(Pai, • • • ,Pad), respectively. We also
define: ra := x - xa, ra := |ra|, na := ra/ra; r12 := xi - x2, r12 := \r12\ (|v|
means here the Euclidean length of the d-vector v). The canonical variables of
the theory consist of matter variables (xa,pa) and field variables (^ij,-K13), where
the space metric jij is induced by the full space-time metric on the hypersurface
ic°=const; its conjugate 7ry can be expressed in terms of the extrinsic curvature of
that hypersurface.
Source terms in the constraint equations written down for two-point-mass
systems are proportional to the d-dimensional Dirac delta functions <S(x — xa). We use
the ADM gauge defined by the conditions (TT = transverse-traceless):
/ (]_2 \4/(<*-2)
The field momentum -k1^ splits into a TT part 7r^T and a rest n^ (traceless but
expressible in terms of a vector), irli = 7pJ' + tt^t. If both the constraint equations
and the gauge conditions are satisfied, the ADM Hamiltonian can be put into its
reduced form:
H(xa,Pa,hj^,TT^T) = - fddxA<P(xa,Pa,hj/,TTl4T). (2)
The PN expansion of the reduced Hamiltonian is worked out up to the 3.5PN order:
2
H = J2ma + HN + H1PN + H2PN + H2.5PN + H3PN + ff3.5PN + 0((v/c)8). (3)
a=l
3. Dimensional regularization of the 3PN Hamiltonian
In Refs. 1 it was shown that the Riesz-implemented Hadamard regularization of
the 3PN two-point-mass Hamiltonian performed in d = 3 space dimensions gives
ambiguous results. The ambiguities were parametrized by two numerical coefficients
called ambiguity parameters and denoted by osmetic and ^static-
Dimensional continuation consists in obtaining the 3-dimensional Hamiltonian
as \iuid~>3 H3Ptf(d), where H3PN[d) is the Hamiltonian computed in d space
dimensions. This can be done straightforwardly if no poles proportional to l/(d— 3) arise
when d —> 3 (or if one shows that these poles can be renormalized away, as happens
in harmonic coordinates4). Reference 2 has shown that out of all terms building
up the Hamiltonian density there are ten terms TA(d), A = 1,..., 10, giving rise to
poles when d —> 3. It was checked that the poles produced by these terms cancel each
other, thus limd-,3 H3P^(d) exists. Moreover, it was shown that for all other terms
the 3-dimensional regularization give the same results as dimensional continuation.
2492
Let Hfpx be the 3PN Hamiltonian obtained in Refs. 1 by using an Hadamard
"partie finie" (Pf) regularization defined in d = 3 space dimensions. To correct this
Hamiltonian one needs to compute the difference AH3p^ := \imd^3 Hsp^(d) —
-^3PN- Only ten terms Ta contribute to Ai^pN, therefore
.10 .10
Aff3PN = lim ddx J2 TA{d) - Pf /d3x ]T TA{3). (4)
"* ^ A=l •* A = l
Below we present three different methods which we used to compute AH^pn- The
details of the 2nd and 3rd method were not published so far. Knowing Ai^pN one
determines the values of both ambiguity parameters: akinetic = 41/24, astatic = 0.
1st method. In Ref. 2 AHspn was computed by means of the analysis of the
local behaviour of the terms Ta around the particle positions x = xa.
2nd method. It is possible to compute all d-dimensional integrals in Eq. (4)
explicitly. To do this one uses the Riesz formula
' Hdr ra Ji rf/2r((a + d)/2)T{{(3 + d)/2)T{ - (a + (3 + d)/2) +p+d
rir2 r(-a/2)r(-/j/2)r((a + /? + 2d)/2) 12 ' [)
and the distributional differentiation of homogeneous functions, e.g.,
f)2 1 / dni ni — fi\ 4ird/2
Pf((d-2)d"""° 6%1)-**ln ^M^-x,). (6)
rt2 V ' r* J dT(d/2-l)
3rd method. Instead of d-dimensional Dirac distributions S one uses d-
dimensional Riesz kernels 6Ea to model point particles:
*(x-xa) = £hmoMx-xa), i(-xa):=*||^- (7)
Then one uses the formula (5) to calculate the integrals in Eq. (4) and, at the end of
the calculation, one takes the limit £\ —> 0, £2 —> 0. No distributional differentiation
is needed.
We have shown that these three methods yield the same final results.
References
1. P. Jaranowski and G. Schafer, Phys. Rev. D 57, 7274 (1998); 63, 029902(E) (2001);
60, 124003 (1999); T. Damour, P. Jaranowski, and G. Schafer, ibid. 62, 044024 (2000);
62, 021501(R) (2000); 63, 029903(E) (2001).
2. T. Damour, P. Jaranowski, and G. Schafer, Phys. Lett. B 513, 147 (2001).
3. L. Blanchet and G. Faye, Phys. Lett. A 271, 58 (2000); Phys. Rev. D 63, 062005 (2001).
4. L. Blanchet, T. Damour, and G. Esposito-Farese, Phys. Rev. D 69, 124007 (2004).
5. Y. Itoh and T. Futamase, Phys. Rev. D 68, 121501(R) (2003); Y. Itoh, ibid. 69, 064018
(2004).
NEW RESULTS AT 3PN VIA AN EFFECTIVE FIELD THEORY
OF GRAVITY
RAFAEL A. PORTO
Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213
NRGR, an Effective Field Theory approach to gravity, has emerged as a powerful tool to
systematically compute higher order corrections in the Post-Newtonian expansion. Here
we discuss in somehow more detail the recently reported new results for the spin-spin
gravitational potential at third Post-Newtonian order.
A new approach, coined NRGR, has been recently introduced as a new technique to
systematically calculate within the Post-Newtonian expansion via an effective field
theory approach. ^3 The purpose of this contribution is to elaborate upon the new
results recently reported4 for the spin-spin potential. Further details will appear in
a forthcoming publication.
The extension of NRGR to include spin effects3 can be achieved by adding rotational
degrees of freedom (e^) in the worldline action. The generalized angular velocity is
given by Q^ = ev -^-, and the spin S^ is introduced as the conjugate momentum.
The form of the world-line action is then fixed by reparameterization invariance,
S = ~ E (/p^^A, + J \s^lvd\q^ , (1)
where A? is the proper length for the g'th worldline. The Papapetrou equations
follow from (l)-3 Higher dimensional terms describing finite size effects have been
left out although its inclusion is straightforward.ll3'4 In order to account for the
correct number of degrees of freedom a so called spin supplementarity conditions
(SSC) is added to the equations of motion (EOM). The most convenient choices are
the covariant, S»vpu = 0, and Newton-Wigner (NW), S»vpv = mS1*0, SSC. Notice
that the latter is not covariant, however it can be shown to have the advantage
that the algebra reduces to a canonical structure (up to subleading corrections)
after Dirac brackets are imposed.5 The leading order spin-spin and spin-orbit effects
were shown to follow from the potentials within the NW SSC.3 The 3PN spin-spin
potential, V^n, was recently obtained,4 so that the spin-spin part of the EOM
followed by means of the traditional Hamilton-Lagrange approach. As we shall see this
is a correct statement up to 4PN where curvature effects in the algebra start to play
a role.
The spin-gravity coupling in (1) can be rewritten by introducing the spin
coefficients, o^fe, as
,ab
1
2
SsPin~-- / SLabU^dX, (2)
2493
2494
with S1b the spin tensor in a local Lorentzian frame denned by the vierbein e£. In
this basis the co-rotating frame is given by e^ = A^(r)e^ with A a Lorentz boost. By
further expanding (2) in the weak gravity limit one obtains the Feynman rules.3'4
Let us emphasize here that the spin tensor appearing in the vertex rules is the one
defined in the local frame, where the NW SSC was chosen a. Before imposing the
SSC one can show that the algebra for the phase space variables [xIJ-,pv, 5£6) is
given by
{x^Va}=8^ {x»,pa} = 6£ (3)
{P°,VP}=0, {x»,x»} = 0, {pa,pp} = \RapabSl\ (4)
{x»,Slb} = 0, {pa,Slb} = 0, {Va,St»} = 0 (5)
{Sf, ScLd} = r]acSbLd + rf>dSaLc - r]adSbLc - r]bcSaLd, (6)
where p11 is related to the canonical momentum by V1 = p11 — \^ab^'h- After
the SSC is enforced a Dirac structure emerges. In flat space-time the NW SSC will
preserve the canonical structure in the reduced space {x\p\ Si), with SlL = eljkS3Lk
the spin three vector. In a curved background however, the algebra turns out to be
(7)
(8)
(9)
(10)
(11)
(12)
with the ellipses representing a series of "curvaturexspin" terms'3. In principle we
should worry about these curvature effects, however we will show by standard power
counting, its effects in the spin-spin EOM are subleading and the canonical
procedure is accurate up to 4PN. The reason is somehow intuitive. To get a correction
coming from the algebra to the Si • S2 piece of the EOM for particle 1, one needs to
consider the S2 part of the spin-orbit Hamiltonian. The latter scales as v3 relatively
to the Newtonian term. We know on the other hand that the spin-orbit EOM does
not receive any corrections at leading order (1.5PN). This is a not trivial
statement given the fact that it could be modified by a non trivial commutator with the
leading order Hamiltonian. Therefore, "algebra corrections" should start at 2.5PN.
To get a correction to the spin-spin EOM we wonld then need to hook up a 1.5PN
spin-orbit Hamiltonian with a 2.5PN algebra term, effectively a 4PN correction. Let
us consider for instance the commutator {xl,xJ} as an example. This commutator
K.^-} =
{x'.xi} =
{V\Vj} =
{x\Si} =
{Vj,Sl} =
{Sh,Si} =
= *} + ...
= 0 + ...
= 0 + ...
= 0 + ...
= 0 + ...
_ ijk ok
aOne could chose to expand the action in terms of S,iV. However, to obtain the EOM from the
potentials one would need to account for a more complicated spin algebra.
hFor example, in the electromagnetic case,5 similar to ours after the identification AM ~ u° S0|,,
the Dirac structure (in the covariant SSC) turns out to be a very cumbersome expression.
2495
in the NW SSC will receive corrections scaling as (schematically) ~ Rx2^z + •••,
with R the Riemann tensor. On the other hand, in the covariant SSC, this bracket
is modified5 to {a;*,a;-7'} = ^-, whose net effect in the EOM is a 1.5PN term,
necessary indeed to prove the equivalence for different SSCs.3 The new term has now an
extra factor scaling as d2hoorx2 at leading order (R ~ d2hoo). In the weak gravity
approximation, /ioo ~ v2, so that the algebra-correction effectively starts at 2.5PN
as we had foreseen0.
Let us add a few words on the NW SSC in a curved background and the spin
choice. The NW condition implies (for each particle)
mSf = Sfpb - S?(pn) = ^Sll(pn) + O^) (13)
where S^n is the spin tensor in the original PN frame (5£6 = e^e^S^), and vl the
three coordinate velocityd. One can also relate both spin tensors (we removed the
pn label for simplicity),
Sft = S\j + Sikh{ - S{khl + ... ~ S¥ + 4^^SiJ + ... (14)
and then transform the EOM in terms of Sl, and hence to the covariant SSC.
As we said above spin-spin subleading effects can be computed regardless of
algebra corrections up to 4PN. This is however not true for subleading spin-orbit
effects at 2.5PN,6,7 where these corrections start to contribute. We will thus finish
this short contribution with yet another approach which will naturally overcome
these difficulties in a more natural fashion.
Going back to the covariant SSC it is easy to show, from Papapetrou equations,
pa = mua - ^-Rp^S^S^u". (15)
2m
Notice that p-u = m on shell (once the SSC is obeyed). One can thus show that
the action (1) is equivalent to the following Routhian,
ft = - £ ^Jmq^qd\q + J isf,Wo6/lU£ - ~Rdeab{xq)SldqStquequl d\
(16)
There is an extra piece, Sf^SLab, n°t shown. This term does not affect the spin
EOM since it is a Casimir operator. However, it enters in the worldline evolution in
the form of a spin dependent mass. The EOM are,
cOther corrections could go as -R-^-t" and can be shown to be subleading.
dDepending on the frame choice the 0(v4) piece will change, however, the leading order condition
stays the same regardless of the choice.
2496
which can be shown to reproduce eq. (15) and Papapetrou equations on shell, e.g.
on the constraint surface S£6p& = 0e. To obtain Post-Newtonian corrections one
calculates 11 perturbatively. Notice that, had we imposed the SSC in (16) one would
get rid of the Riemann term and end up in an approach equivalent to what we
discussed before. We will proceed in a different way and we will impose the SSC
condition after the EOM for (xl,SY) are obtained from (17), while keeping the
power counting rules for spin as before,3 e.g. S£fc ~ vtStjJ. The advantage of this
approach is that one does not have to worry about complicated algebraic structures.
The price to pay is the need of a spin tensor rather than a vector. As an example
let us compute the leading order spin-orbit contribution to the spin EOM f. The
spin-orbit potential is given by (we dropped L for simplicity)
^ = ^P^' {Si0 + S(ktf - 2«*)) +1-2, (18)
with n? — (xi — X2Y ■ The relevant piece of the algebra is the commutator
{S\ Sj0} = eijkSok = vlS° - v^S1 + ..., (19)
which follows from (6) in the covariant SSC. Using (17) one gets,
dSi ( m2\^GN 3 m2GN g _
—— = 21 + — —=-(n x v) xii 5—(6i x n) x vx (20)
at \ mi J r2 rA
with fi the reduced mass and v the relative velocity. This agrees with the known
result after the shift,3
£i^(l-^?)51+iiT1(i?1-51). (21)
Details and higher order computations will appear in a forthcoming paper.
We would like to thank Gerhard Schafer for helpful discussions and bringing to
our attention the subtleties of the algebraic approach. We thank Ira Rothstein for
helpful comments and collaboration. This work was supported by DOE contracts
DOE-ER-40682-143 and DEAC02-6CH03000.
References
1. W. Goldberger and I. Rothstein, Phys. Rev. D 73, 104029 (2006)
2. W. Goldberger and I. Rothstein, Phys. Rev. D 73, 104030 (2006)
3. R. A. Porto, Phys.Rev. D 73, 104031 (2006)
4. R. A. Porto and I. Rothstein, Phys.Rev.Lett. 97, 021101 (2006)
5. A. Hanson and T. Regge, Ann. Phys. (N.Y.) 87, 498 (1974).
6. H. Tagoshi and A. Ohashi and B. Owen, Phys. Rev. D 63, 044006 (2001).
7. G. Faye, L. Blanchet and A. Buonanno, Phys.Rev. D 74 104033 (2006).
8. K. Yee and M. Bander, Phys. Rev. D 48 2797 (1993).
9. P. Jaranowski and G. Schafer, Phys. Rev. D 57 7274 (1998).
JL
be very convenient in Ref.9
fThe leading spin-spin EOM does not include Sa0 and thus follows the exact same steps.
ORBITAL PHASE IN INSPIRALLING COMPACT BINARIES *
MATYAS VASUTHt, BALAZS MIKOCZI* and LASZLO A. GERGELYt
t KFKI Research Institute for Particle and Nuclear Physics
Budapest 114, P.O.Box 49, H-1525, Hungary
^■Departments of Theoretical and Experimental Physics, University of Szeged
Dom ter 9, Szeged H-6720, Hungary
vasuth@rmki.kfki.hu, mikoczi@titan.physx.u-szeged.hu, gergely@physx.u-szeged.hu
We derive the rate of change of the mean motion up to the second post-Newtonian
order for inspiralling compact binaries with spin, mass quadrupole and magnetic dipole
moments on eccentric orbits. We give this result in terms of orbital elements. We also
present the related orbital phase for circular orbits.
Keywords: compact binaries, post-Newtonian expansion, spin, quadrupole moment
Observations by Earth-based gravitational wave observatories are under way aiming
to detect gravitational radiation. Upper limits from interferometer data were already
set on inspiral event rates for both binary neutron stars1 and binaries of 3 — 20 solar
mass black holes.2 The parameters of spinning compact binaries can be estimated
and alternative theories of gravity can also be tested from these measurements.3
An important characteristic of these binaries is the rate of decrease of the orbital
period T due to the energy and angular momentum carried away by gravitational
waves. Here we give the radiative change of the mean motion n = 2tt/T (for eccentric
orbits). We also present the related change occured in the orbital phase (for circular
orbits). In both expressions we include all known linear perturbations for an isolated
compact binary. These are the post-Newtonian (PN), spin-orbit (SO), spin-spin
(SS), self spin (Self, quadratic in the single spins), quadrupole-monopole (QM) and
magnetic dipole-magnetic dipole (DD) contributions.
The expression of the radial period, defined as half of the time elapsed between
the turning points, emerges from generic considerations on the perturbed Keplerian
motion.4'5 Collecting all linear contributions the mean motion has the following form
where 77 = /i/m is the ratio of the reduced mass fi to the total mass m of the binary
system, and £ = —E/fi where E is the conserved energy. Remarkably there are
no explicit spin, quadriipolar and magnetic dipolar contributions in the functional
form of the mean motion. These however contribute implicitly to n through £.
Since the mean motion is a function of E alone, its evolution can be computed as
"Research supported by OTKA grants nos. T046939, TS044665, F049429 and the Janos Bolyai
Fellowships of the Hungarian Academy of Sciences. M.V. and L.A.G. wish to thank the organizers
of the 11th Marcel Grossmann Meeting for support.
2497
2498
(dn/dt) = —l/fj,(dn/d£)(dE/dt). All linear contributions to the secular energy loss
(dE/dt) due to finite size effects are explicitly given in the literature,6-9 in terms of
dynamical constants. The PN contribution is also well-known.10 Employing these
we find the change of the mean motion:
!>„-"•■
l)so=-w/i(1G-!*.)3W ^+N^ • <4»
(5)
dn \ S*i S*2
d*/ss ~32c2m^a2(l-e2)2
x [^ysinK! sinK2cos2(t/)o —"0) +-^scoski cosk2+-^90087] , (6)
dn\
2 2
m
y^Pi [iVio(2-3 sin2«i) +Nn sin2Ki cos 2(ip0-i>i)]
dt/QM 4a2(l-e2)2^
(7)
(^)g^-2GJ2(l2-e2)2^^+^^)' <8>
where we have introduced the notations Sl = (SiCoski + S2COSK2), £l =
[(m2/mi) Si cos «i + (mi/m2) S2 cos K2] and
£7/2 5/2 3
Here Kj and t/>j are the polar and azimuthal angles of the ith spin vector and the
numeric coefficients n^ are:
i
0
1
2
3
j=o
96
292
37
0
1
28016
160248
34650
-5501
2
9408
43120
20916
-1036
3
2128
7936
3510
363
4
1680
7924
4224
291
5
0
16
80
9
6
64
608
552
36
7
-3072
100112
113248
8937
8
-194368
-621536
-264792
-4500
9
65216
211232
91944
1740
10
2888
9660
3897
187
11
288
-7924
-8570
-464
The orbital elements a, e were derived11 from the turning points of the
radial part of the perturbed motion cf. rmax = a (lie). (In these variables n =
mm
(Gm/a3)1/2 [1 + (r? - 9) Gm/2ac2] ).
For Keplerian motions the orbital frequency w = n. Due to the perturbations,
precessions occur in the plane of motion (like periastron advance), and the plane
of motion can also evolve. Therefore the relation of w and n becomes more com-
2499
plicated.12 In the presence of the PN, SO, SS, Self, QM and DD perturbations,
for circular orbits (e = 0) the change in the orbital frequency due to gravitational
radiation is:13'14
du\ _ 96r/m5/3wn/3
~dl
' (743 11 \ 2/3
1 - (,336+ tv(mw)
/34103 13661 59 2 \ . n4/3
v M; ^ 18144 2016 ' 18 / V ;
, (10)
where a = aSs + <rseif +o~qm +cfdd and j3, (TSiS2, <?Seif, &QM, <tdd are the spin-
orbit, spin-spin, self-interaction spin, quadrupole-monopole and magnetic dipole-
dipole parameters.14 For completeness we have added the 2PN and tail
contributions15 and we note that higher order contributions are also known.16,17
In terms of the dimensionless time variable r = rj(tc — t)/5m, denned in terms
of the time (tc — t) left until the final coalescence, the accumulated orbital phase
is <fi = <fic — (5777/77) J (jj(r)dT, where (fic is an integration constant. To second post-
Newtonian order:
3715 55
^8064 + 96'
/ yzfo^yo 284875 looo 9 10a \ -, /s 1 . ,
+ 1 77 H 772 T1/8 > . (11)
14450688 258048 ' 2048 ' 64 I K }
( 9275495 284875 1855 :
" ^14450688 + 258048^ + 2048^
induced bv the finite size effect
The modification induced by the finite size effects SS, Self, QM and DD are all
encoded in a, while the SO contribution is in j3.
References
1. B. Abbott et al, Phys. Rev. D69, 122001 (2004).
2. E. Messaritaki, Class. Quantum Grav. 22, S1119 (2005).
3. E. Berti, A. Buonanno, and C. M. Will, Phys. Rev. D71, 084025 (2005).
4. L. A. Gergely, Z. Perjes, and M. Vasuth, Astrophys. J. Suppl. 126, 79 (2000).
5. L. A. Gergely, Z. Keresztes, and B. Mikoczi, Astrophys. J. Suppl. 167, 286 (2006).
6. L. A. Gergely, Z. I. Perjes, and M. Vasuth, Phys. Rev. D58, 124001 (1998).
7. L. A. Gergely, Phys. Rev. D61, 024035 (2000).
8. L. A. Gergely and Z. Keresztes, Phys. Rev. D67, 024020 (2003).
9. M. Vasuth,Z. Keresztes, A. Mihaly, and L. A. Gergely, Phys. Rev. D68, 124006 (2003).
10. A. Gopakumar and B. R. Iyer, Phys. Rev. D56, 7708 (1997).
11. Z. Keresztes, B. Mikoczi, and L. A. Gergely, Phys. Rev. D72, 104022 (2005).
12. G. Schafer and N. Wex, Phys. Lett. A 174, 196 (1993), Erratum: 177, 461 (1993).
13. L. Kidder, Phys. Rev. D52, 821 (1995).
14. B. Mik6czi, M. Vasuth, and L. A. Gergely, Phys. Rev. D71, 124043 (2005).
15. L. Blanchet, Phys. Rev. D54, 1417 (1996).
16. L. Blanchet, G. Faye, B. R. Iyer, and B. Joguet, Phys. Rev. D65, 061501 (2002);
Erratum: D71, 129902 (2005).
17. L. Blanchet, A. Buonanno, and G. Faye, Phys. Rev. D74, 104034 (2006); Erratum:
D75, 049903 (2007).
GRAVITATIONAL WAVE EMISSION FROM A STELLAR
COMPANION BLACK HOLE IN PRESENCE OF AN ACCRETION
DISK AROUND A KERR BLACK HOLE
PRASAD BASU
Centre for Space Physics, Chalantika-43,
Garia Station road, Kolkata-700084,India pbasu@csp.res.in
S.K. CHAKRABARTI
S.N.Bose National Centre for Basic Science,
J. D block, Sector-3, Klokata-98, India.
and Centre for Space Physics, Chalantika-43,
Garia Station road, Kolkata-700084,India, chakraba@bose.res.zn
SOUMEN MONDAL
R.K.M. R. College, Narendrapur, 24-pgs. Kolkata 700100
and Centre for Space Physics, Chalantika-43
Garia station road, Kolkata-84,India, soumen@bose.res.in
KUSHALENDU GOSWAMI
Department of Physics, Jadavpur University, Kolkata-32, India,
goswami- y2k@yahoo.co.in
We consider a stellar mass black hole orbiting a primary super-massive Kerr black hole
while always staying inside the accretion disk of the primary. We show that due to the
accretion of matter from the disk, which is not necessarily Keplerian, the specific energy
and angular momentum of the companion is modified. This affects the orbital evolution of
the companion which was already lossing angular momentum and energy due to gravity
wave emission. With an illustrative example, we show that the presence of the disk could
significantly change the infall time of the companion towards the central black hole and
modify the characteristic 'chirp' signal of the binary.
1. Introduction
Traditionally, the gravity wave emission from a binary system is studied without
considering the presence of the accretion disk in the system. However, from more
than a decade Chakrabarti3'4 have been pointing out that the disk should have a
significant effect. It is believed that in many of galactic nuclei there is a super-massive
black hole, with mass ~ 107 — 1010MQ (MQ is the mass of the Sun) surrounded
by an accretion disk. This disk may be populated with a large number of small
stars including white dwarfs, neutron stars and black holes which will act as
companions. In this paper, we consider one such stellar black hole as the companion.
Being inside the disk, such a companion will start accreting matter from the disk
and with it, some energy and angular momentum. This is because the disk is likely
to be sub-Keplerian as the matter of the disk may come from stellar winds1 while
the companion will be on an instantaneous Keplerian orbit. The companion will
accrete matter of lower specific angular momentum and therefore, its net angular
2500
2501
momentum would decrease, leading to a faster infall to the black hole. In regions
of radiation pressure or ion pressure domination the angular momentum of the disk
matter is higher and the companion may gain angular momentum from the disk
and its infall would be slower.
In order to incorporate the general relativistic effects of the central super-massive
rotating black hole we use a more useful potential, namely, the Pseudo-Kerr
potential.2 We show that the accretion of matter by the companion from the disk causes
a significant exchange of energy and angular momentum between the disk and the
companion and thus this should be taken into account while interpreting the gravity
wave signals from such systems. Previously, such a computation has been carried
out in Schwarzschild geometry only.4
2. The governing equations
The equation of motion of the companion is given by,
dvr _ d$eff(r,l)
dt dr [ >
where, vr is the radial velocity and $e// is the effective potential taken from
Chakrabarti and Mandal (2006). The angular momentum emission rate due to
gravity wave emission and accretion process are respectively given by,
KdtJgw 5 C ^ ^ '
and
(dL\ -n , ^ 2^Mc (v
where, Mc,lc are the mass and specific angular momentum of the companion and
Idisk is the specific angular momentum of the disk. vrei is the relative azimuthal
velocity between the disk and the companion black hole and as is the sound speed
in the fluid of the disk.
In addition to this, we need to solve the fluid dynamical equations to get the
structure and various flow variables of the disk. The rapidity of the infall of the
companion and the resulting gravitational wave emission are then computed self-
consistently. The results for a single illustrative case are shown.
3. Discussion
In the literature, it is usual to consider binary systems which has no accretion disk.
However works of our group for the first time pointed out3 that the exchange of
energy and angular momentum between the disk and the companion is important.
In the paper also we show that this is very important when the central black hole
is a Kerr black hole. Details would be presented elsewhere. This work is partly
supported by a CSIR fellowship to PB.
2502
Kerr parameter a=0.5
Viscosity parameter a=0.05
r
2 0,4
lo,
Kerr parameter a=0.5
Viscosity parameter a=0.05
log(x)
2.2 2,3 2.4 2.5 2.6 2.7
log(x)
Fig. 1. The ratio of the angular momentum of (left) the disk and the companion and (right)
the angular momentum loss rates due to accretion and due to gravity wave emission are plotted
agaii
Kerr parameter a=0.5
Viscosity parameter a=0.05
0=0.05
Kerr parameter^O.5
2.5e+06
Fig. 2. (left) The number cycles past during the infall is plotted against the radial distance with
and without the presence of the disk. The presence of the disk hasten the merger, (right) The
Mach number vs. logarithmic radial distance of the flow as obtained in the present case. The
centrifugally supported standing shock location is indicated.
References
1. Chakrabarti, S.K.
Co. (Singapore)
2. Chakrabarti, S.K.
3. Chakrabarti, S.K.
4. Chakrabarti, S.K.
1990, Theory of Transonic Astrophysical Flows, World Scientific
and Mondal, S., 2006, MNRAS, 389,976
1993, ApJ, 411, 610
1996, Phys. Rev. D., 53, 2901
THE SECOND POST-NEWTONIAN ORDER GENERALIZED
KEPLER EQUATION *
LASZLO A. GERGELY, ZOLTAN KERESZTES and BALAZS MIKOCZI
Departments of Theoretical and Experimental Physics, University of Szeged,
Dom ter 9, H-6720 Szeged, Hungary
gergely@physx.u-szeged.hu, zkeresztes@titan.physx.u-szeged.hu, mikoczi@titan.physx.u-szeged.hu
The radial component of the motion of compact binary systems composed of neutron
stars and/or black holes on eccentric orbit is integrated. We consider all type of
perturbations that emerge up to second post-Newtonian order. These perturbations are either
of relativistic origin or are related to the spin, mass quadrupole and magnetic dipole
moments of the binary components. We derive a generalized Kepler equation and investigate
its domain of validity, in which it properly describes the radial motion.
Keywords: compact binaries, post-Newtonian expansion, spin, quadrupole moment
Compact binaries composed of neutron stars / black holes are radiating gravitational
waves. The waveform and phase of gravitational waves are strongly influenced by the
orbital evolution of these systems. Before the system reaches the innermost stable
orbit, its evolution can be well described by a post-Newtonian (PN) expansion
about the Kepler motion. As dissipative effects due to gravitational radiation only
enter at 2.5 PN orders, the orbital evolution is conservative up to the 2PN orders.
Even to this order the dynamics is complicated enough not only by the general
relativistic corrections to be added at both the first and second PN orders and by
tail effects, but by finite size effects as well. These include spins, mass quadrupolar
and magnetic dipolar moments.
From among these the spin is the dominant characteristic. The effect of the spin-
orbit coupling on the motion has been considered long time ago,1 and revisited more
recently. 2~9 This contribution suffers from the non-uniqueness in the definition of
the spins, expressed by the existence of at least three different spin supplementary
conditions. The physical results however should be independent of the chosen SSC.
The next contributions (at 2PN) are due to spin-spin coupling. M,10-12 These
include proper spin-spin contributions between the two components as well as spin
self-interactions. An effect of similar size is due to the mass quadrupoles of the
binary components. This is the so-called quadrupole-monopole interaction,1,13'14
representing the effect on the motion of one of the components (seen as a test
mass) in the quadrupolar field of the other component. The quadrupole moment
may either be a consequence of rotation or it may be not. As magnetars with
considerable magnetic field are known, the possibility of the coupling between the
* Research supported by OTKA grants no. T046939, TS044665 and the Janos Bolyai Fellowships
of the Hungarian Academy of Sciences. L.A.G. wishes to thank the organizers of the 11th Marcel
Grossmann Meeting for support.
2503
2504
magnetic dipole moments was also investigated.15,16 With both components having
the magnetic field of 1016 Gauss, the magnetic dipolar contribution provides other
2PN contributions to the dynamics.
Although the above enlisted effects emerge either at 1.5 PN (spin-orbit) or at
2PN orders (spin-spin, quadrupole, possibly magnetic dipole), they all represent
the leading order contributions of the respective type. In this sense they are linear
perturbations of the Keplerian motion. For these perturbations the radial part in
the motion of a compact binary system decouples from the angular motion. With
the aid of the turning points of the radial motion, given as r = 0 both a radial period
and suitably generalized true and eccentric anomaly parametrizations of the radial
motion can be derived.17'18 The eccentric anomaly parameter £ agrees with the
corresponding parameter u of the Damour-Deruelle formalism.19 The true anomaly
parameter \ however is different from the parameter v. The complex counterparts of
these parametrizations have the wonderful property that the overwhelming majority
of the radial integrals can be evaluated simply as the residues in the origin of the
complex parameter plane.
Employing these convenient parametrizations u and x, the radial motion could
be integrated exactly. The result is a generalized Kepler equation:20
n(t-to)=€-etsm£ + F(x;'&o,'tpi) >
2
F (x; tfo, t/>i) = ft sin [X + 2 ty„ ~ ?)] + E ft sin \X + 2 W>o - t/>i)] , (1)
where n, et, ft and fl are orbital elements. Most notably, the true anomaly
parametrization x appears only in combination with the coefficients ft and fl, which
in turn receive contributions only from spin-spin, mass quadrupolar and magnetic
dipolar contributions. These terms also contain the azimuthal angles ipi of the spins
(with 2-0 = ipi + -02 )■ The angle ipo is the argument of the periastron (the angle
subtended by the periastron and the intersection line of the planes perpendicular
to the total and orbital angular momenta, respectively).
Besides the convenient parametrization and integration relying on the use of
the residue theorem, the other main ingredient in obtaining the result (1) was the
introduction of averaged dynamic quantities A and L. These represent averages of
the magnitudes of the Laplace-Runge-Lenz and orbital angular momentum vectors,
respectively. The averages are taken over the angular range defined by one radial
period. Although the quantities A and L are not constant under the spin-spin,
quadrupole and magnetic dipole couplings, their angular average over a radial period
remarkably is (as long as we are considering conservative dynamics).
Another important point to stress is that the orbital elements from Eq. (1)
depend on the relative angle 7 between the spins and the angles re^ of the spins
span with the orbital angular momentum. These in turn evolve, bearing a hidden
time-dependence. However, the precessional motion due to the spin-orbit coupling
does not affect them, while the error made by disregarding the changes due to the
2505
spin-spin interaction are quite small. To see this we note that the lowest order in
which «i 2 occur in Eq. (1) are the spin-orbit terms at 1.5 PN. Their change being
an 1.5 PN effect,7 a variation appears only at 3PN accuracy in the Kepler equation.
The change in 7 is at 1PN,7 however 7 enters only in the spin-spin contributions,
its change becoming significant therefore again at 3PN accuracy. These are smaller
effects (appearing at 0.5 PN higher order) than those occurring from the leading
order radiation reaction. Nevertheless, such changes accumulate over the inspiral.
Therefore the Kepler equation (1) with constant coefficients should be applied with
care. As the magnitude of the still disregarded effects depends on the value of the
post-Newtonian parameter e = Gm/c2r = v2/c2, they are higher during the last
stages of the inspiral. In conclusion the Kepler equation with constant coefficients,
Eq. (1) represents a better approximation in the early stages of the inspiral.
In order to include the general relativistic 2PN contributions, the 2PN terms21-24
given in terms of the ^-parametrization should be also added to the Kepler equation.
Such a Kepler equation represents the complete solution of the radial motion to
2PN orders. We conclude with the remark that a full parametrization of the radial
motion up to 2PN orders, with the inclusion of finite-size effects is possible with the
ensemble of three radial parameters (u = £, v, x)-
References
1. B. M. Barker and R. F. O'Connell, Phys. Rev. D 2, 1428 (1970).
2. L. E. Kidder, C. Will, and A. Wiseman, Phys.Rev. D 47, 4183 (1993).
3. L. E. Kidder, Phys. Rev. D 52, 821 (1995).
4. R. Rieth and G. Schafer, Class. Quantum Grav. 14, 2357 (1997).
5. L. A. Gergely, Z. Perjes, and M. Vasuth, Phys. Rev D 57, 876 (1998).
6. L. A. Gergely, Z. Perjes, and M. Vasuth, Phys. Rev D 57, 3423 (1998).
7. L. A. Gergely, Z. Perjes, and M. Vasuth, Phys. Rev. D 58, 124001 (1998).
8. G. Faye, L. Blanchet, and A. Buonanno, Phys. Rev. D 74, 104033 (2006).
9. L. Blanchet, A. Buonanno, and G. Faye, Phys. Rev. D 74, 104034 (2006).
10. L. A. Gergely, Phys. Rev. D 61, 024035 (2000).
11. L. A. Gergely, Phys. Rev. D 62, 024007 (2000).
12. B. Mikoczi, M. Vasuth, and L.A. Gergely, Phys. Rev. D71, 124043-1-6 (2005).
13. E. Poisson, Phys.Rev. D 57, 5287 (1998).
14. L. A. Gergely and Z. Keresztes, Phys. Rev. D 67, 024020 (2003).
15. K. Ioka and T. Taniguchi, Asrophys. J. 537, 327 (2000).
16. M. Vasuth, Z. Keresztes, A. Mihaly, and L. A. Gergely, Phys. Rev. D 68, 124006
(2003).
17. L.A. Gergely, Z. Perjes, M. Vasuth, Astrophys. J. Suppl. Series 126, 79-84 (2000)
18. L. A. Gergely, Z. Keresztes, and B. Mikoczi, Astrophys. J. Suppl. Series 167, 286-291
(2006).
19. T. Damour and N. Deruelle, Ann. Inst. Henri Poincare A 43 , 107 (1985).
20. Z. Keresztes, B. Mikoczi, and L.A. Gergely, Phys. Rev. D 72, 104022 (2005).
21. T. Damour and G. Schafer, CR Acad. Sci. 77 305, 839, (1987).
22. T. Damour and G. Schafer, Nuovo Cimento B 101, 127 (1988).
23. G. Schafer and N. Wex, Phys. Lett. A 174, 196, (1993); erratum 177, 461.
24. N. Wex, Class. Quantum Gr. 12, 983, (1995).
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Tests of Local Lorentz
Invariance
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THE STANDARD-MODEL EXTENSION AND TESTS
OF RELATIVITY
NEIL RUSSELL
Physics Department, Northern Michigan University,
Marquette, MI 49855, USA
nrussell@nmu. edu
The Standard-Model Extension, or SME, is a general framework for the study of Lorentz
violation in physics. A broad variety of experiments is able to access the SME coefficient
space. Theory and experiments aimed at testing Special Relativity by measuring these
coefficients are discussed.
Lorentz symmetry is a central feature of the existing theories of gravitation and
particle physics. The existence of highly sensitive experiments with the ability to
test Lorentz symmetry at unprecedented levels raises the possibility of discovering
unconventional effects. This is clearly of interest to physicists since it may pave the
way to finding a unified theory of quantum gravity.
A series of publications since 1989 has established a framework, the Standard-
Model Extension, or SME, that provides a detailed description of possible Lorentz
violations in nature in the context of effective field theory. At the basic level, this
work focuses on a variety of theoretical issues, including string theory, and
spontaneous symmetry breaking.1 Much theoretical and experimental effort has been
directed towards the study of Lorentz symmetry in Minkowski space, for which
the effective field theory is an extension of the Standard Model of particle physics.
In flat spacetime, the SME comprises a broad variety of constant coefficients for
Lorentz violation that can in principle be measured.2 These coefficients transform
as conventional Lorentz tensors under observer transformations, but under rotations
and boosts of experimental systems, called particle transformations, they are not
transformed.
An important category of experimental symmetry tests involves searching for
couplings between the electron spin and the Lorentz-violating SME background.
The basic idea is that the radiation released in a transition between different spin
states has frequency that depends on the spin quantization axis and that differs
for particles and antiparticles. Consequently, spectral transitions in atoms with
controlled quantization axes, such as occur in atomic clocks, are well suited to
tests of Lorentz symmetry. To see small variations in the output frequency of a
sensitive clock, one has to compare it to the output of another clock for which the
effects are absent, or at least different. So, such experiments are often called clock-
comparison experiments.3 One of the common scenarios involves monitoring the
outputs for long enough to detect the sidereal effects associated with the rotation
of the apparatus relative to the distant stars. Tests and theoretical investigations
based on these ideas include ones done for hydrogen masers, antihydrogen, noble-gas
masers, space-mounted atomic clocks, Penning traps, and torsion pendula.4
2509
2510
The effects of Lorentz-violation on the electromagnetic sector are described by
19 coefficients at leading order and are amenable to sensitive experimental
investigations. Analysis of birefringence data from cosmological sources has placed stringent
limits on 10 of these, while optical and microwave cavity resonators have placed
limits on the remaining ones.5 Cosmological birefringence tests are based on
distant processes producing the radiation, but offer fantastic sensitivities. Laboratory
cavity experiments have undergone numerous innovations to improve their
experimental reach, including cryogenic cooling, the use of optical sapphire crystals, and
placement on rotating turntables to exploit geometrical properties. Other
investigations involving photons include, for example, ones based on Cerenkov radiation,
synchrotron radiation, Compton scattering, and Doppler-shift experiments.6
Lorentz symmetry has also been tested in the context of various other particles.
For example, in the case of neutrinos, simple models constructed from the SME
coefficients have been found to be consistent with known neutrino data while
offering the advantage of fewer parameters and masses.7 Accelerator-related physics
investigations of Lorentz symmetry include ones with a variety of neutral mesons
and others with muons.8 Further details of Lorentz tests in flat spacetime can be
found in various overview sources.9
The gravitational sector of the Standard-Model Extension consists of a
framework for addressing Lorentz and CPT violation in curved spacetimes, including ones
with torsion.10 The coefficients for Lorentz violation typically vary with position,
adding complexity to the manner in which matter couples to the background. To
set up the framework for the full Standard-Model Extension, the vierbein
formalism can be adopted, since it allows the spinor properties of ordinary matter to be
incorporated. It also has the useful feature of distinguishing naturally between local
Lorentz transformations and general coordinate transformations. Lorentz symmetry
breaking must be either explicit or spontaneous. A study of this topic has shown that
explicit Lorentz violation, in which the breaking occurs in the Lagrangian density,
is incompatible with generic Riemann-Cartan spacetimes. On the other hand,
spontaneous breaking can be successfully introduced in a consistent manner. One of the
far-reaching results associated with spontaneous Lorentz breaking is that it always
goes hand in hand with spontaneous breaking of diffeomorphism symmetry. The
10 possible Nambu-Goldstone modes associated with the six generators for Lorentz
transformations and the four generators for diffeomorphisms have been studied. The
fate of these modes depends on the spacetime geometry and the dynamics of the
tensor field triggering the spontaneous Lorentz violation. The results are consistent
with the known massless particles in nature, the photon and the graviton. An
extensive study has been made of the pure-gravity sector of the SME with the aim of
finding possible experimental consequences. Of particular interest are experiments
involving lunar and satellite laser ranging, laboratory tests with gravimeters and
torsion pendula, measurements of the spin precession of orbiting gyroscopes, timing
studies of signals from binary pulsars, and the classic tests involving the perihelion
precession and the time delay of light. The sensitivity range of these experiments is
parts in 104 to parts in 1015.
2511
References
1. V.A. Kostelecky and R. Potting, Phys. Rev. D 51, 3923 (1995); Phys. Lett. B 381,
89 (1996); Phys. Rev. D 63, 046007 (2001); Nucl. Phys. B 359, 545 (1991); V.A.
Kostelecky and S. Samuel, Phys. Rev. D 39, 683 (1989); Phys. Rev. Lett. 63, 224
(1989); Phys. Rev. D 40, 1886 (1989); Phys. Rev. Lett. 66, 1811 (1991); B. Altschul
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Kostelecky, Phys. Rev. Lett. 84, 1381 (2000); B.R. Heckel et al., Phys. Rev. Lett.
97, 021603 (2006); L.-S. Hou et al, Phys. Rev. Lett. 90, 201101 (2003); D. Colladay
and P. McDonald, Phys. Rev. D 73, 105006 (2006).
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Rev. Lett. 95, 040404 (2005); P. Antonini et al, Phys. Rev. A 71, 050101 (2005);
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V.A. Kostelecky and M. Mewes, Phys. Rev. Lett. 87, 251304 (2001); Phys. Rev. D
66, 056005 (2002); Phys. Rev. Lett. 97, 140401 (2006).
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083003 (2006); CD. Lane, Phys. Rev. D 72, 016005 (2005).
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031902 (2004); Phys. Rev. D 70, 076002 (2004); T. Katori et al, Phys. Rev. D 74,
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37, 1675 (2005); Q.G. Bailey and V.A. Kostelecky, Phys. Rev. D 74, 045001 (2006).
NEW MEASUREMENT OF THE ONE-WAY SPEED OF LIGHT
AND ITS RELATION TO CLOCK COMPARISON EXPERIMENTS
C. S. UNNIKRISHNAN
Gravitation Group, Tata Institute of Fundamental Research, Mumbai - J^OO 005, India
unni@tifr. res. in
www.tifr.res.in
I report the results from the first comparison of the genuine one-way speed of light in
two directions relative to an inertially moving observer. An anisotropy that is first order
in v/c is detected. The implications of the result and its relation to clock comparison
experiments are discussed.
Keywords: One-way speed of light, Special Relativity, Cosmic Relativity, Absolute frame,
Universe.
1. One-way speed of light
The one-way speed of light has never been measured directly in an experiment. The
fundamental assumption of the theory of relativity that the speed of light is an
invariant constant relative to all inertial observers is based on two-way speed
comparisons, as in the Michelson-Morley experiment and in its variations. The difficulty
in measuring the true one-way speed of light lies in the need to pass signals between
spatially separated clocks for synchronization, leading to a logical circularity in the
interpretation of the results.
Recently I have proposed that all known relativistic and kinematical physical
effects are in fact due to the gravitational influence of all the matter in the
universe.1 This theory of 'Cosmic Relativity' is based on direct calculations using the
FRW metric of the observed universe whose physical existence was unknown when
special relativity was formulated. Since such a theory has the isotropic universe as
a preferred absolute frame, it immediately predicts that the one-way speed of light
does depend on the velocity of the observer's frame, though it is independent of the
velocity of the source.
It is well known that any first order anisotropy (dependence on the observer's
velocity) in the speed of light cannot detected by comparing the two-way speed of
light in different directions relative to the moving observer. Also, the first order
anisotropy cannot be detected in an experiment that uses two spatially separated
clocks. In fact, there have been some experiments that looked for the anisotropy
using spatially separated clocks.2 In these experiments, the phase difference between
two clocks was monitored over a stable fiber optic link with the idea that the
changes in the 'preferred frame' velocity of the measurement apparatus(mainly due
to the daily rotation) would cause measurable phase changes if the one-way speed
of light did depend on the velocity of the frame. No phase change was observed
and this was then interpreted as a proof for the isotropy of the speed of light.
But there is a serious flaw in the common analysis of these experiments since well
2512
2513
tested general relativistic effects were not included in the analysis. In an accelerated
reference frame, the spatially separated clocks run at different rates (which can be
interpreted as due to the pseudo-gravitational field equivalent to the acceleration)
and a calculation shows that this exactly compensates the additional phase change
due to the one-way anisotropy of the speed of light.3 Therefore, the null results
in such experiments actually, and ironically, constitute solid proof that the
oneway speed of light is indeed anisotropic relative to moving observers. However, it is
desirable to directly test this without using spatially separated clocks, and without
making the reference platform noninertial.
2. The idea of the measurement
Simple yet rigorous considerations of the measurement of the one-way speed in
situations that we encounter commonly provided a breakthrough in setting up an
experiment to compare the genuine one-way speed of light in two opposite
directions relative to an inertially moving observer.3 Since light travels much faster than
the observer, it is possible to keep the motion of the observer inertial while the
light wavefronts emitted from the moving reference frame loop around in a one-
dimensional path. Then it is an unambiguous prediction of special relativity that
the two wavefronts reach back simultaneously, as analysed from the moving frame.
If the wavefronts have equal speeds relative to the moving reference point, they
are at equal distances from the reference at all instants, and therefore they have to
reach back simultaneously. The distance from start to finish as measured (using a
two-way propagation delay experiment, for example) in the moving frame remains a
constant irrespective of its inertial motion, and therefore the time taken by light for
the round trip should also be independent of the velocity of the frame. This inter-
ferometric experiment is inherently of high precision, with an equivalent temporal
resolution exceeding 10-18 seconds.
3. Experiment and results
The scheme and the set up of the experiment are sketched in the figure 1. The
movable platform is in inertial motion (unlike in the Sagnac configuration) whereas light
wavefronts are looped around to perform a comparison of genuine one-way speeds.
The distances on the space-time diagram to different mirrors etc. rigorously take
into account of the fact that each wavefront propagates along their one-dimensional
directed paths throughout the experiment. If the speeds in the two directions
relative to the moving platform are identical, then the arrival of the two wavefronts after
winding once is simultaneous, and this is tested in the experiment as a function of
the velocity of the inertial observer. The results are plotted in figure 2, for a typical
round trip length of the order of 2 meters. The result indicate that to first order
the one-way speed of light is v — c and v + c relative to the inertial platform moving
at the speed v. The measured anisotropy is numerically identical to the anisotropy
measured in round trip clock comparison experiments. This is easily understood
2514
since the phase is conceptually same as time when the frequency of the reference
oscillator is fixed. However, the physical reason behind the light speed anisotropy is
very different from the reason for the anisotropy in clock time dilations.1 In special
relativity the two are mixed up.
Fig. 1. a) Wavefronts sent at equal speeds relative to an inertially moving observer necessarily
have to reach back simultaneously to the observer, b) The experimental set up c) The indicative
space-tiinc diagram for the propagation of light along the one-dimensional path that loops around
relative to the inertial observer.
■aee-te-
Velocity (m/s)
Pig. 2. Results from the experiment. The one-ways speed of light indeed depends on the velocity
of the inertial observer to first order in v/c.
References
1. C. S. Unnikrishnan, Cosmic Relativity, gr-qc//0406023.
2. T. P. Krisher et al, Phys. Rev. D42, 731, Rapid. Coram, (1990).
3. C. S. Unnikrishnan, Precision measurement of the one-way speed of light: Results and
implications to theories of relativity, to appear iu Proceedings of 'Physical
Interpretations of Relativity Theory' (PIRT -X, Imperial College, London, Ed. M. Duffy, 2006).
TEST OF TIME DILATION WITH A TWO-VELOCITY
ATOMIC CLOCK
G. SAATHOFF1*, S. KARPUK2, S. REINHARDT1, H. BUHR1, T. W. HANSCH3, R.
HOLZWARTH3, G. HUBER2, C. NOVOTNY2, D. SCHWALM1, T. UDEM3, A. WOLF1, M.
ZIMMERMANN3, and G. GWINNER4
1 Max-Planck-Institut fur Kernphysik, D-69029 Heidelberg, Germany
Institut fur Phusik, Universitat Mainz, D-55099 Mainz, Germany
3Max-Planck-Institut fur Quantenoptik, D-85748 Garching, Germany
4Dept. of Physics and Astronomy, University of Manitoba, Winnipeg, MB R3T 2N2, Canada
* Current address: Jila, University of Colorado, Boulder CO, 80309, USA
Guido.Saathoff@colorado.edu
Keywords: Special Relativity; Time Dilation; Doppler Effect
Time dilation is not only one of the most intriguing effects of Special Relativity
(SR) but also one of its early experimental pillars. Following a proposal of Einstein,
Ives and Stilwell1 used the relativistic Doppler shift of optical lines emitted from
Hydrogen canal rays to experimentally confirm time dilation on the percent level.
We report a modern version of this experiment using laser spectroscopy on a beam
of lithium ions in a storage ring (see fig. 1). In forward and backward direction,
the Doppler-shifted frequencies vv and va of a narrow transition of frequency vq are
measured using saturation spectroscopy. To this end, two laser beams are overlapped
accurately parallel and antiparallel to the ion beam and tuned into resonance with
the Doppler-shifted clock transition. In SR, the laboratory laser frequencies at exact
resonance are given by the relativistic Doppler formula, vPjll = 7(l±/3)^o. Here, (3 =
v/c is the ion velocity and 7sr = (1— /32)-1/2 the time dilation factor. Multiplication
of these resonance conditions yields the (3-independent frequency relation vvva =
Vq in case SR holds. A possible Lorentz violating time dilation factor 7 = (1 —
/32)_1'2_Q, where a small, non-zero test parameter a describes the deviation from
SR, alters this frequency relation to vvv^ = ^q(1+2q:/32). Time dilation is thus tested
by comparing the blue- and red-shifted Doppler frequencies with the rest frequency
v0 of the ion. However as the frequency accuracy achieved in our previous experiment
exceeds the precision of the clock transition vq at rest, new measurements were
carried out at two different ion velocities /3siow = 0.03 and /3fast = 0.064. In this
case, the relation for the measured frequencies is independent of vq\
^high^high
„»low„riow = l + M/ftgh - A'low)- (1)
In our experiments at the Max Planck Institute for Nuclear Physics, 7Li+ ions
are accelerated by a tandem van-de-Graaff accelerator and injected into the storage
ring TSR. 7Li+ exhibits the strong 2s 3Si —► 2p 3P2 transition at 548 nm. Through
cooling by a cold electron beam, the ion beam's cr-width is kept at 250 /xin, the a-
divergence at 50 /xrad, and the longitudinal momentum spread at Sp/p = 3.5 x 10~5.
The corresponding Doppler width of 2.8 GHz is narrower than the hyperfme struc-
2515
2516
VIT
blue-shifted laser
vp=(]+P)yv0
4W€
red-shifted laser
vd=(l-P)yv0
Fig. 1. Principle of the Ives-Stilwell experiment: The Doppler shifts of a clock transition in 7Li+,
stored at a velocity fi in a storage ring, are measured by collinear saturation spectroscopy. A
photomultiplier (PMT) records the fluorescence which exhibits a Lamb dip at exact, resonance.
The laser frequencies axe referenced to calibrated hyperfino structure lines of molecular iodine (I2),
which serve as a, clock in the lab frame.
ture splitting of the levels, allowing to probe solely the F = 5/2 —► F = 7/2 two-level
transition. This Doppler broadening is overcome by selecting a narrow velocity class
/? t^ 0 using saturation spectroscopy with two counter-propagating lasers of Doppler-
shifted frequencies va and vp. At exact resonance, which is indicated by a Lamb dip
in the fluorescence spectrum, these laser frequencies are accurately measured by
comparison with calibrated hyperfme structure lines in molecular iodine.
The superb quality of the ion beam allows to crucially limit the influence of
systematic error sources. In our previous experiment2 on a /3fast = 0.064 beam, the
comparison with the rest frequency 1/0 from Ref. 3 was compatible with SR and
resulted in an upper limit for a of \a\ < 2.2 x 10-7. This experiment was limited by
the uncertainty 1\vq/vq = 7 x 10~~10 of the rest frequency, which enters squared in
the the relation vpva = i/'q; it is afflicted by a, though less accurate rest frequency
measurement that differs from Ref. 3 by more than 2a.4 Preliminary analysis of
the current 1/0-independent, two-velocity experiment promises to reach an overall
frequency accuracy of the order of 3 x 10-10 allowing a test of time dilation below
the \ot\ « 10~~7 level. This Doppler shift experiment also limits several parameters
of the Standard Model Extension.5'6
References
E. Ives and G. R. Stilwell, J. Opt. Soc. Am. 28, p. 215 (1938).
Saathoff et al, Phys. Rev. Lett. 91, p. 190403 (2003).
Riis et al, Phys. Rev. A 49, p. 207 (1994).
Rong et al, Eur. Phys. J, D 3, p. 217 (1998).
D. Lane, Phys. Rev. D 72, p. 016005 (2005).
Laboratory Gravity Tests
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ATOM INTERFEROMETRY FOR PRECISION TESTS OF
GRAVITY: MEASUREMENT OF G AND TEST OF NEWTONIAN
LAW AT MICROMETRIC DISTANCES
A. BERTOLDI, L. CACCIAPUOTI*, M. DE ANGELIS, R.E. DRULLINGER, G. FERRARI,
G. LAMPORESI, N. POLI, M. PREVEDELLlt, F. SORRENTINO and G.M. TINO+
Dipartimento di Fisica and LENS - Universita di Firenze,
Istituto Nazionale di Fisica Nucleare, INFM-CNR, Sezione di Firenze
via Sansone 1, 1-50019 Sesto Fiorentino (Firenze), Italy
+ guglielmo.tino@fi.infn.it
www. lens.unifi. it/tino
We describe two experiments where atom interferornetry is applied for precision
measurements of gravitational effects. In the first, we measure the Newtonian gravitational
constant G using an atom interferornetry gravity-gradiometer which combines a rubidium
fountain, a juggling scheme for fast launch of two atomic clouds, and Raman
interferornetry. We show that the sensor is able to detect the gravitational field produced by source
masses and G is measured with better than 10~2 accuracy. In the second experiment,
using ultra-cold strontium atoms in a vertical optical lattice and observing persistent
Bloch oscillations for several seconds, we measure gravity acceleration with micromet-
ric spatial resolution. We discuss the prospects for the study of gravitational forces at
short distances and show that unexplored regions can be investigated in the search for
deviations from Newtonian gravity.
1. Introduction
Recent avarices in atom interferornetry led to the demonstration of different schemes
for fundamental physics experiments and for applications: atom interferornetry was
used for precision measurements of gravity acceleration,l Earth's gravity gradient,2,3
rotations4,5 and h/m.6'7 An overview of basic principles and seminal theoretical and
experimental work can be found in Ref. 8. Atom interferometers are promising
sensors for the investigation of the gravitational interaction such as equivalence
principle tests,9,10 1/r2 law test,11,12 gravitational waves detection13-16 and for possible
applications in geophysics.1'317
Quantum devices based on ultracold atoms show extraordinary features in terms
of sensitivity and spatial resolution, which are important for studies of surfaces,
Casimir effects,18 and searches for deviations from Newtonian gravity predicted by
theories beyond the standard model.19,20 In section 2 we describe the operation of
an atom interferometer conceived for measuring the gravitational constant G and
we report a measurement with better than 10 ~2 accuracy. In section 3, we show that
using laser-cooled strontium atoms in optical lattices, persistent Bloch oscillations
arc observed for about 10 s, and gravity is determined with ppm sensitivity on
'Permanent address: ESA Research and Scientific Support Department, ESTEC, Keplerlaan 1-
P.O. Box 299, 2200 AG Nordwijk ZH, The Netherlands
tPermanent address: Dipartimento di Chimica Fisica, Universita di Bologna, Via del Risorgimento
4, 40136 Bologna, Italy
2519
2520
micrometer scale. We show that this method can improve the sensitivity in the
search of deviations from Newtonian gravity in the micrometer distance range.
2. Measurement of G
After Cavendish first measurement, more than 300 experiments have been performed
to measure G, but the results are not in agreement. In 2002 the recommended
CODATA21 value (G=6.6742(10) x lO"11 m3kg-V2) uncertainty was reduced by
one order of magnitude down to 150 ppm compared to the previous one (CODATA
1998), and this is still much higher than the uncertainty of any other physical
constant. Problems in measuring G with high accuracy arise from the weakness of
the gravitational force, from the impossibility of shielding it and from the difficulty
of realizing well-defined masses and positioning them at well-known distances.
We have applied Raman interferometry techniques with Rb atoms to determine
the Newtonian gravitational constant G.22'23 We implemented a new measurement
scheme aiming to get rid of, or at least to better identify, such systematic effects.
In our experiment freely falling microscopic bodies (atoms) are used as probes of
the gravitational field induced by heavy and well- characterized source masses. The
vertical acceleration is simultaneously measured in two vertically separated
position with two atomic samples, that are launched in rapid sequence with a juggling
method. From the differential acceleration measurements, and from the knowledge
of the added mass distribution, we determine the value of G. The result of another
conceptually similar experiment was recently reported in Ref. 24.
2.1. Experimental apparatus and procedure
The experimental apparatus, described in detail in Ref. 23, is sketched in figure 1.
It consists of a Raman interferometer used as a gravity-gradiometer and two sets
of heavy source masses (SM). Rb atoms are laser-cooled and trapped and launched
upwards into aim long, magnetically shielded tube where the interferometer
sequence takes place. While falling down, they are detected at their passage through
the central vacuum chamber. The two sets of SM are symmetrically arranged around
the tube and can be vertically moved with high precision. The gradiometer requires
two clouds of cold atoms moving with the same velocity at the same time, but
vertically displaced. A vertical separation of 35 cm for atoms launched 60 cm and
95 cm above the MOT results in a launch delay between the two clouds of about
100 ms. The two atomic clouds are prepared using the juggling technique.25
During the ballistic flight of the first cloud of atoms, a second cloud is loaded.
Just before the first cloud falls down in the MOT region, the second one is launched.
Then the first cloud, used as a cold and intense source of atoms, is recaptured, cooled
and launched upwards within less than 50 ins. In our experimental sequence, the
first cloud is launched 60 cm upwards, which leads to a loading time of 650 ms for
the second cloud. In this way, the number of atoms launched in each of the two
clouds used in the gradiometer is 5 • 108. After the launch, the atoms are selected
CivC 1
upper ctoud
apogee
I
Si lower cloud
apogee
pumping
and
detection
Fig. 1. Experimental setup showing the vacuum system, and the two source masses configurations.
The apogees of the atoms trajectories are indicated.
both in velocity and by their nip state. The selection procedure uses vertical beams
so that the state preparation can take place simultaneously on both clouds. After
the selection sequence, the atoms end up in the F=l,mp=0 state with a horizontal
temperature of 4 fiK and a vertical temperature of 40 nK, corresponding to velocity
distribution widths (HWHM) respectively of 3.3 yrec an 0.3
t?rec •
A sequence of three vertical velocity-selective Raman pulses is used to realize
the interferometer. The first (tt/2 pulse) splits the atomic wave packet, the second
(tt pulse) induces the internal and external state inversion and the third (tt/2 pulse)
recombiues the matter waves after their different space-time evolution. Stimulated
Raman transitions are driven by two extended cavity phase^locked diode lasers,
with a relative frequency difference equal to the 87Rb ground state hyperfine
splitting frequency (uh( 87Rb=6.835 GHz) and amplified by a single tapered amplifier. A
detailed description of the laser locking system can be found in Ref. 20.
To compensate for the Doppler shift of the atomic resonance during the atomic
free fall trajectory, the Raman beams frequency difference is linearly swept. The
interferometric sequence is defined in such a way that the tx pulse is sent 5 ins before
the atoms reach the top of their trajectory, when their velocity is still high enough
to discriminate between upwards and downwards propagating Raman beams. For a
Raman beam intensity of 30 mW/cm , the tt pulse lasts 100 /is.
The interferometric phase shifts are detected using the relative phase of the
Raman beams as a reference. To scan the interferometric fringes, a controlled phase
vibration
isolated
mirror
"W ' " <**"
2522
jump 4>l is applied after the tt pulse to the rf signal generated by the low phase noise
reference oscillator. The population of the two hyperfine sublevels of the ground
state after the interferometric sequence is measured using normalized fluorescence
detection. With a typical number of 5 • 104 detected atoms per cloud per state, the
SNRis 60/1.
2.2. Results and discussion of systematics
The main interferometer phase term is the one induced by Earth's gravity
4>{g) = keffg>T2, (1)
with ftkeff being the momentum transferred to the atoms during each Raman pulse.
A gravity gradient determination consists of two vertically separated acceleration
measurements within the interferometer region. If gDW and gvp are the gravity
acceleration values at the height of the lower and upper interferometers the following
relative phase shift is observed
(j)(Ag) = keff (gDW - gup) T2. (2)
A simultaneous realization of these measurements overcomes the stringent limit set
by the phase noise through common mode rejection. The Raman sequence interval
T, as well as the gradiometer sensitivity, can then be increased up to the limit set
by the size of the apparatus.
For the determination of G, in the double differential scheme, the measurements
are repeated twice in the same point, so rotational contributions should cancel out.
Only fluctuations of the launch direction and height within the complete
measurement time can induce such a shift. The results on the SM detection reported here
were obtained using Pb SM but in the final configuration for the G measurement
well characterized W masses will be used. Two sets of masses are used to generate
a well-known gravitational field. Each set is made of 12 identical cylinders,
symmetrically arranged in a hexagonal configuration around the vertical axis of the
atomic fountain. The cylinders have a diameter of 100 mm and a height of 150 mm.
The two sets of masses are placed on two large titanium rings, which in turn are
held by a mount specifically designed for the experiment. A vertical translation
mechanism allows to independently move the two sets of SM with a fine control
of the position on the order of 5 /xm. The SM can be placed at a relative distance
ranging between 4 and 50 cm. SM have been positioned close to the atomic
trajectories in the gradiometer configuration. The change of the local acceleration due to
the added gravitational potential can be measured, thus allowing to determine the
gravitational constant G, once the SM density distribution and their positions are
well-known.
In a first step (Figure 1, configuration 1) the turning point of the upper cloud is
located above the two sets of SM, and the acceleration induced on the atoms is in the
—z direction. The opposite happens for the lower cloud. The differential phase term
2523
0.4"---. 0.5 0.6 0.7
NFli2/N - Lower grav.
1.45
1.4
1.35
1.3
1.25
i : |:rK;*
1 ''"i"7"1"1
0!4
i'
0.5 0.6 0/
NF=2/N-Lower grav/
ri'/f'-
< i ;/■!■■■+ ■+
"M i r-r
H i"*4|
1 2 3 4 5 6 7 8 9 10 11 12 13 14
measure #
Fig. 2. Gravitational phase shift measurements made with Pb cylinders in configuration 1 (empty
squares) and 2 (filled squares) (see Fig. 1). Each data point results from an elliptic fit over 288
gradiometric sequences, with the local oscillator phase step set to 5°. In the two insets above, the
full data set for two measurements in different configurations are shown. The acquisition interval
for each point is 20 minutes.
is then determined for a different position of the SM (Figure 1, configuration 2);
moving them to the external positions with respect to the atoms clouds, the sign of
the induced acceleration is inverted. By evaluating the difference of such consecutive
measurements a reduction of systematic effects27 is achieved, due for instance to
spatially inhomogeneous spurious accelerations, which are constant on the time
scale of SM repositioning. Among these effects, the Earth's gravity gradient g' is
the most important. Other minor contributions are due to inhomogeneous electric
and magnetic fields as well as to inertial forces.
In Figure 2, the differential phase shifts measured for the two sets of Pb cylinders,
alternatively in the two configurations, are reported. Considering the differences
between two consecutive measurements, the resulting phase shift from the whole
data set is 144(5) mrad, which corresponds to a sensitivity of 3 • I0~9g and a relative
uncertainty of 4 • 10~2 in the measurement of G. The total acquisition time was less
than 5 h.
The cylinders for the final G measurement are made of a non-magnetic tungsten
alloy and the characterization tests on these SM are ongoing. The proposed accuracy
of AG/G=W~4 for the final measurement of G can be reached only optimizing
all the parameters so far considered.23'28 W SM will be used, heavier and better
characterized in terms of geometry and density distribution than Pb SM. Atomic
motion's initial parameters also will be critical for the final accuracy. The sensitivity
to initial atomic position and velocity can be dramatically reduced by choosing
the optimum combination of the two SM configuration and atoms' trajectories.
2524
C1-C2 configurations
1e-06
8e-07
6e-07
4e-07
c\P 2e-07
w
E. 0
a -2e-07
-4e-07
-6e-07
-8e-07
-1e-06
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
z(m)
Fig. 3. Simulated acceleration along the vetrical axis in the interferometer region. The Earth's
gravity gradient, W SM and the moving mass of the support have been taken into account. Both
configurations are reported. Atoms trajectories will be in the two regions that are flat in both
configurations, in order to reduce the dependence on initial atomic motion's parameters.
Configuration 1 (Figure 3) will be obtained with the two sets of SM placed as
close as possible. Once the interferometer time T has been chosen (typically T=150
ms), the two atomic trajectories will be selected by maximizing the simulated phase
difference between the two interferometers. After this, atoms will be launched always
up to the same best heights and the interferometer will be realized always at the
same time. Only the SM will be then moved into configuration 2 (Figure 3), that is
chosen in such a way that the new phase difference term (with an opposite sign) can
be as insensitive as possible to the atomic motion's initial conditions. In this way
the interferometers will be realized in those vertical regions where the acceleration
is stationary in both configurations. A less demanding condition on initial atomic
motion's parameters. By optimizing the atoms-masses relative position in this way
AG/G=10-4 can be reached with an initial position uncertainty of 1 mm and an
initial velocity uncertainty of 5 mm/s.
3. Accurate force sensor with micrometric resolution
The confinement of ultracold atoms in optical lattices, regular structures created by
interfering laser beams where the atoms are trapped by the dipole force, provides
clean model systems to study quantum physics problems.29 For example, Bloch
oscillations, predicted for electrons in a periodic crystal potential in presence of a
static electric field30 but not observed in natural crystals, were directly observed
using atoms in an optical lattice.31
In our experiment, laser-cooled 88Sr atoms are trapped in a 1-dimensional
vertical optical lattice. The insensitivity to stray fields and collisions makes Sr in optical
lattices, a candidate also for future clocks,32 a unique sensor for small-scale forces.
The combination of the periodic optical potential and the linear gravitational po-
2525
^d MOT . J^PPf
beams a,oms
probe
beam
2D
optical lattice 1era
beam
Fig. 4. Simplified scheme of the apparatus used to observe Bloch oscillations and to measure
g: Sr atoms are laser cooled and trapped at a temperature of about 400 nK in a red magneto-
optical-trap (MOT). The MOT laser beams are then switched-off and the atoms are transferred
in a vertical 1-dimensional optical lattice generated by a laser beam retroreflected by a mirror;
atoms are confined in series of layers at the maxima of the standing wave by the dipole force. We
measure the momentum distribution of the atoms, after the coherent evolution in the potential
given by the periodic potential plus gravity, by a time-of-flight measurement, after a free fall of 12
ms, using a resonant probe laser beam and absorption imaging on a CCD camera.
tential gives rise to Bloch oscillations at frequency vg given by
mg\L
"* = "2JT (3)
where to is the atomic mass, g is the acceleration of gravity, A/, is the wavelength
of the light producing the lattice, and h is Plancks constant. Since both A^ and
to are well known, the overall force along the lattice axis can be determined by
measuring the Bloch frequency vb- In order to do a force measurement with given
interrogation time, the atomic wavefunction has to undergo a coherent evolution
on the same time timescale. The most common effects limiting the coherence time
for ultracold atoms are perturbations due to electromagnetic fields and atom-atom
interactions. 88Sr is in this respect a good choice because in the ground state it has
zero orbital, spin and nuclear angular momentum that makes it insensitive to stray
electric and magnetic fields that otherwise need to be shielded. In addition, 88Sr has
remarkably small atom-atom interactions;33 this prevented so far the achievement,
of Bose-Einstein condensation for this atom33,34 but becomes an important feature
in experiments where collisions lead to a loss of coherence limiting the measurement
time and the potential sensitivity.
Given the small extension of ultracold Sr atoms confined in optical lattice
potential, and its insensitivity to stray fields and elastic collisions, Sr in optical
lattices results to be a unique sensor for small-scale forces with better performances
and reduced complexity compared to proposed schemes using degenerate Bose35
or Fermi36 gases. This improves the feasibility of new experiments on gravity in
unexplored regions.
2526
3.1. Experimental apparatus
The experimental setup used in this work is schematically shown in Fig. 4. The
method used to produce ultracold Sr atoms was already described in Ref. 37. The
experiment starts with trapping and cooling ~ 5 x 107 88Sr atoms at 3 mK in a
magneto-optical trap (MOT) operating on the 1 So-1 Pi blue resonance line at 461
nin. The temperature is then further reduced by a second cooling stage in a red
MOT operating on the ^o^Pi narrow transition at 689 nm and finally we obtain
~5x 105 atoms at 400 nK. After this preparation phase, that takes about 500 ms,
the red MOT is switched off and a one-dimensional optical lattice is switched on
adiabatically in 50 /is. The lattice potential is originated by a single-mode frequency-
doubled Nd:YV04 laser (XL = 532 nm) delivering up to 350 mW on the atoms with
a beam waist of 200 /xm. The beam is vertically aligned and retro-reflected by a
mirror producing a standing wave with a period A^/2 = 266 nm. The corresponding
photon recoil energy is Er = h2 /2m\2L = kg x 381 nK. As expected from band
theory,38 the amplitude of the oscillation in momentuin space decreases as the lattice
depth is increased. This suggests that in order to measure the Bloch frequency with
maximum contrast the intensity of the lattice laser should be reduced. On the other
hand, reducing the intensity results in a loss in the number of trapped atoms because
of the smaller radial confinement. For this reason, we used a lattice depth of 10 Er.
For a lattice potential depth corresponding to 10 Er, the trap frequencies are 50
kHz and 30 Hz in the longitudinal and and radial direction, respectively. Before
being transferred in the optical lattice, the atom cloud in the red MOT has a disk
shape with a vertical size of 12 /mi rms. In the transfer, the vertical extension is
preserved and we populate about 100 lattice sites with 2 x 105 atoms with an average
spatial density of ~ 1011 cm-3. After letting the atoms evolve in the optical lattice,
the lattice is switched off adiabatically and we measure the momentum distribution
of the sample by a time-of-flight measurement, after a free fall of 12 ms, using a
resonant probe laser beam and absorption imaging on a CCD camera. Fig. 5 shows
time-of-flight images of the atoms recorded for different times of evolution in the
optical lattice potential after switching-off the MOT. In the upper part of the frames,
the atoms confined in the optical lattice can be seen performing Bloch oscillations
due to the combined effect of the periodic and gravitational potential. The average
force arising from the photon recoils transferred to the atoms compensates gravity.
3.2. Data analysis
The images obtained by absorption imaging, as the ones shown in Fig. 5, are
integrated along the horizontal direction and fitted with the sum of two Gaussian
functions. From each image, two quantities are extracted : the first is the vertical
momentum distribution of the lower peak . The second is the width of the atomic
momentum distribution (i.e. the second momentum of the distribution) . We find
that the latter is less sensitive against noise-induced perturbations to the vertical
momentum. We observed ~ 4000 Bloch oscillations in a time t = 7 s. During this
2527
2.4 ms 3.2 ms 4.0 ms 4.8 ms
Fig. 5. Time-of-ftight images of the atoms recorded for different times of evolution in the
optical lattice potential after switcliing-off the MOT. In the upper part of each frame, the atoms
confined in the optical lattice perform Bloch oscillations for the combined effect of the periodic
and gravitational potential. The average force arising from the photon recoils transferred to the
atoms compensates gravity, [n the lower part, untrapped atoms fall down freely under flic effect
of gravity.
time, about 8000 photon momenta are coherently transferred to the atoms.
Oscillations continue for several seconds and the measured damping time of the amplitude
is t ~ 12 s. To our knowledge, the present results for number of Bloch oscillations,
duration, and the corresponding number of coherently transferred photon momenta,
are by far the highest ever achieved experimentally in any physical system.
From the measured Bloch frequency v% = 574.568(3) Hz we determine the
gravity acceleration along the optical lattice g = 9.80012(5) ins-2. The overall estimated
sensitivity is 5 x 10~"6 g and, neglecting the 500 nis preparation of the atomic sample,
we have a sensitivity of 4 x 10~~5 g at, 1 second. We expect that a sensitivity of lO^"7
g can be achieved using a larger number of atoms, and reducing the initial
temperature of the sample. Apart from collisional relaxation, which should contribute to
clecolierence on a minute timescale, the main perturbation to quantum evolution
is represented by vibrations of the retro-reflecting mirror.39 Minor contributions to
decoherence may come from the axial momentum dispersion of the lattice at 10~6
due to its radial extension.
3.3. Testing the Newtonian gravity law
The micrometric spatial extension of the atomic cloud in the vertical direction,
and the possibility to load it into the optical potential at micrometric distance
from a surface, makes the scheme we demonstrated particularly suitable for the
investigation of forces at small spatial scales. The possibility of investigating the
gravitational force at small distances by atomic sensors was proposed in Ref. 11,
discussed in detail in Ref. 40, and preliminary demonstrated in Ref. 41. Deviations
from the Newtonian law are usually described assuming a Yukawa-type potential
v{r) = ^G!HU^l{l + aerr/x} (4)
r
where G is Newton gravitational constant, m\ and ni? are the masses, r is the
distance between them. The parameter a gives the relative strength of departures
from Newtonian gravity and A is its spatial range. Experiments searching for
possible deviations have set bounds for the parameters a and A. Recent results using
2528
inicrocantilever detectors lead to extrapolated limits a ~ 104 for A ~ 10 /jm and
for distances ~ 1 /an it was not possible to perform direct experiments so far.19,20
The small size and high sensitivity of the atomic probe allows a direct, model-
independent measurement at distances of a few /tm from the source mass with no
need for modeling and extrapolation as in the case of macroscopic probes. This
allows to directly access unexplored regions in the a — A plane. Also, in this case
quantum objects are used to investigate gravitational interaction.
Our results indicate that our Sr atoms when brought close to a thin layer can
be used as probe for the gravitational field generated by the massive layer.42 If we
consider, in fact, a material of density p and thickness d, the added acceleration of
gravity in proximity of the source mass is a = 2-nGpd so that when d ~ 10 /an and
p ~ 10 g/cin3) as for tungsten crystals the resulting acceleration is a ~ 4 x 10-11
ms-2. Measuring v& at a distance ~ 4 /an away from the surface would allow
to improve the constraint on a by two orders of magnitude at the corresponding
range A ~ 4 /im. Spurious non-gravitational effects (Van der Waals, Casimir forces),
also present in other experiments, can be reduced by using an electrically conductive
screen and performing differential measurements with different source masses placed
behind it. Moreover, by repeating the same experiment with the 4 stable isotopes (3
bosons, 1 fermion, with atomic mass ranging from 84 to 88), we can further discern
among gravitational and other forces.
Acknowledgments
This work was supported by Istituto Nazionale di Fisica Nucleare, LENS, Ente
Cassa di Risparmio di Firenze.
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10. S. Dimopoulos, P. Graham, J. Hogan and M. Kasevich, arXiv:gr-qc/0610047 (2006).
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Firenze, 2001 (I. Ciufolini, D. Dominici, L. Lusanna eds., World Scientific, 2003). Also,
Tino G. M., Nucl. Phys. B 113, 289 (2003).
2529
12. G. Ferrari, N. Poli, F. Sorrentino and G. M. Tino, Phys. Rev. Lett. 97, p. 060402
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13. C.Borde, G.M.Tino and F.Vetrano, 2004 Aspen Winter College on Gravitational
Waves, http://www.ligo.caltech.edu/LIG0-web/Aspen2004/pdf/vetrano.pdf.
14. Chiao, Y. Raymond, Speliotopoulos and D. Achilles, Journal of Modern Optics 51(6-
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15. A. Roura, D. Brill, B. Hu, C. Misner and W. Phillips, Phys. Rev. D 73, p. 084018
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18. M. Antezza, L. P. Pitaevskii and S. Stringari, Phys. Rev. Lett. 95, p. 113202 (2005).
19. J. C. Long, H. W. Chan, A. B. Churnside, E. A. Gulbis, M. C. M. Varney and J. C.
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20. S. J. Smullin, A. A. Geraci, D. M. Weld, J. Chiaverini, S. Holmes and A. Kapitulnik,
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22. J. Stuhler, M. Fattori, T. Petelski and G. M. Tino, J. Opt. B: Quantum Semiclass.
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23. A. Bertoldi, G. Lamporesi, L. Cacciapuoti, M. D. Angelis, M. Fattori, T. Petelski,
A. Peters, M. Prevedelli, J. Stuhler and G. M. Tino, Eur. Phys. J. D 40, p. 271
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39. Independent measurements with an accelerometer at the level of the retro-reflecting
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DEVELOPMENT OF ACCELEROMETER PROTOTYPE FOR TESTING
THE EQUIVALENCE PRINCIPLE IN FREE FALL
V. IAFOLLA, D. LUCCHESI, V. MILYUKOV, S. NOZZOLI and F. SANTOLI
Istituto di Fisica dello Spazio Interplanetario INAF, Via Fosso del Cavaliere 100
Rome 00133, Italy
I.I. SHAPIRO, E.C. LORENZINI1, M.L. COSMO2, J. ASHENBERG and
P.N. CHEIMETS
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street,
Cambridge, MA 02138, USA
S. GLASHOW
Boston University, 590 Commonwealth Avenue,
Boston, MA 02215, USA
Progresses in the development of a free-fall test of the Principle of Equivalence (PE) are
reported with particular emphasis on the work related to development of the differential
accelerometer prototype and its laboratory tests. The PE experiment is planned to be
carried out in free-fall conditions, inside a capsule (Einstein elevator) released from a
stratospheric balloon. The accuracy goal for the experiment is a few parts in 10 with an
integration time of about 25 s. This accuracy, if reached, would imply an improvement of
two orders of magnitude in testing the PE with respect to the state of the art in this field.
1 Introduction
The state-of-the-art accuracy for Principle of Equivalence (PE) tests with laboratory
experiments on the ground is now several parts in 1013 [1]. The performance of
experiments for PE tests in free-fall conditions in the Earth's gravitational field promises
to be significantly better because free fall removes the key limitations of laboratory
experiments. In fact the free-fall conditions eliminate the seismic noise and increase the
strength of the gravitational field in which test bodies fall by about 3 orders of magnitude.
Furthermore, the masses of the test bodies in weightlessness conditions can be
substantially heavier than in terrestrial laboratory. Thus, the ultimate accuracy goal for
space-based experiments are presently estimated to be 4-5 orders of magnitude better than
the state of the art, with potential accuracy of the Eotvos ratio 8g/g in the range 10"17-
10" . The seismic noise in orbit is replaced by the noise sources of the space
environment, which require complex isolation systems such as drag compensation in
order to achieve the expected improvements in the experiment accuracy.
1 Presently at the University of Padova, Dept. of Mechanical Engineering, Padua, Italy
Presently at the Italian Space Agency, Rome, Italy
2530
2531
There are several obstacles that need to be overcome before a space mission
materializes. First, space-bound detectors cannot be tested in the laboratory at the
accuracy expected in space. Second, the inaccessibility of the hardware in space prevents
the fine tuning and improvements expected for a sensor operating in free fall conditions
for the first time. An alternative to the free fall in space is vertical free fall inside a drag-
shielding capsule released from a balloon flying at a stratospheric altitude.
2 Vertical Free Fall Technique
The free-fall conditions for our experiment (General Relativity Accuracy Test or GReAT)
will be obtained by utilizing a capsule lifted to an altitude of over 40 km by means of a
stratospheric balloon, then released to free fall while at the same time the differential
acceleration detector housed into a package is released from the top of the capsule to free
fall in vacuum. The detector package will experience picogravity conditions, thanks to
the vacuum, during the free-fall phase before it reaches the bottom of the capsule which is
slightly decelerated by the thin atmosphere at those altitudes. The instrument package can
fall freely for about 25-30 s (depending on the capsule ballistic coefficient) inside the
capsule as it spans the 2-m length of the vacuum chamber. The whole capsule is itself a
cryogenic dewar and the detector package is cooled near liquid helium temperature to
provide temperature stability and uniformity and to reduce the Brownian noise of its proof
masses. The detector package is set to spin at a frequency of 0.5 Hz about an horizontal
axis before release by a spin/release system that supports the package at two support
points in a spit arrangement. At release, which occurs in almost ideal weightless
conditions, both support points are quickly withdrawn by the release mechanism, leaving
the package spinning about its axis at the desired signal frequency. The spin/release
system has been studied so as to reduce the spurious components of the rotational velocity
to a negligible level. Once the instrument package reaches the chamber's floor it is caught
by a trapping system to avoid damage and the capsule is decelerated by a parachute
system for recovery and later reflights. A drag-shielded vertical free-fall from a
stratospheric balloon retains most of the advantages of an orbital free fall except for the
longer integration time. In addition, the balloon-released system allows the repetition of
the experiment at reasonable intervals with possible adjustments/improvements of the
experimental hardware. This technique seems to have the potential for improving
significantly the accuracy in testing repeatedly the Principle of Equivalence at an
affordable cost.
3 Differential Accelerometer
The differential accelerometer is a key part of the experimental apparatus and it must have
the required sensitivity to detect the differential accelerations associated with a possible
PE violation. The basic requirements that the differential accelerometer must satisfy for a
test of the Equivalence Principle with an accuracy of a few parts in 1015 are: low
resonance frequencies and Q factors of order 105 for the two mechanical oscillators that
are at the heart of the detector; cryogenic temperature to reduce the Brownian noise; proof
masses with second-order spherical inertia ellipsoid and higher-order sphericity; and
2532
accurate construction in terms of shape, material homogeneity, and coincidence of their
centers of mass.
From the point of view of the motion of the proof masses, differential accelerometers
can be designed to exploit: (a) a purely translational motion; (b) a combination of linear
and rotational motion; or (c) a purely rotational motion.
Differential accelerometer prototypes of type (b), operating at room temperature,
were developed and used to test key aspects of the experiment in the laboratory. Key
experimental issues are the quick abatement of transient oscillations after detector's
release and the ability (expressed by the common-mode rejection factor) of the
differential accelerometer to reject accelerations that acts commonly on the two proof
masses.
The proof masses of the first prototype that we built had a four-finger shape and were
interpenetrated in such a way to achieve a close coincidence of their centers of mass to
reduce the effects of gravity gradient forces and rotational motion on the differential
output signal. One proof mass is made all of aluminum while the other proof mass can
have inserts of another material (e.g., platinum) to become sensitive to differential
acceleration that violates the PE. Each sensing mass was constrained by torsional arms to
rotate about an axis passing through the pivot arms. The value of the mechanical resonant
frequency was about 10 Hz.
One proof mass of a differential accelerometer of type (c) is made from two different
materials placed on opposite side of the pivot axis, through the center of mass, so that a
PE violation will generate a pure torque about the pivot axis. Such a proof mass will be
highly insensitive to any linear acceleration including those associated with gravity-
gradient forces, but will react to torques. The other proof mass is also a purely rotational
proof mass that is made all of one material so that it is also insensitive to any PE
violations. This proof mass, however, will move inertially like the other proof mass (if
the resonant frequencies of the two oscillators are well matched) so that by measuring the
differential rotations of the two proof masses, the inertial motion of the two masses is
strongly attenuated and the differential rotation due to PE violation is 'highlighted' out of
the motion.
As already mentioned an important parameter for the differential accelerometer is the
common-mode rejection factor. This factor expresses the ability of the differential
accelerometer to attenuate common accelerations (linear or rotational depending on the
configurations of the detector) acting on the differential accelerometer. A common-mode
response is different from a differential-mode response (i.e. the response to a PE
violation) because in the former case the two sensing masses move in phase, while in the
latter case they move out of phase. Common-mode disturbance on the accelerometer can
be reduced by several orders of magnitude. An attenuation of about 10 4 has already been
obtained in the laboratory experiments over the desired (narrow) frequency band for the
prototype differential detectors.
4 Accuracy Goal of Experiment
An error budget for the GReAT experiment, that considers the strengths and frequency
content of the main noise sources, has been computed [2]; we will point to the principal
2533
ones in the following. The Earth's gravity gradient produces error signals which are well
above the expected experimental accuracy. However, thanks to the instrument spin, the
diagonal components of the Earth's gravity gradients are modulated at twice the spin
frequency and the off-diagonal components are at negligible level for a spin axis that is
within a degree from the horizontal plane (referred to the local gravity vector). A
violation signal, which would appear at the spin frequency, is therefore discernible from
this noise components. The gravity gradients generated by the capsule are lower than
those generated by the Earth and the main components of them are also at twice the
rotation frequency. Higher-order mass coupling terms of concern (i.e., even terms) are
also modulated at twice the spin frequency and even multiples of that frequency if the
spin axis of the detector is aligned with the pivot axis of each proof mass, as in our latest
detector's conceptual design [3]. All the noise components with frequencies well
separated from the signal frequency a$ do not affect the measurement so long as they do
not exceed the dynamic range of the instrument and can be filtered out of the output
signal by frequency analysis.
The free-fall technique, inside the co-falling capsule, provides an environmental
acceleration at a level well below 10" m/s2 (~1012 g). The common-mode rejection
factor at a level of 10" makes the effect of the environmental acceleration negligible.
Without going into details, we are presently estimating our detector's threshold sensitivity
at the level of about lxlO"14 g/Hz"2 at a temperature below 10 K. This value of the
threshold sensitivity leads to an accuracy of 5 parts in 1015 in testing the Principle of
Equivalence with a 95% confidence level over the duration of the detector's free fall.
The GReAT experiment seems a good compromise between the proposed satellite
experiments (which could reach even higher accuracy) and classic ground experiments.
Our experiment could potentially improve significantly the present accuracy level of PE
tests and provide the option of repeating the experiment at periodic intervals with the
affordable cost of a balloon flight.
References
1. S. Baessler, B.R. Heckel, E.G. Adelberger, et al, Phys. Rev. Lett. 83, 3585 (1999).
2. V. Iafolla et al, Review of Scientific Instruments, 69(12), 4146 (1998).
3. E.C. Lorenzini et al., "Detector Configurations for Equivalence Principle Tests with Strong
Separation of Signal from Noise," XXVIII Spanish Relativity Meeting ERE 2005, Oviedo,
Spain, A1P Conference Proceedings, 841, 502-506 (2006).
MEASUREMENT OF THE GRAVITATIONAL CONSTANT G
HINRICH MEYER, ULF KLEINEVOSS and HELMUT PIEL
University of Wuppertal,
Wuppertal D-4S10S, Germany
hinrich.meyer@desy. de
A Gravimeter based on a RF-Cavity and two 560 kg field masses is used to determine
an absolute value of the gravitational constant, G. The field masses change the length
of the cavity which is proportional to a change of the resonance frequency determined
with very high precision. The value for G obtained is found to be in very good agreement
with the world average.
1. Experimental details
The field masses consist of cylinders of brass about 40 cm long and 40 cm in
diameter and with a weight of 560 kg each. Between the two masses is a micro-wave cavity
of 20 cm diameter and 24 cm length. The two ends of the cavity facing the field
masses are made of spherical copper mirrors, each suspended by two tungsten wires
of 0.2 mm thickness and about 2.60 cm length thus forming two pendulums. The
gravitational pull of the field masses at a distance of k 1 m moves the pendulum
masses by a small amount « 10_8m and changes the resonance frequency of the
cavity. At a frequency of 22 GHz this change of length corresponds to a frequency
shift of « 1 kHz. The field masses are moved between two positions in 20 minute
intervalls imposing a frequency modulation which can be measured with high
precision. The pendulums are placed in a vacuum container which is kept at « 10~4
torr.
Changes of the local gravitational potential due to moving masses, even at
distances much larger then 1 m usually have much longer timescales than the 20
minutes intervalls of the move of the field masses and are easily subtracted.
The RF source is a HP-8340B Synthesized Sweeper stabilized by a Rb-Standard
and the DCF-77 station signal. The waveguides are not mechanically connected to
the cavity; the RF is supplied crossing a small ~ 1 mm gap suitably matched to
keep losses and reflections small. The cavity is operated at various frequencies in
the 22-23 GHz range. The sweep supplies four measurements on both sides of the
resonance peak and when averaged provides one frequency measurement typically
three times a second.
The change of the length of the cavity Ab is proportional to the change in the
resonance frequency A/.
Ab = [3-Af
The value of (3 depends on the geometry of the cavity and on the resonance
frequency and can be calculate from cavity theory with sufficient precision.
2534
2535
G is determined from Ab through Newtons law with M the field masses and
ujo = l/g the pendulum frequency (/ = pendulum length, g « 9.8cm/sec2)
M-G = Wo2(/(r1)^/(r2))
where the two functions /(r) contain the geometry of the masses M and the
pendulum masses, the cavity length b and the distances ri2 of the field masses at the
two measurement positions. The integration over the mass distributions (geometry)
basically weighted with the 1/r2 law are performed numerically. At larger distances
compared to the dimensions of the masses one rapidly approaches point mass
geometry. A typical set of measurements is shown in Fig. la, with the frequency difference
(kHz) displayed as function of time (h). Between 7.4 and 7.6 h the influence of an
earth quake (Azores) is easily recognized. Long term smooth drifts mainly due to
changes of temperature in the local environment are parametrized by a polynominal
function and subtracted.
2. Results
Series of measurements have been taken at various distances between 915 mm and
1500 mm, also two values of the resonance frequency (22 and 23 GHz) have been
chosen. Typically 200 cycles have been taken at each run. The distribution of values
for G from each cycle in a run are nicely gaussian with very few outliers cut at 3.5a.
Table 1. Individual values for G [IQ-^u^k^sec-2],
measured at different distances r.
position r [mm] G23GHZ G22GHz
915 6.67444 ± 0.00099 6.67485 ± 0.0021
6.67461 ± 0.00340
945 6.67299 ± 0.00150 6.67422 ± 0.0026
6.67318 ± 0.00150
985 6.67430 ± 0.00095 6.67490 ± 0.0019
6.67519 ± 0.0049
1095 6.67536 ± 0.00140
1300 6.67264 ± 0.00380
1500 6.67332 ± 0.00720
The individual values for G obtained are shown in table 1 and are displayed
in Fig. lb as function of distance. No deviation from an exact 1/r2 dependence is
observed at a level of about 10~3. Therefore all values for G combined result in
G = (6.6742 ± 0.0005) • lO^Wkg^sec-2
The main systematic error results from corrections due to tilts of the tower
dominantly under the influence of temperature changes. The error on this systematic
effect is estimated at about a factor of 2 of the statistical error and thus is the main
limitation in this phase of the experiment.
2536
» 23 GHz
• 22 GH=
1000 1200 1400 1600
Time [hi r [mm]
Fig. 1. (left) A typical set of frequency measurements, (right) Individual values of G as function
of distance r.
3. Future
The whole experimental setup has been moved to DESY recently. At the new
location much better ground stability, temperature controll and alignment is available.
Measurement runs will resume in the next future and we hope for improvements in
the error budget by about a factor of 3.
Acknowledgments
I like to thank S. Schubert for help with the manuscript. This experiment has been
supported by the Deutsche Forschungsgemeinschaft DFG, Bonn under the grant
Me 1577/1-5; 08107107.
^4
B 6.68
6,67
SOLAR RADIUS AT MINIMUM OF CYCLE 23
COSTANTINO SIGISMONDI
ICRA & University of Rome La Sapienza, Piazzale Aldo Moro, 5 00185 Rome, Italy
* sigismondi@icra.it www.icra.it/solar
Observations of Baily beads in French Guyana, during 2006 September 22 annular eclipse,
have been made to measure solar radius around solar minimum activity of cycle 23. The
correction to standard solar radius at unit distance (1 AU) 959.63" to fit observations is
Aflg = —0.01" ± 0.17". Sources of errors are outlined in view of relativistic accuracies.
Keywords: Sun, Astrometry, Eclipses, Solar Diameter, Solar Variability
1. Introduction
Baily beads are visible only near the centerline of annular or total solar eclipses1.
At ant/umbral's borders the beads' series equals approximately the duration of an-
nularity/totality. The scope of an observative champaign is to record the maximum
number N of beads' events, identifying their UTC of dis/appearance, and Watts'
angle (counterclockwise from lunar North pole) in the atlas of lunar limbs2, now
available in Winoccult - Baily Beads 3.1.2 program3 of eclipses simulation. Merging
or divisions of beads are discarded to avoid black-drop like events, affected by
instrumental astigmatisms4. Videorecording apparatus A telescope Meade ETX
70 with orange photo filter Tamron Y2A (73% transmittance), projecting solar
image (0 > 10 cm) on a white screen, and a SONY DCR-TRV9E handycam with
800000 pixel CCD.
2. UTC timing
Handycam's internal clock timing has been compared with UTC by filming GPS
Garmin II plus screen, computer screen with Dimension 4 synchronizing software,
Kourou's ESA space base watches, and by audiorecording time radio signals (only in
Italy). Due to temperature variations between Italy and locations of eclipses, timing
made at the beginning and at the end of the trip have to be carefully extrapolated
to annularity. Due to delays of GPS and computer screen with respect to real UTC
time, a further sistematic delay of our control watches has been also considered.
3. Bead identification
In Watts' atlas random errors in the heigth of limb's features are within ±0.2".
With N beads, the statistical uncertainty on solar radius correction is reduced of
a factor V^V, if the features' Watts angles are correctly identified. Beads A and M
in table 1 are the more uncertainly identified. Watts' profile there does not show
significant valleys.
2537
2538
Table 1. Baily beads recorded in September 22, 2006 annular eclipse
Bead
A
B
C
D
E
F
G
H
I
J
K
L
M
Average
Average l
(*)
'all)
UTC
9:49:30.9
34.9
35.0
35.5
35.8
35.8
36.0
9:55:19.7
20.5
20.5
21.6
25.0
26.4
event
apparition
apparition
apparition
apparition
apparition
apparition
apparition
disapparition
disapparition
disapparition
disapparition
disapparition
disapparition
W. A. [°]
256.3
267.9
269.8
270.8
272.8
282.4
284.5
97.4
105.5
107.0
116.2
85.8
122.3
residuals ["J
0.00
0.55
0.69
0.73
0.90
0.83
0.38
-0.97
-0.76
-0.66
-0.59
-0.56
-0.45
vq ["/s] corr. residuals ["]
0.36
0.38
0.41
0.45
0.45
0.45
0.45
-0.45
-0.45
-0.45
-0.40
-0.43
-0.38
-0.64
-0.10
0.04
-0.08
0.25
0.20
-0.24
-0.33
-0.14
-0.05
-0.02
0.09
-0.21
-0.01±0.17
-0.07±0.23
4. Data set and analysis
In Table 1 are Baily beads measured on September 22, 2006 annular eclipse, with the
eclipsed Sun at 7° above the eastern Ocean's horizon at Les Roches (Kourou, French
Guyana) latitude 5°9'42.6" N longitude 52°37'41.5" W altitude 3 m above sea
level. The residuals have been calculated with the Morrison-Appleby5 systematical
correction to lunar profile. The residuals are computed as the difference Sun-Moon
limb according to solar VSOP87A and lunar DE200 ephemerides at the time of
observed dis/appearance. The relative velocity of the solar limb v© is computed by
the formula vQ = vorb, Moon + vpar. Moon - vorb. © = 0.493" x sm(P.A. - 28.8°) +
0.028" x sin(P.A) - 0.042" x sin(P.A - 23.5°), where P.A.=W.A.+21.86° for that
eclipse. Corrected residuals are obtained by minimizing standard deviation applying
first a correction in lunar longitude A long. = —1.29 s x vorb. Moon and after a
correction in watch display ATwatch[= UTC — Twatch = —0.01 s] x vG. The average
(*) is calculated eliminating beads A and M, the more uncertainly identified.
5. Errorbars discussion
The correction in lunar longitude has been calculated using all 13 beads identified
from video record. Limiting the computation to 11 beads the value is
A long. = —1.43 s x worb. Moon- A correction in lunar latitude A lat. = 0.11" x
s'm(P.A. — 90° — 28.8°) does not improve significantly the final errorbar on Ai?.©.
The contribution of lunar latitude correction is very small, because the beads are
nearly equatorial, and ranging from Delta lat. = 0.11" to -0.73" an does not
change. In order to keep low the number of fitting parameters, I prefer to not use
lunar latitude correction to lunar ephemerides. The final correction is almost only
to lunar longitude.
2539
The statistical uncertainty on solar radius correction is or = 0.17" and it
corresponds to 2 part over 104 of the whole radius. For solar oblateness an accuracy 20
times more is required. With available data we improved that value discarding beads
A and M, that are the more uncertain as W.A. identification. Otherwise or = 0.23"
and AR = —0.07". Going to eclipse path's limits (in this case the limits were in the
Amazon forest and in the Ocean), would have increased the number of beads and
improving the statistical uncertainty. In future lunar limb data better than Watts'
atlas will eliminate that source of random error. Polar beads are already used to
compare solar radius corrections in different eclipses because libration in latitude
is near zero during eclipses and the limb profile is there nearly the same. The ideal
condition is after a Saros cycle of 18.03 years, when the libration is the same also
in longitude, in this case random errors from lunar limbs become systematical.
6. Conclusion
At solar minimum the radius reaches its maximum value, after an oscillating cycle
of 11 years6. Within our errorbar this maximum Rq = 959.63" + A i?Q = —0.01" ±
0.17" is consistent with the average value at unit distance so that
1. such oscillations are within A RQ = ±0.17" or/and
2. there is a secular trend of shrinking for which the maximum now corresponds to
average value calculated in 20t/l century.
Acknowledgments
Thanks to Prof. Albert Picciocchi for assistance in Guyana and to Prof. Remo
Ruffini for funding this mission. Special thanks to dr. Chiara Melchiorre, Silvia
Pietroni, Micol Benetti, Paolo Fermani, Antonella Mastrobuono, Irene di Palma
and Marco Innocenti for their contributions in this project and fruitful discussions.
References
1. Fiala, A., Dunham, D., Sofia, S., Variation of the solar diameter from solar eclipse
observations, 1715-1.991 Solar Physics 152 97-104 (1994).
2. Watts, C. B., The Marginal Zone of the Moon, Astronomical Papers prepared for the use
of the American Ephemeris and Nautical Alm,anac (United States Government Printing
Office, Washington) XVII Washington D. C. (1963).
3. Herald, D., http://www.lunELr-occultations.com/iota/occult3.htm (2007).
4. Pasachoff, J., Schneider, G., Golub, L., The black-drop effect explained, Proc. IAU
Coll. 196, D. Kurtz & G. Bromage eds., Cambridge University Press (2004).
5. Morrison & Appleby,Mon. Not. R. Astr. Soc. 196, 1013 (1981).
6. Thuillier, G., Sofia, S., M. Haberreiter, Past, present and future measurements of the
solar diameter, Advances in Space Research 35, 329-340 (2005).
THE NEWTONIAN GRAVITATIONAL CONSTANT:
MODERN STATUS AND PERSPECTIVE OF NEW DETERMINATION
VADIM MILYUKOV
Sternberg Astronomical Institute, Moscow University,
Moscow, 119992, Russia
JUN LUO
Center for Gravitational Experiments, Huazhong University of Science and Technology
Wuhan, 430074, P.R.China
The Newtonian gravitational constant G together with Planck's constant Tl and the speed of light c
are the fundamental constants of nature. Due to the weakness of gravity the accuracy of G is
essentially below the accuracy of other fundamental constants. New measurements on the accuracy
level of 10-30 ppm are rather desirable. The history and current status of the experiments for the
determination of the gravity constant are reviewed. The new experiment for the G measurement,
which is carried out in the framework of collaboration of Russia and China on the pointed accuracy
level, is reported.
1 Introduction
The Newtonian gravitational constant G together with Planck's constant % and the speed
of light c are the fundamental constants of nature which represent the fundamental limits:
c is the maximal speed, % is the minimal angular momentum and G is the gravitational
radius of unit mass (the maximal radius of the sphere for relativistic gravitational
collapse).
Due to the weakness of gravitational interaction an accuracy of experimental
determination of G is essential below an accuracy of other fundamental constants,
progress occurs slowly enough: the error value decreases approximately 10 times per
century [1]. The modern history (last 25-30 years) of the G determination contents large
number of laboratory experiments, however discrepancies of results surpass noticeably
the confidential level. Till now there are no the convincing explanations to such a large
discrepancy of gravitational constant's values determined in various experiments. Thus,
the problem of a gravitational constant, including all its aspects, is still actual, its
significance for fundamental science is difficult to overestimate.
2 Modern History of G Determination
The modern history has started from the tree experiments, performed in 70th of last
century. It were the experiment in France reported in 1972, the experiment of Moscow
University, reported in 1979 [2], and the American experiment, reported in 1982 [3]
(Table 1). The system of values of fundamental constants CODATA 1986 (Committee on
Data for Science and Technology) contained G value with relative accuracy 128 ppm and
based mainly on the value obtained in [3].
2540
2541
Within 90th a number of laboratory experiments on the measurement of the
Newtonian gravitation constant were done with relative accuracy about of 100 ppm and
less. The part of them is summarized in Table 1, including the HUST experiment [4].
Nevertheless, the discrepancies between the values of the gravitational constant obtained
in these experiments remained enough large. As a result of such scattering of G values,
COD ATA should increase significantly an uncertainty and recommended in 1998 value of
G = (6.673±0.010) xlO"" rnkg'c2, with a relative error of 1500 ppm. I.e. "uncertainty of
knowledge" of G has increased almost in 10 times!
Table I. The best world experiments on the measurement of G and
CODATA values
Authors, year of publication
Facy and Ponticis, 1972
Sagitov, Milyukov, et al., 1979
Luther and Towler, 1982
CODATA 1986
Karagioz, Izmailov, 1996
Bagley and Luther, 1997
CODATA 1998
Jun Luo, et al., 1999
Fitzgerald and Armstrong, 1999
Gundlach and Merkowich, 2000
Quinn, Speake et all, 2001
Schlamminger et all., 2002
CODATA 2003
Armstrong and Fitzgerald, 2003
[2]
[3]
[4]
[5]
[6]
[7]
f8]
GxlO""
m3kg"'s"
6.6714
6.6745
6.6726
6.67259
6.6729
6.6740
6.673
6.6699
6.6742
6.674215
6.67559
6.67407
6.6742
6.67387
STD
xlO"11
m3kg"'s"
0.0006
0.0008
0.0005
0.00085
0.0005
0.0007
0.010
0.0007
0.0007
0.000092
0.00027
0.00022
0.0010
0.00027
ppm
90
120
75
128
75
105
1500
105
105
14
41
33
150
41
After 2000 some new results have been published, which had a relative error, less
than 50 ppm. These are experiments of Washington University with a relative error of 14
ppm [5], University of Birmingham with a relative error of 41 ppm [6], University of
Zurich with a relative error of 33 ppm [7], and the Measurement Standards Laboratory
(New Zealand) with an uncertainty of 40 ppm [8]. However these results are not also
intersect within confidential intervals. The new G value recommended CODATA in 2002,
is based on the data accessible on the end of 2002 and is equal to 6.6742x10"" m3kg~ c
with a relative accuracy of 150 ppm. We would like to emphasize the following fact: the
value of G=6.6745x10"" m3kg~'c2 with a relative accuracy of 120 ppm has been obtained
in Moscow University in 1978. After 25 years, in 2003, CODATA recommends value of
the gravitational constant, practically coincides with our "old" value!
Thus, the knotty problem of G measurements, which we have for present time, makes
actual the performance of new experiments at a level of relative accuracy of 10-30 ppm.
3 New Experiment on G Determination
The new experiment on measurement of gravitational constant at a level of accuracy
of 10-30 ppm is prepared in the framework of international cooperation between SAI
MSU (Russia) and HUST (China), which have a good experience in this field [2,4]. The
new experiment will be done on the HUST experimental setup by using time-of-swing
2542
method. The new design of experiment has to greatly reduce the G uncertainties: (1) a flat
plate torsion balance, which the rectangular glass block coated with gold, has less
vibration modes and improves the stability of the period as well as minimizes the
uncertainty of inertial momentum; (2) the spherical source masses minimize the
uncertainties of the eccentricity of the mass center from geometrical one; (3) both the test
and source masses are set in a vacuum vessel to facilitate measuring the relative positions;
(4) remote control of the torsion system lowers environment disturbances.
The first preliminary set of the experiments on the G measurement was done. The
principal contributions to the error budget were estimated. The total contribution of the
geometrical and mass parameters of the torsion balance is 5.8 ppm. The total contribution
of the source masses is 2.5 ppm. The oscillation of the torsion balance is monitored by an
optical lever system, the output signal is sampled at a rate of 2 Hz with a frequency
stability below 2xl0~9/day. The torsion period is about 586.08 s and the typical quality
factor of the torsion balance is about 1930. The change of the period due to the "near"
and "far" positions of the source masses is about 4.25 s, which could be distinguished
with uncertainties all within 0.05 ms. This uncertainty would contribute 16.6 ppm to G
value for an individual measurement. However, the statistical variation of A(co2) for 6 sets
of experimental data contributes only an uncertainty of 5.2 ppm in the error budget. These
first experiments are shown that the final result of the value of the Newtonian
gravitational constant has to be on the accuracy level of 11 ppm.
Acknowledgments
This work is supported by the Russian Foundation for Basic Research (grant No 05-02-
39014) and the National Basic Research Program of China (grant No: NSFC10121503).
References
1. G. T.Gilles, Rep. Prog. Phys. 60, 151 (1997).
2. M.U. Sagitov, V.K. Milyukov, et. al, Dokladi AN USSR, 245, 567 (1979).
3. G.G. Luther and W.R. Towler, Phys. Rev. Lett. 48, 121 (1982).
4. Jun Luo, Zhong-Kun Hu, Xiang-Hui Fu, et. al, Phys. Rev. D59, 042001 (1999).
5. J.H Gundlach and S. M. Merkowich, Phys. Rev. Lett. 85, P. 2869 (2000).
6. T.J. Quinn, C.C. Speake, S.J Richmann, et al, Phys. Rev. Lett. 87, 111101 (2001).
7. St. Schlamminger, E. Holzschuh, W. Kundig, Phys. Rev. Lett. 89, 161102 (2002).
8. T. R. Armstrong and M.P. Fitzgerald, Phys. Rev. Lett. 91, 201101-1 (2003).
RELATIVISTIC ASTROMETRY WITH GAIA
ADVANCES IN THE RAMOD PROJECT
B. BUCCIARELLI, M. T. CROSTA, M. G. LATTANZI and A. VECCHIATO
INAF - Astronomical Observatory of Torino,
strada Osservatorio 20, 10025 Pino Tonnese (TO), Italy
G. PRETI and F. DE FELICE
Department of Physics, University of Padova,
via Marzolo 8, 35131 Padova, Italy and
INFN - Sezione di Padova
defelice @pd. infn.it
Aim of the RAMOD project is to solve the general relativistic ray-tracing problem in the
gravitational field of the Solar System to the accuracy of a micro-arcsecond in the
measurements of angles. The project consists in the construction of a family of models with
increasing complexity and accuracy each one acting as test bed for the more advanced
ones. The models are operated by a numerical code having a multimodular structure
which allows one to activate specific functions according to the need. Here we discuss
the latest contribution to the model structure consisting in a new modulus conceived to
analyze the error budget and determine the stellar positions.
Keywords: General Relativity; astrometric models.
1. Introduction
Modern space technology will soon provide stellar imaging with an accuracy of
a micro-arcsecond (/xas). At this level one has to take into account the general
relativistic effects on light propagation arising from metric perturbations due not
only to the bulk mass but also to the rotational and translational motion of the
bodies of the Solar System and to their multipole structure.
Aim of the RAMOD project is to develop a general relativistic astrometric model
which wonld enable us to deduce, to the microarcsecond accuracy, the astrometric
parameters of a star in our Galaxy from observations taken by a satellite like Gaia.
Up to now we have produced several relativistic astrometric models with
increasing accuracy The first two, termed RAMODl1 and RAMOD2,2 have been essential
touchstones of comparison for the more advanced many-body model RAMOD34
where the astrometric problem is tackled in the presence of geometry perturbations
due to the bodies of the Solar System. Here again we consider first a static case
corresponding to an accuracy of the milliarcsecond. However, snch an accuracy is
not enough for the modern space astrometry, hence we further extended RAMOD3
into a dynamical model accurate to a microarcsecond, which means retaining terms
of the order of 1/c3. This is RAMOD45 which has been succesfully tested.
The above model produces a set of coupled second order partial differential
equations (the master equations) whose integration requires appropriate boundary
conditions which are fixed at the observation in terms of the observables. For Gaia,
they are the coordinate position of its trajectory and the direction of the incoming
2543
2544
light ray with respect to the spatial axes of a frame comoving with the satellite.
The problem of defining the boundary conditions has been solved up to 1/c3
in two side models termed RAMODINOl6 and RAMODIN02.7 The final result of
this analysis was an analytical relation between the observables and the boundary
contitioiis where the satellite is identified with an appropriae tetrad fully compatible
with the motion and attitute specifications of Gaia.
The observables and the satellite attitude will have some kind of uncertainty
which causes an error of the solutions. The knowledge of these errors is as important
as that of the solutions themselves. In a recent work8 we investigated this problem
for the case of the observables; the extension to the satellite attitude will be discussed
elsewhere. Hereafter Greek indeces run form 0 to 3 while latin indeces run fron 1 to
3 corresponding to Carthesian-like coordinates (x, y, z).
2. The error budget
The master equations of RAMOD4,5 read:
_ = -VPPdohij ~ PP [dihkj - -dkh
1 - 1
- -tktdih00 - t (dihko + d0hki - dkh0i) + -dkh00 . (1)
where a is a parameter along the light ray and the metric perturbations ha(j are
at least of the order of 1/c2. The unknowns in (1) are the spatial components
of the space-like vector £ which physically identifies, at each point of the light
trajectory, the line of sight of a local baricentric observer. Equations (1) only admit
a numerical solution in the form t((j) = I%(I(ao),dpha[}(a)), where Ik((Jo) = IkQ)
are the components of the vector field I &t the time of observation cjq. Evidently the
boundary values £(0) can be expressed in terms of the metric coefficients at the time
of observation cto and of the observables ea, i.e. the direction cosines of the incoming
light ray with the satellite spatial frame {Ea}- The latter fixes the satellite attitude.
In other words (see Ref. 8 for details) ^0) = ^(0)(ea, E&, hai3(a0)). For Gaia it is
enough to consider the following approximate solution
{0)ik(a) = £k{0) + ^ Fk(I{o),dpha0(a'))da'. (2)
A numerical integration shows that the above solution 10,^(0) differs from the full
solution £k(a) by an amount which ranges from 5 x 10-6 arcsec for Sun-skimming
rays to 5 x 10-15 arcsec for rays passing near Jupiter surface. At the angular distance
from the Sun of about 3 degrees the difference goes down to few 10~8 arcsec, so the
approximate solution is accurate enough and we can exploit this semplification to
overcome the difficulty we would have encounterd in fixing the error budget.
Following Ref. 8 we have applied standard variational method to analyze how
statistical errors of the boundary conditions arising from uncertainties in the
observables propagate to the solutions. The main result of our analysis is given by the
2545
following expression:
8xl{a*) = 8EQ) J da exp
O"0
{ofiljda'
(3)
where (o)Wk{£(o),dpha0) = Hkn(£(O),dpha0(a)) with
r)Tk
nkn(£(a),dpha0(a))=w-. (4)
Here xl(a*) would be the position of the star if we knew the value of the emission
parameter a*; the latter however can be deduced as follows.
3. The stellar position
Let a star be observed from within the GAIA satellite when the latter was at position
Xo on its orbit with respect to the baricenter of the Solar System; at this moment
the quantities ea{xo) fixed the instantaneous local line of sight. Let the same star
be observed later when the satellite was at Xo + Axo on its orbit with ea(xo + Axo)
being the new corresponding observables. If we treat the quantity
S'ea=ea(x0 +Ax0)-ea{x0) (5)
as a (small) variation of the observables then we can apply the variational method
having in mind that now the variations of the boundary values R0) are given by
S'£\o) = ^(ojM^o + Ax0); ha0{xo + Ax0)} - ^0)[ea(xo); ha0{xo)] (6)
corresponding to a non zero variation - Axo in fact - of the point of observation
while the emission point is being fixed instead. Under these conditions the emission
parameter a* is solution of the equation
0 = AxJ0 + S'£^0) / da" exp
J an
H\do'
(O)T- j
(7)
Analytical considerations concerning the consistency of this equation with expected
results is carried on in.8 A numerical investigation shows that the above equation
is able to provide a very accurate determination of the stellar position.
References
1. F. de Felice, M. G. Lattanzi, A. Vecchiato, P. L. Bernacca, A&A 332, 1133 (1998).
2. F. de Felice, M. G. Lattanzi, A. Vecchiato, A&A 373, 336 (2001).
3. A. Vecchiato, M. G. Lattanzi, B. Bucciarelli, M. T. Crosta, F. de Felice and M. Gai,
A&A 399, 337 (2003).
4. F. de Felice, M. T. Crosta, A. Vecchiato, B. Buciarelli and M. G. Lattanzi, ApJ 607,
580 (2004).
5. F. de Felice, A. Vecchiato, M. T. Crosta, B. Bucciarelli and M. G. Lattanzi, ApJ 653,
1552 (2006).
6. D. Bini and F. de Felice, Class. Quantum Grav. 20, 2251 (2003).
7. D. Bini, M. T.Crosta and F. de Felice, Class. Quantum Grav. 20, 4695 (2003).
8. F. de Felice and G. Preti, Class. Quantum Grav. 23, 5467 (2006).
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Clock and Space Tests
of Gravity
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DYNAMICAL CLOCK SYNCHRONIZATION IN EINSTEIN'S
THEORY: IMPLICATIONS FOR THE ACES MISSION OF ESA
LUCA LUSANNA
Sezione INFN di Firenze, Polo Scientifico,
Via Sansone 1, 50019 Sesto Fiorentino (FI), Italy
lusanna@fi. infn. it
The ACES (Atomic Clock Ensemble in Space) mission1 will operate a new
generation of atomic clocks in the microgravity environment of the ISS (International
Space Station). Fractional frequency stability and accuracy of few parts in 10~16
will be achieved. The on-board time base, distributed on Earth via a microwave link,
will be used to perform space-to-ground as well as ground-to-ground comparison of
atomic frequency standards (in the first mission only two-way frequency shifts will
be measured). Based on these comparisons, ACES will develop applications in time
and frequency metrology, universal time scales, global positioning and navigation,
geodesy and gravimetry.
To realize these achievements all the aspects of the mission have to be modeled
on the most advanced understanding of Special (SR) and General (GR,)
Relativity near the Earth, taking into account relativistic effects at the order 1/c32 (the
GR effect of the gravitational red-shift generated by the geoid and the connected
Shapiro time delay, of the order of few picoseconds, will show up) and not only at
the order 1/c2 like in GPS (Global Positioning System)3 and in DSN (Deep Space
Network) of NASA at JPL Ref. 4, which governs the motion of satellites. The
precision of the atomic clocks involved in the ACES mission raises a set of interconnected
problems to be clarified by assuming the validity of GR as a working hypothesis
(after their clarification one can look at deviations from GR). This implies that the
needed reference systems and the underlying metrology's notions must be defined at
the 1.5 Post-Newtonian order in accord with the IAU conventions,5'6 where there is
the definition of the (quasi-inertial) BCRS (Barycentric Celestial Reference System)
[centered on the solar system barycenter, with space axes kinematically non-rotating
with respect to some fixed stars, and a time axis (the barycenter world-line)
employing a coordinate time scale TCB] and GCRS (Geocentric Celestial Reference
System) [centered at the geocenter, with spaces axes kinematically non-rotating
with respect to BCRS, and a time axis (the geocenter world-line) employing a
coordinate time scale TCG].
Both TCB and TCG are connected in conventional ways to the proper time
standard, the SI atomic second (9,192,631,770 cycles of the radiation
corresponding to the ground state hyperfine transition of Cesium 133 [BIPM1998]). However,
BCRS and GCRS are not quasi-inertial systems of SR but are non-inertial systems
of GR, because to both of them in the given 4-coordinates is associated a 4-metric
tensor, solution of Einstein's equations in harmonic coordinates (so that both TCB
and TCG are harmonic time coordinates) at the 1.5 PN (i.e. Post-Newtonian at the
2549
2550
order 1/c3) approximation.
Another non-inertial aspect to be taken into account is the rotation of the Earth,
which requires rotating reference frames and adapted time scales connected to IERS
(International Earth Rotation and Reference System Service).7 In particular ITRS
(International Terrestrial Reference System, BIPM) is defined from GCRS by a
spatial rotation leading to a quasi-Cartesian system and uses TCG as coordinate time.
ITRS has terrestrial latitude, longitude and height given with respect to a reference
ellipsoid (an oblate spheroid best fit of the geoid, i.e. a gravitational equipotential
surface). The standard reference ellipsoid WGS84 (World Geodetic System 1984) is
the basis for the coordinates obtained from GPS.3'4
This state of affairs requires a rethinking of SR and GR, which emphasizes the
role of non-inertial frames centered on accelerated observers with their associated
notion of instantaneous non-Euclidean 3-spaces. Namely Einstein's convention for
the synchronization of distant clocks is no more sufficient, since it only identifies the
instantaneous Euclidean hyper-planes of an inertial system centered on an inertial
observer.
While in Newtonian physics space and time are absolute notions, in SR only
space-time (with its conformal structure identified by incoming and outgoing rays
of light) is absolute. Any notion of instantaneous 3-space and of spatial distance
is observer- and frame-dependent, since it is determined by the arbitrary choice
of a convention for the synchronization of distant clocks done by a time-like
observer. Given the observer and the convention, a M0ller-admissible 3+1 splitting of
Minkowski space-time (and therefore a (in general) non-inertial frame centered on
the observer) is obtained.8 It is convenient to use radar 4-coordinates (r. <jr) adapted
to the 3+1 splitting: r is observer proper time and ar are curvilinear 3-coordinates
on each equal-time 3-surface Sr with origin on observer's world-line.
In the framework of parametrized Minkowski theories,9 the dynamics of every
isolated system admitting a Lagrangian formulation is formulated in such a way
that the change of the clock synchronization convention is a gauge transformation,
so that any admissible convention is gauge equivalent to Einstein's one. The Wigner-
covariant rest-frame instant form of dynamics is associated with the inertial 3+1
splitting whose instantaneous 3-spaces are orthogonal to the conserved 4-momentum
of the isolated system.
In particular in Ref. 8 there is the definition of the simplest family of 3+1
splittings of Minkowski space-time, whose instantaneous 3-spaces are hyper-planes
endowed with differentially rotating 3-coordinate systems (rigid rotations are
forbidden by M0ller conditions in SR and GR), which could be used to model Earth's
rotation in GCRS with a covariant treatment of the Sagnac effect and a
reformulation of the SR part of the results of Ref. 2. If the ACES mission will be successful,
it will open the path to the future determination of the one-way time transfer from
Earth to ISS: this will allow to determine the non-inertial SR deviation from
Einstein's convention for clock synchronization at the order 1/c3.
The treatment of the previous effects in the framework of GR, where only non-
2551
inertial frames exist due to the equivalence principle, can be done by using the
rest-frame instant form of metric and tetrad gravity reviewed in Ref. 9. In this
framework it is possible to show10 that any solution of Einstein's equations in a
given 4-coordinate system dynamically determines an associated 3+1 splitting of
the Einstein space-time, namely a global non-inertial frame centered on some non-
inertial observer, in accord with the fact the whole chrono-geometrical structure of
Einstein's space-times is dynamical: the line element is determined by the 4-metric
solution of Einstein's equations. As a consequence there is a dynamical convention
for clock synchronization and a set of dynamical instantaneous 3-spaces emerging
also from the 1.5 PN solution used in the IAU conventions. The resulting non-
Euclidean 3-spaccs differ from the hyper-planes TCG = const, of GCRS by terms
of the order 1/c3. This introduces a further GR. deviation (besides the SR one) from
Einstein's convention for clock synchronization.11
References
1. C. Salomon et. al., A Search for Variations of Fundamental Constants by using Atomic
Fountain Clocks, C.R.Acad.Sci.Paris t.2 Se'rie 4, 1313 (2001) (physics/0212112); see
the talks at the Workshop Advances in Precision Tests and Experimental
Gravitation in Space (Firenze, September 28/30, 2006) (http://www.fi.infn.it/GGI-grav-
space/egs-w.html).
2. L. Blanchet, C. Salomon, P. Teyssandier and P. Wolf, Relativistic Theory for Time
and Frequency Transfer to Order 1/c3, Astron. Astrophys. 370, 320 (2000).
3. N. Ashby, Relativity in the Global Positioning System, Living Reviews in Relativity
(2003-1) (http://www.livingreviews.org).
4. T.D. Mover, Formulation for Observed and Computed Values of Deep Space Network
Data Types for Navigation (John Wiley, New York. 2003).
5. M. Soffel, S.A. Klioner, G. Petit, P. olf, S.M. Kopeikin, P. Bretagnon, V.A. Brumberg,
N. Capitaine, T. Damour, T. Fukushima, B. Guinot, T. Huang, L. indegren, C. Ma, K.
Nordtvedt, J. Ries, P.K. Seidelmann, D. Vokroulicky', C. Will and Ch. Xu, The IAU
2000 Resolutions for Astrometry, Celestial Mechanics and Metrology in the Relativistic
Framework: Explanatory Supplement, Astron. J., 126, pp.2687-2706, (2003) (astro-
ph/0303376); G.H. Kaplan, The IAU Resolutions on Astronomical Reference Systems.
Time Scales and Earth Rotation Models, U.S. Naval Observatory circular No. 179
(2005) (astro-ph/0602086).
6. G. Petit and P. Wolf,Relativistic Theory for Time Comparisons: a Review, Metrologia,
42, S138-S144, (2005).
7. IERS Conventions (2003), eds. D.D. McCarthy and G. Petit, 1ERS TN 32 (2004),
Verlag des BKG.
8. D. Alba and L. Lusanna, Generalized Radar JrCoordinates and Equal-Time Cauchy
Surfaces for Arbitrary Accelerated Observers (2005), submitted to Int. J. Mod. Phys.
D (gr-qc/0501090); Simultaneity, Radar JrCoordinates and the 3+1 Point of View
about Accelerated Observers in Special Relativity (2003) (gr-qc/0311058).
9. L. Lusanna, The Chrono-geometrical Structure of Special and General Relativity: a
Re-Visitation of Canonical Geometrodynamics, Lectures given at the 42nd Karpacz
Winter School of Theoretical Physics, "Current Mathematical Topics in Gravitation
and Cosmology," Ladek, Poland, 6-11 February 2006 (gr-qc/0604120).
10. D. Alba and L. Lusanna, The York Map as a Shanmugadhasan Canonical Transforma-
2552
tion in Tetrad Gravity and the Role of Non-Inertial Frames in the Geometrical View
of the Gravitational Field (2006), submitted to Gen. Rel. Grav. (gr-qc/0604086); L.
Lusanna and M. Pauri, Dynamical Emergence of Instantaneous 3-Spaces in a class of
Models of General Relativity, to be published in Relativity and the Dimensionality of
the World, ed. A. van der Merwe (Springer Series Fundamental Theories of Physics)
(gr-qc/0611045).
11. See my talk at the SIGRAV Graduate School on Experimental
Gravitation in 5pace(Firenze, September 25-27, 2006) (http://www.fi.infn.it/GGI-grav-
space/egs-s.html).
STEP PROTOTYPE DEVELOPMENT STATUS
C. MEHLS, C. BAY ART, J. BOWER, B. CLARKE, C. COX, D. GILL, D. STRICKER,
N. VORA, S. WANG, P. ZHOU, R. TORII, P. WORDEN and D. DEBRA
Hansen Experimental Physics Laboratory, Stanford University, 445 Via Palou
Stanford, CA 94305-4085, USA
H. DITTUS
ZARM, University Bremen, Am Fallturm
28359 Bremen, Germany
F. LOEFFLER
PTB Braunschweig, Bundesallee 100
38116 Braunschweig, Germany
STEP, the Satellite Test of the Equivalence Principle [1], proposes to test the
Equivalence Principle to a part in 1018 by comparing the free-fall acceleration of
cylindrical shaped test masses [2] in Earth orbit. Magnetic bearings constrain the test
mass motion to their axis of symmetry [3]. The displacement of the test masses is
measured using a DC SQUID and superconducting coils [4], enabling a displacement
sensitivity as small as 10"15 m. In combination with a small spring stiffness a differential
acceleration sensitivity of 10"18 g is achievable. Residual satellite acceleration is reduced
to better than 10" 4 g by compensating satellite drag forces with thrust provided by
helium gas.
We report on recent progress in the development of STEP prototype flight
accelerometers, in particular the development of the high precision quartz housing for the
engineering inner accelerometer and the testing of SQUID and capacitive readout
systems using 'brass board' accelerometer prototypes.
1 Components and Assembly of the Engineering Model Inner Accelerometer
The housing aligns the magnetic bearings of the inner and outer test masses through
precision-machined line contacts between housing components. It also provides a
standard for the test mass position in the axial direction. The alignment combined with
the precision achieved in patterning the bearing circuits on the quartz substrates
ultimately determines how well the test masses will reject radial disturbances. The
accumulated dimensional errors of the actual parts have been calculated, and showed that
requirements on housing alignment (for CMRR of lCf4) and concentricity have been met.
The quartz housing, having a small coefficient of thermal expansion, provides a
stable reference frame to measure the test mass position, and subsequently calibrate the
SQUID. Ideally the centers of the capacitive and magnetic sensors (SQUID) coincide.
Due to the small machining tolerances down to lum for single quartz components the
accumulated tolerances result in a separation of only 2um for an actual assembly
combination, less than the required 5um separation.
2553
554
Finally, we have for the first time assembled and disassembled a nearly complete set
of quartz housings (5 components) for an engineering inner accelerometer (fig. 1).
Figure 1. Assembled quartz housing of engineering model inner accelerometer (left), and engineering model
components showing the gold coated eapacitive sensing electrodes (right).
The accelerometer is also equipped with capacitance sensors machined in the quartz
housing, which can measure test mass displacement in all six degrees of freedom (fig.l).
Electrodes and other surfaces facing tlie test mass are coated with a 500nm gold layer.
2 SQUID Readout System
The acceleration measurement of the test masses is accomplished by superconducting
coils on opposite sides of the test masses. The coils are connected to a DC-SQUID and
provide also the axial constraint force, produced by the current trapped in the coils.
Displacements of tlie test masses result in changing currents which are detected by the
SQUID. Larger currents give higher displacement sensitivity, but also increase tlie spring
stiffness, leading to smaller test mass displacements for a given acceleration. The optimal
on-orbit current trapped in the coils is 10mA [5].
We have manufactured coils on the quartz housing substrate consisting of 6 turns of
lOOum wide Nb/Au traces of 400nm/50nm thickness. A 50um PblnAu wire bond closes
tlie return path to the current leads. The measured coil inductance is 3uH. Critical current
measurements showed a transition at much smaller currents than measured just for the
traces, which have a critical current of up to 1 A, depending on surface condition [6]. We
found that the weak link is tlie wrap around of the traces down to the side of the coil
substrate. After rounding and polishing the edges a critical current of up to 100mA was
achieved. The superconducting joints had critical currents of at least 35mA, which is
sufficient to trap tlie required current.
3 Capacltlve Position Detector
We have build a complete 'brass board model' inner accelerometer, with the same
nominal dimensions and features as tlie engineering model, to test all components of the
2555
capacitive sensor and its possible interference with the SQUID readout system. All
electrode surfaces are gold coated and separated by 0.5mm wide 0.5mm deep grooves.
Electrical connection is made by specifically designed spring connectors soldered to
coaxial cables. A caging mechanism, designed to hold the test mass during transportation
and launch, is used to position the test mass in the center of the cavity.
Measured capacitances of the various sense electrodes to the test mass are between
lpF and 12pF, which is 10% to 30% (axial / radial sense electrodes) higher than the
calculated capacitances (if edge effects are neglected). The measured total capacitance is
about 25%) higher than the sum of all single capacitances, which is roughly V8pF,
including caging pins. The reason for the discrepancies is currently under investigation.
Injecting signals into different electrodes showed that capacitive coupling to the
pick-up coil is increasing with decreasing distance from the coil. The coupling is
expected to disappear after a charge control layer is applied on top of the coil.
In a different set-up, the test mass was moved along the axial direction within the
electrode housing, and showed the expected linear variation of capacitance [7].
4 Summary
We have assembled the precisely machined quartz housing for an inner accelerometer.
Specifications on alignment, axial position, and concentricity have been met. The
capacitive sensing circuitry was defined, and capacitive cross coupling to the SQUID coil
studied. We will soon start investigations of the cross coupling with an operating SQUID
sensing system at 4K. The capacitive readout system will be tested to a higher precision
using the engineering model and a precisely machined Nb test mass.
Acknowledgments
This work was supported by NASA through Marshall Spaceflight Center under
cooperative agreement #NNM04AA18A-04.
References
1. J. Mester, R. Torii, P. Worden, N. Lockerbie, S. Vitale, and C. W. F. Everitt, Class.
Quantum Grav. 18, 2475 (2001).
2. N. A. Lockerbie, X. Xu, and A.V. Veryaskin, Class. Quantum Grav. 13 A91-A95
(1996).
3. P. Worden, PhD Thesis, Stanford University.
4. H.J. Paik, JAppl. Phys. 47, 1168 (1976).
5. O. Clavier, PhD Thesis, Stanford University.
6. J. Bower, C. Mehls, N. Vora, R. Torii, and T. Kenny, ASC 2006 (2006).
7. P. Ambekar, PhD Thesis, Stanford University.
ON STELLAR SYSTEM TESTS OF THE
COSMOLOGICAL CONSTANT
MAURO SERENO* and PHILIPPE JETZERt
Institut fur Theoretische Physik, Universitat Zurich,
Winterthurerstrasse 190, CH-8057 Zurich, Switzerland
* sereno@physik. unizh. ch
T jetzer@physik. unizh. ch
The understanding of the cosmological constant A is one of the most outstanding
topic in theoretical physics. On the observational side, the cosmological constant
is motivated only by large scale structure observations as a possible choice for the
dark energy. In fact, when fixed to the very small value of ~ 10-46km~ , A, together
with dark matter, can explain the whole bulk of evidence from cosmological
investigations. In principle, the cosmological constant should take part in phenomena on
every physical scale but due to its very small size, a local independent detection of
its existence is still lacking. Measuring local effects of A would be a fundamental
confirmation and would shed light on its still debated nature, so it is worthwhile to
investigate A at any level.
The influence of the cosmological constant on the gravitational equations of
motion of bodies with arbitrary masses can be discussed with a perturbation approach
and eventually the two-body problem can be solved.1 Due to the anti-gravity effect
of the cosmological constant, the barycenter of the system drifts away. The relative
motion is like that of a test particle in a Schwarzschild-de Sitter space-time with a
source mass equal to the total mass of the two-body system. Hence, we can use this
last metric to consider local effects of A.
The main effect of the cosmological constant on a bound gravitational system is
the precession of the pericenter on the orbital motion. We determined observational
limits on the cosmological constant from measurements of the periastron advance in
stellar systems, in particular binary pulsars and the solar system.1 Based on accurate
planetary ephemerides properly accounting for the quadrupole moment of the Sun
and for major asteroids, the best constraint comes from Mars and Earth, A < 1 —2 x
10_36km~ . Due to the experimental accuracy, observational limits on A from binary
pulsars are still not competitive with results from interplanetary measurements in
the solar system. Accurate pericenter advance measurements in wide systems with
orbital periods > 600 days could give an upper bound of A < 10~34 — 10_33km-2, if
determined with the accuracy performed for B1913+16, i.e. 5lu > 10~6 deg/years.
For some binary pulsars, observations with an accuracy comparable to that achieved
in the solar system could allow to get an upper limit on A as precise as one obtains
from Mars data.
The effect of A on the precession of a gyroscope, the change in the mean
motion of a massive body and the gravitational redshift can be analyzed as well in the
framework of the Schwarzschild -de Sitter metric.2 As it could be expected from a
dimensional argument, relative variations due to A always goes as ex A(o,3/rg)(a/rg)1,
2556
2557
i = {0,1,...}, with a the typical physical length of the system and rg = GM/c2
the typical gravitational radius of the massive source. An analysis of anomalies in
the mean motion provides limits at the same order of magnitude. Measurements
of gyroscope precession of the Moon, via laser ranges, or of satellite, such as the
Gravity Probe B mission, fall short in constraining L. Beyond the solar system,
limits competitive with Earth precession data could come from gravitational redshift
measurements in white dwarfs.
The bound on A from Earth or Mars perihelion shift is nearly ~ 1010 times
weaker than the determination from observational cosmology but it still gets some
relevance. The cosmological constant might be the non perturbative trace of some
quantum gravity aspect in the low energy limit. A is usually related to the vacuum
energy density, whose properties depends on the scale at which it is probed. So that,
in our opinion, it is still interesting to probe A on local scales. In fact, these tests can
probe the universal origin of the cosmological constant on very different scales. Any
detection of perturbations in the orbital motion in a bound gravitational system,
either the solar system or a binary pulsar, probes A on a scale of the order of the
astronomical unit. On the other hand, the relevant length scale in measurements
of gravitational redshift is the distance to the source, which is of order of < 102 pc
for galactic white dwarfs. The experiments we have considered cover a range in
distance of nearly seven orders of magnitude, which help in filling the gap between
local systems and the cosmological distances.
Measurements of periastron shift should be much better in the next years. New
data from space-missions should get a very high accuracy and might probe spin
effects on the orbital motion. A proper consideration of the gravito-magnetic effect
in these analyses could also play a central role to improve the limit on A by several
orders of magnitude. Near-future technology should allow to improve bounds by
nearly five orders of magnitude, the crucial step being radio ranging observations of
solar system outer planets. Beyond the solar system, together with future
measurements of periastron advance in wide binary pulsars, gravitational redshift of white
dwarfs could provide bounds competitive with Earth and Mars data.
Acknowledgments
M.S. is supported by the Swiss National Science Foundation and by the Tomalla
Foundation.
References
1. Ph. Jetzer and M. Sereno, Phys. Rev. D73, 044015 (2006)
2. M. Sereno and Ph. Jetzer, Phys. Rev. D73, 063004 (2006)
THE LENSE-THIRRING EFFECT AND THE PIONEER
ANOMALY: SOLAR SYSTEM TESTS
LORENZO IORIO*
Viale Unita di Italia 68, 70125, Bari (BA), Italy
lorenzo.iorio@libero.it
We report on a test of the Lense-Thirring effect with the Mars Global Surveyor orbiter
and on certain features of motion of Uranus, Neptune and Pluto which contradict the
hypothesis that the Pioneer anomaly can be caused by some gravitational mechanism.
1. The Lense-Thirring effect
Up to now the Lense-Thirring effect1-5 has been only tested in the terrestrial
gravitational field with the LAGEOS satellites.6~10 Although the relativistic predictions are
not in disagreement with the results of such tests, their realistic accuracy has always
been controversial.8,11~13 Recent advances in planetary ephemerides14 have made
meaningful to compare the relativistic predictions for the Lense-Thirring effect of
the Sun on the inner planets of the Solar System10'16 to the least-squares estimated
corrections to the perihelia rates of such celestial bodies14. There is no contradiction
between them; although the errors are still large so that also a zero-effect cannot
be ruled out, the hypothesis of the existence of the solar gravitomagnetic field is in
better agreement with the data16, fn April 2004 the GP-B spacecraft17'18 has been
launched to measure the Schiff precession19 of the spins of four superconducting
gyroscopes carried onboard: the expected accuracy is « 1%. The field of Mars has
recently yielded the opportunity of performing another test20 of the Lense-Thirring
effect. Almost six years of range and range-rate data of the Mars Global Surveyor
(MGS) orbiter, together with three years of data from Odyssey, have been used in
order to precisely determine many global properties of Mars21. As a by-product,
also the orbit of MGS has been very accurately reconstructed21. The average of the
RMS overlap differences of the out-of-plane part of the MGS orbit amounts to 1.613
m over an about 5-years time span (14 November 1999-14 January 2005). Neither
the gravitomagnetic force was included in the dynamical models used in the data
reduction, nor any empirical out-of-plane acceleration was fitted, so that the RMS
overlap differences entirely account for the martian gravitomagnetic force. The
average out-of-plane MGS Lense-Thirring shift over the same time span amounts just
to 1.610 m: a discrepancy of 0.2%. The error has been preliminarily evaluated as20
0.5%. Let us. finally, note that the MGS test is based on a data analysis done in a
completely independent way with respect to the author of Ref. 20, without having
gravitomagnetism in mind at all.
* Fellow of the Royal Astronomical Society
2558
2559
2. The Pioneer anomaly
The Pioneer anomaly22^24 is an unexpected, almost constant and uniform extra-
acceleration Apio directed towards the Sun of (8.74 ± 1.33) x 10~10 m s~2 detected
in the data of both the Pioneer 10/11 probes after 20 AU. It has attracted much
interest because of the possibility that it is a signal of some failure in the
currently known laws of gravitation25'26. If the Pioneer anomaly was of gravitational
origin, it should then fulfil the equivalence principle and an extra-gravitational
acceleration like Ap10 should also affect the motion of any other object moving in
the region in which the Pioneer anomaly manifested itself. Uranus, Neptune and
Pluto are ideal candidates to perform independent and clean tests of the
hypothesis that the Pioneer anomaly is due to some still unexplained features of gravity.
Indeed, their paths lie at the edge of the Pioneer anomaly region or entirely reside
in it because their semimajor axes are 19.19 AU, 30.06 AU, and 39.48 AU,
respectively, and their eccentricities amount to 0.047, 0.008 and 0.248. Under the action
of Aploy whatever physical mechanism may cause it, their perihelia would secularly
precess at unexpectedly large rates. For Uranus, which is the only outer planet
having completed a full orbital revolution over the time span for which modern
observations are available, the anomalous perihelion rate is —83.58 ± 12.71 arc-
seconds per century. E.V. Pitjeva in processing almost one century of data with
the EPM2004 ephemerides27 also determined extra-rates of the perihelia of the
inner14 and outer28 planets as fit-for parameters of global solutions in which she
contrasted, in a least-square way, the observations to their predicted values
computed with a complete suite of dynamical force models including all the known
features of motion. Thus, any unmodelled force as Api0 is entirely accounted for
by the perihelia extra-rates. For the perihelion of Uranus she preliminarily
determined an extra-rate of +0.57± 1.30 arcseconds per century. The quoted uncertainty
is just the mere formal, statistical error: the realistic one might be up to 10 — 30
times larger. Even if it was 50 times larger, the presence of an unexpected
precession as large as that predicted for Uranus would be ruled out. It is unlikely
that such a conclusion will be substantially changed when further and extensive
re-analysis29'30 of the entire Pioneer 10/11 data set will be carried out since they
will be focussed on what happened well before 20 AU. This result is consistent
with the findings of Ref. 31 in which the time-dependent patterns of a cos 6 and
S induced by a Pioneer-like acceleration on Uranus, Neptune and Pluto have been
compared with the observational residuals determined by Pitjeva27 for the same
quantities and the same planets over a time span of about 90 years from 1913
(1914 for Pluto) to 2003. While the former ones exhibited well defined
polynomial signatures of hundreds of arcseconds, the residuals did not show any
particular patterns, being almost uniform strips constrained well within ±5
arcseconds over the data set time span which includes the entire Pioneer 10/11
lifetimes.
2560
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Lense-Thirring Test with SLR and the GRACE Gravity Mission, in Proceedings of
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Noomen et al. (NASA Goddard, 2003).
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from Laser Ranging to Artificial Satellites, in Nonlinear Gravitodynamics. The Lense-
Thirring Effect, eds. R. Ruffini and C. Sigismondi (World Scientific, 2003).
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13. L. Iorio J. Geodesy 80, 128 (2006b).
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15. L. Iorio Astron. Astrophys. 431, 385 (2005a).
16. L. Iorio Planet. Space ScL, at press, gr-qc/0507041 (2007).
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Gyroscope Performance, in Proc. Int. School Phys. "Enrico Fermi" Course LVI, ed.
B. Bertotti (Academic Press, 1974).
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24. M.M. Nieto and J.D. Anderson Class. Quantum Grav. 22, 5343 (2005).
25. M.-T. Jaekel and S. Reynaud Class. Quantum Grav. 22, 2135 (2005).
26. J.R. Brownstein and J.W. Moffat Class. Quantum Grav. 23, 3427 (2006).
27. E.V. Pitjeva Sol. Sys. Res. 39, 176 (2005b).
28. E.V. Pitjeva paper presented at 26th meeting of the IAU, Joint Discussion 16, #55,
22-23 August 2006, Prague, Czech Republic, (2006a); private communication (2006b).
29. S.G. Turyshev, V.T. Toth, L.R. Kellogg et al. Int. J. Mod. Phys. D 15, 1 (2006a).
30. S.G. Turyshev, M.M. Nieto and J.D. Anderson EAS Publication Series 20, 243
(2006b).
31. L. Iorio and G. Giudice New Astron 11, 600 (2006).
THE EQUIVALENCE PRINCIPLE AND ITS TESTS IN THE
CONTEXT OF GRAVITY, QUANTUM MECHANICS AND
COSMOLOGY
C. S. UNNIKRISHNAN
Gravitation Group, Tata Institute of Fundamental Research, Mumbai - 400 005, India
* E-mail: unni@tifr.res.in
www.tifr.res.in
After a brief review of some results pertaining to the equivalence principle in the
context of gravity and quantum mechanics, I discuss the important relation between the
equivalence principle and the matter filled universe. I show that the universality of free
fall is surprisingly robust in this context even if the gravitational constant is material
dependent.
Keywords: Equivalence Principle, Inertia, Quantum Mechanics, Casimir energy, Cosmic
Relativity, Gravitomagnetism, Universe.
1. The Equivalence Principle, gravity and quantum mechanics
The Equivalence Principle (EP) derives its empirical basis from the universality of
free fall (UFF). The postulated equivalence of the inertial and the gravitational
mass, and the implied equivalence of gravity and an accelerated frame, lead to the
correct theory of gravity. Possibilities of small violations of the EP and UFF in the
context of physics beyond the standard model are vigorously pursued in experiments
on the earth and in space, as evident from papers presented in these sessions.
There have been questions raised about the validity of the EP in the context
of quantum dynamics. Rigorous answers confirming the validity of the UFF and
the equivalence of physics in a uniform gravitational field and in an accelerated
frame for quantum dynamics, to the extent tested by experiments in the classical
context, were discussed earlier.1 The validity of the EP in the quantum context
follows simply from calculating the quantum propagator in the gravitational field
and in an equivalent accelerated frame.
Though the UFF has been tested for a variety of different elements, and forms
of energy, quantum mechanical energy in the zero point modes of fields and the
Casimir energy of the zero point modes in constrained conducting geometries remain
elusive from a direct test. The difficulty in testing the UFF for Casimir energy arises
from the fact that its contribution to the total rest energy of the test body is less
than 10-24 or so, whereas even planned experimental tests are limited to a relative
precision of 10~17. However, the fact that the Casimir energy can be converted
to kinetic energy by simply letting the conducting boundaries to fall towards each
other by the Casimir force, for example, allows a two-step test of the EP for the
Casimir energy.1 Since EP is tested for kinetic energy of matter in the laboratory
(this is of the order of binding energies in test masses) it would become possible to
construct a perpetual motion machine if the Casimir energy did not obey the UFF.
Therefore, by combining energy conservation requirement with existing results from
2561
2562
the tests of the EP, one can conclude that the Casimir energy between conducting
plates indeed obeys that UFF and EP good to a few percent or so. Going beyond
this requires improved precision in the Casimir effect experiments.
2. The Equivalence Principle and the Universe
My main theme in this paper is to establish that the EP is a consequence of the
gravitational interaction with all the matter in the universe.2 While this had been
already indicated by the Mach's principle, and by the work of D. Sciaina,3 I sketch
a convincing treatment that establishes this fact in the context of the new and
relatively complete knowledge of the properties of the observed universe. On the way
we show that Newton's law of motion is a relativistic gravitational law arising from
cosmic gravitomagnetism, and that UFF remains valid even if different materials
interact gravitationally with a material dependent gravitational constant.
In a frame that is uniformly moving with velocity V with respect to the CMBR
or the average matter distribution of the universe with nearly critical density, the
FRW metric is anisotropic and there are off-diagonal elements, gor, representative of
gravitoinagnctic potentials, equal to V/c. This immediately implies that an
acceleration of such a frame will lead to a time dependent off-diagonal metric element and
a corresponding time dependent gravitomagnetic potential or a nonzero Christoffel
symbol equivalent to a 'classical force'. The time dependence might arise from
either the magnitude of the velocity or its direction changing with time. In the former
case we get the Newton's law from cosmic gravitomagnetism, and the latter case
corresponds to the familiar centrifugal force. The Coriolis force comes out as the
equivalent of the Lorentz force law in electromagnetisin.
A time varying vector potential generates the force that opposes the motion
dV __„
F = —Grrig—-— = —Gmg a (1)
Here mg is the gravitational mass (charge). This reactive force can be identified as
the Newton's second law, now derived from relativistic cosmic gravity by identifying
rrii = —Grng\ It is this aspect that was discussed by Sciama in the context of the
origin of inertial forces and the Mach's principle.3 The inertial mass is simply the
gravitational mass scaled by the cosmic gravitational potential. In the language of
the gravitational potential of the universe, the inertial mass is
to; = —^mg (2)
and the ratio of the inertial and gravitational masses is $/c2, determined by
average matter density and other properties of the universe. Thus the mystery of this
universal ratio is completely solved. Since $/c2 = 1 for a critical universe, one can
see that to,; = mg. What is even more interesting is that the necessary equivalence
of rrii and mg in an experimental situation can be shown even without assuming the
universality of the gravitational constant. For this, let us assume that the effective
2563
gravitational constant for interaction of the test bodies A and B depends on some
properties of the body. Then equation (2) will read as
m,i(A) = -T^m9 = KGApmg(A)
rrii(B) = -2—mg = nGBpmg(B) (3)
where I have indicated that the effective gravitational interaction of the two bodies
with all the matter in the universe are different, by labeling the potential with the
different effective gravitational constants in parenthesis, p is the average density
of the universe and k indicates a proportionality factor that is common for both
equations. Thus the ratio rrii/mg could be different for the two test bodies and this
difference is proportional to the assumed difference in the gravitational coupling
constants. However, in an experiment, what is measured directly is not the difference
in the ratio of the inertial and gravitational masses of two bodies. The ratio rrii/mg
for the two test bodies is compared by comparing the accelerations of the two
test bodies in a gravitational field g. Since the long range interaction of the test
bodies with the source mass has different effective gravitational constants, we write
the gravitational field seeir by the two test bodies as aGAg and aG^g (aG is unity
when there is no such material dependence). The accelerations of the two test bodies
are
rrig(A) _ aGAgrng(A) _ ag
aA = aGAg
rrii(A) KGApm,g(A) np
mg(B) aGBgmg(B) ag
aB = aGBg — = — — = — (4)
mi(B) nGBpmg\B) up
Therefore we get the important result that
7] = = 0 (5)
^average
asserting the complete validity of the weak equivalence principle.
3. Summary
I have shown that the EP and UFF are consequences of the gravitational
interaction with all the matter in the universe. Newton's law of motion is a relativistic
gravitational law arising from cosmic gravitomagnetism and the ratio of the
inertial and gravitational masses is essentially the cosmic gravitational potential. As a
consequence the UFF remains valid even if the gravitational constant is material
dependent.
References
1. C. S. Unnikrishnan, Mod. Phys. Lett. A 17, 1081 (2002).
2. C. S. Unnikrishnan, Cosmic Relativity, gr-qc/0406023.
3. D. Sciama, MNRAS 113, 34 (1953).
THE FLYBY ANOMALY
CLAUS LAMMERZAHL and HANSJORG DITTUS
ZARM, University of Bremen, am Fallturm, 28359 Bremen, Germany
laemmerzahl@zarm. uni- bremen. de, dittus @zarm. uni- bremen. de
At various occasions a significant unexplained velocity increase by a few mm/s of
satellites after an Earth swing-by has been observed what is called the flyby anomaly. We
discuss the validity of these observations and discuss general features.
1. The observations
According to information from1-3 the observed flybys are listed in Table 1. The data
can be put into diagrams where the velocity increase can be plotted as a function
of the orbital eccentricity e, see Fig.l. Though from four data points it is much too
early to draw any serious conclusion one may speculate that if the velocity increase
really is due to an unknown gravitational interaction, then (i) the effect should
goes down with increasing eccentricity, and (ii) should go down for an eccentricity
approaching e = 1 because no effect has been observed for bound orbits.
The main problem is not just the limited number of flybys for which sufficiently
precise data are publicly available so that the anomaly can be seen at all. Even
these available data suffer from low cadence (the anomaly often appears between
two data points) and so far only allow an anomaly in the speed, but not in the
direction of motion etc. to be identified. Precise data at a much higher cadence
of all the motion parameters of the spacecraft prior to, during and after the flyby
would allow a qualitatively improved analysis.
2. Error analysis
This velocity increase must be due to an anomalous acceleration of the order
10~4 m/s2. This is 10~5 of the Newtonian acceleration (also the anomalous
Pioneer acceleration is of the order 10~5 of the Newtonian acceleration).
An analysis of possible mismodeling of the calculations should cover (i)
atmospheric modeling, (ii) ocean tides, (iii) if the spacecraft becomes charged, then it
Table 1. Observed flybys (rp = pericentre, e = eccentricity, Voo = velocity at infinity, Au =
velocity increase). a too low orbit with too large atmospheric drag, b thruster activities. (We
thank J.D. Anderson, J.K. Campbell and T. Morley for providing us with the relevant data.)
Mission
Galileo
Galileo
NEAR
Cassini
Stardust
Rosetta
Hayabusa
MESSENGER
agency
NASA
NASA
NASA
NASA
NASA
ESA
Japan
private
year
Dec 1990
Dec 1992
Jan 1998
Aug 1999
Jan 2001
Mar 2005
May 2004
Aug 2005
rp [km]
959.9
303.1
538.8
1173
5950
1954
3725
2347
Uoo [km/s]
8.949
8.877
6.851
16.01
??
3.863
??
4.056
e
2.47
2.32
1.81
5.8
1.327
??
1.36
Av [mm/s]
3.92 ±0.08
no reliable dataa
13.46 ±0.13
0.11
no reliable data°
1.82 ±0.05
no data available
~0
2564
2565
NEAR
0 12 3 4 5 6
Fig. 1. The velocity increase Av as function of the eccentricity and of the perigee.
may experience an additional force due to the Earth's magnetic field, (iv) also the
interaction of a hypothetical magnetic moment of the spacecraft with the Earth's
magnetic field may give an additional force, (v) ion plasma drag, (vi) Earth albedo,
and (vii) Solar wind. It has been shown2'4 that all errors in these models are orders
of magnitude below the observed effect.
3. General approach to describe a modified particle dynamics
It can be shown that within an approach for a general space-time metric governing
clocks and a general equation of motion for massive particles (path structures) a
relativistic approximation of the equation of motion is given by4
dh
= diU + {dlh0
djhi)x3
+ x2diV + xlV + Tl + T)x> + T)kx>x* +
1)
where the first terms is the usual Newtonian acceleration and the second term
the Lense-Thirring effect. The other terms are additional terms beyond standard
General Relativity. In this approach the Universality of Free Fall is respected though
gravity cannot be transformed away locally. The V term which can be motivated
by a running coupling constant5 proportional to the distance which can be used
to describe the constant anomalous Pioneer acceleration. If we assume that the
coefficients Tl-k depend on the Newtonian gravitational potential only, then by
combinatorical reasons, they van lead to additional accelerations
GMr1
v r
= (A2i+ A22
GMrH
GM rlr ■ r
A22
GMr\_
A
31"
■A.
GMr1
32"
A
GMr1
33"
(2)
(3)
(4)
only, where r]_
(r-r)/r2 is the component of the body's velocity orthogonal
to the connecting vector r, and the Aij are some numerical factors. In general, this
equation of motion does not respect energy conservation.
The first term associated with An amounts to a redefinition of the gravitational
constant. The A22 term describes an additional acceleration in direction of the
2566
velocity. It is largest at perigee where for the flyby situation leads to an acceleration
of the order 10-4 m/s . The A21 term vanishes at perigee and leads for the Pioneer
scenario to an anomalous acceleration of 10~9 m/s which, however, is position
dependent and, thus, cannot explain the anomalous Pioneer acceleration. The higher
order terms are too small to be of relevance in the flyby and Pioneer scenarios.
4. Future flybys
In the near future there will be three flybys, all by Rosetta3
• Rosetta: flyby on 13 November 2007 (pericentre altitude 4942 km).
• Mars flyby of Rosetta on 25 February 2007.
• Rosetta: flyby on 13 November 2009 (pericentre altitude 2483 km).
We strongly suggest that due to the lack of explanation of the flyby anomaly one
should use these opportunities in order to carry through a better observation of the
Rosetta flybys. A better data basis then will enable one to establish a correlation
between the observed velocity increase and the orbital parameters like eccentricity,
perihelion distance to the Earth, perihelion velocity, or inclination. In particular, a
continuous observation (Doppler tracking, ranging, positioning, and perhaps other
data from the spacecraft like temperature, pressure, etc) also should give hints to
the particular direction of the local acceleration and also on the strength and, thus,
to the position dependence of the anomalous force. Furthermore, a Mars flyby would
provide an excellent augmentation of the Earth flybys. Since Mars possesses other
conditions than the Earth (weaker atmosphere, almost no magnetic field, other
gravitational field, lower thermal radiation, etc.) many competing effects can be
ruled out. Therefore the effect, if it will be observed also at Mars, then will turn
out to be universal and beyond any doubt and will become an extremely important
science case.
We like to thank O. Preuss and S. Solanki for discussions and the German
Aerospace Centre for financial support.
References
1. J.D. Anderson and J.G. Williams. Long-range tests of the equivalence principle. Class.
Quantum Grav., 18:2447, 2001.
2. P.G. Antreasian and J.R. Guinn. Investigations into the unexpected delta-v increase
during the Earth Gravity Assist of GALILEO and NEAR. In ., editor, Astrodynam-
ics Specialist Conf. and Exhibition, pages paper no 98-4287. American Institute of
Aeronautics and Astronautics, Washington, 1998.
3. T. Morley. Private communication. .
4. C. Lammerzahl, O. Preuss, and H. Dittus, Is the physics of the Solar system really
understood? in Lasers, Clocks, and Drag-Free, eds. H. Dittus, C. Lammerzahl, and
S.G. Turyshev (Springer-Verlag, Berlin 2007).
5. M.-T. Jaekel and S. Reynaud. Gravity tests in the solar system and the Pioneer
anomaly. Mod. Phys. Lett., A 20:to appear, 2005.
GRAVITY TESTS AND THE PIONEER ANOMALY
MARC-THIERRY JAEKELt
Laboratoire de Physique Theorique, ENS, UPMC, CNRS,
Paris, F-75231, FRANCE
*jaekel@lpt.ens.fr http://www.lpt.ens.fr
SERGE REYNAUD*
Laboratoire Kastler Brossel, CNRS, ENS, UPMC,
Pans, F-75252, FRANCE
t reynaud@spectro.jussieu.fr http://www.spectro.jussieu.fr
Validity of general relativity has been confirmed at distance scales ranging from the
millimeter to the size of planetary orbits. But windows remain open for potential violations
at shorter or longer scales. The anomalous acceleration recorded on Pioneer 10/11 probes
on their escape trajectories outwards the solar system might constitute a first hint that
gravity laws should be modified at large scales.
Keywords: General relativity; gravity tests; Pioneer anomaly
Experimental tests of gravity show a good agreement with General Relativity (GR)
at all scales ranging from laboratory to the size of the solar system.1_4 However
there exist a few anomalies which may be seen as challenging GR. Anomalies in the
rotation curves of galaxies or in the relation between redshifts and luminosities can
be accounted for by considering dark matter and dark energy but they can as well
be thought of as consequences of modifications of GR at large scales.
The anomalous acceleration recorded on Pioneer 10/11 probes might point at
some anomalous behaviour of gravity at a scale of the order of the size of the solar
system.5'6 The observation of such an effect has stimulated a significant effort to find
explanations in terms of systematic effects on board the spacecraft or in its
environment but this effort has not met success up to now.7 The Pioneer anomaly remains
the subject of intensive investigation because of its potential implications.8 n New
missions have been proposed12 and efforts have been made for recovering data
associated with the whole duration of Pioneer 10/11 missions and submitting them
to new analysis.13'14
These observations involve Doppler tracking data of the two probes. They show
an anomalous acceleration ap ~ 0.8 nm s~2 directed towards the Sun with a roughly
constant amplitude over a large range of heliocentric distances 20 AU < rp <
70 AU (the symbol AU stands for the astronomical unit). Besides this secular term,
the recorded anomalous acceleration also shows diurnal and annual modulations
which could also be the consequence of some not yet understood artefact. Note that
secular and modulated anomalies can hardly be due to the same artefact. The main
result presented here is that modulated as well as secular anomalies are a natural
prediction of post-Einsteinian metric extensions of GR.
The present extended abstract summarizes publications which have investigated
the capability of metric extensions of GR to account for the Pioneer anomaly while
2567
2568
remaining compatible with other gravity tests performed in the solar system. A
main property of such 'post-Einsteinian' extensions is to preserve the very core of
GR, where gravity identifies with the metric tensor gM„ and motions are described
by geodesies. In particular, the weak equivalence principle, one of the most
accurately verified properties in physics, is preserved. This does not mean that there
can be no violations of this principle but only that such violations are too small
to account for the large Pioneer anomaly (of the order of one thousandth of the
Newton acceleration at the place explored by Pioneer probes).
Nonetheless, the metric may differ from its GR standard form so that
observations may show deviations from standard expectations. Such metric extensions of
GR have first been introduced in the context of a linearized treatment of
gravitation fields15'16 and then discussed with non linearity taken into account.17 Recently,
more precise and detailed investigations of the Pioneer observations have been
published, improving the preliminary results of previous papers and changing some of
their conclusions.18 All these calculations are based upon the assumption of a static
and isotropic metric (the effects of rotation and non sphericity of the Sun are
disregarded). Therefore the metric fields are given by two functions goo and grr of a
single variable, the radius r,
ds2 = g00(r)c2dt2 + grr(r) (dr2 + r2d02 + r2 sin2 6dip2) (1)
with the metric written in terms of Eddington isotropic coordinates.
Recently,18 the anomalous acceleration recorded in Pioneer data has been
calculated by representing the Doppler tracking observables in terms of propagation
time delays. With this representation, the influences of metric perturbations on
probe motion on one hand, and link propagation on the other hand, are treated in a
natural and consistent manner. As a result of these calculations, modulated as well
as secular anomalies are naturally predicted by post-Einsteinian metric extensions
of GR. As a matter of fact, the Doppler observable not only depends on the motion
of the Pioneer probe but also on the perturbation of electromagnetic propagation
along the up- and down-links. As the paths followed by these links are themselves
modulated by motions of the stations, the anomalous Doppler acceleration is
expected to contain diurnal and annual modulations.
When the metric extensions of GR are considered from a phenoinenological
point of view, anomalous observations in the solar system could tell us that the two
functions goo and grr entering (1) deviate from their standard expressions. Possible
deviations are conveniently described by two "sectors" corresponding to deviations
6goo and S (googrr)- The first sector represents an anomaly of the Newton potential2
while the second sector may be seen as an extension of PPN phenomenology1 with
a scale dependent parameter 7. The existence of two sectors opens an additional
phenomenological freedom with respect to a mere modification of the Newton
potential as well as with the PPN framework where 7 is constant. This provides new
possibilities for accomodating Pioneer-like anomalies with other gravity tests. 15~18
As the secular anomaly, the annual anomaly is a natural consequence of the
2569
presence of a second potential, being produced by propagation along the up- and
down-links. This anomaly is strongly correlated with the effect of a change of the
trajectory of the probe, allowed by the fact that range observables were not
available for Pioneer 10/11 missions. The behaviour is qualitatively reminiscent of the
observations of annual anomalies which have been reported,6 but it is only after
a quantitative comparison, taking into account all the details known to be
important for data analysis,6 that it will be possible to decide whether or not the metric
extensions of GR fit the Pioneer observations.
These conclusions constitute motivations for new experiments in the solar
system. Clearly, experiments with ranging capabilities will offer qualitatively better
perspectives than Pioneer observations which were performed without such
capabilities. Missions going to the borders of the solar system12 will either confirm or
disprove the existence of the anomaly at such long distances. Comparison with the
theoretical expectations presented here will give an answer to the question whether
such an anomaly may have a metric origin, with a metric departing from the GR
prescription. This idea could also be tested on a shorter time scale by adding specially
designed instruments on planetary probes going to Mars, Jupiter, or Saturn, the
reduction of the explored heliocentric distance being compensated by a potentially
large improvement of the measurement accuracy.
References
1. CM. Will, Theory and Experiment in Gravitational Physics (Cambridge Univ. Press,
1993); Living Rev. Rel. 4, 4 (2001).
2. E. Fischbach and C. Talmadge, The Search for Non Newtonian Gravity (Springer,
Berlin, 1998).
3. E.G. Adelberger, B.R. Heckel and A.E. Nelson, Ann. Rev. Nucl. Part. Sci. 53, 77
(2003).
4. M.-T. Jaekel and S. Reynaud, Int. J. Mod. Phys. A20, 2294 (2005) and references
therein.
5. J.D. Anderson et al , Phys. Rev. Lett. 81, 2858 (1998).
6. J.D. Anderson et al , Phys. Rev. D 65, 082004 (2002).
7. J.D. Anderson et al , Mod. Phys. Lett. A17, 875 (2003).
8. M.M. Nieto and S.G. Turyshev, Class. Quantum Grav. 21, 4005 (2004).
9. S.G. Turyshev, M.M. Nieto and J.D. Anderson, 35th COSPAR Scientific Assembly
gr-qc/0409117.
10. O. Bertolami and J. Paramos, Glass. Quantum Grav. 21, 3309 (2004); see also astro-
ph/0408216 and gr-qc/0411020.
11. C. Lammerzahl, O. Preuss and H. Dittus, in Proc. 359th WE-Heraeus Seminar on
Lasers, Clocks, and Drag-Free Technologies for Future Exploration in Space and Tests
of Gravity (Springer, Berlin, 2006) p 75.
12. H. Dittus et al , Trends in Space Science and Cosmic Vision 2020 ESA Spec. Pub.
588, 3 (2006).
13. M.M. Nieto and J.D. Anderson, Class. Quantum Grav. 22, 5343 (2005).
14. S.G. Turyshev, V.T. Toth, L.R. Kellogg et al , Int. J. Mod. Phys. D15, 1 (2006).
15. M.-T. Jaekel and S. Reynaud, Mod. Phys. Lett. A20, 1047 (2005).
16. M.-T. Jaekel and S. Reynaud, Class. Quantum Grav. 22, 2135 (2005).
17. M.-T. Jaekel and S. Reynaud, Class. Quantum Grav. 23, 777 (2006).
18. M.-T. Jaekel and S. Reynaud, Class. Quantum Grav. 23, 7561 (2006).
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Astrometry
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A NICE TOOL FOR RELATIVISTIC ASTROMETRY:
SYNGE'S WORLD FUNCTION
PIERRE TEYSSANDIER
Dept SYRTE, CNRS/UMR-8630
Observatoire de Paris, 61 avenue de I'Observatoire, F-75014 Paris, France
Pierre. Teyssandier@obspm.fr
CHRISTOPHE LE PONCIN-LAFITTE
Lohrmann Observatory, Dresden Technical University, Mommsenstr. 13, D-01062 Dresden,
Germany
christophe.le-poncin-lafitte@tu-dresden.de
We give a brief outline of a general method enabling to solve the problems of relativistic
astrometry with the aid of the so-called Synge's world function.
1. Introduction
In a foreseeable future, it will be indispensable to get a fully relativistic treatment
of the angular distance between two light sources beyond the first order in the
Newtonian gravitational constant G, especially in the areas of space astrometry
and highly precise tests of general relativity (GR) like the Laser Astrometric Test
Of Relativity (LATOR) mission.1 We give here a brief account of a method of
calculation which spares the trouble of integrating the differential equations of the
null geodesies.2'3 This method is based on the so-called Synge's world function.4
Space-time is assumed to be covered by some global coordinate system xa = (x° =
ct,xl) such that goo > 0. We set x = (x1 , x2,x3), x.y = 5%ix%yi and \y — x\ =
2. Angular distance as measured by an arbitrary observer
Let r and Y' be two light rays received at point x0 = (ct0, x0) by an observer 0{u)
moving at x0 with a unit 4-velocity u. The signature of the metric being (H ),
the angular separation <fiu between Y and V as measured by 0{u) is defined by5
ga0lal'0
COS(j)u = 1
0 <(/)„< 7T,
XgnuuH»){gp(TuPl'°
where /" and I'13 are vectors respectively tangent to Y and Y' at point x0 = (ct0, x0).
Noting that / and /' are null vectors, it is easily seen that Eq. (1) is equivalent to6
\goo + 2gokPk+gkiPkPl)gl3(l'i-km-hy
sm — = —-
2 4
where
(2)
Zo = 1'z< = £' * = 1>l< = i> r={**)x=-c{-w) • (3)
2573
2574
The next section shows how (k)Xo may be derived from Synge's world function.
3. Synge's world function and relativistic astrometry
Let xa and xb be two points of space-time. Assume that there exists one and only
one geodesic path Tab joining them. Denote by A the unique affine parameter on
Tab such that A^ = 0 and Ajg = 1. Synge's world function is the invariant bifunction
£1(xa,xb) defined by
1 f1 dx^ dxu
n(xA,xB) = -J g^(xa(A)) —— dA, (4)
the integral being taken along Tab- The relevance of this function in the
determination of the angular separation comes from the following properties.3
Property 1. The covariant components of the vectors tangent to the geodesic
path Tab ai xa and ig respectively, are given by
( dxv\ dn . , / dxu\ dn .
{9^^\)A = -dxJ{XA>XB)> {9^^\)B=dx^{XA>XB)- (5)
Property 2. The world function fl(xa, %b) satisfies the Hamilton-Jacobi
equations
7;9a/3(xA)jr^(xA,XB)—^-(xA,xB) = £l(xA,xB), (6)
1 oxA (jx'A
7)ga0{xB)TT-^{xA,XB)—-Q-{xA,XB) =£l{xA,XB). (7)
z OXB (jx'B
Property 3. Two points xa andxs are joined by a light ray (i.e. a null geodesic)
if and only if the condition
n(xA,xB) = 0 (8)
is fulfilled.
Let xe = (cte, Xg) be the coordinates of the emission point of the light ray I\
Solving for te the equation obtained by substituting (cte,xe) for xa and (ct0,xe)
for xb into Eq. (8) yields the travel time t0 — te of a photon between (cte, xe) and
(ct function of xe, t0 and x0. So we can put
l0 Ze lr\Xe, t0, X0). [&)
We call Tr(xe,t0,x0) the reception time transfer function. Differentiating the
identity
iLyCt0 ClryXe^ 60, X0j7 Xe, CZ0: XqJ = U V^UJ
w.r.t. t0 and xl0, and then taking into account Eqs. (5), it is easily seen that
j \ Olr\Xe: £0, X0)
dxi
0lr[Xei t0, X0)
dt0
11
2575
So relativistic astroinetry reduces to the determination of a single function
Tr(xe, t0, x0) which may be derived from CI(xa, xb)-
We showed3 that Eqs. (6) or (7) enable to determine CI(xa,xb) and %(xe, t0, x0)
by iterative procedures when the metric is given by a generalized post-Minkowskian
expansion. Consider, e.g., a metric field generated by a single static spherically
symmetric body of mass M
1-2^^+2/3 1 ^ I +0(c-5)
dsz
0\2
„ GM
1 + 27-5- +
(dx
i*m +^
5ij dx1 dx3,
(12)
where /3, 7 and 5 are post-Newtonian parameters (they are all equal to 1 in GR). Our
explicit calculation of U(xa,xs)3 led to an expression as follows for the reception
time transfer functiona
1 {xei x0)
Xr,
+ fr + 1>GMln
+ -
GlMl\xn-xe
■4/3 + 87 + 35
^e 1 TQ -\- \X0 Xe
¥e 1 To yE>o ^e
(13)
k\frzr.
2„2
e o
. arccos
1 Xv ■ XCl
re.
(1+7)2
rer0 + (xt
up to the order of c-6. This formula generalizes a result obtained in GR.7 The
second-order terms added to the usual Shapiro time delay will be very useful in
highly precise tests of general relativity.
4. Conclusion
We demonstrated that the theoretical value of the angular separation between two
light sources can be determined when Synge's world function is known. We are
now studying how the time transfer function Tr(xe,t0, x0) can be directly obtained
without calculating Q(xa,xb)-
References
1. S. G. Turyshev, M. Shao and K. Nordtvedt, gr-qc/0601035.
2. B. Linet and P. Teyssandier, Phys. Rev. D66, 024045 (2002).
3. C. Le Poncin-Lafitte, B. Linet and P. Teyssandier. Class. Quantum Grav. 21, 4463
(2004).
4. J.-L. Synge, Relativity: The General Theory (North-Holland, 1964).
5. M. Soffel, Relativity in Astrometry, Celestial Mechanics and Goedesy (Springer-Verlag,
1988).
6. P. Teyssandier and C. Le Poncin-Lafitte, gr-qc/0611078.
7. V. A. Brumberg, Kinematics Phys. Celest. Bodies 3, 6 (1987).
Note that this function does not depend on the reception time, so we drop the subscript r.
LUNAR LASER RANGING: A SPACE GEODETIC TECHNIQUE
TO TEST RELATIVITY
JURGEN MULLER
Institut fur Erdmessung (IfE), Leibniz University of Hannover
Schneiderberg 50, SO 167 Hannover, Germany
mueller@ife.uni-hannover. de
Lunar laser ranging (LLR) has routinely provided observations for more than 36 years.
The main benefit of this geodetic technique is the determination of many parameters of
the Earth-Moon dynamics (e.g. orbit and rotation of the Moon or lunar physics) and
the test of metric theories of gravity. LLR data analysis determines gravitational physics
quantities such as the equivalence principle, any time variation of the gravitational
constant, and several metric parameters. We give an overview of the recent status of our
LLR analysis procedure and present new results for some relativistic quantities.
1. Introduction
LLR observations began shortly after the first Apollo 11 manned mission to the
Moon in 1969. The LLR data are collected as normal points, i.e. the combination of
lunar returns obtained over a certain time span. Out of « 1019 photons sent per pulse
by the transmitter, less than 1 is statistically detected at the receiver; this is caused
by several factors, e.g., energy loss (i.e., the 1/R4 law), atmospherical extinction and
geometric reasons (rather small telescope apertures and reflector areas). These poor
conditions are the main reason, why only a few observatories worldwide are capable
of laser ranging to the Moon. To study the dynamics of the Earth-Moon system (e.g.
Earth orientation or the secular increase of the Earth-Moon distance: 3.8 cm/year),
LLR data acquired since 1970 are analysed at IfE, where the main goal is the test
of relativity (e.g. strong equivalence principle, time-variable gravitational constant,
metric parameters), cf. Muller et al. (2006).
2. LLR Modeling
The existing LLR model is fully relativistic and is complete up to first post-
Newtonian (1/c2) level (see e.g. Muller and Nordtvedt 1998 or Muller 2000, Muller
et al. 2006 and references therein). It uses the Einstein's general theory of
relativity. The basic observation equation is defined in the Barycentric Celestial
Reference System (BCRS). Therefore, all quantities have to be transformed in this
reference frame which requires consistent relativistic transforniations, the so-called
generalized Lorentz transformations from the Geocentric Celestial Reference System
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(GCRS) for the Earth and from a GCRS-like selenocentric system for the Moon.
The Earth-Moon vector is obtained by numerical integration of the relativistically
defined equations of motion. Corresponding relativistic equations are applied to
describe the rotational motion of the Moon. Finally, when modeling the pulse travel
time, besides atmospheric effects also (relativistic) transformations into the right
time system and the light time equation (Shapiro effect) have to be considered.
3. Analysis and Results
In the case of LLR many tests of possible modifications of general relativity can be
performed. Here, only a selection of post-Newtonian parameters to be determined
by LLR analyses is shown, for a more complete description see Miiller et al. (2006)
- values for general relativity are given in parentheses:
(1) Strong equivalence principle (EP) parameter rj (= 0). A violation of the EP
would show up as a displacement of the lunar orbit towards the Sun.
(2) Time variation of the gravitational coupling parameter G/G (= 0 yr_1).
(3) Geodetic (or de Sitter) precession S7gp of the lunar orbit (~ 1.92 "/cy).
(4) a\ (= 0) and 0:2 (= 0) parametrizing 'preferred frame' effects in metric gravity.
The lunar measurements contain the summed signal of all effects in one, so
that the separation of the individual effects is a big challenge. Many relativistic
effects produce a sequence of periodic perturbations of the Earth-Moon range (e.g.
annual, monthly, nodal and combinations of them). These periodicities support the
separation of the various signal parts (Miiller et al. 2006).
The EP-parameter 77 benefits most from highest accuracy over a sufficient long
time span (e.g., one year) and a good data coverage over the synodic month.
In combination with the recent value of the space-curvature parameter 7cassini
(7 — 1 = (2.1 ± 2.3) • 10~5) derived from Doppler measurements to the Cassini
spacecraft (Bertotti et al. 2003), the non-linearity parameter /3 can be determined
by applying the relationship rj = 4/3 — 3 — 7cassini (note that using the EP test to
determine (3 assumes that there is no composition-induced EP violation and that
the contribution of further PPN parameters like ot\ and 0:2 can be neglected). Even
the assumption of a reduced accuracy for 7cassini in the order of 10-4 (see Kopeikin
et al. 2006) would hardly change this result.
The estimate for the temporal variation of the gravitational constant benefits
most from the long time span of LLR data and has experienced the biggest
improvement over the past years. For the estimation of the de Sitter precession of the
lunar orbit, a Coriolis-like term is added to the equation of motions, which adds the
precession effect as predicted by Einstein for a second time. The preferred-frame
parameters ct\ and 012 can either be determined by extending the equations of
motion or by adding analytical terms to the Earth-Moon distance. In both cases quite
similar results are obtained. Recent determinations are given in Table 1.
2578
Table 1. Determined values for the relativistic quantities and their
realistic errors.
Parameter
Equivalence Principle parameter rj
Metric parameter (3 — 1 from r\ = 4/3 — 3 — 7Cassini
Time varying gravitational constant G/G [yr-1]
Differential geodetic precession Qqp - ^deSit [" /cv]
'Preferred frame' parameter ct\
'Preferred frame' parameter a.2
Results
(6 ±7)- 10"4
(1.5 ±1.8) ■ 1CT4
(6 ±8)- 10"13
(6 ±10) ■ 10-3
(-7±9)-10"5
(1.8 ±2.5) ■ 1CT5
In addition, further metric parameters like the space-curvature parameter 7,
the Yukawa coupling parameter a and others can be determined from LLR data.
Results for all relativistic parameters obtained from the IfE analysis are given in
Muller et al. (2006). The realistic errors are comparable with those obtained in other
recent investigations, e.g., at JPL (see Williams et al. 2004, 2005). To exploit the
full available potential of LLR, the theoretical models as well as the measurements
require further optimization.
In conclusion, LLR has become the strongest tool for testing Einstein's theory of
gravitation in the solar system (e.g., tests of the equivalence principle, time-variable
gravitational constant), no violations of general relativity have been found so far.
Acknowledgments
It is a pleasure to thank S. Turyshev and J. Williams (both JPL) as well as M. Soffel
and S. Klioner (TU Dresden) for many fruitful discussions. This work has partially
been funded by Deutsche Forschungsgemeinschaft (DFG grant MU1141/6-2).
References
1. Bertotti, B., L. less, and P. Tortora: A test of general relativity using radio links with
the Cassini spacecraft. Nature 425, 374-376, 2003.
2. Kopeikin, S., I. Vlasov, G. Schafer, and A. Polnarev: The orbital motion of Sun and
a new test of general relativity using radio links with the Cassini spacecraft, in print
2006, gr-qc/0604060.
3. Muller, J.: FESG/TUM, Report about the LLR Activities. ILRS Annual Report 1999,
M.Pearlman, L.Taggart (eds.), 204-208, 2000.
4. Muller, J. and K. Nordtvedt: Lunar laser ranging and the equivalence principle signal.
Physical Review D, 58, 062001, 1998.
5. Muller, J., J.G. Williams, and S.G. Turyshev: Lunar Laser Ranging Contributions
to Relativity and Geodesy. In: Proceedings of the Conference on Lasers, Clocks, and
Drag-free, ZARM, Bremen, 2005, in print 2006, gr-qc/0509114.
6. Williams, J.G., S.G. Turyshev, and D. H. Boggs.: Progress in lunar laser ranging tests
of relativistic gravity. Phys. Rev. Lett., 93, 261101, 2004, gr-qc/0411113.
7. Williams, J.G., S.G. Turyshev, and D. H. Boggs.: Lunar Laser Ranging Tests of the
Equivalence Principle with the Earth and Moon. In proceedings of 'Testing the
Equivalence Principle on Ground and in Space', Pescara, Italy, September 20-23, 2004, C.
Laemmerzahl, C.W.F. Everitt and R. Ruffini (eds.), to be published by Springer Ver-
lag, Lect. Notes Phys., 2005, gr-qc/0507083.
APOLLO: NEXT GENERATION LUNAR LASER RANGING
T. W. MURPHY, Jr.*, E. L. MICHELSEN and A. E. ORIN
CASS/0424; University of California, San Diego
9500 Oilman Drive, La Jolla, CA, 92093-0424, USA
* tmurphy@physics.ucsd.edu
J. B. BATTAT and C. W. STUBBS
Physics Department; Harvard University
18 Hammond Street, Cambridge, MA, 02138, USA
E. G. ADELBERGER, C. D. HOYLE and H. E. SWANSON
Physics Department; University of Washington
Box 351560, Seattle, WA, 98195-1560, USA
APOLLO (the Apache Point Observatory Lunar Laser-ranging Operation) is anew effort
in lunar laser ranging that uses the Apollo-landed retroreflector arrays to perform tests
of gravitational physics. APOLLO achieved its first range return in October, 2005, and
began its science campaign the following spring. The strong signal (> 2500 photons in a
ten minute period) translates to one-millimeter random range uncertainty, constituting at
least an order-of-magnitude gain over previous stations. One-millimeter range precision
will translate into order-of-magnitude gains in our ability to test the weak and strong
equivalence principles, the time rate of change of Newton's gravitational constant, the
phenomenon of gravitomagnetism, and the inverse-square law.
Keywords: Lunar Ranging; Solar System Tests
1. Overview
Lunar Laser Ranging (LLR) has a long history of providing many of our strongest
tests of gravity.1 LLR currently provides the the best tests of the following
gravitational parameters, at the indicated levels of precision:
• Strong Equivalence Principle (SEP) to 4 x 10-4,
• Weak Equivalence Principle (WEP) to 10~13,
• Time-rate-of-change of the Gravitational constant (G/G) to 10-12 per year
• Gravitomagnetism (basis of frame dragging) to 0.1%
• Geodetic precession to 0.35%
• Best test of 1/r2 to 10~10 times the strength of gravity at ~ 109 length
scales
LLR thus far has not seen deviations from the expectations of general relativity.
The state-of-the-art in 2005 was 2 cm range precision, usually accomplished
in an observing period lasting a few tens of minutes, and collecting 5-50 photons
of returned laser energy. Typical performances of the two routine LLR stations in
France (OCA) and Texas (MLRS) have been a return rate of 0.01 and 0.002 photons
per pulse to the larger Apollo 15 array, respectively. At 10 Hz pulse repetition rate,
this corresponds to one photon every 10 and 50 seconds, respectively.
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The LLR error budget is typically dominated by uncertainty associated with the
tilt of the retroreflector array normal relative to the line of sight. These tilts—up to
about 7° in each axis —are caused by "optical" librations of the moon. Even if the
tilt is known precisely, the range measurement is spread temporally, with a peak-to-
peak uncertainty in the ballpark of a tan6° « 0.1a, where a is the array dimension
of roughly one meter. In a root-mean-square sense, the resulting 30-50 mm range
uncertainty can be averaged to 1 mm uncertainty by gathering 900-2500 photons.
This number is well outside the grasp of the OCA or MLRS stations.
A new lunar ranging apparatus, APOLLO (the Apache Point Observatory
Lunar Laser-ranging Operation), has begun operation in southern New Mexico on a
mountaintop at an elevation of 2780 m. Using a 3.5 m telescope aperture and
taking advantage of good atmospheric image quality ("seeing"), APOLLO is capable of
receiving multiple photons per pulse. Details of the apparatus can be found online.2
2. APOLLO Project Status
The summer of 2005 saw most of the hardware and software come together in an
integrated system at the observatory. We achieved our first unambiguous range results
in October 2005, reaping about 2,400 photons in a period less than 30 minutes. This
is more than enough photons to provide statistical averaging at the one-millimeter
level. The McDonald Laser Ranging System collected a similar number of lunar
return photons over a three year period from 2000-2002. Also of note is that this initial
success was achieved near full moon, when other stations are unable to acquire the
range signal against the lunar background.
As of July 2006, APOLLO has accomplished much in its first half-year of
operation:
• >2,000 lunar return photons within 10 minutes (on several occasions);
• peak rates of > 0.5 photons per pulse over half-minute intervals (on two
occasions);
• as many as 8 return photons have been seen in a single pulse (plus many
7's, 6's, etc.);
• about half of the return photons in strong runs arrive in multi-photon
packets;
• full-moon ranging does not represent a significant challenge;
• typical acquisition time for each reflector is less than a few minutes.
An example run is shown in Figure 1. We have demonstrated the capability
of collecting sufficient numbers of photons to achieve one-millimeter precision on
timescales less than ten minutes.
APOLLO range measurements in October, November, December, and January
were processed by the analysis group at JPL. A solution for the APOLLO station
position was found that resulted in range deviations at the 0.1 ns level,
corresponding to 1-2 cm. This level of imprecision was not inconsistent with knowledge of our
2581
time (minutes)
1---1 — ' -____T_____T___ _ _____
1 100 ps bins
time offset ins)
Fig. 1. Example Apollo 15 time series (top) showing photon return time (vertically) within a 40 ns
portion of the 100 ns range gate. The lunar return is evident against the background photons. The
width, more clearly seen in the histogram (below), is consistent with the temporal spread of the
reflector array. The asymmetric tail is due to photo-electron diffusion in the APD device.
system performance during that time. Known causes of systematic error were
removed in March 2006, so that data starting in April 2006 i'epresent what we believe
to be the first unbiased, differential measurements from APOLLO.
APOLLO is poised at the edge of a. data campaign unlike anything in the history
of lunar ranging. Early work on the 2.7 meter telescope at McDonald Observatory-
approached single-photon-per-pulse performance, but at 0.3 Hz repetition rate and
3 ns pulse width. The APOLLO return rate is at least two orders-of-maguitude
higher than currently operating LLR stations (helped some by a higher repetition
rate), so that order-of-magnitude gains in physics seems feasible. Project status
updates are available on the APOLLO website.2
References
1. Williams, J. G., Newhall, X. X., & Dickey, J. O., "Relativity parameters determined
from lunar laser ranging," Physical Review D, 53, 6730, (1996)
2. http://physics.ucsd.edu/~tanirphy/apollo/
METRIC EXTENSIONS OF GENERAL RELATIVITY AND
GRAVITY TESTS IN THE SOLAR SYSTEM
SERGE REYNAUDt
Laboratoire Kastler Brossel, CNRS, ENS, UPMC,
Paris, F-75252, FRANCE
t reynaud@spectro.jussieu.fr http://www.spectro.jussieu.fr
MARC-THIERRY JAEKEL*
Laboratoire de Physique Theorique, ENS, UPMC, CNRS,
Paris, F-75231, FRANCE
ijaekel@lpt.ens.fr http://www.lpt.ens.fr
The anomalous acceleration recorded on Pioneer 10/11 probes on their escape trajectories
outwards the solar system might constitute a first hint that gravity laws should be
modified at large scales. But the modification needed to accomodate the Pioneer anomaly
has to remain compatible with other gravity tests in the solar system. This question is
discussed in the framework of metric extensions of General Relativity.
Keywords: General relativity; gravity tests; Pioneer anomaly
The anomalous acceleration recorded on Pioneer 10/11 probes might point at some
anomalous behaviour of gravity at a scale of the order of the size of the solar
system.1'2 Despite significant efforts devoted to this purpose, it has not been possible
up to now to find any satisfactory explanations in terms of a systematic effect on
board the spacecraft or in its environment.3 Further efforts are presently made for
submitting to a new analysis the data recently recovered for the whole duration
of Pioneer 10/11 missions.4,5 Missions going to the borders of the solar system6 to
confirm, or infirm, Pioneer observations are also proposed to the space agencies.
Meanwhile, it remains important to study whether or not the Pioneer anomaly is
compatible with the fact that existing tests of gravity in the solar system show good
agreement with General Relativity (GR).7~10 This question can be investigated in
a quantitative manner in the framework of metric extensions of GR. Such 'post-
Einsteinian' extensions preserve the very core of GR with gravity identified with
the metric tensor g^v and motions described by geodesies. In particular, the weak
equivalence principle, one of the most accurately verified properties in physics, is
preserved. However the metric may differ from its standard (GR) form so that
observations may show deviations from standard expectations.
The metric extensions of GR have been introduced in the context of a linearized
treatment of gravitation fields11'12 and then discussed with non linearity taken into
account.13-14 In these papers, the spacetime in the solar system is represented by the
static and isotropic metric defined by two functions goo and grr of a single variable,
the radius r.
ds2 = g0o(r)c2dt2 + grr(r) (dr2 + r2df)2 + r2 sin2 Odkp2) (1)
2582
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with the metric written in terms of Eddington isotropic coordinates. Its components
g^v can be described as sums of standard GR expressions and deviations Sg^. As
GR is a good effective description of gravity in the solar system, the deviations are
necessarily small (l^g^l <C 1) so that variations can be calculated at first order. We
also convene that the deviations vanish at the radius of Earth orbit on which or in
the vicinity of which the most accurate experiments are performed.
It thus remains to study the effects of variations with the radius r of the
anomalous metric components 6goo and 5grr- From the point of view of phenomenology,
the variations are conveniently separated as two sectors corresponding to the effects
of 6goo(r) and 6 (.googw) (»")• The first sector represents an anomaly of the Newton
potential8 while the second sector describes an extension of PPN phenomenology7
with a scale dependent parameter 7. The existence of two sectors opens an additional
phenomenological freedom with respect to models where only the Newton potential
is modified as well as those where 7 differs from unity but remains constant.
Modifications of the Newton potential, i.e. anomalies in the first sector according
to the terminology of the preceding paragraph, have been investigated in numerous
papers. Interpreting the Pioneer anomaly as reflecting such an effect leads to an
anomalous potential 6goo varying roughly as r in the range of heliocentric distances
(20 to 70 AU) where the anomaly has been registered. If this dependence also holds
at smaller radii,2 or if the anomaly follows a simple Yukawa law,10 one deduces
that it cannot have escaped detection in the more constraining tests performed
with martian probes.15 17 Brownstein and Moffat have explored the possibility that
such a linear dependence is cut off within the orbital radius of Saturne.18 Iorio and
Giudice19 as well as Tangen20 have then claimed that the ephemeris of outer planets
were accurate enough to prevent the presence of the required linear dependence in
the range of distances explored by the Pioneer probes. This argument has however
been contested by the previous authors.18
Anyway, this argument only deals with metric anomalies in the first sector and
disregards those in the second sector. The discussion of the compatibility of metric
modifications with observations performed in the solar system requires a greater
care, accounting for the presence of the two sectors as well as for possible scale
dependences. Preliminary discussions have been presented12 for the case of deflection
experiments on electromagnetic sources passing behind the Sun,21-23 and for
planetary tests such as the advance of perihelion.13 More complete discussions will be
given in forthcoming papers which will in particular deal with the fact that effects
in the two sectors are superposed in most observables. This is for example the case
for the Pioneer-like anomaly which was recently calculated.14 The expression found
there for the anomalous acceleration improves and corrects the preliminary results
obtained in preceding papers.
The latter conclusions constitute motivations for a renewed analysis of the
gravity observations in the solar system. It appears quite interesting to look for related
signatures in other experiments. Finding, or not finding, such signatures will bring
information of interest on the existence of metric anomalies in the solar system. For
2584
example, deflection experiments such as GAIA24 or LATOR25 will have a largely
improved accuracy, thus allowing one to test whether or not the deflection behaves
as expected from GR when the angular distance to the Sun varies.
The presence of anomalous metric components can also be detected in planetary
tests. In particular, the perihelion precession of planets can be used as a sensitive
probe of the value and variation of the two potentials with r, the orbital radius of
the planet. Like the first anomalous potential, the second one could in principle be
present at the long distances explored by Pioneer probes, but not at the smaller
distances corresponding to the radii of inner planets. This entails that it would
be extremely interesting to track with accuracy the motions of small bodies which
may have significant radial velocities while being at large heliocentric distances.
This possibility of testing GR by following small bodies can be considered as a
further fundamental challenge for GAIA.24
References
1. J.D. Anderson et al , Phys. Rev. Lett. 81, 2858 (1998).
2. J.D. Anderson et al , Phys. Rev. D 65, 082004 (2002).
3. J.D. Anderson et al , Mod. Phys. Lett. A17, 875 (2003).
4. M.M. Nieto and J.D. Anderson, Class. Quantum Grav. 22, 5343 (2005).
5. S.G. Turyshev, V.T. Toth, L.R. Kellogg et al , Int. J. Mod. Phys. D15, 1 (2006).
6. H. Dittus et al , in Trends in Space Science and Cosmic Vision 2020 ESA Spec. Pub.
588, 3 (2006).
7. CM. Will, Theory and Experiment in Gravitational Physics (Cambridge Univ. Press,
1993); Living Rev. Rel. 4, 4 (2001).
8. E. Fischbach and C. Talmadge, The Search for Non Newtonian Gravity (Springer,
Berlin, 1998).
9. E.G. Adelberger, B.R. Heckel and A.E. Nelson, Ann. Rev. Nucl. Part. Sci. 53, 77
(2003).
10. M.-T. Jaekel and S. Reynaud, Int. J. Mod. Phys. A20, 2294 (2005) and references
therein.
11. M.-T. Jaekel and S. Reynaud, Mod. Phys. Lett. A20, 1047 (2005).
12. M.-T. Jaekel and S. Reynaud, Class. Quantum Grav. 22, 2135 (2005).
13. M.-T. Jaekel and S. Reynaud, Class. Quantum Grav. 23, 777 (2006).
14. M.-T. Jaekel and S. Reynaud, Class. Quantum Grav. 23, 7561 (2006).
15. R.D. Reasenberg, I.I. Shapiro, P.E. MacNeil et al , Astrophys. J. Lett. 234, L219
(1979).
16. R.W. Hellings et al , Phys. Rev. Lett. 51, 1609 (1983).
17. J.D. Anderson et al , Astrophys. J. 459, 365 (1996).
18. J.R. Brownstein and J.W. Moffat, Class. Quantum Grav. 23, 3427 (2006).
19. L. Iorio and G. Giudice New. Astron. 11, 600 (2006).
20. K. Tangen, gr-qc/0602089.
21. L. less et al , Class. Quantum Grav. 16, 1487 (1999).
22. B. Bertotti, L. less and P. Tortora, Nature 425, 374 (2003).
23. S.S. Shapiro et al , Phys. Rev. Lett. 92, 121101 (2004).
24. A. Vecchiato, M.G. Lattanzi and B. Bucciarelli, Astron. Astrophys. 399, 337 (2003).
25. S.G. Turyshev et al , Trends in Space Science and Cosmic Vision 2020 ESA Spec.
Pub. 588, 11 (2006).
MEASUREMENT OF THE PPN-7 PARAMETER WITH
A SPACE-BORN DYSON-EDDINGTON-LIKE EXPERIMENT
A. VECCHIATO*, M. GAI, M. G. LATTANZI and R. MORBIDELLI
INAF - Astronomical Observatory of Torino,
strada Osservatorio 20, 10025 Pino Torinese (TO), Italy
* E-mail: vecchiato@oato.inaf.it
We explore the possibility of measuring the 7 parameter of the Parameterized Post-
Newtonian (PPN) formalism with an Earth-orbiting satellite and looking as close as
possible to the solar limb. The technique is inspired to that used during the solar eclipse
of 1919, when the gravitational bending of the light was measured for the first time.
Simple estimations suggest that even a low-cost satellite could reach the 10~6 level of
accuracy with ~ 106 observations of relatively bright stars at about 2° from the Sun.
Further simulations with different magnitude limited star samples, uniformly distributed
on the ecliptic plane, show that this result could be reached with only 20+20 days of
measurements. A quick look at the real star densities suggest that this result could be
greatly improved by observing particularly crowded regions near the galactic center.
Keywords: PPN parameters; Space experiments; Astrometry.
1. Historical background and scientific rationale
The bending of the light path due to the gravitational pull of massive bodies is
one of the best known effects introduced by the General theory of Relativity (GR).
Historically, the very first experiment devoted to the testing of GR, during the solar
eclipse of 1919, by Dyson, Eddington and collaborators,1 was based on this effect.
The result of this experiment confirmed the forecasts of GR within a 10% accuracy.
The same kind of measurements were used for several decades after 1919 but,
despite the many attempts conducted by different teams, its accuracy could not be
improved.2 These kind of experiments failed basically because of the short
observation time (limited by the eclipse duration) and the background noise due to the
solar corona, which limited the number and the accuracy of the observations.
These difficulties directed the scientists toward other testing approaches, which
involve completely different kind of observables or different observing conditions.
In the jargon of the PPN formalism,3 the solar eclipse experiments proved the
PPN parameter 7 to be ~ 1 ±0.1, while the best estimation achieved to date is that
of the Cassini mission, which reached the 10~5 level of accuracy using the derivative
of the Shapiro effect.4
2. Motivations and concept for an astrometric space experiment
A promising effort in progress is represented by the ESA mission Gaia,5 which is
believed capable of reaching the 10-7 level on 7 by the end of the next decade.6
Nonetheless, it appears intriguing to think of a space-based verion of the Dyson-
Eddington experiment. Technology has been greatly improved after the last eclipse
experiment of 1973. Moreover, Gaia looks at about 45° away from the Sun, and
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its measure of 7 will be the result of a complex process, in which the relativistic
parameter is one among millions of unknowns, some of which might also be
correlated with 7 itself. This does not diminish the potential of the Gaia measurement,
however it marks a point in favor of a more direct measurement which has basically
a single unknown. On the other hand, there cannot be any significant improvement
in a simple repetition of a solar eclipse experiment, so the problem is how to keep
only the best of the original concept.
A satellite-based observatory would virtually have no limits on the
observation time; moreover, the number of potential targets would be greatly increased.
Therefore we focused on the basic idea of developing a space-born version of the
Dyson-Eddington test. Additionally, we constrained the instrument performance to
the budget of a low-cost mission. After a preliminary assessment, and also taking
advantage of some of the techniques studied for Gaia,7 we focused on the following
measurement concept: the instrument is built around a Fizeau interferometer with a
dual field of view (FOV), in order to observe simultaneously the two desired regions
on opposite sides of the solar limb. The arcs between the stars in a FOV and those
in the other one are measured first with the Sun in between and then far away, and
then compared to measure the light deflection suffered by these objects.
All telescope mirrors are monolithic, in order to reduce as far as possible the
differential effects within the instrument, with a FOV of approximately 7' x T. The
field separation is implemented by a beam combiner in front of the primary mirror,
folding the optical axis in two different directions on the sky, separated by a base
angle of about 4°. The elementary precision for a 100 s exposure of a V = 13 mag
source is a ~ 0.3 mas and scales with magnitude approximately as io°'2(m"m°).
3. Estimation of the measurement performance
If Aa is the light bending of a light source at angular distance a from the center
of the Sun, it is easy to derive that <t7/7 — 2(j&a/A.a on the hypothesis that the
errors on the determination of a and of the observer's position are negligible w.r.t.
that of the light deflection. This is accurate to about 10% in our case.
This means that, since at 2° away from the Sun the deflection is Aa ~ 0"2, and
given the measurement precision of the instrument, each 100 s measurement of a
V = 13 star could give a ~ 10~3 estimation of 7, and so an interesting value of
§j r^ 10-6 could be reached with about 106 observations of relatively bright stars.
Using the star counts from the GSCII catalog8 we showed that, in about 20 days
of observations, a satellite based on the above mission concept would accumulate
about half a million of observations of stars up to F = 16 per FOV and about a
million up to F = 17. They are average star counts, i.e. they are deduced considering
a uniform distribution of the real star counts on the sky region of interest for
the satellite. Being these far from the actual observing conditions, in the 20 days
considered, the real situation could be much different, according to the local stellar
density of the pointed region. We set up a simulation to verify our estimations.
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A relativistic astrometric model which almost perfectly fits our present needs was
already prepared for a series of works on the Gaia mission.6 A series of simulations
with this relativistic model, modified according to the new mission profile, confirmed
our preliminary assessment. In particular, they indicated that a satellite orbiting at
1 AU from the Sun, by repeated 100 s exposures of stars up to V = 16 for 20 days
(plus 20 more after six months), could reach an accuracy on 7 of about 3 • 10-6, in
the case of average star counts for the sky regions of interest.
However, the Sun crosses some very crowded regions near the galactic center,
where the local stellar density can be more than 100 times the average. They are
located in particular at the ecliptic longitude range of 260° < A < 290°. equivalent
to about one month of observations. Since the number of observation N increases
linearly with the number of stars, the potential accuracy in the real case could be
up to 10 times better, that is a1 ~ 10-7.
4. Conclusions
Preliminary work shows that a low-cost satellite could measure the PPN 7
parameter with a 10~6 level of accuracy. This could be done with 20+20 days of
measurements just considering the average stellar densities of the sky regions swept
by the satellite. Actual densities, however, suggest that a 10-time better accuracy
is achievable using particularly crowded regions close to the galactic center. Work
is now in progress to further assess the performance of this mission concept with a
more detailed error model, with the investigation of the best data reduction strategy,
and of possible ways of improving the performance of the instrument itself.
References
1. F. W. Dyson, A. S. Eddington and C. Davidson, Phil. Trans. R. Soc. A 220, 291 (1920).
2. M. H. Soffel, Relativity in Astroraetry, Celestial Mechanics and Geodesy (Springer-
Verlag, Berlin Heidelberg New York, 1989).
3. C. M. Will, Living Rev. Relativity 2 (2001), [Online article]: cited on December 14,
2006, http://www.livingreviews.org/Articles/Volume4/2001-4will/.
4. B. Bertotti, L. less and P. Tortora, Nature (London) 425, 374 (2003).
5. M. A. C. Perryman, K. S. de Boer, G. Gilmore, E. H0g, M. G. Lattanzi, L. Lindegren,
X. Luri, F. Mignard, O. Pace and P. T. de Zeeuw, Astron. Astrophys. 369, 339 (2001).
6. A. Vecchiato, M. G. Lattanzi, B. Bucciarelli, M. Crosta, F. de Felice and M. Gai,
Astron. Astrophys. 399, 337 (2003).
7. D. Busonero, M. Gai, D. Gardiol, M. G. Lattanzi and D. Loreggia, The Astro Optical
Response Model, in ESA SP-576: The Three-Dimensional Universe with Gaia, eds.
C. Turon, K. S. O'Flaherty and M. A. C. Perryman (2005).
8. A. Spagna, M. G. Lattanzi, B. McLean, B. Bucciarelli, R. Drimmel, G. Greene,
C. Loomis, R. Morbidelli, R. Pannunzio, R. Smart and A. Volpicelli, Exploiting Large
Surveys for Galactic Astronomy, 26th meeting of the IAU, Joint Discussion 13, 22-23
August 2006, Prague, Czech Republic, JD13, #49 13 (2006).
RELATIVISTIC LIGHT DEFLECTION NEAR GIANT PLANETS
USING GAIA ASTROMETRY
G. ANGLADA-ESCUDE1, S.A. KLIONER2 and JORDI TORRA1
1 Dept. d'Astronomia i Meteorologia, Universitat de Barcelona, Barcelona, 08028, Spain
2Lohrmann-Observatorium, Technische Universitat Dresden, Dresden, 01069, Germany
Relativistic light deflection effects in high-accuracy astrometric observations close to
planets of the solar system are analyzed using real star catalogue and appropriate
relativistic modelling. The gravitational deflection effects involve deflection due to monopole
and quadrupole gravitational fields and due to translational motion of the corresponding
planet. The data reduction scheme incorporates the bayesian analysis as a robust way
to estimate the magnitude of the effects as well as the confidence levels for the fitted
values.
Keywords: general relativity — light deflection — solar system
1. Gaia as a tool for fundamental physics
High-accuracy space astrometry enables one to test General Relativity with
unprecedented accuracy. The Gaia mission recently adopted by ESA is an astrometric
survey that will perform astrometric measurements of all celestial objects up to
stellar magnitude V ~ 20 with an accuracy of up to a few microarcseconds. Here,
we discuss the possible results of light deflection measurements obtained close to
the planets of the solar system.
The satellite continuously scans the sky in such a way that the full celestial
sphere is covered after 6 months of observations. The initial conditions for such a
scanning law will determine the whole sequence of observations during the mission.
The Gaia satellite has two telescopes with a single focal plane consisting of a mosaic
of CCDs covering a field of view of 0.7° x 0.7°. The elementary measurements are the
instants of transit of each star through the CCDs. The expected one-dimensional
astrometric accuracy is a = Am 10°-2<V-15) for V > 12, where V is the visual
magnitude of the observed star, and Am = 100 fias. The expected accuracy for
stars with V < 12 no longer depends on the brightness and remains ~ 30 /xas.
2. Physical model and parameters
According to the standard relativistic model for high-accuracy astrometry1 the light
deflection 53 of the photon is given by
5a=-(l + -y)63pN(t?) + e6ffQ(t?) , (1)
tr =t0bs-arc~1 |xobs - xa(tr ) | , (2)
where c is the speed of light, x0ts is the position of the observer, and xa is that
of the gravitating body A. Deflection 53 contains the post-Newtonian effect of the
mass monopole 53pn and the effect of the mass quadrupole 53q . Here 7 is the well-
known PPN parameter and e is an ad hoc parameter. The deflection 53 depends on
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the positions of the gravitating bodies xa- The position of each gravitating body
A must be evaluated2,3 at the corresponding retarded instant t^ given by (2). Here
we introduce one more ad hoc numerical parameter ar to see how accurately that
retardation can be measured. The numerical values of 7, e and ar are equal to 1 if
the observed light deflection is fully consistent with General Relativity.
3. Simulation and data reduction
Guide Star Catalog 2.3.14 is used for this study to simulate realistic distribution
of stars around planets of the solar system at the moments of time when Gaia can
observe sufficiently close to them. Since the quadrupole light deflection decreases
very rapidly with increasing angular distance, the accuracy of e strongly depends
on the availability of a few observations of bright stars close to the planets. How
many such observations we have depends on the initials conditions of the scanning
law of the satellite. Two possible scenarios are considered. In the first one the initial
conditions are chosen randomly. In the other, they are optimized to obtain a better
measure of the Jovian quadrupole deflection.
Our data reduction approach consists in an iterative least-square solution to
obtain the optimal parameter values and the Monte Carlo integration of the Bayesian
Probability Distribution Function5 in order to provide the confidence levels and
correlations between the parameters.
4. Results
As expected observations around Jupiter produce the best results given in Table 1.
Each effect has different dependence on the angular distance ip between the planet
and the star: inonopole ~ ip~l, retardation ~ i>~2, quadrupole ~ i/>-3. Different
statistical behavior is clearly seen on Fig. 1. While 7 always improves when stars at
larger ij; are considered, this is not the case for ay and e. Using fainter stars always
improve the accuracy. This may not be the case with real data, since faint stars
(V > 17) could be strongly affected by systematic errors.
The ephemeris errors for planets can seriously influence the obtained values,
especially for e and ay. Random periodic signals of a few hundreds of kilometers
have been introduced. In our data analysis we introduce two free parameters to
improve the ephemeris position of the planet. A shift of position in the direction of
the velocity of the body has the same influence on observations as the retardation
parameter ay, so that such errors in the ephemerides prevent the determination of
ar better than some limit. Here we made a realistic assumption that the ephemeris
guarantees positional uncertainties of ~ 100 km for Jupiter.
5. Concluding remarks
The measurements of Jupiter monopole deflection will provide an estimation of 7
at the same level ~0.1% as HIPPARCOS provided for the Sun.6 The measurement
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Table 1. Expected values and formal standard deviations for Jupiter
Scenario
(7> ± <*-i
(ar) ± aacT
(e)±*e
Standard 1.0012 ± 0.0013 1.0005 ± 0.0030 1.20 ±0.28
Optimized J2 0.9995 ± 0.0008 0.9989 ± 0.0022 0.97 ± 0.08
0.007
0.006
0,005.
0,004
0.001
0.002
0.001
0
I ' 1
^ \
^'■V
1
1 ' 1
■ '6
• --•8'
■—■ 10-
•- -• iy -
■ - 25' -
-
^^ISct^te^t^
1,1,
s\
. 1
1 1 ■
■ '6
»-• 8' -
•-— w
•--• 15 ■ .
• - 25"
' ' T ' '
Fig. 1. Standard deviations (vertical axes) as a function of the considered maximal angular
separation (in arcminutes) and limiting magnitude (horizontal axis) for 7, ar and e (left to right).
of 7 using only Jupiter observations is an important consistency test for General
Relativity. The effect of the translational motion of Jupiter will be measured with
an accuracy better than aar ~ 0.2%, that is, two orders of magnitude better than
previous results.7 Depending on the final scanning law, the quadrupole deflection
from Jupiter will be measured with an accuracy of up to f 0%. This will be the first
direct measurement of the quadrupole deflection.
The monopole deflection can be reliably measured also for Saturn, Uranus,
Neptune and Mars. For Saturn the retardation coefficient can also be obtained with a
good accuracy, but not the quadrupole deflection.
Acknowledgements. The work G.A. and J.T. is supported by Spanish MCyT
grant PNE-2003-04352. G.A. is grateful for the assistance grant from the MGll
organizing committee. S.K. was partially supported by the BMWi grant 50 QG 0601
awarded by the Deutsche Zentrum filr Luft- und Raumfahrt e. V. (DLR).
References
1. S. A. Klioner, A J 125, 1580 (2003).
2. S. M. Kopeikin and G. Schafer, Phys.Rev.D 60, 124002 (1999).
3. S. A. Klioner and M. Peip, A&A 410, 1063 (2003).
4. A. Spagna, M. G. Lattanzi, B. McLean, et al. Exploiting Large Surveys for Galactic
Astronomy, 26th General Assembly of the IAU, Joint Discussion 13, Prague, Czech
Republic, JD13, #49 (2006).
5. D. MacKay, Information Theory, Inference, and Learning Algorithms (Cambridge
University Press, 2003).
6. M. Froeschle, F. Mignard and F. Arenou, in ESA SP-402: Hipparcos - Venice '97, 49
(1997).
7. E. B. Fomalont and S. M. Kopeikin, ApJ 598, 704 (2003).
8. M. T. Crosta and F. M. Mignard, Class.Quant.Grav. 23, 4853 (2006).
ASTROMETRICAL MICROLENSING WITH RADIOASTRON*
ALEXANDER F. ZAKHAROV
National Astronomical Observatories of Chinese Academy of Sciences, 20A Datun Road,
Chaoyang District, Beijing, 100012, China
Institute of Theoretical and Experimental Physics, 117259, Moscow, Russia
Bogoliubov Laboratory for Theoretical Physics, JINR, 141980 Dubna, Russia and
Center of Advanced Mathematics and Physics, National University of Science and Technology,
Rawalpindi, Pakistan
zakharov@itep. ru
It is well-known that gravitational lensing is a powerful tool in the investigation of the
distribution of matter, including that of dark matter (DM). Typical angular distances
between images and typical time scales depend on the gravitational lens masses. For
the of microlensing, angular distances between images or typical astronietric shifts are
about 10^5 — 10 as. Such an angular resolution will be reached with the space-ground
VLBI interferometer, Radioastron. It is known that in gravitationally lensed systems the
probability (the optical depth) to observe microlensing is relatively high, therefore, for
example, such gravitationally lensed objects, like CLASS gravitational lens B1600+434,
look the most suitable to detect astrometric microlensing, since features of photometric
microlensing have been detected in these objects. However, to directly resolve these
images and to directly detect the apparent motion of the knots, the Radioastron sensitivity
would have to be improved, since the estimated flux density is below the sensitivity
threshold, alternatively, they may be observed by increasing an integration time,
assuming that a radio source has a typical core — jet structure and microlensing phenomena are
caused by the superluminal apparent motions of knots. In the case of a confirmation (or
a disproval) of claims about microlensing in gravitational lens systems, one can speculate
about the microlens contribution to the gravitational lens mass. The basic targets for
microlensing searches should be bright point-like radio sources at cosmological distances.
In this case, an analysis of their variability and a solid determination of microlensing
could lead to an estimation of their cosmological mass density. Moreover, one could not
exclude the possibility that non-baryonic dark matter could also form microlenses if the
corresponding optical depth were high enough. Astrometric microlensing due Galactic
MACHOs is not very important because of low optical depths and long typical time
scales. Therefore, the launch of the space interferometer Radioastron will give excellent
new facilities to investigate microlensing in the radio band, allowing the possibility not
only to resolve microimages but also to observe astrometric microlensing.
Microlensing studies with the forthcoming Radioastron space mission are discussed
in brief,1 see also papers2 for more detailed discussion. As it was noted earlier, there
are non-negligible chances to observe mirages (shadows) around the black hole at
the Galactic Center and in nearby AGNs in the radio-band (or in the mm-band)
using Radioastron (or Millimetron) facilities. Since a shadow size should be about
50 /xas for the black hole in the Galactic Center and analyzing the shadow size and
shape one could evaluate the spin and charge of the black hole.3
Microlensing was discussed in number of papers.4 Note that an astrometric
displacement of distant image due to light bending by gravitational field of microlenses
is called astrometric microlensing and the effect could be detectable with optical as-
*This research has been partially supported by the National Natural Science Foundation of China
(NNSFC) (Grant # 10233050) and National Basic Research Program of China (2006CB806300).
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trometric mission like SIM, Gaia and radio projects like VERA (VLBI Exploration
of Radio Astroinetry) and Radioastron. If we assume that microlenses are located
in our Galaxy and typical time scales for astrometric microlensing is double time
to change an image position displacement from ^threshold to maximal displacement
#max- So, for ^threshold = 10 /ias a typical time scale is about tastromet ~ 20 years
and for threshold = l^as it is about tastromet ~ 200 years.2
To prove the microlensing hypothesis for variability of a distant quasar, the
source have to have the following properties from a list of perspective targets of
VSOP or Radioastron missions (or from its extended version):
a) A source should demonstrate signatures of microlensing which are different from
typical features for scintillations at time scales < 3-5 years (that is an estimated
time of Radioastron mission);
b) A compact core for the source should have size < 40 /ias and flux density should
be higher than Radioastron thresholds.2
From theoretical point of view there is a possibility to detect microlensing for
both core and bright knots. In this case the two situations will be characterized by
different time scales.
First, one have to out that gravitational lensed systems are the most perspective
objects to search for microlensing. Astrometric microlensing could be detected in
the gravitational lens system such as B1600+434 in the case if a proper motion
of source, lens and an observer are generated mostly by a superluminal motion of
knots in jet.2 In this case if there is microlensing of core in the B1600+434 system
for example, then astrometric microlensing in the system could be about should be
about 20 - 40 [ias5 and the Radioastron interferometer will have enough sensitivity
to detect such an astrometric displacement.
Second, in principle microlensing for distant sources could be the only tool to
evaluate fli from microlensing event rate.6 To solve this problem with the
Radioastron interferometer one should analyze variabilities of compact sources with a core
size < 40 lias to fit the most reliable model for variabilities of the sources such as
scintillations, microlensing etc.
Therefore, one could say that astrometric microlensing (or direct image
resolution with Radioastron interferometer) is the crucial test to confirm (or rule out)
microlens hypothesis for gravitational lensed systems and for point like distant
objects. Astrometric microlensing due to MACHO action in our Galaxy is not very
important for observations with the space interferometer Radioastron, since first,
probabilities are not high; second, typical time scales are longer than estimated life
time of the Radioastron space mission.
Thus, just after the Radioastron launch it will be the first chance to detect
microlensing by a direct way. A number of point like bright sources at cosmological
distances and gravitational lensed systems with point like components
demonstrating microlens signatures is not very high and the sources should be analyzed by
the careful way to search for candidates where microlens model is preferable in
comparison with alternative explanations of variabilities.
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References
1. A.F. Zakharov, these proceedings, the COOl session.
2. A.F. Zakharov, Astron. Reports 50, 79 (2006); A.F. Zakharov, Intern. J. Mod. Phys.
D (accepted).
3. A.F. Zakharov, A.A. Nucita, F. De Paolis, G. Ingrosso, New Astronomy 10, 479
(2005); A.F. Zakharov, A.A. Nucita, F. De Paolis, G. Ingrosso, Retro gravitational
lensing for Sgr A* with Radioastron, in Proc. of the 16th SIGRAV Conference on
General Relativity and Gravitational Physics, eds. G. Vilasi, G. Esposito, G. Lambiase,
G. Marmo, G. Scarpetta, (AIP Conference Proceedings, 2005) 751, p. 227; A.F.
Zakharov, A.A. Nucita, F. De Paolis, G. Ingrosso, Observational Features of Black Holes,
in Proc. of the XXVII Workshop on the Fundamental Problems of High Energy and
Field Theory, ed. V.A. Petrov (Institute for High Energy Physics, Protvino, 2005)
p. 21; gr-qc/0507118; A.F. Zakharov, A.A. Nucita, F. De Paolis, G. Ingrosso,
Measuring parameters of supermassive black holes, in Proc. of XXXXth Rencontres de
Moriond "Very High Energy Phenomena in the Universe", eds. J. Tran Thanh Van
and J. Dumarchez, (The GIOI Publishers, 2005) p. 223; A.F. Zakharov, A.A. Nucita,
F. De Paolis, G. Ingrosso, Shadows (Mirages) Around Black Holes and Retro
Gravitational Lensing, in Proc. of the 22nd Texas Symposium on Relativistic Astrophysics
at Stanford University, SLAC-R-752, eds. P. Chen, E. Bloom, G. Madejski, V. Pet-
rosian, SLAC-R-752, eConf:C041213, http://www.slac.stanford.edu/econf/C041213,
paper 1226 (2005); A.F. Zakharov, F. De Paolis, G. Ingrosso, A.A. Nucita, Astron. &
Astrophys. 442, 795 (2005); A.F. Zakharov, A.A. Nucita, F. De Paolis, G. Ingrosso,
Shadow Shapes around the Black Hole in the Galactic Centre, in Proc. of "Dark
Matter in Astro- and Particle Physics" (DARK 2004), eds. H.V. Klapdor-Kleingrothaus
andD. Arnowitt, (Springer, Heidelberg, Germany, 2005), p. 77; A.F. Zakharov, F. De
Paolis, G. Ingrosso, A.A. Nucita, Measuring the black hole parameters from space, in
Gravity, Astrophysics, and Strings'05, Proc. of the 3rd Advanced Workshop, eds. P. P.
Fiziev and M. D. Todorov, St. Kliment Ohridski University Press, Sofia, 2006, p. 290.
4. A.F. Zakharov, Gravitational Lensing and Microlensing, (Janus-K, Moscow, 1997);
A.F. Zakharov, M.V. Sazhin, Physics-Uspekhi 41, 945 (1998); E. Kerins, MACHOs
and the clouds of uncertainty, in Cosmological Physics with Gravitational Lensing,
Proceedings of the XXXVth Rencontres de Moriond, eds. J. Tran Than Van, Y. Mel-
lier, M. Moniez, (EDP Sciences, 2001), p. 43; K. Griest, Baryonic Dark Matter and
Machos, in "Dark Matter in Astro- and Particle Physics", Proc. of the International
Conference DARK-2002, eds. H.V. Klapdor-Kleingrothaus, R.D. Villier (Springer-
Verlag Heidelberg, 2002), p. 62; A.F. Zakharov, Gravitational Microlensing and Dark
Matter Problem: Results and Perspectives, Publ. Astron. Obs. Belgrade 75, 27; astro-
ph/0212009; A.F. Zakharov, Gravitational microlensing and dark matter problem in
our Galaxy: 10 years later, in Proc. of the Eleven Lomonosov Conference on
Elementary Particle Physics, ed. A.I. Studenikin (World Scientific, Singapore, 2005) p. 106;
astro-ph/0403619; A.F. Zakharov, Gravitational microlensing: results and perspectives
in brief, Letters to Physics of Particles and Nuclei (accepted), astro-ph/0610857.
5. M. Treyer, J. Wambsganss, Astron. & Astrophys. 416, 19 (2004).
6. A.F. Zakharov, L. C. Popovic, P. Jovanovic, 2004, Astron. & Astrophys. 420, 881
(2004); A.F. Zakharov, L. C. Popovic, P. Jovanovic, Contribution of microlensing to
X-ray variability of distant QSOs, in Gravitational Lensing Impact on Cosmology,
Proc. of the IAU Symposium, eds. Y. Mellier and G. Meylan, 225, (Cambridge, UK,
Cambridge University Press, 2005) p. 363.
ASTEROIDAL OCCULTATION OF REGULUS: DIFFERENTIAL
EFFECT OF LIGHT BENDING
COSTANTINO SIGISMONDI and DAVIDE TROISE
ICRA & University of Rome La Sapicnza, Piazzdle Aldo Moro, 5 00185 Rome, Italy
* sigismondi@icra.it www. icra.it/solar
Asteroid 166 Rhodope moved at 14.4 milliarcsec/s during the occultation of Regulus of
October 19, 2005. We made a 25 Hz frame rate video (resolution 0-6 mas per frame)
near centerline in Vibo Valentia, Italy. Stellar and asteroidal diameters and relativistic
light bending by solar field are outlined. The 0.16 mas differential effect of light bending
(star-asteroid) is recovered fitting 7 observations with asteroid spherical model.
Keywords: Occultations, Fresnel diffraction, Gravitational light bending, Stellar diameter
1. Historical Review
First asteroidal occultation was observed in 1958,: that one of Regulus, a Leonis,
by asteroid 166 Rhodope on October 19, 2005 was predicted in 20042 and it has
been the first event with a bright star. Besides the mutual occultations of planets
observed by Kepler3 with his master Michael Maestlin in 1590-1591 (Venus over
Mars, and Mars over Jupiter) there are no news of such observations in the history.
Stellar occultations are used in planetary investigation: Uranus' rings discovery
was during the occultation of SAO 1586874 and a lunar occultation measured in
radio domain was used to establish the quasi-stellar nature of quasar 3C273.5 In
asteroidal occultations high spatial resolution information on the objects involved
is contained in the occultation light curve. This allows to obtain one-dimensional
spatial resolutions far beyond the diffraction limit of the observing telescope, limited
only by temporal resolution of observations.
2. Regulus Occultation: generalities and observations
Regulus is a B8 giant rapidly rotating star,6 it lies almost exactly on the ecliptic,
and it is oblate with axes 1.25 x 1.65 mas. It is a my =1.35 magnitude star.
Penumbral and umbral phases
The duration of the phase of penumbra depends on the angular diameter of the star
and on the angular velocity of the asteroid combined with Fresnel diffraction.
Fresnel diffraction
The star is nearly pointlike and at infinite, therefore wavefronts are parallel and
each point is a source of spherical waves (Huygens' principle). In presence of a seini-
infmite obstacle, perpendicular to the wavefronts, in the region behind the obstacle
there are still zones of positive interference with some amount of light. On a screen
posed at distance D behind the obstacle the luminous intensity drops to half of the
unobstructed value at 0 lateral distance, and it goes to zero at the Fresnel distance
d ~ y/D x A/2. After the first zero, there are few other bumps rapidly decreasing
with d. Rhodope was at D = 460 Gm from us. Then d = 371.5 meters for a 600nm
wavelenght (good sensitivity for CCD receptors). Regulus angular width along the
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M-
& 0.0"
-0.5'
-O.S 0.0 OS
~Aa coeft (mas)
Fig. 1. Left: Regukis and the path (arrow) of occulting asteroid seen from centcrliue, adapted
from [6]. Bight: Regulus occultation light curve at 25 Hz frame rate obtained with an handycam.
occulting path is ~ 1.28 mas, then the equivalent diameter at 460 Gin is 2855
meters or 7.7 cl. Consequently there are no fringes at all.
The role of stellar diameter
At distance d from the predicted centerlinc the duration of occultation Tocc x vx =
C'd, the length of the chord at d from diameter. All obKcrvational data are available
at Euraster website' and 8 are plotted in table 1. Each observation is provided with
position and UTC synchronization. Those data contributed to establish the diameter
of the asteroid to 65.06 ± l.C Km (before the estimate was 35 km). According to
the predictions umbral phase duration was expected to be 1.09 s, while we observed
1.92 s and maximum duration was 2.03 s.
During the occultation a thin cloud uncovered Regulus, so the final luminosity
of the star is larger than the initial one. The occultation starts at 1.24 s of our
timescale when stars' luminosity begins to drop and it ends at 3.44 s with a noisy
restoration of the full luminosity. The edge of the geometric shadow is at 50%
intensity of the Regulus light, since the light from the occulting body doesn't have
any effect because it is 14 magnitudes fainter. Our video has a high dark level and
the 14 magnitude drop of star light is not visible. Looking in our data for heavy
light drops or rises (without noise bumps) we have such features from 1.24 to 1.44 s
and from 3.36 to 3.44 s. Consequently penumbra phases last 0.20 s (more affected by
the cloud) and 0.08 s, averagely 0.14±0.06 s. From this measurement the diameter
of Regulus is 2.0 ± 0.9 mas consistent with the expected value of 1.28 mas.
Geometrical circumstances
On October 19, 2005 the elongation of Regulus from the Sun was x. = 56°. From
ephemerides the vectorial composition of orbital velocities yield a relative motion of
Rhodope with speed vx ~ 32.1 Km/s perpendicular to the hue of sight. Rhodope
was at 2.56 AU from the Sun and 3.07 AU from the Earth. On the Earth's surface
the velocity of the asteroidal shadow is approximately oriented on the parallel at
38° North and it is v = vi /cos(l — Z0), being l0 = 57°17'30" East of Greenwich the
longitude where Regulus culminated at 4:24:30 UTC. Near Regulus the ecliptic has
an inclination of +19.76° with respect to celestial parallels, from East to West.
feifsite $mtm »&*««*
2596
Table 1. Asteroidal occultation best fit: radius 32.53 Km, centerline 201 m North, \2 =6.9.
Observers
W. Nobel
C. Sigismondi,
D. Troise,
D. Montagnese
R. Goncalves
A. Ayiomamitis
D. Dunham
O. Farago
D. Nye
D. Dunham
Lat / Long [°]
38.54 /-1.85
38.68 / 16.10
37.92 / -8.24
38.30 / 23.74
38.06 / -6.24
38.50 / -3.50
38.17 / -8.49
37.95 / -6.23
c.l. distance [Km]
8.09 South
8.80 S
28.83 S
8.29 S
30.27 S
3.48 S
2.29 S
41.09 S
tocc [s]/semi-lengths [Km]
1.94 ±0.04 / 31.1 ±0.6
1.92 ±0.04/30.8 ±0.6
0.96 ±0.04/15.4 ±0.6
1.95 ±0.05/31.3 ±0.8
0.50 ±0.1/ 8.0 ± 1.6
2.02 ±0.02/ 32.4 ±0.3
2.03 ±0.04/32.6 ±0.6
no occ. (not included in the fit)
3. General Relativity and Occultations
Thanks to the optimal astrometry for bright stars we can get asteroidal orbital data
with maximum exactness. In our case many measurements allow to fix the orbit up
to 1/25 s, i.e. 1300 m in space. For an orbit radius of 2.5 AU this is an accuracy of
6 parts over 1010. Perihelion precession for Mars (1.5 AU) is about 1" per century,
i. e. 1 part over 106 of its orbit. The annual change is of 1 part over 108.
Gravitational light bending: differential effect
This effect is given by the equation 5\ = 4GMQ/c2r tan(%/2)
for the dipolar solar field. At x = 56° and the observer's position r=l AU 5x =15.4
mas. This bending shifts radially away from the Sun (on the ecliptic in this case)
the stars' apparent position. Asteroidal apparent position is also deviated by the
gravitational field of the Sun by a similar smaller amount. The differential effect
(calculated with r=2.65 AU and x = 161.8°) is Sx =0.48 mas, and the Northern
component is 5xn =0.16 mas, corresponding to 362 m at 460 Gm. It explains the
difference of 201 m of the best fit with the predicted centerline position.
Acknowledgments
This work is in memory of Raymond Dusser who explained to us several topics in
asteroidal occultations. Thanks also to Steve Preston. Thanks to Danilo Montagnese
who hosted us in Vibo Valentia, providing also to us the only one videocamera in
Italy which not failed to record the event after a night of trials.
References
1. Dunham, D.W., http://www.iota.jhuapl.edu/mpl66ol7.htm (2005)
2. Denissenko, D., http://hea.iki .rssi.ru/~denis/special.html (2004)
3. Kepler, J., Ad Vitellionem Paralipomena, quibus Astronomite Pars Optica Traditur
Frankfurt (1604)
4. Sinvhal, S.D.; et al. IAU Circ. 3061 Occultation of SAO 158687 by [Iranian Satellite
Belt (1977)
5. Leinert, C. et al., Lunar occultation of the quasar 3G273 observed on Calar Alto (2002)
6. McAlister, H.A.; et al. , Ap. J. 628, 439-452 (2005)
7. http://www.euraster.net/results/2005/index.html#1019-166 (2005)
TESTING GENERAL RELATIVITY BY ASTROMETRIC
MEASUREMENTS CLOSE TO JUPITER, THE REAL
GAREX- PART II
MARIA TERESA CROSTA, DANIELE GARDIOL, MARIO G. LATTANZI and
ROBERTO MORBIDELLI
Astronomical Observatory of Turin - INAF
Via Osservatorio 20, Pino Torinese 10025, Italy
crosta@oato.inaf.it
The ESA astrometric mission Gaia will be able to carry out general relativistic tests by
means of both global and differential astrometric measurements. Global tests will be done
through the full astrometric reconstruction of the celestial sphere, while the differential
experiments will be implemented in the form of repeated Eddington-like measurements,
i.e., comparing the evolution of relative distances in stellar fields observed in the vicinity
of a giant planet like Jupiter. Results based on simulated observations show that Gaia can
provide, for the first time, the measurement of the bending effect due to the quadrupole
moment with a 3<r confidence level. New simulations of the differential experiments which
utilize selected fields from the GSCII catalogue and a realistic error model, show how to
further improve the Gaia ability to detect the quadrupole light deflection.
1. Introduction
The payload design for the next space astrometry mission Gaia (approved in 2000
as a cornerstone within the European Space Agency science program1) allow to
observe stellar sources very close to Jupiter's limb. The light deflection produced
by an oblate planet on grazing photons has been simulated for a Gaia-like mission
for the first time in Crosta&Mignard.2 This initial study is part of a wider project
called GAia Relativistic Experiment (GAREX), which aims to study in depth all
the possibilities to test General Relativity (GR) with Gaia measurements. Gaia will
mainly carry out light deflection experiments, divided into (i) global astrometry, in
particular highly accurate determinations of the PPN parameter 7a by observing
the change in stellar positions at different angular distances from the barycentre
of the solar system; (ii) small field experiments, investigating light propagation
by means of differential measurements of stellar positions near the planets. The
paper of Crosta&Mignard,2 based on a crude Galaxy model, proved that Gaia is
capable of detecting the quadrupole light bending due to a planet, a relativistic
effect predicted by GR but never observed. For the GAREX equation model we
derived a vectorial formulation of the light bending which contains the monopole
contribution (parameterized by 7, along the radial direction towards the centre of
the planet) plus the quadrupole one (radial and orthoradial) in the static case. The
quadrupole deflection has been parameterized by introducing a new parameter e,
called Quadrupole Efficient Factor (QEF), which should be equal to one in GR.
Results show that the monopole deflection can be determined13 to 10-3, while the
aThe PPN parameter 7 indicates the amount of space-time curvature produced by a unit rest-mass,
assumed equal to one in General Relativity, the standard theory of gravity in the PPN formalism.
bThis result is better than the one already achieved by Hipparcos in the case of the Sun.
2597
2598
quadrupole light deflection will be detectable for the first time with a 3-ct confidence
level. Most importantly, the simulation gave clues on how to design an optimal
strategy to carry out this experiment in the case of a real stellar distribution. In
fact, a bright 3-6 arc-minute wide open cluster around Jupiter gives better estimates
than those obtained with fainter uniformly distributed stellar backgrounds. In the
following section we describe briefly the preliminary results obtained by using GSCII
data and a more realistic error model.
2. Towards the real Garex
With the new simulation we generated ten thousands continuous fields (3 times
per day) from 2011 to 2020 using ephemeris of Jupiter as observed from Gaia. This
number assures a sufficiently fine sampling for searching the best candidate scenarios
and, consequently, to place requirements on the initial phase for a good optimization
of the scanning law in order to observe the selected fields. Star counts were extracted
from the GSCII data base in areas of about 0.5 x 0.5 square deg (approximately
the size of the field of view of Gaia) and centered around each generated equatorial
coordinate of Jupiter. F-band magnitudes were used, which are close to the G-
band magnitudes measured with Gaia. For this simulation, we decided to run the
experiment with the same theoretical formulation of the effect, as in the initial
paper,2 namely no gravito-dynamical influence of Jupiter was inserted. This tells
us to what extent QEF is measurable with a more realistic observing scenario.
Examples of good candidate fields Among the many generated fields, we
selected two examples of good observing scenarios for the GAREX experiment.
Table 1 shows the results obtained after 100 Montecarlo runs for the parameters
7 and e, both assumed equal to one, in fields observed on 1st April 2014 and on
20th February 2019 (when Jupiter is crossing the galactic plane towards the galactic
centre).
Table 1. Background field (n*) around Jupiter for different F magnitude limits on 1st April
2014 (columns 1-2) and on 20th February 2019 (columns 3-4), at different Jupiter's radii (Rj)-
< 7 >
< e >
n* = 11052, F < 20
whole field
1.00±4.21 X 10~3
1.00± 0.19
n* = 10, F < 15
3Rj
1.00±1.10 x 10-3
1.01±0.29
n*=14402, F < 20
whole field
1.00±3.4 X 10-4
0.98±0.09
n*=206, F < 15
13Rj
1.00±7.4 x 10~4
1.02±0.12
Realistic error model The model for the Gaia astrometric error versus star
magnitude is obtained from a simulation, taking into account the most relevant noises.
However, the most important effect for bright magnitudes is CCD saturation. At
magnitude 12 (13 in the selected Gaia configuration) the PSF begins to become sat-
2599
urated. For this reason part of the signal is lost, and the astrometric error increases
with respect to the non saturated case. Montecarlo simulations show that using an
appropriate centroiding algorithm it is still possible to achieve good performances
on partially saturated images. In this case the astrometric error can be described, as
function of magnitude, y the following approximated formula a = 109(F), where the
function g(F) is given in Gardiol.3 For a complete transit we have 9 independent
measurements and the final error is divided by 3 times the square root of the mean
number of observations per star.
3. Discussion
The results presented indicate that the accuracy of our approach is close to that
obtained with global fits used in Angladaet al..AAs confirmation of the statistical
results already obtained in Crosta&Mignard2 when we looked for the best
configurations, it is enough to select background fields which include a few bright star
close to Jupiter to produce the best results. If we include the background noise
in the error model, the experiment is still possible, but not too close to Jupiter.
In fact, with background noise (about 50000/10000 photoelectrons from 1.25" to
6"from the Jupiter's limb), we obtain the results shown in table 2. Therefore, it
Table 2. Background field close to Jupiter on 20th
February 2019
n*=6, F < 15, 3Rj n*=3,F < 14, l-2Rj
< 7 > 1.001± 0.0148 0.9899±0.0214
< e > 0.8428±1.5822 1.1800± 2.0600
appears that the best way to detect the quadrupole light bending effect is to choose
optimal configurations during the mission operational life. Further work will take
into account an even improved description of the observing scenario by including
the details of instrumental/technical effects (e.g. how to compare the two
observations with/without Jupiter), and those associated with the stellar fields (e.g. proper
motions). The final task will be to apply the complete formula for the relativistic
model,which includes all relevant relativistic effects at the level of the Gaia accuracy.
References
1. The three-Dimensional Universe with Gaia, 2004, ESA-SP-576
2. M.T. Crosta and F. Mignard, 2006, Class. Quantum Grav. 23,4853-4871
3. D. Gardiol, GAlA-CH-TN-INAF-DG-001-1, tec.note on Gaia Livelink
4. G. Anglada, S. Klioner, J. Torra, in Proc. of the Eleventh Marcel Grossmann Meeting
on General Relativity, edited by H. Kleinert, R.T. Jantzen and R. Ruffini, World
Scientific, Singapore, 2007
RELATIVISTIC TESTS FROM THE MOTION OF THE ASTEROIDS
D. HESTROFFER, S. MOURET and J. BERTHIER
IMCCE, UMR CNRS 8028, Observatoire de Paris, Paris, F-75014 Prance
hestrojfer@imcce.fr, mouret@imccefr,berthier@imcce.fr
F. MIGNARD
Observatoire de la Cote d'Azur, CNRS
he Mont Gros, BP 4229, 06304 Nice, France
francois.mignard@obs-nice.fr
Because of their negligible mass and size, asteroids act as particle test in the
gravitational field of the Sun (or the solar system at large); hence the knowledge of their orbit
can provide local tests of general relativity. In addition to the "3D census of the Galaxy"
the space-mission Gaia will enable, this ESA astrometric mission will provide highly
accurate positions of a large number of solar systems objects. Given the expected nominal
precision —ranging from a few milli-arcseconds for the faintest bodies down to sub-mas
precision for the brightest ones— one can expect to perform the classical perihelion
precession test of the GR from the motion of the asteroids. We present preliminary results
of a variance analysis involving realistic simulations of a subset of asteroids including
Near-Earth objects and main-belt asteroids that will be observed by Gaia. These show
the formal precision achievable for the joint determination of the Solar J2 together with
the PPN parameter /3, as well as the precision for G/G and the link of the dynamical
reference frame to the kinematically non-rotating conventional ICRS.
Keywords: GAIA; astrometry; PPN; asteroids; NEOs; orbit; variance analysis.
1. Introduction
One of the first successful test of General Relativity (GR) at the beginning of
last century lies in the explanation of the well known—and up to this date—much
debated problem: the anomalous perihelion advance of Mercury. Indeed, after having
taken into account the precession of the equinoxes, all planetary perturbations and
other effects on the orbit of Mercury, the computed positions could not match the
observed ones, while the same methods applied to the other planets was able to
predict with good accuracy the positions. It was argued that the motion of Mercury
could be perturbed by an unknown massive planet orbiting closer to the Sun, or that
Mercury could depart from the Newtonian inverse-square law of gravity. General
Relativity on the other hand set a landmark by giving for this precession the value of
43.5 arcesc/cy, one of the "classical test" of GR.1 Attempts to derive such test from
the perihelion drift of the asteroid Icarus - which was for long one of the highest
known eccentric-orbit Near-Earth object (NEO) - were unsuccessful partly because
of the too large observational and systematic errors.2^ In any case, however, such
test performed on a single body cannot separate the effect due to general relativity
from other badly modeled perturbations of the orbit yielding to a perihelion drift.
While the effects of planetary perturbations can be known with enough accuracy,
the precession brought about by the Sun quadrupole J2 cannot be disentangled
from the GR effect. Thus, to better separate the PPN parameter (3 and J2, one
2600
2601
must rely on the observation of a set of test particles that cover a wide range in
the (e, a) plane. In addition the analysis of the orbit also provides a test for the
variation of the constant of gravity G and allows to relate reference frames. We give
here the expected performances of this investigation based on a variance analysis
of the parameter fitting.
Gaia is a cornerstone mission of the European Space Agency to be launched
in late 2011. The main objective of this astrometric mission is to provide a 3-
dimensional census of the Galaxy. However, in addition to the observation of stars,
galaxies and quasars, Gaia will also observe a large number of « 300,000 solar
system objects, all brighter than magnitude V < 20. These are mainly asteroids in
the main belt but also NEOs (given by the astorb.dat catalogue5). Each source
will be observed about 60 times during the 5-year mission with single-observation
position in the range of 0.3 — 5 mas—depending on the magnitude and velocity—
enabling an analysis of subtle effects. One will hence be able to measure the small
secular drift of the orbital elements, in particular the argument of the perihelion
(ui ~ 6tt
rap
(27 - (3 + 2) +
R%
■h
neglecting the asteroid's orbit
a5/2 (1-e2) v" ' ^ """ > "T" a7/2 (l_e2)2
inclination), and similarly the longitude of the node il, and the mean anomaly M.
One should note that the known population of NEOs is still incomplete with only
2/3 of the largest bodies discovered so far. In order to provide an estimation of the
GR test we have considered a simulated population that is statistically6 realistic in
terms of orbit and absolute magnitude distributions.
2. Orbits improvement and tests of GR
20
.9
mag
O
"
16
1S0
a
0
-100
V
e
I 5°
" fc
f\
t
s
m
$
&
>k\
1
&
if
T '■
I; J
A ,B ,C Hi
gA .B 0C Hi
Eccentricity, i
Fig. 1. Distributions of the asteroids considered in the simulation. Left: in the (e, a) plane of their
orbital elements. Sensitivity of the orbit to the relativistic perihelion precession is shown by the
solid curves. The dotted curves show the separation between the known population of asteroids
and a simulated set of a complete catalogue of NEOs. Right-, plots of the sensitivity for 3 different
simulation sets vs. magnitude (top) and number of observation (bottom).
2602
A discrepancy in the observed positions from the predicted one as given by the
equation of motion in the GR can be attributed to a correction to PPN parameters.
Assuming that the PPN 7 is known with enough accuracy from other experiments
(e.g. light deflection of the stars observed by Gaia7), one can derive in a direct
manner the couple (/3, J2). Moreover one can introduce in the fitted parameters two
vectors for a rotation W and rotation rate W between the dynamically non-rotating
reference frame to which the equations of motion are referred and a kinematically
non-rotating frame in which the observed directions are given, as well as a possible
variation of the gravitational constant G/G. We consider 3 sets of simulated NEOs
population for our computations (see Fig. 1). No strong variation of the formal
standard deviation have been observed between the results based on these different
populations, suggesting that one should find NEOs to perform the tests to the
precision given in Table 1. Depending on the kind of solution foreseen (only one of
(/3, J2), or both parameters) the precision change little, showing that the parameters
are well separated. Although the two parameters are correlated we have found that
the system is well conditioned. The fit will directly yield the solar quadrupole at the
10-8 level precision; and also provide a 2 a detection at the 5 x 10-11 rad/yr level
for a possible rotation-rate of the supposed kinematically "non-rotating" frame.
Table 1. Formal precision on simultaneous determination of all global parameters
(1 /^as ss 4.85 X 10-12 rad). Extreme values are given for the measure of /3 and J2
and their correlation p(f), J2) depending on the NEOs data set (see text).
[xlO-4]
2-5
■h
[xl0~8]
0.5- 1.5
[-]
0.11 -0.85
G
[yr-1]
2 x 10~12
O.K.z)
[Mas]
5-5-14
[^as/yr]
1-1-5
References
1. C. M. Will, Theory and Experiment in Gravitational Physics (Cambridge UP, 1993).
2. J. J. Gilvarry, Physical Review 89, 1046(March 1953).
3. I. I. Shapiro, M. E. Ash and W. B. Smith, Physical Review Letters 20, 1517(June 1968).
4. G. Sitarski, Astronomical Journal 104, 1226 (1992).
5. E. Bowell, K. Muinonen and L. H. Wasserman, A Public-Domain Asteroid Orbit Data
Base, in IAU Symp. 160: Asteroids, Comets, Meteors 1993, eds. A. Milani, M. di
Martino and A. Cellinol994.
6. W. F. Bottke, A. Morbidelli and R. Jedicke et al., Icarus 156, 399 (2002).
7. F. Mignard, Relativistic effects from HIPPARCOS and GAIA missions, in MGM #11,
Berlin, 23-29 July 2006, 2006.
Quantum Gravity
Phenomenology
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EFFECTIVE VACUUM REFRACTIVE INDEX FROM GRAVITY
AND PRESENT ETHER-DRIFT EXPERIMENTS
M. CONSOLI
Istituto Nazionale di Fisica Nucleare, Sezione di Catania,
Catania, 95123 Italy
A simple re-analysis of the data published by two present ether-drift experiments
provides non-zero daily averages for the amplitude of the signal. The two experimental
values, A0 ~ (10.5 ± 1.3) ■ 10-16 and A0 ~ (12.1 ± 2.2) • 10-16 respectively, are in good
agreement with each other and with the theoretical prediction (9.7 ± 3.5) ■ 10~16 (see
Phys. Lett. A333 (2004) 355 and N. Cim. 119B (2004) 393) formulated in the framework
of a flat-space picture of gravity.
1. Basic formalism and experimental data
In this contribution, I will summarize the results of Ref.1 where a re-analysis of the
data reported in Refs.2,3 , for the anisotropy of the speed of light in the vacuum,
was presented. The basic measured quantity is the relative frequency shift of two
rotating optical resonators at a given time t (wrot being the rotation frequency)
—-Q = S(t)sm2ujIott + C(t) cos 2wroti (1)
For brief observations (2-3 days) S(t) and C(t) can be expressed as
S(t) = S0 + Ssl sinr + Scl cost + Ss2 sin(2r) + Sc2 cos(2r) (2)
C(t) = Co + Csl sin t + Ccl cos r + Cs2 sin(2r) + Cc2 cos(2r) (3)
t = Wsidt being the sidereal time of the observation in degrees and ws;d ~ 23fe^6,. In
this framework, the published data are a set of values for the elementary coefficients
Co, Csi, Cci, CS2, Cc2 and for their S'-counterparts. All relevant numerical values are
reported in Tables 1-5 of Ref.1 . The main point for the re-analysis consists in
rewriting the frequency shift in the equivalent form
^^ = A(t) cos(2Wrot* " 20o(*)) (4)
with C{t) = A(t) cos 280(t), S(t) = A(t) s'm200(t) and where 60(t) represents the
instantaneous direction of a hypothetical ether-drift effect in the plane of the
interferometer. Introducing the two-way speed of light in the vacuum
c(6)=c(l-^(A + Bsm20)) (5)
the amplitude of the signal can be expressed as
A(t) = \\B\^-, (6)
2605
2606
v(t) being the magnitude of the projection of the cosmic Earth's velocity in the
plane of the interferometer. Analogously to Eqs.(2) and (3), one finds
A(t) = A0 + Ai sin r + A2 cos t + A3 sin(2r) + A4 cos(2r) (7)
Since A0 was not explicitly given by the authors of Ref.2'3 , in Ref.1 its value
was deduced from the published data using simple algebraic identities. In this way,
averaging over the 15 observation periods of Ref.2 , one finds
A0 ~ (10.5 ± 1.3) -1(T16 (8)
in good agreement with the value obtained from the data of Ref.3
A0 ~ (12.1 ±2.2)- 10'16 (9)
2. An effective refractive index for the vacuum
It is interesting that the two experimental values in Eqs.(8) and (9), besides being in
agreement with each other, are also well consistent with the theoretical prediction
that can be obtained from Ref.5
4h-^|th4-(9-7±3-5)-10"16 (10)
2 cz
This was formulated, in connection with the vacuum anisotropy parameter4 |5|th ~
42 • 10~10, after inserting the average cosmic velocity (projected in the plane of
the interferometer) vq = (204 ± 36) km/s that derives from a re-analysis5 of the
classical ether-drift experiments. Due to this rather large theoretical uncertainty, the
different locations of the various laboratories and any other kinematical property of
the cosmic motion can be neglected in a first approximation.
The theoretical prediction for the anisotropy parameter was obtained starting
from the close analogy that one can establish between General Relativity and a fiat-
space description where gravity represents a long-distance perturbation of a medium
that modifies the masses (and with them the physical space-time units) by also
generating an effective refractive index for the vacuum. This alternative approach,
see for instance Wilson6 , Gordon7 , Rosen8 , Dicke9 , Atkinson10 , Puthoff11 and
even Einstein himself12 , before his formulation of a metric theory of gravity, in
spite of the deep conceptual differences, produces an equivalent description of the
phenomena in a weak gravitational field. For an apparatus placed on the Earth's
surface (but otherwise in free fall with respect to any other gravitational field) both
approaches predict the weak-field, isotropic form of the metric
ds2 = c2dt2(l - ^-) - (1 + ^-)(dx2 + dy2 + dz2) = c2dr2 - dl2 (11)
G being Newton's constant and M and R the Earth's mass and radius. Here dr and
dl denote respectively the elements of "proper" time and "proper" length in terms
of which, in General Relativity, one would again deduce from ds2 = 0 the same
2607
•/vvacuum L 2d \ )
universal value ^ = c. However, in the flat-space approach the condition ds2 = 0
is interpreted in terms of an effective refractive index for the vacuum
2GM
c2R
Therefore, differently from General Relativity, in the flat-space approach light can
be seen isotropic in only one reference frame13 , say E. The ether-drift experiments
can then clarify whether E coincides with the Earth's frame or with a hypothetical
preferred frame. In the former case, corresponding to no observed anisotropy, the
equivalence between General Relativity and the gravitational-medium picture would
persist. In the latter case, using Lorentz transformations to connect E to the Earth's
frame, one predicts an anisotropy parameter4,5
|5|th~3(ACacuum-l)~42-10-10 (13)
whose apparent observation seems to uniquely single out the flat-space scenario.
3. Summary and conclusions
On the basis of the alternative re-analysis of Ref.1 , the data seem to support both
a) the existence of a preferred frame and b) a flat-space description of gravity. The
novelty of this conclusion emphasizes the importance of comparing different points
of view and approaches to the data to finally achieve a full understanding of the
underlying fundamental physical problem.
Acknowledgements
I thank Giovanni Amelino-Camelia and Evelina Costanzo for useful discussions.
References
1. M. Consoli and E. Costanzo, arXiv.gr-qc/0604009, submitted to Eur. Phys. J. C.
2. S. Herrmann, et al., Phys. Rev. Lett. 95 (2005) 150401.
3. P. Antonini, et al., Phys. Rev. A71 (2005) 050101(R).
4. M. Consoli, A. Pagano and L. Pappalardo, Phys. Lett. A318 (2003) 292.
5. M. Consoli and E. Costanzo, Phys. Lett. A333 (2004) 355; N. Cim. 119B (2004) 393.
6. H. A. Wilson, Phys. Rev. 17 (1921) 54.
7. W. Gordon, Ann. Phys. (Leipzig) 72 (1923) 421.
8. N. Rosen, Phys. Rev. 57 (1940) 150.
9. R. H. Dicke, Int. School "Enrico Fermi", Varenna 1961, Academic Press 1962, p.l.
10. R. D'E. Atkinson, Proc. R. Soc. 272 (1963) 60.
11. H. E. Puthoff, Found. Phys, 32 (2002) 927.
12. A. Einstein, Ann. der Physik 35 (1911) 898, On the influence of gravitation on the
propagation of light, English translation in The Principle of Relativity, Dover
Publications, Inc. 1952, page 99.
13. A. M. Volkov, A. A. Izmest'ev and G. V. Skrotskij, Sov. Phys. JETP 32 (1971) 686.
QUANTUM GRAVITY EFFECTS IN ROTATING BLACK HOLES
M. REUTER and E. TUIRAN
Institute of Physics, University of Mainz, D-55099 Mainz, Germany
1. Introduction
The effective average action has been used for detailed studies of the nonpertur-
bative renormalization behavior of Quantum Einstein Gravity, in particular in the
context of the asymptotic safety scenario [1,2]. As a first application of the scale
dependent Newton constant derived in [1], quantum corrections to the Schwarzschild
spacetime were discussed in [3]. Indications were found that, due to quantum
effects, the Hawking evaporation process stops once the mass of the black hole is of
the order of the Planck mass [3, 4]. In this note we report on some aspects of the
corresponding analysis for Kerr black holes [5].
2. Renormalization Group Improvement
The (truncated) renormalization group (RG) equation for the average action
provides us with a running Newton constant G (k) where the mass parameter k sets
the scale of the "coarse graining" which has been performed. Technically it is
implemented as an infrared cutoff in the underlying functional integral over all metrics. In
the improvement approach of [3] one tries to relate k to the geometrical properties
of the system under consideration. More concretely, one sets up a correspondence
k = k(P) between scales and spacetime points P. A still rather general ansatz for
this correspondence is k oc 1/d(P), where d{P) = jc \/\ds2\ is the proper length
of a spacetime curve C related to P which is computed with respect to the classical
metric. This ansatz is manifestly diffeomorphism invariant, and thanks to its
nonlocal character it is potentially capable of mimicking certain (not explicitly known)
nonlocal terms in the average action. In [5] various choices for C are discussed, for
instance a radial path from the center of the black hole to P. Using standard Boyer-
Lindquist (BL) coordinates, d(P) becomes a function d(r,9). It turns out [5] that
within the expected domain of reliability of this approach the #—dependence of the
invariant distance is inessential and dm d(r) depends on the radial coordinate only.
Asymptotically, d(r —> oo) « r.
For a "semi-quantitative" analysis we used the approximation for G (r) given
by G(k) = Go/ (l + wGok2) [3]. It entails the position dependent Newton
constant G(r) = G0d2 (r) / (d2 (r) + wG0). Here w and w are positive constants,
and Go = M^ is the standard Newton constant. The "RG improvement"
consists in substituting Go —> G (r) in the classical Kerr solution. The resulting
metric of the RG improved Kerr spacetime reads, in BL coordinates, dsf =
- (Ap"2) [dt - asm2 edip}2 + (p~2 sin2 0) [(r2 + a2) dip - adt}2 + (/^A"1) dr2 +
p2dO . The general structure of this metric, as well as the abbreviations p2 =
2608
2609
r2 + a2cos2#, a = J/M, are as in the classical case. The only place where G (r)
appears is in A = r2 — 2G (r) Mr + a2.
3. Critical Surfaces of the Improved Kerr Metric
The spacetime described by ds2 has two infinite redshift surfaces (goo = 0) at radii
r = rs± (0) given by r2 - 2G (r) Mr + a2 cos2 0 = 0. We denote them by S±. The
outer one, S+, is the static limit surface. Furthermore, the spacetime has two event
horizons (grr = 0, A = 0) with radii r = r± to be obtained from r2 — 2G (r) Mr +
a2 = 0. We denote the inner (outer) horizon by H- (H+). In Fig. 1 we plot the
radii r± and rs± (0 = ?) in the equatorial plane for the approximation d(r) = r,
and we compare them to the classical values (w=Q) . The upper and lower branches
of the curves correspond to S+, H+ and S~, H-, respectively. We observe that for
small enough M the horizons coalesce and then disappear, and similarly for S± at
even lower masses. The coalescence of H± and S± occur for M of the order of MP\
where the applicability of the method becomes questionable. It can be safely applied
for r ^> /pi if M ^> mpi and a <^i MGq where the quantum corrections are small.
15
12.5
10
r(M) 75
5
2.5
0
0 2 4 6 8
M
Fig. 1.
Radial coordinates of the critical surfaces at the equatorial plane vs. mass in Planck units
and their improved counterparts. Dashed lines represent the static limit surfaces S±, solid
lines the event horizons H±. The thicker lines correspond to the classical surfaces.
4. Antiscreening and Smarr's Formula
Since the improved Kerr metric is known explicitly, we can compute its Einstein
tensor and write it in the form G^u = SttGoT^, thus defining an effective energy-
momentum tensor Tf^ for the quantum fluctuations which drive the renormalization
group evolution. Nevertheless, improved vacuum black holes are in many respects
quite different from classical ones in presence of matter. The reason is that T^
does not have any of the standard positivity properties which are crucial in black
hole thermodynamics, for instance. Corrections to the mass Mh and angular
momentum Jh of the Kerr black hole coming from the "pseudo-matter" described by
T^ can be calculated by performing the Komar integrals at the event horizon:
2610
MH = - (87tG0)_1 §VatPdSal3 , JH = (WnGor1 §Va(f>0dSa0. Here t'3 and <$P are
the Killing vectors associated to stationarity and axial symmetry, respectively. One
finds:
Jv
1 — arctan [
V
JG(r+) | M2r2hG'(r+)G(r+)
Go Go^
a ^
1 -
,G'(r+)(r2. + a2)"
' aG(r+)
2MG(r+) +
arctan
a
( a
(2)
For the case of the mass, (1) tells us that, due to quantum fluctuations, the
classical mass M is decreased to a value Mr < M for every possible running of the
Newton's constant [5]. This can be interpreted due to the antiscreening character of
quantum gravity [1]. Remarkably enough, Smarr's formula still holds in its classical
form Mh = 2S7H-/H + kA/ (4ttGo). For the quantum corrected black hole, the
horizon's angular frequency, surface gravity, and area are given by flu — a/ (r\ + a2),
k = (r+ - 2M) \G{r+) + r+G'{r+)\ / {r2+ + a2) and A = 4tt {r\ + a2). The classical
appearance of these formulas (except for the G'-term in k) is deceptive: The radius
r+ = r+ (a, M) depends on the parameters of the black hole, and this relationship
is modified by the renormalization effects.
5. Summary
We explained how quantum gravity effects in the spacetiine of rotating black holes
can be taken into account by a RG improvement of the classical Kerr solution. We
discussed the Black hole's critical surfaces as well as the "gravitational dressing"
of its mass and angular momentum. Further properties of the improved Kerr black
hole, in particular its thermodynamics and Penrose process, will be described in ref.
[5].
References
M. Reuter, Phys. Rev. D57 (1998) 971 and hep-th/9605030.
O. Lauscher and M. Reuter, Phys. Rev. D 65 (2002) 025013 and hep-th/0108040, Phys.
Rev. D 66 (2002) 025026 and hep-th/0205062, Class. Quant. Grav. 19 (2002) 483 and
hep-th/0110021; M. Reuter and F. Saueressig, Phys. Rev. D 65 (2002) 065016 and
hep-th/0110054; A. Bonanno, M. Reuter, JHEP 02 (2005) 035 and hep-th/0410191.
A. Bonanno and M. Reuter Phys. Rev. D 62 (2000) 043008 and hep-th/0002196.
A. Bonanno and M. Reuter, Phys. Rev. D 73 (2006) 083005.
M. Reuter and E. Tuiran, in preparation.
LORENTZ SYMMETRY FROM LORENTZ VIOLATION
IN THE BULK
ORFEU BERTOLAMI1'2.* and CARLA CARVALHO1'2^
1 Departamento de Fisica, Instituto Superior Tecnico, Avenida Rovisco Pais 1,
1049-001 Lisboa, Portugal
Centro de Fisica dos Plasmas, Instituto Superior Tecnico, Avenida Rovisco Pais 1,
1049-001 Lisboa, Portugal
* orfeu@cosmos.ist.utl.pt
t ccarvalho@ist. edu
We consider the mechanism of spontaneous symmetry breaking of a bulk vector field to
study signatures of bulk dimensions invisible to the standard model confined to the brane.
By assigning a non-vanishing vacuum expectation value to the vector field, a direction is
singled out in the bulk vacuum, thus breaking the bulk Lorentz symmetry. We present the
condition for induced Lorentz symmetry on the brane, as phenomenologically required,
noting that it is related to the value of the observed cosmological constant.
1. Introduction
Braneworld scenarios have changed our view of the extra dimensions. The various
models predict that gravity in our braneworld can exhibit significant deviations from
that described by Einstein's general relativity. In particular, in string theory inspired
scenarios which assume that the background bulk spacetime is anti-de Sitter, it is
possible to cancel out any 4-dimensional brane contribution to the cosmological
constant (see e.g. [1] and references therein). Although not on its own a solution for
the cosmological constant problem, it is suggestive that braneworld scenarios might
be an important feature of a consistent description of the world.
It is therefore relevant to investigate the implications of the braneworld
scenarios to the formulation of fundamental symmetries, another fundamental ingredient
of the physical description. Lorentz symmetry, being from the phenomenological
point of view one of the most well and stringently tested symmetries of physics, is
particularly suitable to test the relation between bulk and brane symmetries as a
possible signature for the existence of extra dimensions.
The possibility of violation of Lorentz invariance has been extensively discussed
in the recent literature (see e.g. [4]) and in particular its astrophysical implications
have been studied.5 Furthermore, a connection between the cosmological constant
and the violation of Lorentz invariance has been conjectured in the context of the
string field theory.6
In this contribution we report on a recent study whose motivation was to
understand the way spontaneous Lorentz violation in the bulk is related to Lorentz
symmetry on the brane.2 We consider a bulk vector field coupled non-minimally to
the graviton which, upon acquiring a non-vanishing expectation value in the
vacuum, introduces spacetime anisotropics in the gravitational field equations through
the coupling with the graviton.3 After deriving the equations of motion in the bulk,
we project them parallel and orthogonal to the surface of the brane. The brane is
2611
2612
assumed to be a distribution of ^-symmetric stress-energy about a shell of
thickness 25 in the limit <5 —> 0. Derivatives of quantities discontinuous across the brane
will generate singular distributions on the brane which relate to the localization of
the stress-energy This relation is encompassed by the matching conditions across
the brane obtained by the integration of the corresponding equation of motion in
the direction normal to the brane. The matching conditions provide the boundary
conditions on the brane for the bulk fields, thus constraining the parallel projected
equations to produce the induced equations on the brane. Spontaneous symmetry
breaking is then treated by assuming that the bulk vector field acquires a non-
vanishing expectation value which reflects on the brane the breaking of the Lorentz
symmetry in the bulk.
2. Bulk Vector Field Coupled to Gravity
Aiming to examine the gravitational effects of the breaking of Lorentz symmetry
in a braneworld scenario, we consider a bulk vector field B with a non-minimal
coupling to the graviton in a five-dimensional anti-de Sitter space. The Lagrangian
density consists of the Hilbert term, the cosmological constant term, the kinetic and
potential terms for B and the B-graviton interaction term, as follows
£
1
*(5)
R-2A + ^BfiB"Ri
1
(IV
-BIU,B'"' -V{B»Bli±b2),
(1)
where B^v = V^-B,, — V^B^ is the tensor field associated with B^ and V is the
potential which induces the spontaneous global symmetry breaking when the B
field is driven to the minimum at B^B^ ± b2 = 0, b2 being a real positive constant.
8ttGn = Mpl: Mpi is the five-dimensional Planck mass and £ is a
Here,
V(5)
dimensionless coupling constant that we have inserted to track the effect of the
interaction. In the cosmological constant term A = A(5) + A(4) we have included
both the bulk vacuum value A(5) and that of the brane A(4), described by a brane
tension a localized on the locus of the brane, Am) = aS(N).
The Einstein equation is given by:
1
"(5)
G)iu + AgM„ - £L
JflV
■&,
(IV
l-T
(2)
where
g^BWRp. - (B^BPRp, + RWBPBV)
(3)
S^ = - [V^Vp{ByBP) + VvVpiB^B") - V2(BMB,) - g^VPVa{B>>B°)\ (4)
are the contributions from the interaction term and
<-\iv
BwBvp + AV B^BV + sM„
1
BpaB<>° - V
(5)
2613
is the contribution from the vector field for the stress-energy tensor. For the equation
of motion for the vector field B, we find that
VMB,
2V'BI1 + 2£,BVRIIV = 0.
(6)
V (V„B„
where V = dV/dB2. Projecting the equations parallel (A) and orthogonal (N) to the
surface of the brane, we proceeded to integrate them in the normal direction to the
extract the matching conditions. These conditions constrain the parallel projected
equations to yield the induced equations on the brane. The general features of this
procedure have been previously discussed.7
When the bulk vector field B acquires a non-vanishing, covariantly conserved3
vacuum expectation value by spontaneous symmetry breaking, the bulk vacuum
acquires an intrinsic direction determined by (Ba) , thus inducing the breaking
of the Lorentz symmetry in the bulk. In order to obtain a vanishing cosmological
constant and ensure that Lorentz invariance holds on the brane, we take the Einstein
equation induced on the brane and impose respectively that
A
(5)
^
2(£-l))Kv
(7)
and that
1
*(5)
2KACKBC- [~+^~l)KABK
;9ab
(*
1
2
ind)
2KCDKCD~(l-2(£~l))K<
(I - 2 + l) (<B*> <B*> ^B
^nd) + (BB) (BC) R%d)
(^ + 2)KACKBD(BC)(BD)
:9ab
(Bc) (BD) R,
(ind)
CD
2{Z-l)KCEKED(Bc)(BD)
(8)
which for £ = 1 reduce to the results presented in [2].
3. Discussion and Conclusions
In this contribution we examine the spontaneous symmetry breaking of Lorentz
invariance in the bulk and its effects on the brane. For this purpose, we considered a
bulk vector field subject to a potential which endows the field with a non-vanishing
vacuum expectation value, thus allowing for the spontaneous breaking of the Lorentz
symmetry in the bulk. This bulk vector field is directly coupled to the Ricci tensor
so that, after the breaking of Lorentz invariance, the breaking of this symmetry is
transmitted to the gravitational sector. We assign a non-vanishing vacuum
expectation value to the component of B parallel to the brane (the generality of this
procedure has been discussed in [8]). We observe that there is a connection between
2614
the vanishing of the cosmological constant and the reproduction of the Lorentz
invariance on the brane. The conditions above were enforced so that the higher
dimensional signatures encapsulated in the induced geometry of the brane cancel
the Lorentz symmetry breaking inevitably induced on the brane, thus
reproducing the observed geometry. Naturally, the first condition, Eq. (7), can be modified
to account for any non-vanishing value for the cosmological constant induced on
the brane. A much more elaborate fine-tuning, however, is required for the Lorentz
symmetry to be observed on the brane, as expressed by the condition Eq. (8). We
believe that this is a new feature in braneworld models, as in most such models
Lorentz invariance is a symmetry shared by both the bulk and the brane. Notice
that a connection between the cosmological constant and Lorentz symmetry had
been conjectured long ago.6 We shall examine further implications of this
mechanism in a forthcoming publication where we will also discuss the inclusion of a bulk
scalar field.9
References
1. O. Bertolami, "The Adventures of Spacetime", gr-qc/0607006.
2. O. Bertolami and C. Carvalho, Phys. Rev. D74 (2006) 084020.
3. V.A. Kostelecky, Phys. Rev. D69 (2004) 105009; R. Bluhm and V.A. Kostelecky,
Phys. Rev. D71 (2005) 065008; O. Bertolami and J. Paramos, Phys. Rev. D72 (2005)
044001.
4. CPT and Lorentz Symmetry III, Alan Kostelecky, ed. (World Scientific, Singapore,
2005); O. Bertolami, Gen. Rel. Gravitation 34 (2002) 707; O. Bertolami, Lect. Notes
Phys. 633 (2003) 96, hep-ph/0301191; D. Mattingly, Liv. Rev. Rel. 8 (2005) 5, gr-
qc/0502097; R. Lehnert, "CPT- and Lorentz-symmetry breaking: a review", hep-
ph/0611177.
5. H. Sato, T. Tati, Prog. Theor. Phys. 47 (1972) 1788; S. Coleman and S.L. Glashow,
Phys. Lett. B405 (1997) 249; Phys. Rev. D59 (1999) 116008; O. Bertolami and
C. Carvalho, Phys. Rev. D61 (2000) 103002.
6. O. Bertolami, Class. Quantum Gravity 14 (1997) 2785.
7. M. Bucher and C. Carvalho, Phys. Rev. D71 (2005) 083511.
8. O. Bertolami and C. Carvalho, "Brane Lorentz Symmetry from Lorentz Breaking in
the Bulk", gr-qc/0612129.
9. O. Bertolami and C. Carvalho, in preparation.
QUANTUM GRAVITY AND SPACETIME SYMMETRIES
RALF LEHNERT
Center for Theoretical Physics
Massachusetts Institute of Technology
Cambridge, MA 02139
rlehnert@lns.mit.edu
Small violations of spacetime symmetries have recently been identified as promising
Planck-scale signals. This talk reviews how such violations can arise in various approaches
to quantum gravity, how the emergent low-energy effects can be described within the
framework of relativistic effective field theories, how suitable tests can be identified, and
what sensitivities can be expected in current and near-future experiments.
1. Introduction
One of the most intriguing open questions in current physics research concerns the
structure of spacetime at the Planck length Lp. While tremendous theoretical efforts
have been devoted to this subject, there is a major obstacle for experimental work:
the diminutive size of Lp. A propitious avenue to attack this problem is provided
by ultrahigh-precision tests of symmetries that hold exactly in present-day physics
but might be violated at a more fundamental level.
In this context, violations of Lorentz and CPT invariance have recently been
found to be promising signatures for Planck-length effects:1'2 These symmetries are
pillars of established physical laws, so that any violation of them would indicate
qualitatively novel physics. In addition, Lorentz and CPT tests are among the most
precise null experiments that can be preformed with present or near-future
technology. Many of these tests have Planck reach. We also mention that a number of
approaches to underlying physics can lead to small Lorentz and CPT breakdown,
as will be briefly discussed later in this talk.
Lorentz and CPT symmetry are closely intertwined in the CPT theorem, which
roughly states states that a local, unitary, relativistic point-particle quantum field
theory is CPT invariant. One may wonder whether CPT and Lorentz invariance
can be broken independently in such a field-theoretical context. The answer to this
question lies in Grccnberg's "anti CPT theorem:" under mild technical assumptions,
such as unitarity, CPT violation is always associated with Lorentz breakdown.3 We
remark that the opposite, namely Lorentz breaking implies CPT violation, is false.
An explicit example for these results is given by the Standard-Model Extension,
which is discussed in the next section.
2. Standard-Model Extension
For the identification and analysis of Lorentz and CPT tests, a theoretical framework
for Lorentz and CPT violation is needed. Over the last decade, such a framework,
called the Standard-Model Extension (SME), has been developed.4 This section
reviews the cornerstones of the SME.
To maintain relative independence of the (unknown) underlying physics, the
2615
2616
SME is constructed to be as general as possible while preserving physically desirable
features. We first use the fact that, on practical grounds, we need a model valid at
length scales much larger than Lp. It is then reasonable to assume that Lorentz- and
CPT-violating effects can be described by an effective field theory.a The second basic
idea is that all of presently established physics should be recovered for vanishing
Lorentz and CPT violation. The desired framework is thus a Lagrangian field theory
£sme, such that
•Csme = -Csm + -Ceh + SC , (1)
where £sm and £eh are the usual Standard-Model and Einstein-Hilbert La-
grangians, respectively. Lorentz- and CPT-breaking effects are contained in SC
For the construction of SC, a third ingredient is needed: coordinate
independence. This fundamental principle simply states that coordinate systems are
mathematical tools, and as such they should leave unaffected the actual physics. It follows
that SC must be a coordinate scalar. A sample term contained in 5C is ipjsfitp, where
ip is a fermion field in £sm and W a small external nondynamical 4-vector violating
both Lorentz and CPT symmetry. In the SME, 6M is a coefficient to be determined
by experiment. Such coefficients are assumed to be generated by underlying physics.
Some examples are given in the next section.
To date, numerous experimental Lorentz and CPT tests have been analyzed
within the SME.5 Studies of cosmic radiation have been a particularly popular
class of Lorentz tests.6 The idea is that the one-particle dispersion relations contain
additional Lorentz-breaking terms from SC The resulting modifications in particle-
reaction thresholds would become apparent or more pronounced at high energies,
and they might therefore be observed in cosmic rays. An example of such an effect
is vacuum Cerenkov radiation.7 If derived within the SME, these dispersion-relation
corrections are compatible with underlying dynamics. However, the purely kinemat-
ical approach of postulating modified dispersion relations has also been considered.8
3. Sample mechanisms for Lorentz breaking
The tensorial coefficients for Lorentz and CPT violation contained in the SME can
be generated in a variety of approaches to more fundamental physics. This section
lists sample theoretical ideas that have been developed in this context.
Spontaneous Lorentz and CPT breakdown in string theory. — From a theoretical
perspective, spontaneous symmetry violation (SSV) is an attractive mechanism for
Lorentz and CPT breaking. SSV is well established in condensed-matter physics,
and in the electroweak model it is associated with mass generation. The basic idea
is that a symmetric zero-field configuration is not the lowest-energy state. Nonzero
vacuum expectation values (VEVs) are energetically favored. In string field theory, it
has been demonstrated that SSV can trigger VEVs of vector and tensor fields, which
aEffective field theories have been tremendously successful in particle and condensed-matter
physics. The conventional Standard Model itself is usually viewed as an effective field theory,
so that an effective-field-theory description of leading-order Lorentz and CPT violation would
seem natural. Moreover, discrete backgrounds, as might be expected for quantum-gravity effects,
are known to be compatible with effective field theory, at least in solid-state physics.
2617
would then be identified with the Lorentz- and CPT-breaking SME coefficients.9
Nontrivial spacetime topology. - This approach considers the possibility that
one of the usual three spatial dimensions is compactified.10 On observational
grounds, the compactification radius would be very large. Note that the local
structure of flat Minkowski space is preserved. The finite size of the compactified
dimension leads to periodic boundary conditions, which implies a discrete momentum
spectrum and a Casimir-type vacuum. It is then intuitively reasonable that such a
vacuum possesses a preferred direction along the compactified dimension.
Cosmologically varying scalars. — A varying scalar, regardless of the mechanism
driving the spacetime dependence, typically implies the breakdown of translational
invariance.11 Since translations and Lorentz transformations are closely intertwined
in the Poincare group, it is unsurprising that the translation-symmetry violation
can also affect Lorentz invariance. Consider, for instance, a system with varying
coupling £(x) and two scalar fields <\> and $, such that the Lagrangian includes a
kinetic-type term ^(x)dfJ,(j>dfJ,^>. A suitable integration by parts generates the term
— (<9M£) 0<9M<I> while leaving unaffected the physics. It is apparent that the external
nondynamical gradient <9M£ can be identified with a coefficient of the SME.
Acknowledgments
This work is supported by the U.S. Department of Energy under cooperative
research agreement No. DE-FG02-05ER41360 and by the European Commission
under Grant No. MOIF-CT-2005-008687.
References
1. See, e.g., V.A. Kostelecky, ed., CPT and Lorentz Symmetry III, World Scientific,
Singapore, 2005.
2. See also G. Amelino-Camelia et al, AIP Conf. Proc. 758, 30 (2005) [arXiv:gr-
qc/0501053].
3. O.W. Greenberg, Phys. Rev. Lett. 89, 231602 (2002) [arXiv:hep-ph/0201258].
4. D. Colladay and V.A. Kostelecky, Phys. Rev. D 55, 6760 (1997) [arXiv:hep-
ph/9703464]; Phys. Rev. D 58, 116002 (1998) [arXiv:hep-ph/980952lj; V.A.
Kostelecky and R. Lehnert, Phys. Rev. D 63, 065008 (2001) [arXiv:hep-th/0012060]; V.A.
Kostelecky, Phys. Rev. D 69, 105009 (2004) [arXiv:hep-th/0312310].
5. See, e.g., D. Mattingly, Living Rev. Rel. 8, 5 (2005) [arXiv:gr-qc/0502097].
6. T. Jacobson, S. Liberati, and D. Mattingly, Phys. Rev. D 66, 081302 (2002)
[arXiv:hep-ph/0112207].
7. R. Lehnert and R. Potting, Phys. Rev. Lett. 93, 110402 (2004) [arXiv:hep-
ph/0406128]; Phys. Rev. D 70, 125010 (2004) [arXiv.hep-ph/0408285]; C. Kaufhold
and F.R. Klinkhamer, Nucl. Phys. B 734, 1 (2006) [arXiv:hep-th/0508074].
8. G. Amelino-Camelia, Int. J. Mod. Phys. D 11, 35 (2002) [arXiv:gr-qc/0012051]; R.
Lehnert, Phys. Rev. D 68, 085003 (2003) [arXiv:gr-qc/0304013].
9. V.A. Kostelecky and S. Samuel, Phys. Rev. D 39, 683 (1989); V.A. Kostelecky and
R. Potting, Nucl. Phys. B 359, 545 (1991).
10. F.R. Klinkhamer, Nucl. Phys. B 578, 277 (2000) [arXiv:hep-th/9912169].
11. V.A. Kostelecky, R. Lehnert, and M.J. Perry, Phys. Rev. D 68, 123511 (2003)
[arXiv:astro-ph/0212003]; R. Jackiw and S.-Y. Pi, Phys. Rev. D 68, 104012
(2003) [arXiv:gr-qc/0308071]; O. Bertolami et al., Phys. Rev. D 69, 083513 (2004)
[arXiv:astro-ph/0310344].
LORENTZ INVARIANCE VIOLATION IN HIGHER ORDER
ELECTRODYNAMICS
DENNIS LOREK
Institute for Theoretical Physics, University of Bremen, Otto-Hahn-Allee, 28359 Bremen,
Germany
and
ZARM, University of Bremen, Am Fallturm, 28359 Bremen, Germany
lorek@zarm.uni-bremen.de
CLAUS LAMMERZAHL
ZARM, University of Bremen, Am Fallturm, 28359 Bremen, Germany
laemmerzahl@zarm.uni-bremen.de
A generalization Standard Model Extension by allowing higher order derivatives in the
extended Maxwell equations imply a modification of the dispersion relation and, thus,
of the propagation of light, and also of the Coulomb potential which predicts shifts of
energy levels in hydrogen atoms. A comparison with experiments gives estimates on the
Lorentz violating terms.
1. Introduction
Though up to now no finally worked out theory of Quantum Gravity exists all
approaches like loop quantum gravity (LQG), string theory and non-commutative
geometry suggest small violations of Lorentz invariance.1_3 Owing to the lack of
specific predictions general phenomenological approaches has been worked out like the
Standard Model Extension4 and models with even allow charge non-conservation.5
In these cases a violation of Lorentz invariance came in through a modification of
the constitutive law. However, as suggested by the low energy limit of, e.g., LQG,
the effective Maxwell equations6 contain beside an arbitrary constitutive tensor also
higher order derivatives. This is what we are considering within a general framework.
2. Higher order Maxwell equations
The standard dynamics of the electromagnetic field is described by the homogenous
and inhomogeneous Maxwell equations
4-7T
d0Fa^ = O, dpF"? =-—f. (1)
where Fap is the field-strength tensor. From these equations it follows that light rays
propagate along null-geodesies, a point charge gives an electric Coulomb potential
and a point-like magnetic moment gives a magnetic dipole field.
Modifications of these ordinary Maxwell equations have been formulated within
the SME where a violation of the Lorentz invariance is encoded in the constitutive
tensor ka^1& which is assumed to be constant and possesses the symmetries kaf3lS =
k{af3]~tS = kaf3[~fS] and ka{p-fs] = q The resuiting extension of Maxwell equations,
4-7T
d0Fa(3 = 0, d0FaP + dp (ka0^Fl5) = ~—ja (2)
2618
2619
in terms of a general constitutive law has already been discussed extensively.8'9
Another approach5 relaxes the strong symmetry conditions on the constitutive tensor
and allows charge non-conservation.
Here we generalize these scheme even further by allowing higher order
derivatives, a feature which has been suggested by the low energy limit of LQG.6 The
most general modification of the inhomogeneous Maxwell equations, which is still
linear in F, is given by
dppaP + dp (JZ ^]bSWd«) F^ = ~T]n ' (3)
where m is given by the product of i indices and the sum is of arbitrary order
N. With i equals to zero, we have the ordinary SME model, with i equals to one,
we get one more index corresponding to first derivatives, with i equals to two, we
include second derivatives, and so forth. Furthermore, A is totally symmetric in all
indices nf, in this case, charge conservation is secured and the Maxwell equations
can be derived from a Lagrangian. Higher order Maxwell equations have also been
considered in connection to questions of reality and causality of the theory.7
N
With the definition Ga$ = Fa$ + J2 X[^]h&]nid^FlS the Maxwell equations
acquire their ordinary form
A-jr
dpFaP = 0, df3Ga/3 = ja . (4)
There is an analogy between the Lorentz violating electrodynamics in vacuum and
the conventional situation in homogeneous anisotropic media. It is possible to
introduce D- and if-fields through
Do = G0j ; Hi = leokiGkl . (5)
Obviously, the effective Maxwell equations retain their ordinary structure, where
higher derivatives are included in D and H. Since these Maxwell equations are
more general than previous phenomenological models, we now have to look anew
for ways how to confront the equations with experiments.
3. Observational implications
A first route is to determine the wave equation for the electromagnetic field and to
calculate the dispersion relation which has the structure
w = [l+p{k)±a{k)]k, (6)
where p, a ~ Yl Pi(k)\k\\ Yla'i{^)\^Y are sums of terms depending on the direction
of propagation k and on powers of the modulus |A;|. Therefore, we obtain not only
birefringence and an anisotropic propagation of light but also a higher order
dispersion. Today's most precise birefringence estimate8 give^ an upper limit of 2 • 10~32.
Adequate isotropy experiments are Michelson-Morley interferometric experiments.
2620
The non-homogeneity of the dispersion can be confronted with astrophysical time-
of-flight bounds10 and may be tested in gravitational wave interferometers.11
A second route is to derive the modified Coulomb potential for a point charge
/i \ N
*=£(;+**) -<* ^z^r^ (7)
^ ' i=0
5-terms were neglected. As one can read off, a point-like charge creates additional
electric multipole fields. In particular, with i = 1 and i = 2 we obtain additive
dipole- and quadrupole-fields, where the dipole and quadrupole are given by pl ~
q ■ Xl(1) and Qlm ~ q ■ Ate, respectively. This results in modifications of the energy-
levels in hydrogen atoms. The straightforward calculated perturbation operator for
an additive dipole causes a decrease of both the hyperfine splitting and the Lamb
shift. Since there are no observed anomalies with an accuracy of12 10"15, the Lorentz
invariance violating coefficient A(i) has to be smaller than 10~18m.
Similar calculations can be carried through for the quadrupole. However, we
may also use results from Particle Physics. Since the discovery of non-conservation
of parity in the weak interactions, it has become of interest to investigate the
possible existence of an electric dipole moment of the elementary particles. From the
most precise measurement13 we obtain |Am| < 1.6 • 10~29m. Moreover, from
spatial isotropy tests14 we can derive that the A(2)-coefficient has to be smaller than
10-30m2.
These results were received using the modified Coulomb potential, which has
not been considered in phenomenology as yet. Furthermore our model includes the
feature that a point charge creates a magnetic field, which yields a Zeeman splitting
in the hydrogen atom, and a magnetic moment creates an electric field which yields
yet another effect.
References
1. D.Giulini, C. Kiefer, C. Lammerzahl (eds.), Quantum Gravity, Springer, Berlin
Heidelberg (2003).
2. G. Arnelino-Camelia, J. Kowalski-Glikman (eds.), Planck Scale Effects in Astrophysics
and Cosmology, Springer, Berlin Heidelberg (2005).
3. C. Lammerzahl, Appl. Phys. B 634, 551 (2006); ibid B 634, 563 (2006).
4. D. Colladay, V. A. Kostelecky, Phys. Rev. D 58, 116002 (1998).
5. C. Lammerzahl, A. Macias and H. Miiller, Phys. Rev. D 71, 025007 (2005).
6. J.Alfaro, H. A. Morales-Tecotl, L. F. Urrutia, Phys. Rev. D 65, 103509 (2002).
7. S.A. Martinez, R. Montemajor, and L. Urrutia, Phys. Rev. D 74, 065020 (2006).
8. V.Alan Kostelecky, M. Mewes, Phys. Rev. Lett. 87, 251304 (2001).
9. V.Alan Kostelecky, M. Mewes, Phys. Rev. D 66, 056005 (2002).
10. G. Amelino-Camelia et al. Nature 393, 763 (1998).
11. G. Amelino-Camelia, C. Lammerzahl, Class. Quantum Grav. 21, 899 (2004).
12. T.Udem, R. Holzwarth, T. W. Hansen, Physik Journal 1, 39 (2002).
13. B.C.Regan et al., Phys. Rev. Lett. 88, 071805 (2002).
14. S. K. Lamoreaux et al, Phys. Rev. A 39, 1082 (1988).
HUBBLE MEETS PLANCK: A COSMIC PEEK
AT QUANTUM FOAM
Y. JACK NG
Institute of Field Physics, Department of Physics and Astronomy,
University of North Carolina, Chapel Hill, NC 27599-3255, USA
yjng@physics.unc. edu
If spacetime undergoes quantum fluctuations, an electromagnetic wavefront will acquire
uncertainties in direction as well as phase as it propagates through spacetime. These
uncertainties can show up in interferometric observations of distant quasars as a decreased
fringe visibility. The Very Large Telescope and Keck interferometers may be on the verge
of probing spacetime fluctuations which, we also argue, have repercussions for
cosmology, requiring the existence of dark energy/matter, the critical cosmic energy density,
and accelerating cosmic expansion in the present era.
Keywords: detection of quantum foam, holography, critical energy density, dark
energy/matter
1. Quantum Fluctuations of Spacetime
Conceivably spacetime, like everything else, is subject to quantum fluctuations. As
a result, spacetime is "foamy" at small scales,1 giving rise to a microscopic structure
of spacetime known as quantum foam, also known as spacetime foam, and entailing
an intrinsic limitation SI to the accuracy with which one can measure a distance I. In
principle, 51 can depend on both I and the Planck length lp = ytnG/c3, the intrinsic
scale in quantum gravity, and hence can be written as 51 > ?1_Q?p, with a ~ 1
parametrizing the various spacetime foam models. (For related effects of quantum
fluctuations of spacetime geometry, see Ref. 2.) In what follows, we will advocate the
so-called holographic model corresponding to a = 2/3, but we will also consider the
(random walk) model with a = 1/2 for comparison. The holographic model has been
derived by various arguments, including the Wigner-Saleckar gedankan experiment
to measure a distance3 and the holographic principle.4'5 (See my contribution to the
Proceedings of MGIO.6) Here in the two subsections to follow, we use instead (1) an
approach based on quantum computation, and (2) an argument over the maximum
number of particles that can be put inside a region of space respectively.
1.1. Quantum Computation
This method7'8 hinges on the fact that quantum fluctuations of spacetime manifest
themselves in the form of uncertainties in the geometry of spacetime. Hence the
structure of spacetime foam can be inferred from the accuracy with which we can
measure that geometry. Let us consider a spherical volume of radius I over the
amount of time T = 2l/c it takes light to cross the volume. One way to map out
the geometry of this spacetime region is to fill the space with clocks, exchanging
signals with other clocks and measuring the signals' times of arrival. This process of
mapping the geometry is a sort of computation; hence the total number of operations
(the ticking of the clocks and the measurement of signals etc) is bounded by the
2621
2622
Margolus-Lcvitin theorem9 in quantum computation, which stipulates that the rate
of operations for any computer cannot exceed the amount of energy E that is
available for computation divided by irh/2. A total mass M of clocks then yields, via
the Margolus-Levitin theorem, the bound on the total number of operations given by
(2Mc2/nh) x 211 c. But to prevent black hole formation, M must be less than lc2/2G.
Together, these two limits imply that the total number of operations that can occur
in a spatial volume of radius I for a time period 21/c is no greater than ~ (l/lp)2.
(Here and henceforth we neglect multiplicative constants of order unity, and set
c = 1 = h.) To maximize spatial resolution, each clock must tick only once during
the entire time period. And if we regard the operations partitioning the spacetime
volume into "cells", then on the average each cell occupies a spatial volume no less
than ~ I3/(I2/tp) = ll2?, yielding an average separation between neighboring cells
no less than l1//3lp . This spatial separation is interpreted as the average minimum
uncertainty in the measurement of a distance I, that is, SI > ll'3lp .
Parenthetically we can now understand why this quantum foam model has come
to be known as the holographic model. Since, on the average, each cell occupies a
spatial volume of Up, a spatial region of size I can contain no more than I3 /(Up) =
(l/lp)2 cells. Thus this model corresponds to the case of maximum number of bits
of information I2 jlp in a spatial region of size I, that is allowed by the holographic
principle,10 acording to which, the maximum amount of information stored in a
region of space scales as the area of its two-dimensional surface, like a hologram.
It will prove to be useful to compare the holographic model in the mapping of
the geometry of spacetime with the one that corresponds to spreading the spacetime
cells uniformly in both space and time. For the latter case, each cell has the size
of (Pip)1/4 = lYl2lp both spatially and temporally, i.e., each clock ticks once in
the time it takes to communicate with a neighboring clock. Since the dependence
on I1/2 is the hallmark of a random-walk fluctuation, this quantum foam model
corresponding to 51 > (Up)1//2 is called the random-walk model.11 Compared to
the holographic model, the random-walk model predicts a coarser spatial
resolution, i.e., a larger distance fluctuation, in the mapping of spacetime geometry. It
also yields a smaller bound on the information content in a spatial region, viz.,
(l/lP)2/(l/lp)1/2 = (l2/l2p)3/4 = (l/lP)3/2.
1.2. Maximum Number of Particles in a Region of Space
This method involves an estimate of the maximum number of particles that can be
put inside a spherical region of radius I. Since matter can embody the maximum
information when it is converted to energetic and effectively massless particles, let
ns consider massless particles. According to Heisenberg's uncertainty principle, the
minimum energy of each particle is no less than ~ l—1. To prevent the region from
collapsing into a black hole, the total energy is bounded by ~ l/G. Thus the total
number of particles must be less than (l/lp)2, and hence the average interparticle
distance is no less than ~ ll^3lp . Now, the more particles there are (i.e., the
2623
shorter the interparticle distance), the more information can be contained in the
region, and accordingly the more accurate the geometry of the region can be mapped
out. Therefore the spatial separation we have just found can be interpreted as the
average minimum uncertainty in the measurement of a distance /; i.e., 51 > ll'3lp .
Two remarks are in order. First, this minimum 51 just found corresponds to the
case of maximum energy density p ~ (llP)~2 for the region not to collapse into a
black hole, i.e., the holographic model, in contrast to the random-walk model and
other models, requires, for its consistency, the critical energy density which, in the
cosmological setting, is (H/lp)2 with H being the Hubble parameter. Secondly, the
numercial factor in 51, according to the four different methods alluded to above, can
be shown to be between 1 and 2, i.e., SI > l1/3^ to 2ll/3lP/3.
2. Probing Quantum Foam with Extragalactic Sources
The Planck length lp ~ 10-33 cm is so short that we need an astronomical (even
cosmological) distance / for its fluctuation SI to be detectable. Let us consider light
(with wavelength A) from distant quasars or bright active galactic nuclei.12,13 Due to
the quantum fluctuations of spacetime, the wavefront, while planar, is itself "foamy",
having random fluctuations in phase13 A0 ~ 2tt51/ X as well as the direction of the
wave vector14 given by A0/27T. a In effect, spacetime foam creates a "seeing disk"
whose angular diameter is ~ A0/27T. For an interferometer with baseline length D,
this means that dispersion will be seen as a spread in the angular size of a distant
point source, causing a reduction in the fringe visibility when A(/)/2ir ~ X/D. For
a quasar of 1 Gpc away, at infrared wavelength, the holographic model predicts a
phase fluctuation A(j> ~ 2tt x 10~9 radians. On the other hand, an infrared
interferometer (like the Very Large Telescope Interferometer) with D ~ 100 meters has
X/D ~5x 10~9. Thus, in principle, this method will allow the use of interferome-
try fringe patterns to test the holographic model! Furthermore, these tests can be
carried out without guaranteed time using archived high resolution, deep imaging
data on quasars, and possibly, supernovae from existing and upcoming telescopes.
The key issue here is the sensitivity of the interferometer. The lack of observed
fringes may simply be due to the lack of sufficient flux (or even just effects originated
from the turbulence of the Earth's atmosphere) rather than the possibility that
the instrument has resolved a spacetime foam generated halo. But, given sufficient
sensitivity, the VLTI, for example, with its maximum baseline, presumably has
sufficient resolution to detect spacetime foam halos for low redshift quasars, and in
principle, it can be even more effective for the higher redshift quasars. Note that
the test is simply a question of the detection or non-detection of fringes. It is not a
question of mapping the structure of the predicted halo.
aUsing k = 2-7T/A, one finds that, over one wavelength, the wave vector fluctuates by Sk =
2-7r<5A/A2 = kSX/X. Due to space isotropy of quantum fluctuations, the transverse and
longitudinal components of the wave vector fluctuate by comparable amounts. Thus, over distance I, the
direction of the wave vector fluctuates by Akx/k = £<5A/A ~ 81/X.
2624
3. From Quantum Foam to Cosmology
In the meantime, we can use existing archived data on quasars or active galactic
nuclei from the Hubble Space Telescope to test the quantum foam models.14
Consider the case of PKS1413+135,15 an AGN for which the redshift is z = 0.2467.
With / w 1.2 Gpc and A = 1.6/im, we13 find A0 - 10 x 2tt and 10"9 x 2tt for
the random-walk model and the holographic model of spacetime foam respectively.
With fl = 2.4m for HST, we expect to detect halos if A0 ~ 10"6 x 2tt. Thus, the
HST image only fails to test the holographic model by 3 orders of magnitude.
However, the absence of a quantum foam induced halo structure in the HST
image of PKS1413+135 rules out convincingly the random-walk model. (In fact,
the scaling relation discussed above indicates that all spacetime foam models with
a < 0.6 are ruled out by this HST observation.) This result has profound
implications for cosmology.7,14'16 To wit, from the (observed) cosmic critical density in the
present era, a prediction of the holographic-foam-inspired cosmology, we deduce that
p ~ Hq/G ~ (RhIp)2, where Ho and Rh are the present Hubble parameter and
Hubble radius of the observable universe respectively. Treating the whole universe
as a computer,7,17 one can apply the Margolus-Levitin theorem to conclude that the
universe computes at a rate u up to pRH ~ RhIJ>2 for a total of (Rn/lp)2
operations during its lifetime so far. If all the information of this huge computer is stored
in ordinary matter, then we can apply standard methods of statistical mechanics to
find that the total number I of bits is {R2H/l2P)3/i = {Rh/Ip)3/2- It follows that each
bit flips once in the amount of time given by I/u ~ (RhIp)1'2- On the other hand,
the average separation of neighboring bits is (R3H /1)1'3 ~ (RhIp)1/2. Hence, the
time to communicate with neighboring bits is equal to the time for each bit to flip
once. It follows that the accuracy to which ordinary matter maps out the geometry
of spacetime corresponds exactly to the case of events spread out uniformly in space
and time discussed above for the case of the random-walk model of quantum foam.
Succinctly, ordinary matter only contains an amount of information dense enough
to map out spacetime at a level consistent with the random-walk model. Observa-
tionally ruling out the random-walk model suggests that there must be other kinds
of matter/energy with which the universe can map out its spacetime geometry to a
finer spatial accuracy than is possible with the use of ordinary matter. This line of
reasoning then strongly hints at the existence of dark energy/matter independent of
the evidence from recent cosmological (supernovae, cosmic mircowave background,
gravitational lensing, galaxy configuration and clusters) observations.
Moreover, the fact that our universe is observed to be at or very close to its
critical energy density p ~ (H/lp)2 ~ (Rh^p)~2 must be taken as solid albeit
indirect evidence in favor of the holographic model because, as aforementioned,
this model is the only model that requires the energy density to be critical. The
holographic model also predicts a huge number of degrees of freedom for the universe
in the present era, with the cosmic entropy given by16 / ~ HRH/lp ~ (Rn/lp)2-
Hence the average energy carried by each bit is pR3H /I ~ RJj1. Such long-wavelength
2625
bits or "particles" carry negligible kinetic energy. Since pressure (energy density)
is given by kinetic energy minus (plus) potential energy, a negligible kinetic energy
means that the pressure of the unconventional energy is roughly equal to minus its
energy density, leading to accelerating cosmic expansion as has been observed. This
scenario is very similar to that for quintessence.
How about the early universe? Here a cautionary remark is in order. Recall
that the holographic model has been derived for a static and flat spacctime. Its
application to the universe of the present era may be valid, but to extend the
discussion to the early universe may need a judicious generalization of some of the
concepts involved. However, there is cause for optimism: for example, one of the
main features of the holograpahic model, viz. the critical energy density, is actually
the hallmark of the inflationary universe paradigm. Further study is warranted.
Acknowledgments
This work was supported in part by the US Department of Energy and the Bahnson
Fund of the University of North Carolina.
References
1. J.A. Wheeler, in Relativity, Groups and Topology, eds. B.S. DeWitt and CM. DeWitt
(Gordon & Breach, New York, 1963), p. 315. Also see S.W. Hawking et al., Nucl.
Phys. 170, 283 (1980); A. Ashtekar et al., Phys. Rev. Lett. 69, 237 (1992); J. Ellis et
al., Phys. Lett. B 293, 37 (1992).
2. L. H. Ford, Phys. Rev. D51, 1692 (1995); B. L. Hu and E. Vergaguer, Living Rev.
Rel. 7, 3 (2004).
3. H. Salecker and E.P. Wigner, Phys. Rev. 109, 571 (1958); Y.J. Ng and H. van Dam,
Mod. Phys. Lett. A9, 335 (1994); A10, 2801 (1995). Also see F. Karolyhazy, Nuovo
Cimento A42, 390 (1966).
4. Y. J. Ng, Phys. Rev. Lett. 86, 2946 (2001), and (erratum) 88, 139902-1 (2002).
5. Y. J. Ng, Int. J. Mod. Phys. Dll, 1585 (2002).
6. Y. J. Ng, in Proc. of the Tenth Marcel Grossman Meeting on General Relativity, eds.
M. Novello et al. (World Scientific, Singapore, 2005), p. 2150.
7. S. Lloyd and Y.J. Ng, Sci. Am. 291, # 5, 52 (2004).
8. V. Giovannetti, S. Lloyd and L. Maccone, Science 306, 1330 (2004).
9. N. Margolus and L. B. Levitin, Physica D120, 188 (1998).
10. G. 't Hooft, in Salamfestschrift, eds. A. Ali et al. (World Scientific, Singapore, 1993),
p. 284; L. Susskind, J. Math. Phys. (N.Y.) 36, 6377 (1995). Also see J.D. Bekenstein,
Phys. Rev. D7, 2333 (1973); S. Hawking, Comm. Math. Phys. 43, 199 (1975).
11. G. Amelino-Camelia, Mod. Phys. Lett. A9, 3415 (1994); Nature 398, 216 (1999).
12. R. Lieu and L. W. Hillman, Astrophys. J. 585, L77 (2003); R. Ragazzoni, M. Turatto,
and W. Gaessler, Astrophys. J. 587, LI (2003).
13. Y. J. Ng, W. Christiansen, and H. van Dam, Astrophys. J. 591, L87 (2003).
14. W. Christiansen, Y. J. Ng, and H. van Dam, Phys. Rev. Lett. 96, 051301 (2006).
15. E. S. Perlman, et al., 2002, Astro. J. 124, 2401 (2002).
16. M. Arzano, T. W. Kephart, and Y. J. Ng, arXiv:gr-qc/0605117.
17. S. Lloyd, Phys. Rev. Lett. 88, 237901-1 (2002).
EVOLUTIONARY REFORMULATION OF QUANTUM GRAVITY
GIOVANNI MONTANPt
* ICRA—International Center for Relativistic Astrophysics
Dipartimento di Fisica (G9), Universita di Roma, "La Sapienza",
Piazzale Aldo Mora 5, 00185 Rome, Italy
^ENEA-C.R. Frascati (U.T.S. Fusione),
via Enrico Fermi 45, 00044 Frascati, Rome, Italy
montani@icra.it
We present a critical analysis of the Canonical approach to quantum gravity, which
relies on the ambiguity of implementing a space-time slicing on the quantum level. We
emphasize that such a splitting procedure is consistent only if a real matter fluid is
involved in the dynamics.
Keywords: Quantum gravity. Schrodinger dynamics.
General Relativity is a background independent theory which identify the
gravitational interaction into the metric properties of the space-time and this peculiar
nature makes very subtle even simple questions about its quantization. To deal with
a canonical method for the fields dynamics necessarily involves the notion of a
physical time variable, whose conjugate momentum fixes the Hamiltonian function.
Already in the context of classical General Relativity, the task of recovering a physical
clock acquires non-trivial character, depending on the local properties of the space-
time. However, a well grounded algorithm devoted to this end was settled down by
Arnowitt-Deser-Misner (ADM) in1 It consists of a space-time slicing based on a one
parameter family of spacelike hypersurfaces T?t, defined via the parametric
representation t'J' = i^(i, xl) (/i = 0,1,2,3 and i = 1,2,3). In what follows, we denote
the set of coordinates {t, x1} by xfi, in order to emphasize that the slicing procedure
can be recast as a 4-diffeomorphism, i.e. ds2 = gllv(tp)dtlldiv = gfli?(xp)dx11dxv'. The
main issue of adopting the coordinates xfi is that they allow to separate the 4-metric
tensor into six evolutionary components, which determine the induced 3-metric
tensor hij of the hypersurfaces and four Lagrangian multipliers, corresponding to the
lapse function N and to the shift vector Nl. These non-evolutionary variables have
a precise geometrical meaning, given by the relation dtt,L = Nn^1 + NlditfJ' (where
glwn,1n1' = 1 and gliun,ldit1' = 0), n^{tp) denoting the orthonormal vector to the
family Ef. The classical dynamics of the 3-metric h^ is governed, in vacuum, by
the following set of equations
kH
GlwnV = --7= = 0 (1)
2v^
kH
G^dit" = —7= = 0 (2)
2V"
G^di^djt" = GZJ = 0 , (3)
2626
2627
h being the 3-metric determinant and GILV the Einstein tensor (H and Hi are called
the super-Hamiltonian and the super-momentum respectively). The first two lines
above correspond to constraints for the initial values problem and they play a crucial
role in the canonical quantization of the system, while the last line fixes the evolution
of the 3-metric and it is lost on the quantum level. As a consequence, the canonical
quantum dynamics of the gravitational field is characterized by the so-called frozen
formalism.2ln fact the dynamics of a generic state | g^) =| hij, N, Nl) is provided
by the requests
pN | h^, N, N*) = 0, pN, | hij, N, N*) = 0 (4)
H | h^, N, Nl) = 0, Hi | hij, N, N*)=0, (5)
Pn and pNi being the momenta operators associated to N and N' respectively. The
four operator constraints listed above are the quantum translation of the diffeomor-
phisms invariance of the theory and they can be summarized by the Wheeler-DcWitt
equation3 H | {hij}) = 0, where by {hij} we denote a class of 3-geomctries. The
frozen formalism consists of the independence that the states acquires from N and
Nl, i.e. the quantum picture is the same on each spacelike hypersurfaces.
This non evolutionary character of the Wheeler-DeWitt approach is striking in
contrast with the Einstein equations which predict a 3-metric field evolving over the
slicing. In what follows, we argue that this absence of a proper time in canonical
quantum gravity is connected to the inconsistency of the 3+1-splitthig referred to a
quantum (vacuum) space-time. As issue of this criticism, we outline a time-matter
dualism and provide an evolutionary re-formulation of the canonical paradigm for
the gravitational field quantization.
Let us assume to have solved the quantum gravity problem in the framework
of generic coordinates t^, having determined a complete set of orthonormal states
I guv)a on which a given configuration | g^) can be decomposed as | giW) =
Ylaca I 9tiv)a- Now, assigned a 4-vector nM, its norm n = g^/n^n" (and therefore
its timelike character too) can be established only in the sense of expectation values
on the state | g^), having the form (n) = ^2aca(n)a = s(f). This field s is
clearly a random scalar one, whose dynamics is induced by the quantum behavior
of the 4-metric g)iv. By the diffeomorphism invariance, we deal with a scalar field
s(tp{x'')) = s(t,x*) on the slicing picture too. Analogous considerations hold for the
quantity rn = gfll/n'J'ditv (which states the timelike nature of nM in the coordinates
a;'4) and leads to conclude that its expectation value (n,) = Yla ca{ni)a = Sj(i'')
define in turn a random vector si(tp(xi1)) = Si{t,xl) living on the 3-hypersurfaces
Ej. The outcoming of these four degrees of freedom {s, s^} indicates that, for
a quantum space-time, the slicing picture preserves the number of evolutionary
variables, because we pass from </M„ in the system t'1 to {hij, s, s,-} in the splitting
coordinates x^. In this respect N and Nt simply give the components of the vector
n'J' in the 3+1-scheme. Now, the evolutionary behavior of ten variables (right the
number of 4-metric components) implies that the super-Hamiltonian and the super-
2628
momentum constraints (1) are violated in the sense of expectation values, so that
(kH/2\/h) = e and (kHi/2\<rh) = q^. Here e and qt denote a 3-scalar and a 3-
vector field respectively. Their presence comes out because of the equation Gij = 0,
which classically ensure the existence of constraints, are lost on a quantum level.
By other words, if we quantize the gravitational field before the slicing procedure
is performed, then the quantum translation of the 3+1-picture can no longer be
recovered and the frozen formalism is overcome.
The physical issue of the analysis above, leads to a time-matter dualism within
the context of an evolutionary quantum gravity. In fact, the following two statements
take place on the quantum and classical level respectively.
i) The non-vanishing behavior of the super-Hamiltonian and the super-momentum
expectation values implies that the corresponding operators do not annihilate the
states of the theory (like in the Wheeler-DeWitt approach) and therefore we have
to deal with a schrodinger quantum dynamics of the gravitational field. More
precisely, in the coordinates x^ the state acquires a dependence on the label time,
i.e. it reads | t, hij) and it obeys the Schrodinger equation
ihdt | t, h^) =n\t, hij) = J d3x {NH + NlHt\ \ t, h(j); (6)
this equation provides the time evolution of the 3-metric states along the slicing and,
once fixed the proper operator ordering to deal with an Hermitian Hamiltonian, then
a standard procedure defines the Hilbert space.
ii) The classical WKB limit for h —-> 0 maps the Schrodinger dynamics above into
the relaxed Hamiltonian constraints, which contain e and qi.4 By using the relations
(1), the classical limit is recognized to be General Relativity in presence of an Eckart
fluid,5 i. e.
G^v = k (-en^nu + ntlqv + q^ii^), q^ = qih^djt^ ; (7)
above, /jJJ denotes the inverse 3-inetric and the 4-vector q^ has the physical meaning
of heat conductivity. Here n11 plays the role of 4-velocity, according to the request
of a physical slicing which preserves the light cone on a quantum level too.
It is relevant to stress that the energy density of the Eckart fluid is positive in
correspondence to the negative part of the super-Hamiltonian spectrum. Therefore,
showing that such a region is predicted by the quantum dynamics acquires here a
key role.
Having this idea in mind, we adopt more convenient variables to express the
3-metric tensor, i.e.
hij = T]i/3Uij , (8)
with rj = h1'4 and detUij = 1.
Expressed via these variables, the super-Hamiltonian reads
2629
3 2c2k 1
H = ~-^c2kp\ + -^ruiku,ipl3pkl - ^V2V{uij, V77, Vuy), (9)
where pv and pli denote the conjugate momenta to 77 and uy respectively, while
the potential term V comes from the 3-Ricci scalar and V refers to first and second
order spatial gradients.
In this picture, the eigenvalue of the super-Hamiltonian operator takes the
explicit form
Axe = {idbw ~ ^ivAu~ iv2y{u^ Vr/; Vt%)}X£ = £x£ m
A S S .
l\u=UikUjl- 7 • (11)
duij duki
From a qualitative point of view, the existence of solutions for the system (10)
with negative values of £ can be inferred from its Klein-Gordon-like structure.
However, a more quantitative analysis is allowed by taking the limit 77 —+ 0, where
the system (10) admits an asymptotic solution. In fact, in this limit, the potential
term is drastically suppressed with respect to the A„ one and the dynamics of
different spatial points decouples, so reducing the quantization scheme to the local
minisuperspace approach. It is easy to see that such approximate dynamics admits,
point by point in space, the solution
X£ = L£(v,p)Gp2(uij), (12)
i and Gp2 satisfying the two equations respectively
[ 1 S2 | 32p2 \ __£l_ (13)
[ hck2 5rj2 hck2rj2 J
AUGP2 = -p2Gp2 . (14)
As far as we take 1 = y/rj9{r]) and we consider the negative part of the spectrum
£ = — I £ |, the function 0 obey the equation
1 ™ ■ l 86 ^'H^V = 0 (15)
hck2 5rj2 hck2rj Srj \ hck2if
q2 = ± (1 - 128p2) £' = hck2£. (16)
Thus, we see that a negative part of the spectrum exists in correspondence to
the solution
0(V, £', p) = A/^vT^W + BJ^i^inv), (17)
2630
where J±q denote the corresponding Bessel functions, while A and B are two
integration constants.
The above analysis states that the Eckart energy density always has a (quantum)
range of positive value, (associated to the negative portion of the super-Haniiltonian
spectrum) near enough to the "singular" point rj = 0.
However, the correspondence between e and £/2rj2 can occur only after the
classical limit of the spectrum is taken.
We conclude this analysis, observing that to give a precise physical meaning to
this picture, the following three points (elsewhere faced) have to be addressed.
i) The existence of a stable ground level of negative energy has to be inferred or
provided by additional conditions, ii) The spatial gradients of the dynamical
variables and therefore the associated super-momentum constraints, are to be included
into the problem and treated in a consistent way. iii) The physical nature of the
limit 77 —-> 0 has to be clarified within a cosmological framework.6
We conclude by observing that reliable investigations6'7 provide negative
components of the super-Hamiltonian spectrum which are associated to the constraint
he
T4
£\<w (18)
"pi
where lPi = \/^r denotes the Planck length.
As shown in,7 this range of variation for the super-Hamiltonian eigenvalue
implies, when an inflationary scenario is addressed, a negligible contribution to the
actual Universe critical parameter. In fact, estimating the critical parameter
associated to the new matter term, say fig, we get
O£<o(^|~O(l0-60), (19)
i?o ~ O (l028cm) being the present radius of curvature of the Universe.
Therefore, by above, we see that the predictions of an evolutionary quantum
cosmology phenomcnologically overlap those ones of the Wheeler-DeWitt approach.
References
1. R. Arnowitt, S. Deser and C, W. Misner, in Gravitation: an introduction to current
research, (1962), eds I. Witten and J. Wiley, New York.
2. K. Kuchar, in Quantum Gravity II, a second Oxford symposium, (1981), eds C. J.
Isham et al., Clarendom Press., Oxford,
3. B. S. DeWitt, Phys. Rev. , (1967), 160, 1113.
4. G. Montani, Nucl. Phys. B, (2002), 634, 370.
5. C. Eckart, (1940), Phys. Rew., 58, 919.
6. M. V. Battisti and G. Montani, Phys. Lett. B, (2006), 637, 203.
7. G. Montani, Int. Journ. Mod. Phys. D, (2003), 12, n. 8, 1445.
KERR'S GRAVITY AS A QUANTUM GRAVITY ON THE
COMPTON LEVEL*
ALEXANDER BURINSKII
Gravity Research Group, NSI Russian Academy of Sciences,
B. Talskaya 52, Moscow 115191, Russia
bur@ibrae.ac.ru
The Dirac theory of electron and QED neglect gravitational field, while the corresponding
to electron Kerr-Newman gravitational field has very strong influence on the Compton
distances. It polarizes space-time, deforms the Coulomb field and changes topology. We
argue that the Kerr geometry may be hidden beyond the Quantum Theory, representing
a complimentary space-time description.
1. Introduction
The Kerr-Newman solution displays many relationships to the quantum world. It
is the anomalous gyromagnetic ratio g = 2, stringy structures and other features
allowing one to construct a semiclassical model of the extended electron1-4 which
has the Compton size and possesses the wave properties. Meanwhile, the quantum
theory neglects the gravitation at all. The attempts to take into account gravity are
undertaken by superstriiig theory which is based on the space-time description of
the extended stringy elementary states: Points —> Extended Strings, and
also, on the unification of the Quantum Theory with Gravity on Planckian level of
masses Mpi, which correspond to the distances of order 10-33 cm.
Note, that spin of quantum particles is very high with respect to the masses. In
particular, for electron S = 1/2, while m « 10~22 (in the units G = h = c = 1).
So, to estimate gravitational field of spinning particle, one has to use the Kerr,
or Kerr-Newman solutions,5 contrary to the ordinary estimates based on spherical
symmetric solutions.
Performing such estimation, we obtain a striking contradiction with the above
scale of Quantum Gravity !
Indeed, for the Kerr and Kerr-Newman solutions we have the basic relation
between angular momentum J, mass m and radius of the Kerr singular ring a :
J = ma. Therefore, Kerr's gravitational field of a spinning particle is extended
together with the Kerr singular ring up to the distances a = J/m = h/2m ~ 1022
which are of the order of the Compton length of electron 10-11 cm., forming a
singular closed stringa. Since a >> m, this string is naked (no event horizon of
black hole). In the Kerr geometry, in analogy with string theory the 'point-like'
Schwarzschild singularity turns into an extended string of the Compton size.
Note, that the Kerr string is not only analogy. It was shown that the Kerr
singular ring is indeed the string,8 and, in the analog of the Kerr solution to low
"Talk at the QG1 session of the MG11 meeting, partially supported by RFBR grant 07-08-00234.
aSee also.1-6-8
2631
2632
energy string theory,9 the field around the Kerr string is similar to the field around
a heterotic string.10 It is an Alice topological string,2'4 and the Kerr space exhibits
a change of topology on the Compton distances. Therefore, the Kerr geometry
indicates essential peculiarities of space-time on the Compton distances, and the use
of Kerr geometry for estimation of the scale of Quantum Gravity gives the striking
discrepancy with respect to the ordinary estimations based on the Schwarzschild
geometry.
There appears the Question: "Why Quantum Theory does not feel such
drastic changes in the structure of space time on the Compton distances?" How can
such drastic changes in the structure of space-time and electromagnetic field be
experimentally unobservable and theoretically ignorable in QED?
There is, apparently, unique explanation to this contradiction. We have to
assume that the Kerr geometry is already taken into account in quantum theory and
play there an important role. In another words, the Kerr geometry is a
complimentary (dual) space-time description of quantum processes.
Fig. 1. Skeleton of the Kerr spinning particle in the rest frame: the Kerr singular ring and two
semi-infinite singular half-strings which are determined by two null-vectors of polarization of a free
electron.
Indeed, the local gravitational field at these distances is extremely small, for
exclusion of an extremely narrow vicinity of the Kerr singular ring forming a closed
string of the Compton radius. This closed Kerr string is presumably the source of
quantum effects.
Such point of view coincides with the old conjecture on the Kerr spinning particle
as a model of electron, a 'microgeon' model, where the spin and mass of electron are
related with e.m. and spinor excitations of the Kerr closed string.1_3 The compatible
with the Kerr geometry'aligned' excitations2'3 have a peculiarity in the form of two
extra semi-infinite singular half-strings, as it is shown on fig.l.
Excitations of the Kerr circular string of the Compton size a = h/m have the
wave lengths A = ^, and, as usual in string theory, generate the mass m = E =
Hc/X and spin of particle J = ma = h/2. In the same time, the waves induced by
excitations on the axial strings carry de'Broglie periodicity.2,3
Vacuum polarization near the singular strings leads to the formation of a false
2633
Fig. 2. Image of the dressed Kerr spinning particle.
vacuum, so there has to be a phase transition near the sources,4 and the Kerr
spinning particle turns out to be dressed, taking the form shown on fig.2.
One of the often discussed objections against the Compton size of electron is
the argument based on the experiments on the deep inelastic scattering of electron
which demonstrates its almost point-like structure. Explanation of this fact may be
divided onto two parts:
a) the point like exhibition of the structure of electron may be related with the
complex representation of the Kerr source which is point-like from the complex point
of view.2,11 Working in the momentum space, one can feel namely this point-like
structure. On the real space-time slice it is realized as a contact interaction of the
'axial' strings;2
b) the space-time Compton extension of electron has also been observed in the
low-energy experiments with a coherent resonance scattering of electron.12 In this
relation, the experiments with polarized electrons has to be the most informative.
Finally, one can mention the obtained recently multiparticle Kerr-Schild
solutions13 which show that theory of electron is to be multiparticle one, indeed.
References
1. A.Burinskii, Sov. Phys.JETP, 39(1974)193.,
2. A. Burinskii Phys.Rev. D 70, 086006 (2004); hep-th/0406063.
3. A. Burinskii, Grav.&Cosmol.lO, (2004) 50; hep-th/0403212, hep-th/0507109.
4. A. Burinskii,J. Phys. A, 39 6209 (2006); gr-qc/0606097.
5. G.C. Debney, R.P. Kerr, A.Schild, J. Math. Phys. 10(1969) 1842.
6. W.Israel, Phys. Rev. D2 (1970) 641;
7. C.A. Lopez, Phys. Rev. D 30 313 (1984).
8. A. Burinskii, Phys. Rev. D 68 105004 (2003); hep-th/0308096.
9. A. Sen, Phys. Rev. Lett. 69 1006 (1992).
10. A. Burinskii, Phys. Rev. D 52 5826 (1995); hep-th/9504139.
11. A. Burinskii, Kerr geometry beyond the Quantum Theory, gr-qc/0606035.
12. V.B. Berestetsky, E.M. Lifshitz, L.P. Pitaevsky, "Quantum Electrodynamics ( Course
Of Theoretical Physics, 4)", Oxford, Uk: Pergamon ( 1982).
13. A.Burinskii, Grav.&Cosmol.l2,(2006) 119; gr-qc/0610007; Int. J. Geom. Meth. Mod.
Phys., iss.2 (2007) (to appear); hep-th/0510246.
A LINK BETWEEN GENERAL RELATIVITY
AND QUANTUM MECHANICS
KJELL ROSQUIST
Stockholm University
AlbaNova University Center
10691 Stockholm, Sweden
kr@physto. se
For a number of reasons including having a Dirac g-factor g = 2, the most probable approximation
for the exterior gravitational and electromagnetic field of the electron is the Kerr-Newman solution to
the Einstein-Maxwell equations. It is shown that the Kerr-Newman solution when used as the exterior
Einstein-Maxwell field for the electron gives rise to a standard statistical measuring uncertainty in
the position of the particle. The size of the uncertainty is the Compton wavelength. The uncertainty
therefore coincides with that which is usually inferred for the electron in the context of relativistic
quantum mechanics.
1. Introduction
The purpose of this contribution is to point out a possible connection between general
relativity and quantum mechanics. The four non-zero moments of the electron (mass, charge,
spin angular momentum and magnetic dipole moment) are accurately represented by the
Kerr-Newman solution of the Einstein-Maxwell field equations. The particular solution
with the parameters of the electron is not a black hole, it has neither horizon nor ergo
region. By applying the Kerr-Newman solution to the exterior classical field of the electron,
there emerges an uncertainty in the position of the electron. This comes about as a standard
statistical measuring uncertainty which depends on properties of the Kerr-Newman
solution together with the inequality a » e » m where a = S/m is the specific spin angular
momentum. We emphasize that we do not consider Kerr-Newman or any other solution of
the classical Einstein-Maxwell equations as a complete model for the electron or other
elementary particles. However, in the exterior region where classical physics applies, by the
correspondence principle, the fields should indeed satisfy the Einstein-Maxwell equations.
To aid readers we will use a step-by-step procedure introducing first an uncharged particle
without spin, then charge will be added and finally the spin angular momentum will be
taken into consideration.
We are using geometric units1 with c = 1, G = 1 and 4neo = 1. However, sometimes
we reinstate Newton's constant G to be able to take the limit G —> 0. For a particle which is
uncharged and without spin, we may reasonably assume that there is vacuum in the exterior.
Since the Schwarzschild metric is the only spherically symmetric vacuum solution to the
Einstein equations, it is reasonable to use it for the exterior3 gravitational field. Now let
the particle have also charge, but still no spin. In this case, the Schwarzschild metric is no
longer an exact solution of the Einstein-Maxwell field equations. It is then more appropriate
to use the Reissner-Nordstrom solution which is the unique spherically symmetric charged
aWe use the term exterior field in this context to emphasize that we are not considering the fields all the way
"inside" the particle.
2634
2635
generalization of the Schwarzschild metric given by
gRN = ~f(r)dt2 + /(r)-'dr2 + dr2(d62 + sinW) (1)
where
2GM GO2
f(r) = 1 + -f- . (2)
r rL
and where M is the mass and Q is the electric charge. When Q > M as is the case for all
charged microscopic systems, then f(r) > 0 for all r > 0 implying that there is no horizon
in the geometry. This is the overextreme Reissner-Nordstrom solution which therefore does
not represent a black hole. Instead, the curvature singularity at r = 0 is naked. This means
that the general relativistic description of such a particle breaks down already classically
near the singularity. Our interest here is to discuss what happens in regions which are not
affected by this breakdown of the classical theory. The curvature tensor has two independent
components which can be represented by the curvature invariants23
4(24 -Mr + 2Q2
^V=7T> ^2 = ^ (3)
where R^v is the Ricci tensor and ^ is a certain linear combination of the Weyl tensor
components. The g-forceb on a static (r, 6, cf> constant) object in this geometry is
Q2-Mr
/static —
2 _ j/f2\
q\ iei(e2-M^)
+ O(r) as r —> 0
(4)
r2 Jr2 _ IMr j- CP- ^ liilvii
2Q4
with the sign referring to the positive r-direction.
The expression (4) for the g-force shows that the electric charge gives rise to an effective
negative mass —\Q\ at short distances (r < rciass). Gravity is therefore repulsive at small
distances in this geometry. The transition from attractive to repulsive gravity occurs at
r = Q2/M, which corresponds to the classical electron radius rc!ass = e2/me ~ 3 X 10"13cm.
Although, rdass is well below the Compton scale Ae = 4 X 10""cm where quantum effects
become important, it is still noteworthy that general relativity predicts that gravity changes
its character at that scale. The electromagnetic field is given by the vector potential A =
-(Q/r)dt. Taking the limit G —> 0 gives back flat space via (1) and the radial coordinate
r is then the standard spherical radius. Therefore, in that limit, the electromagnetic field
reduces to the Coulomb field. We will see later that more drastic effects appear when the
spin is taken into account leading to a modification of the electromagnetic field even in the
G -» 0 limit.
The third and final step is to take into account also the spin of the electron. Despite the
quantum character of the spin S, it couples to the orbital angular momentum L in such a
way that the total angular momentum J = L + S is conserved while L and S are not in
general separately conserved. This means that it is not only possible but indeed mandatory
bThe g-force is by definition the acceleration (or force per unit mass) of a test particle which does not carry any
non-gravitational charges.
2636
to consider the spin as an angular momentum which contributes to the gravitational field of
the electron. In addition, the electron also carries a magnetic moment^ which is related to
the spin by/u/e = S/m plus a small anomalous part. This relation shows that the electron has
the g-factor g = 2. We must therefore consider a solution of the Einstein-Maxwell equations
which possesses not only mass and charge but also spin and magnetic moment in the right
combination. The simplest solution by far which satisfies these requirements is the Kerr-
Newman Einstein-Maxwell field. It should be noted that the very fact that the Kerr-Newman
solution has the g-factor g = 2 means that the solution has spin angular momentum rather
than orbital angular momentum. There are a number of other reasons for using the Kerr-
Newman solution in this context, some of which were given in Ref.4. The fact that the
overextreme (M2 < Q2 + a2) Kerr-Newman solutions have the same multipole structure as
the underextreme Kerr-Newman solutions may indicate that they can serve as final states
or at least quasi-stable intermediate states (cf. contribution5 by this author to session GT7
in these volumes). Perhaps the strongest argument in favor of the Kerr-Newman solution
as a candidate for exterior fields is the finiteness of its electromagnetic Lagrangian (see
below). This is in sharp contrast to the divergence of the Coulomb Lagrangian. With the
finite Lagrangian it becomes possible to compute interactions between two or more Kerr-
Newman fields without the need for cut-offs.
The Kerr-Newman electrostatic potential with respect to static observers is given by4
<DE = uaAa = Qr (5)
V(r2 - 2GMr + GQ2 + a2 cos26)(r2 + a2 cos26»)
In the limit G —> 0 this goes over into
Qr
°e = ~T^\ Ta ■ (6>
rl + a1 cos16
This potential deviates from the Coulomb form at the Compton scale4 corresponding to
about 500 fm for the electron. The deviation is most pronounced along the spin axis. Since
the proton g-factor (gp = 5.59) differs from two, the Kerr-Newman field cannot be an
accurate representation of its exterior Einstein-Maxwell fields. However, we assume that
the electromagnetic field of the proton is approximately given by the Kerr-Newman solution
with the specific spin adjusted by the g-factor, (gv/2)a. Computing the resulting electric
field one finds a characteristic length scale of ~ 1 fm corresponding to the conventional
radius of the proton.4
Although the Einstein-Maxwell field of the Kerr-Newman solution has curvature, the
pure gravitational forces in the region of interest are small due to the smallness of the ratio
m/e for elementary particles. Also, the effective curvature radius of the gravitational field
is macroscopic so one may to a high degree of approximation go in the limit G —> 0 when
considering interactions. Evaluating the electromagnetic Lagrangian
for the Kerr-Newman field gives £, = 0. This behavior is in sharp contrast to the situation
for a Coulomb field. The finiteness is a special property of the particular form of the
infinite sequence of multipoles in the Kerr-Newman solution. Changing any number of finite
2637
moments destroys the finiteness of the Lagrangian. Given two Kerr-Newman fields one can
calculate their interaction by considering the sum of their electromagnetic fields, something
which is possible in the limit G —> 0. In this approach, the fields are displaced by a distance
d and are in general rotated and moving with arbitrary velocity. Limiting consideration to
fields with aligned spins and which are moving in the spin direction, the combined field is
F„y(z) = mF^v(z - d/2) + <2>F^v(z + d/2) (8)
where only the z dependence has been emphasized. This leads to the interaction potential0
QQ'\d\
d2 + (a + a')2
where (a, Q) and (a', Q') are the respective parameter values of the two fields (the masses
do not contribute since we have taken the G —> 0 limit). The kinetic energy of the full
Lagrangian contains velocity dependent terms which must be calculated before a complete
evaluation of the motion can be performed. The form of the kinetic energy in this
approximation also shows that the effective mass of the field has a size of the order m&g ~ am.
The physical explanation of this observation is that most of the mass is contained in the
electromagnetic field and is therefore spread out to a certain characteristic radius. When
the distance between the fields is larger than this radius, the G = 0 approximation breaks
down due to inertial effects since meg starts to increase towards m. Having these caveats in
mind, some conclusions can nevertheless be drawn from the form of the potential.
The Kerr-Newman disk
The Kerr-Newman metric can be written in the form
gKN = ~h(M°)2 + hr\Mxf + (M2)2 + (M3)2 , (10)
where
r2 + a2
and Ma represents an orthonormal Minkowski frame in boosted oblate spheroidal
coordinates. The oblate spheroidal (r, 6, </>) coordinates are related to Cartesian coordinates by
x = po sin 6 cos <p
y = po sin 6 sin <f> (12)
z = rcos.6,
where p2, = r2 + a2. The boost is in the 0-direction and is given by v = -a sin 6/po. Setting
r = 0 in the spheroidal coordinates corresponds to the disk x2 + y2 = a2 in the equatorial
plan z = 0. This is the Kerr-Newman disk. It follows that the spheroidal coordinate r is a
measure of the distance to the Kerr-Newman disk. Note that the disk has radius a and that
its size is therefore determined by the spin per unit mass.
CI am indebted to L. Samuelsson for help with the calculation of the potential.
2638
The Kerr-Newman disk and position uncertainty
Let us now discuss what kind of interactions are involved when we measure the position
of an electron. The electron has four (known) characteristics which can in principle be
involved. They are the two gravitational moments (mass and spin) and the two
electromagnetic moments (charge and magnetic dipole). Although the gravitational force on the
electron can be measured, for example in a Millikan type setup, the electron's own
gravitational field is way to small to be measured. Also, when we measure the electron's mass or
spin it is only through the inertial effects of the corresponding kinetic energy terms.
Therefore, any observation of an electron is done by means of some electromagnetic interaction
between the electron and a measuring device. This implies that the position of an electron
can only be inferred through its electromagnetic field.
Consider first a Coulomb field for the electron. Then the position of the electron can in
principle be determined to arbitrary accuracy (classically) by measuring the electric field
strength via the Coulomb relation E = ke2/r2. It is assumed that k and e are already known
to sufficient accuracy. The strength of the electric field is then a good measure of the
distance to the particle. If the electric field is instead of the Kerr-Newman type, the field
strength is not a measure of a distance to a particular point. Rather, a meridional projection
of the Kerr-Newman electric field has a dumbbell-like structure (cf. Ref.6). It follows that a
measurement of the electric field strength only tells us that we are near the disk, not whether
we are near the disk center or the perimeter or somewhere in between. Therefore, unless
we have some other independent information, there will be an uncertainty in the position
(defined as the center of the disk) which will be at least of order Ar ~ 2a = h/m. This is
precisely equal to the relativistic quantum uncertainty in the position of an electron (see for
example the first chapter of the classic QED textbook7 in the Landau & Lifshitz series).
Acknowledgement
The author has benefited from useful discussions with G. Amelino-Camelia when writing
up this contribution. This work was carried out with support from the ICRANet network.
References
1. C. W. Misner, K. S. Thome and J. A. Wheeler, Gravitation (Freeman, San Francisco, USA, 1973).
2. C. Cherubini, D. Bini, S. Capozziello and R. Ruffini, Int. J. Mod. Phys. D 11, p. 827 (2002),
(related online version: gr-qc/0302095).
3. C. Wfitrich, On time machines in Kerr-Newman spacetime, Master's thesis, Philosophisch-
naturwissenschaftlichen Fakultat der Universitat Bern (1999).
4. K. Rosquist, Class. Quantum Grav. 23, 3111 (2006), (related online version: gr-qc/0412064).
5. K. Rosquist, Some physical consequences of the multipole structure of the Kerr and Kerr-Newman
solutions (2006), Contribution to the GT7 session of the 11 th Marcel Grossmann Meeting on
General Relativity.
6. D. Lynden-Bell, A magic electromagnetic field, in Stellar astrophysical fluid dynamics, eds. M. J.
Thompson and J. Christensen-Dalsgaard (Cambridge University Press, Cambridge, 2003) p. 369.
7. V. B. Berestetskii, E. M. Lifshitz and L. P. Pitaevskii, Quantum electrodynamics, 2nd edn. (Perg-
amon Press Ltd., Oxford, England, 1982).
SPACETIME FLUCTUATIONS AND INERTIA
ERTAN GOKLU and CLAUS LAMMERZAHL
ZARM Universitdt Bremen, Am Fallturm, 28359 Bremen, Germany
ABEL CAMACHO and ALFREDO MACIAS
Universidad Autonoma Metropolitan a-1ztapalapa, Physics Department, Mexico city,
09340 Mexico D.F., Mexico
The effects upon the Klein-Gordon field of nonconformal stochastic metric fluctuations
are analyzed. We characterize the stochastic properties of the fluctuations by gaussian
white noise. These fluctuations lead to an effective mass which is different from the 'bare'
mass. We show that our model also implies violation of the weak equivalence principle.
Finally, we give rough estimates about the magnitude of the space-time fluctuations.
1. Introduction
In Quantum Gravity Phenomenology the problem of lack of experimental predici-
tions from approaches to quantum general relativity is attacked by looking for
possible detectable effects. For instance the search for additional noise sources in gravity-
wave interferometers was considered.1 In our approach we assume that quantum
gravity corrections emerge as nonconformal stochastic fluctuations of the metric.2
2. Nonconformal metric fluctuations
The spacetime metric undergoes nonconformal stochastic fluctuations
9llv{x) = diag [e^x\ -e«x\ -e«x\ -e^% (1)
where a; is a spacetime point. The first and second moments can be obtained if we
make the claim that in the average the Minkowskian metric should be reproduced
(e^r/oo) (x) = 7700, (eCVij) 0*0 = Vij- (2)
Then one yields
(iP2){x,x') = a215\x-x') (C2)(x,x')=al64{x-x') (3)
and
W(x)=0, (O(i)=0, (d^)(x)=0, (d^)(x)=0. (4)
The averaging procedure {..} is carried out by integration over a spacetime volume
occupied by the particle while using a weight function f(x), defined on supports
Ax and At lying between Planckian and quantum mechanical scales. We make the
assumption that g^^ix) - thus ip(x) and £(x) - varies over spacetime scales which are
small compared to the typical wavelength of a scalar field <pj (x) and we can write
{(Jnv4>j) = {9tiv)4>3 (high-frequency fluctuations). Futhermore we assume that the
amplitudes of the fluctuations are small \ip\ -C l,|Cl "C 1, which defines a preferred
frame.
2639
2640
3. Modified Klein-Gordon equation
We calculate the modified Klein-Gordon equation in fluctuating metric
(5)
where we labeled each particle with the index j and V = {dx,dv,dz).
Approximating the exponential functions and calculating the average we get
m2c4
O = 520J-c2V20J+-^^, (6)
where rh = (1 + gl 8 g2 J m and c = ( 1 + gl 8"2 J c.
A non-relativistic expansion of the modified Klein-Gordon equation yields a
modified Schrodinger equation
im*>=■£ i1+^+^0v2^ (7)
which can also be written as (accounting for the modified inertial mass rh)
h2
ihdtifij = -—rVVi- (8)
2m
Hence, we may speak of a bare inertial mass m and a experimentally detectable
inertial mass rh which shows a stochastic behavior inherited from the features of
the metric which is also true for the speed of light 5.
4. Violation of the weak equivalence principle
We introduce the Newtonian potential C/(x) yielding
0 = dUi - c2e~( (V - |Q V% + e*^£fc - 2^-U^. (9)
Establishing the classical correspondence of the operators and approximating the
square root (concerning positive energy and neglecting terms of the order c-4) leads
us to
{E)=rhlc2-^2U + ^, (10)
where c? = (c2e^~^\ ,fn2 = ^m2e2^~^) and rh\ s (m2e^~^^ (averaging procedure
after series expansion of the exponential functions). The energy is given by the
classical Hamilton function H which leads directly to the Hamiltonian equations of
motion. We obtain
a=?|VC/(x). (11)
In equation (6) the mass term rh2 is the square of the inertial mass, which can be
identified here with fh\ leading to
rtii = fh\ = rh and mg = rri^. (12)
2641
Hence we get for the gravitational mass
i + d±£2 (13)
The ratio of the gravitational and inertial mass is therefore dependent on the flu-
cuations of spacetime and shall be a dependent on the type of particle. Hence the
weak equivalence principle is violated.
5. Modified Maxwell equations
We consider the Maxwell equations in curved spacetime in vacuum and the modified
equations - after averaging - read
0=V-E, 0=VxB('l + ^ + (722)j^(E. (14)
This can be rewritten according to D and H
In this context the second moments of the fluctuations a\ may be identified with
the coefficients appearing in the photonic sector of the minimal SME3 (we have
a preferred frame). If we compare our coefficients with results from astrophysical
sources4we can estimate
(15)
■a2<l(T32. (16)
6. Conclusions
Our model of spacetime fluctuations leads to modified inertial and gravitational
masses which are affected by the properties of the underlying stochastic process
(gaussian white noise). This allows us to interpret the mass parameter m as a
'bare' mass. Due to the dependence on the type of particle and on nonconformal
fluctuations of the metric the weak equivalence principle is violated. Finally, by
introducing small fluctuations defining a preferred frame a violation of Lorentz
invariance occurs, allowing us to compare the results of tests of light propagation
with our fluctuation amplitudes.
Acknowledgments
It is a pleasure to thank Hansjorg Dittus for discussions. This work has been
supported by Deutsches Zentrum fur Luft- und Raumfahrt (DLR).
References
1. Amelino-Camelia, G. Nature 398, (1999) 216.
2. Camacho, A. Gen. Rel. Grav. 35, (2003) 1839.
3. Kostelecky, V. A. and Mewes, M., Phys. Rev. D 66, (2002) 056005.
4. Kostelecky, V. A. and Mewes, M., Phys. Rev. Lett. 87, (2001) 251304.
QUANTUM GRAVITY IN CYCLIC (EKPYROTIC) AND MULTIPLE
(ANTHROPIC) UNIVERSES WITH STRINGS AND/OR LOOPS
T. J. CHUNG
The University of Alabama in Huntsville
Huntsville, AL 35899, USA
This paper addresses a hypothetical extension of ekpyrotic and anthropic principles, implying cyclic
and multiple universes, respectively. Under these hypotheses, from time immemorial (/ = -co) , a
universe undergoes a big bang from a singularity, initially expanding and eventually contracting to
another singularity (big crunch). This is to prepare for the next big bang, repeating these cycles
toward eternity (/ = +oo), every 30 billion years apart. Infinity in time backward and forward
(t = +go ) is paralleled with infinity in space (x/= + <x>) , allowing multiple universes to prevail,
each undergoing big bangs and big crunches similarly as our own universe. It is postulated that
either string theory and /or loop quantum gravity might be able to substantiate these hypotheses.
Recently, the cyclic or ekpyrotic model has been reported [1-2]. Without invoking
superluminal inflation theory proposed in [3-4], the cyclic model addresses the
cosmological horizon, flatness and monopole problems and generates a nearly scale-
invariant spectrum of density perturbations. In this model, 11-dimensional M-theory is
used, showing that the eleventh dimension collapses, bounces and re-expands and
reducing to a weakly coupled heterotic string theory. This suggests the transition from
contraction to expansion, with the universe undergoing an endless sequence of epochs
which begin with a big bang and end in a big crunch.
The anthropic model [5-9] stipulates an existence of multiple universes or many-
world interpretation [10-14], although no rigorous physical or mathematical
justifications are available at this time. The anthropic principle can be studied by means
of string theory [5-8]. Hopefully, it may become possible to determine the number of
vacua with each particular property such as the cosmological constant, Higgs mass or
fine structure constant [5,8]. Structure and complexity of multiple universes may be
predicted from the outcome of quantum accidents over the course of their histories [6]. In
this approach no boundary histories of the universe depend on what is being observed,
contrary to the usual idea that the universe has a unique, observer independent history. A
concept of subuniverses is examined in [7]. It is speculated that the various subuniverses
may be (1) different regions of space, (2) different eras of time in a single big bang, (3)
different regions of spacetime, or (4) different parts of quantum mechanical Hilbert
space. In the many-world interpretation (MWI) the collapse of the quantum wave is
avoided. There is no experimental evidence in favor of collapse and against the MWI.
World is a nonlocal concept, but it avoids action at a distance and, therefore, it is not in
conflict with the relativistic quantum mechanics. The multiple parallel universes are non-
communicating in the sense that no information can be passed between them.
Hypotheses in Resolution of Time (Fig. 1): From the observations above, it is
postulated that our universe began from time immemorial, (t = -<&). Every 30 billion
2642
years apart, there were big bangs, which will continue likewise forever, toward / = +<» .
Between big bangs, in any one of these 30 billion year periods, the earth with human
beings as well as all other astronomical objects would emerge, with the universe
expanding initially and subsequently contracting, but eventually disappear into a
singularity of black hole (big crunch), preparing for the next big bang. Thus human
activities are confined, isolated, and discontinuous in time (/ = +co ) from one big bang to
another.
30 Billion years
t
Last big bang occurred 15 b years ago
The solar system 10 b years after big bang
Today, the solar system is 5 b years old
The solar system will collapse in 5 b years
+00
Universe contracting
Universe
Next big bang
Fig. 1 Cyclic ( Ekpyrotic) universes
X = —00 -4 -Jf +■ x ~ +00
r\ / r, °
One of these
spheres may be
our universe
y — —oo
Fig. 2 Multiple (Anthropic) univeres. There are infinitely many universes of different sizes randomly
scattered throughout the space. Each universe undergoes cyclic big bangs as shown in Fig. 1 at different
times and different places.
Hypotheses in Resolution of Space (Fig. 2): Infinity in space (*/=±co)
accommodates infinite number of universes. Each universe has its own big bang, its own
solar system, and an earth like our own. This implies that an infinite number of earths
with their inhabitants prevail throughout the space. Big bangs occur in different
universes at different times. Thus each earth has human beings with varying degree of
civilization. Would any two of these civilizations communicate and exchange
information across more than 1023 light years away? Unfortunately, if the message is
2644
deliverable, it will be delivered trillions upon trillions of years later, but by then it will be
delivered not to the one originally intended but to a distant future big bang generation.
All astronomical objects belonging to a universe will be accounted for when merging
into a singularity of black hole at the end of a big bang generation, preparing for the next
big bang. Thus a universe is confined, isolated, and discontinuous in space (x;= +co)
from one universe to another.
Concluding remarks: This paper represents frustration of the past and perhaps
enthusiasm for the future. Difficulties of quantum gravity for the past 70 years have
brought frustration to every one. Will the hypotheses of infinities in spacetime proposed
in this paper lead us to identify new directions to follow with enthusiasm? Will the
string theories and/or loop quantum gravity lead us to a new destination?
References
1. Khoury, J., Ovrut, B. A., Steinhardt, P. J., and Turok, N. "The Ekpyrotic Universe:
Colliding Branes and the Origin of the Hot Big Bang", hep-th/0103239.
2. Steinhardt, P. J. and Turok, N. "Cosmic Evolution in a Cyclic Universe", hep-
th/0111098.
3. Guth, A. H. (1981) Phys. Rev. D23, 347.
4. Linde, A. D. (1982) Phys. Lett. 108B, 389.
5. Susskind, L. "The Anthropic Landscape of String Theory", hep-th/0302219.
6. Hawking, S. W. and Hertog, T. (2006) "Populating the Landscape: A top-down
Approach", Phy. Rev, D 73, 123527.
7. Weinberg, S. "Living in the Multiverse", hep-th/0511037.
8. Susskind, L. "Supersymmetry Breaking in the Anthropic Landscape", hep-
th/0405189.
9. Barrow, J. D. and Tipler, F. J. (1986) The Anthropic Cosmological Principle. Oxford
Univ. Press.
10. Barrett, J. A. (1999) The Quantum Mechanics of Minds and World, Oxford
University Press.
11. Barvinsky, A. O., and Kamenshchik, A. Y. "Preferred Basis in Quantum Theory and
the Problem of Classification of the Quantum Universe" Physical Review D52, 743-
757.
12. Tegmark, M., (1998) "The Interpretation of Quantum Mechanics: Many Worlds or
Many Word?", Fortschritte der Physik 46, 855-862.
13. Everett, H. (1957) "Relative State Formulation of Quantum Mechanics", Reviews of
Modern Physics, 29, 454-462.
14. DeWitt, B. and Graham, N. (1973) eds. The Many-Worlds Interpretation of
Quantum Mechanics, Princeton University.
Quantum Fields
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QUANTUM LIOUVILLE THEORY WITH HEAVY CHARGES*
PIETRO MENOTTI
Dipartimento di Fisica, Universita di Pisa and
INFN Sezione di Pisa
menotti@df.unipi.it
ERIK TONNI
Scuola Normale Superiore, Pisa and
INFN, Sezione di Pisa
e.tonni@sns.it
We develop a general technique for solving the Riemann-Hilbert problem in presence of
a number of " heavy charges" and a small one thus providing the exact Green functions
of Liouville theory for various non trivial backgrounds. The non invariant regularization
suggested by Zamolodchikov and Zamolodchikov gives the correct quantum dimensions;
this is shown to one loop in the sphere topology and for boundary Liouville theory and to
all loop on the pscudosphere. The method is also applied to give pcrturbative checks of
the one point functions derived in the bootstrap approach by Fateev Zamolodchikov and
Zamolodchikov in boundary Liouville theory and by Zamolodchikov and Zamolodchikov
on the pseudosphere, obtaining complete agreement.
1. Introduction
Liouville theory has attracted a lot of interest as an example of quantum conformal
field theory1 and for its applications to model string theory and to brane theory.
Remarkable results have been obtained within the bootstrap approach,2'4 which
starting from some assumptions provides exact results for a few interesting
correlation functions.
Here we address the problem to recover the conformal quantum Liouville field
theory from the functional integral procedure understood in the usual sense in which
one starts from a stable background and then one integrates over the quantum
fluctuations. As it is well known, a quantum field theory is specified not only by an
action but also by a regularization and renormalization procedure.
Both on the sphere topology formulated on the Riemann sphere, on the
pscudosphere and obviously in the conformal boundary case, the Liouville action has to
be supplemented by boundary terms. For definiteness we shall illustrate here the
conformal boundary case. The action in presence of sources is given by
S
r.N
IH
— drddfcf) + fie
n
2b4>
d2( +
or
In
d\ (1)
1
2?r7
n
d~in
d(
d(
c - Cn c - c»
N
Y^ &l log e\
"Presented by Pietro Menotti
2647
2648
where the integration domain re = T\ Un=i 7™ is obtained by removing N
infinitesimal disks 7„ = {|C — Cn\ < £n} from the simply connected domain T and
(j> w — an log |£ — Cn\2 for Q —>, Qn. Q = 1/6 + 6 and A; is the extrinsic curvature of
the boundary dT, defined as
i=SsKk*§-k*3s)' C(A)ear (2)
where A is the parametric boundary length, i.e. d\ = y dQdQ. It is possible to write
action (1) as the sum of a classical part and quantum action. One notices that
due to Q ^ 1/6 the above written action is not exactly invariant under conformal
transformations. In8'9 it was found that if one starts from Q = 1/6 and adopts an
invariant regularization procedure one does not reach a theory invariant under the
full conformal group. This is similar to the result of6 . The reason is that in such
an approach the cosmological term e2b^ acquires weight (1 — 62,1 — 62) instead of8
(1,1) as required by the full infinite dimensional conformal invariance.
The regularization suggested at the perturbative level in4 in the case of the
pseudosphere provides the vertex functions with the correct quantum dimensions1
at the first perturbative order AQ = a(l/b + b — a). In10 is was explicitly proven
that such a result stays unchanged to all orders perturbation theory. In particular
the weight of the cosmological term becomes (1,1) as required by the invariance
under local conformal transformations. These calculations correspond to a double
perturbative expansion in the coupling constant and in the charge of the vertex
function.
Here we use a more powerful approach which allows to resuin infinite classes
of graphs9 . We start from the background generated by finite charges, i.e. "heavy
charges" in the terminology of3 . This means that we consider the vertex operators
Van(zn) = e2""^"' with an = rjn/b and r/„ fixed in the semiclassical limit 6 —> 0.
This has the remarkable advantage to give the resummation of infinite classes of
usual perturbative graphs. In order to do that however one needs the exact Green
function on a non trivial background.
In the case of a single heavy charge, by solving a Rieinann-Hilbert problem in
presence of the given heavy charge and an infinitesimal one we are able to compute
such exact Green function on such a background in closed form in terms of
incomplete Beta functions and such a Green function is used to develop the subsequent
perturbative expansion in the coupling constant 6.
After such a result is accomplished one is faced with the non trivial task of
computing a functional integral constrained by the boundary conditions imposed
by action (1).
The background generated by a single charge is stable only in presence of a
negative value of b2/iB- We compute the Green function on such a background
satisfying the correct conformally invariant boundary conditions and such a Green
function is regularized at coincident points by simply subtracting the logarithmic
divergence. For the sphere and conformal boundary case one obtains the correct
2649
quantum dimensions to one loop in such background improved perturbation theory.
The presence of a negative boundary cosmological constant imposes to work with the
fixed boundary length I constraint and to compare our results with the ones given in5
also the fixed area A constraint is introduced. It is possible to factorize the functional
integral in a term resulting from the boundary length and area constraints and an
unconstrained functional on functions satisfying the correct conformal invariant
boundary condition. We compute such functional integral through the technique of
varying the charges and the invariant ratio A /I2. The one loop result on the one
source background obtained in this way is11
Z(mA,l) = e-*^.0/* A _J_ ^§^(l + 0(&2)) (3)
where So(rj; A, l)/b2 is the classical action without the bulk and boundary
cosmological terms, computed on the one source background. Eq.(3) agrees with the
expansion of the fixed area and boundary length one point function derived through
the bootstrap method in5 and for which there was up to now no perturbative check.
Applying similar techniques in the pseudosphere case one obtains for the one
point function
< V-'M > = '-*"*" m-w-w ('+ 0(°» (4)
where Sci is the full classical action and the one loop expression for the two point
function due to a finite charge and an infinitesimal one. Eq.(4) agrees with the
expansion of the bootstrap result, while the expression for the two point function
is consistent with the existing results of the standard perturbation approach and
agrees with the exact two point function when one vertex is given by the
degenerate field V_I/mm. On the other hand adopting the invariant regularization for the
Green function at coincident points one finds an expression which disagrees with
the degenerate two point function (V-i^2b){x)VE/b(y)) on the pseudosphere.
References
1. T.L. Curtright and C.B. Thorn, Phys. Rev. Lett. 48 (1982) 1309; G. Jorjadze and
G. Weigt, Phys. Lett. B581 (2004) 133
2. H. Dorn, H.J. Otto, Nucl.Phys.B429:375-388,1994, J. Teschner,Phys.Lett.B363:65-
70,1995,
3. A.B. Zamolodchikov and ALB. Zamolodchikov, Nucl. Phys. B477 (1996) 577
4. A.B. Zamolodchikov and ALB. Zamolodchikov, hep-th/0101152;
5. V. Fateev, A.B. Zamolodchikov and ALB. Zamolodchikov, hep-th/0001012;
J. Teschner, hep-th/0009138.
6. E. D'Hoker, D.Z. Freedman and R. Jackiw, Phys. Rev. D28 (1983) 2583.
7. P. Menotti and E. Tonni, Phys. Lett. B586 (2004) 425.
8. P. Menotti and E. Tonni, Nucl. Phys. B707 (2005) 321.
9. P. Menotti and G. Vajente, Nucl. Phys. B709 (2005) 465.
10. P. Menotti and E. Tonni, Phys. Lett. B633 (2006) 404; JHEP 0606:020,2006
11. P. Menotti and E. Tonni, JHEP 0606:022,2006
ON THE PATH INTEGRAL IN NON-COMMUTATIVE (NC) QFT
CHRISTOPH DEHNE*
Institut fur Theoretische Physik, Universitat Leipzig,
Postfach 100 920, D - 04009 Leipzig
As is generally known, different quantization schemes applied to field theory on NC
spacetime lead to Feynman rules with different physical properties, if time does not
commute with space. In particular, the Feynman rules that are derived from the path
integral corresponding to the T*-product (the so-called naive Feynman rules) violate
the causal time ordering property.
Within the Hamiltonian approach to quantum field theory, we show that we can
(formally) modify the time ordering encoded in the above path integral. The resulting
Feynman rules are identical to those obtained in the canonical approach via the Gell-Mann-
Low formula (with T—ordering). They preserve thus unitarity and causal time ordering.
1. Introductory remarks on set-up of QFT on NC spacetime
In the last 15 years, much work and effort has been devoted to the construction and
study of quantum field theories on NC spacetime. The increase in research activity
in this field can be traced back to the appearance of the seminal work by Doplicher,
Fredenhagen and Roberts,1 to an important discovery in string theory2 and last,
but not least, to its relation to non-commutative geometry,3 in general.
The nowadays most popular idea how to implement the non-commutativity of
spacetime in field theory is based on the Weyl-Moyal correspondence. The formerly
pointwise product between fields fi(x) and J2{x) is then replaced by the so-called
star product:
(A * /2)(a;) := [eM^dldy)h{x)h{y)}v=x. (1)
Here, Q^v is defined via [x^x,,] =: iO^t; x^, xv are coordinate operators; 6^ is
a real, antisymmetric, constant matrix (d = 1+3). The field theoretic change to a
physical system with a, say $3 self-interaction is then given by the following action:
/I 2
d4x(-d^ * d^(x) - ^-$ * $(a;) - |$ * $ * $(a;)). (2)
Since the star product is cyclic under the trace (J d4x / * g(x) = J d4x g * f(x),
f,g e SCR4)), it follows that the quantum theory of the kinetic part is the free
theory of ordinary quantum field theory. However, as for the interacting theory, a
perturbative expansion of Green's functions leads to Feynman rules that depend on
the starting point for quantization and are no longer equivalent. In the following,
we will see how a (slightly) different set-up of the generating functional formula
(path integral) leads to Feynman rules with different physical properties!
* Christoph.Dehne@itp.uni-leipzig.de
2650
2651
2. Path integral in NC QFT corresponding to T* ordering
The easiest way to set up the path integral formula for the kind of non-local model
considered here is to take over the formula of the generating functional Z(J) from
the local case and replace in the interaction term £int(<&) the local field products
by the star products (The free theory remains unchanged.). The resulting formula
is then given by
Z[J] = exp [i JdAzCmt(j~-),} exp [~ Jd4x Jd4yJ(x)Ac(x - y)J(y)}, (3)
where £int(<&)* reads for our before mentioned example $ * $ * $(x) (without
factors) and Ac(z) := J ,%Jj4 'T^^Tte *s the causal propagator of the free field.
A perturbative expansion and a subsequent setting to zero of the external sources
J(x) leads to the so-called naive Feynman rules.4 For example, the NC analogon of
the "fishgraph" in momentum space reads
-1 f dAk l + cos^A^p")
It is important to note that the same Feynman rules are derived within the canonical
approach by starting from the Gell-Mann - Low formula and applying the T*-
operator. The latter is denned as follows:6 All time derivatives associated to the
star product act after the time ordering has been carried out (multiplication by
step function.). Although these Feynman rules preserve the properties of the action
related to the spacetime symmetry, one can show that these Feynman rules violate
causal time ordering.
3. Path integral in NC QFT corresponding to T-ordering
Since, as stated in the section before, the naive Feynman rules violate causal time
ordering, one may wonder whether it is possible to modify the derivation of the
above formula for the generating functional Z(J) such that the resulting Feynman
rules preserve causality. It turns out that such a modification is possible by means
of the introduction of derivative shift brackets:
Z[J] =exp {ijd4z[£mt(j^—)]^z] exp [~ Jd4x JdAyJ(x)TA+(x - y)J(y)].
(5)
Here, TA+(x-y) is denned by i}(x0~y°)A+(x-y)+^{y°-x°)A+{y-x), A+(x-y)
is the positive frequency solution of the Klein-Gordon equation and {)(x0 - y°)
Heavyside's step function. (^L))^ means the following: For each time-ordered
configuration (A+(x - y) or A+(y - x)), shift all time derivatives associated with
60i through the step function which is to the right of this shift bracket. Then, realize
the time ordering by multiplying with a step function.
Finally, the resulting Feynman rules are the same as those of old-fashioned
perturbation theory (OTO).5 The latter are derived by starting from the Gell-
Mann - Low formula and applying the T-operator (T -operator: All time deriva-
2652
tives associated with the star product act before the time ordering is applied.).
For example, the fishgraph amplitude now reads ((a, b,c) := aAb + aAc + bAc,
a A o := -~—-):
2^ / / r(1 + )(1 + )* (91+92 -p)
CV .-'(-PA! •91+.92+)„-l<-PA2'<Jl+''J2+)-, /<p „~ '(-J>A, .«1 - .92- ) -'(-PA2 .91 - .92- ) %
L p° - wfl - wj2 + ie -p° - uig1 - w,-2 + ie J '
(6)
where p± := (±wp, p1,p2, p3)T. It has been shown that these Feynman rules maintain
unitarity. By construction, they preserve also causal time ordering.
4. Summary and outlook
In this article, we tried to clarify that, within the Hamiltonian approach (We start
from a Hamilton density TC with it := $.), the time ordering is not rigidly
implemented in the path integral.
We close this article by commenting on an aspect that has only been mentioned
at the end of the talk. As the time ordering in the path integral seems to be better
understood, one can then try to take over all formal manipulations from the Wick
rotation of local quantum field theory. However, it is not clear whether one should
also rotate d°l (i e {1, 2, 3}). It turns out that a nonlocal generalization of reflexion
positivity can be derived and that 8°l has to be rotated to ±i90\ correspondingly,
in order to assure reflexion positivity. These interesting findings and further results
will be reported on in future publications.7
Acknowledgements
The author is grateful to Prof. Sibold for constructive criticism and to Prof. Belinski
for giving the opportunity to present results at the 11th Marcel Grossmann meeting.
References
1. S. Doplicher, K. Fredenhagen, J. E. Roberts, Commun. Math. Phys. 172 (1995) 187.
2. A. Connes, M. R. Douglas, A. Schwarz, JHEP 02 (1998) 003 (arXiv:hep-th/9711162);
M. R. Douglas and C. Hull, ibid. 02 (1998) 008 (arXiv:hep-th/9711165); N. Seiberg
and E. Witten, ibid. 09 (1999) 032 (arXiv:hep-th/9908142); V. Schomerus, JHEP
9906 (1999) 030 (arXiv:hep-th/9903205).
3. A. Connes, J. Lott, Nucl. Phys. Proc. Suppl. 18B (1991) 29; V. Gayral, J. M. Gracia-
Bondia, B. Iochum, T. Schucker, J. C. Varilly, Commun. Math. Phys. 246 (2004) 569
(arXiv: hep-th/0307241).
4. J. Gomis, T. Mehen, NP B591 (2000) 265 (arXiv:hep-th/0005129).
5. Y. Liao, K. Sibold, Eur. Phys. J. C 25 (2002) 469 (arXiv:hep-th/0205269); Y. Liao,
K. Sibold, Eur. Phys. J. C 25, 479 (2002) (arXiv:hep-th/0206011).
6. P. Heslop, K. Sibold, Eur. Phys. J. C 41 (2005) 545 (arXiv:hep-th/0411161).
7. C. Define, to appear.
AN IRREDUCIBLE FORM FOR THE ASYMPTOTIC EXPANSION
COEFFICIENTS OF THE HEAT KERNEL OF FERMIONS
S. YAJIMA, M. FUKUDA and S. TOKUO
Department of Physics, Kumamoto University, 2-39-1 Kurokami, Kumamoto 860-8555, Japan
yajima@sci.kumamoto-u.ac.jp
S.-I. KUBOTA
Computing and Communications Center, Kagoshima University, 1-21-35 Koorimoto,
Kagoshima 890-0065, Japan
Y. HIGASHIDA
Takuma National College of Technology, 551 kohda, Takuma-cho, Mitoyo,
Kagawa 769-1192, Japan
Y. KAMO
Radioisotope Center, Kyushu University, 3-1-1 Maidashi, Higashi-ku,
Fukuoka 812-8582, Japan
We consider the asymptotic coefficients of the heat kernel for a fermion of spin |
interacting with all types of non-abelian boson fields, i.e. totally antisymmetric tensor fields,
in even dimensional Riemannian space. The coefficients are decomposed by irreducible
matrices which are the totally antisymmetric product of the 7-matrices. The form of the
coefficients given in our method is useful to evaluate some fermionic anomalies.
The heat kernel1 plays a very important role in both mathematics and physics,
motivated by studying one-loop quantities (such as the effective action, £ function,
Green functions, anomalies, etc.) in quantum field theory and supergravity. The
heat kernel K (x, x') for a fermion of spin | in even d dimensions defined by
^tK{d\x,x'-t) = -HK{d\x1x';t), (1)
Kl"l(x,x';0) = l\h(x)\-i\h(x')\-iS{d>(x,x'), (2)
where S (x,x') is the d-dimensional invariant ^-function, 1 = {5ab} the unit
matrix for the spinor, and h = det/ta,,, in which ha^ is a vielbein. Here H is the
second order differential operator, corresponding to the square of the Dirac operator
p in the case of the fermion tp,
H
X
Z=^v[V»,Vv}+rV»Y + Y\ [Dll,D„]i, = AllI/i>, (3)
where uiab^ is the Ricci's coefficient of rotation. We consider the fermion interactions
with the totally antisymmetric tensors in the Lagrangian. Therefore, the Dirac
p2
z -
V
= D^ + X,
■V^-Q^Q",
p = Y
Q„ =
iv^v^+i^r + r2,
-vM + yM,
= \{l^Y},
[D^D,
L>M=VM
vMv =
s]ip = k^ii.
+
d.
Qm.
+ i-
lb
fi!ab
2653
2654
operator contains the coupling of the totally antisymmetric products of 7-matrices,
d d
y = £7''1-M%1...w = 5>(i)v(i), 7^-^=7[^...7«], (4)
3=0 j=o
where 7(°) = 1. Here V^...^ is real (pure imaginary) when (—1)5J'0'+1) is even
(odd), due to the hermiticity of the Dirac operator. The quantities Q^, X, A^ and
their derivatives with respect to D^ are expressed with the irreducible matrices,
(d-2)/2
Q»= £ ^^^D + l^ + l)!120^,)),
d d
X = £7«X(j), A^ = £7('>A(j>„. (5)
j=o j=o
The components X^) and Ayw are represented by V^^, the curvature tensor
Rappv and their derivatives with respect to V^ in the tensorial form.
The differential equation (1) of the heat kernel for the fermion interacting with
the general boson fields is not solvable strictly. Therefore the heat kernel is usually
calculated by using De Witt's ansatz,2 automatically satisfying (2),
T,rww / n Al/2(x,x') /a(x,x')\ ^k , ,. „ , ,
K(dKx,x'-t) ~ ^,/exp [-^f2) £a,(a;.a;')i*, (6)
where a(x,x') and A(x, x') are a half of square of the geodesic distance and the
Van Vleck-Morette determinant between x and x', respectively, and aq(x,x') are
bispinors called as the Hadamard-Minakshisundaram-DeWitt-Seeley (HMDS)
coefficients. Note that the coincidence limit of a<j is Hirv^^. ao(x, x') = [ao](x) = 1, and
the metric tensor in curved space is gliv = ha^h1'vr\ab with r/ab = diag(—1, • • • , — 1).
In order to evaluate the anomalies in 2n dimensions, the coincidence limit
[a„](a;) (n > 1) of the HMDS coefficients are required.3 The lowest five coefficients
have been calculated in several methods.4 The coefficients contain many 7-matrices.
since [aq] are expressed by products of X, A^v and their derivatives with respect
to D^, containing the contribution of all types of background fields. Therefore, the
trace calculation of [aq] becomes complicated at higher orders, because the number
of terms of [a(]] exponentially increases as the q grows. In order to simplify the
calculation in evaluation of fermionic loop effects, it is useful to obtain the components
of the coefficients with respect to irreducible matrices of the products of 7-matriccs,
because the trace of product of 7-matrix factors and [aq] is easily performed.
The products of X. A;i„ and their derivatives with respect to D^, e.g. D^X —
\7flX + [Qfj.- X], can be expressed by the (anti)commutators of the components X^,
A(jw„ of the quantities with respect to the irreducible matrices, because the product
of 7-rnatrix valued quantities U, W such as X and K.^v can be always separated
2655
into a commutator [U, W]- and an anticommutator [U, W}+,
1 ' J± ^ 2^U< x> u - k)\{j - k)\kr
i+j-2k<d
(£/M1...Mt(i_fc|^1"^li_fc)±(-l)fca+«W^1...;it(i_fc|^"^|i_fc))
±{U^W), (7)
d
j=0
(d-2)/2 d f min[2i+l,j]
U U\ k (2Z + 1-*)!(y-*)!(*-!)!
7(2'+1+J'-2fc)(v,1...Mt_l(2l+i-M^M1-''fc-1|i-*)
-(-l)fc2+^/1-'"=-1(j_fc|V,1...;it_1|2J+1_fc)
min[2i'J'] f07U I
+ V ( i)ik(k+i) (2f)-'j!
fc^ l J (2Z-A)!C/-A:)!A!
7M(2'+i-2fc)(v,1..w_1(2J-fc|^1-"fc-1|J--fc)
_('_n'£2+i/7^i"^fc-i,. ,,y ,„, ,,
k L) u (J-k\vfJ,i---iJ,k-1\2l-k)
V"'Jr-i^fc(fc+1) (2; + 1)!j'!
£<0 { ' {2l-k)\{j-k)\k\
7(2'+i-2fc)(v,Ml...Mt_l(2J-fc|^1-"fc-1|J-fc)
-(-l)fc3^1-Mfc-1(i-fc|^1...Mt_1|2J-fc)) I- (8)
By repeating the application of these relations, [aq] are derived by the irreducible
matrices and the (anti)commutators in the tensorial form. In calculation of the
loop diagrams, the trace of the non-abelian boson fields A, B over the gauge group
reduce, due to tr[A, £?]_ =0. We have verified the facts on [02] in 4 dimensions.5
References
1. J. Schwinger, Phys. Rev. 82, 664 (1951).
2. B. S. DeWitt, Dynamical Theory of Groups and Fields (Gordon and Breach, 1965).
3. L. N. Chang and H. T. Nieh, Phys. Rev. Lett. 53, 21 (1984); H. T. Nieh, Phys. Rev.
Lett. 53, 2219 (1984); Yu N. Obukhov, Nucl. Phys. B212, 237 (1983).
4. A. E. M. van de Ven, Class. Quantum Grav. 15, 2311 (1998); S. Yajima et al, Phys.
Rep. Kumamoto Univ. 12, 39 (2004); hep-th/0011082.
5. S. Yajima, S.-I. Kubota, Y. Higasida, M. Fukuda, S. Tokuo and Y. Kamo, Class.
Quantum Grav. 23, 1193 (2006).
min[2Jj
QUANTUM ANOMALIES FOR GENERALIZED EUCLIDEAN
TAUB-NEWMAN-UNTI-TAMBURINO METRICS*
MIHAI VISINESCU and ANCA VISINESCU
Department of Theoretical Physics,
Institute for Physics and Nuclear Engineering,
Magurele, P.O.Box MG-6, Bucharest, Romania
mvisin, avisin@theory.nipne.ro
We investigate the gravitational and axial anomalies with regard to quadratic constants
of motion for the Euclidean Taub-Newman-Unti- Tamburino (Taub-NUT) space and its
generalizations as was done by Iwai and Katayama. The generalized Taub-NUT metrics
exhibit in general gravitational anomalies. This is in contrast with the fact that the
standard Taub-NUT metric does not exhibit gravitational anomalies, which is a
consequence of the fact that it admits Killing-Yano tensors forming Stackel-Killing tensors as
products. For the axial anomaly, interpreted as the index of the Dirac operator, the role
of Killing-Yano tensors is irrelevant. We compute the index of the Dirac operator for the
generalized Taub-NUT metrics with the APS boundary conditions and find these metrics
do not contribute to the axial anomaly for not too large deformations of the standard
Taub-NUT metric.
1. Introduction
In order to study the geodesic motions and the conserved classical and quantum
quantities for fermions on curved spaces, the symmetries of the backgrounds proved
to be very important. We mention that the following two generalization of the
Killing (K) vector equation have become of interest in physics:
(1) A symmetric tensor field K^,,,^ is called a Stackel-Killing (S-K) tensor of
valence r if and only if
^,.*.;A)=0. (1)
The usual Killing (K) vectors correspond to valence r = 1 while the hidden
symmetries are encapsulated in S-K tensors of valence r > 1.
(2) A tensor f^...^ is called a Killing-Yano (K-Y) tensor of valence r if it is totally
antisymmetric and it satisfies the equation
ffj.1...{fJ.r-\) = 0- (2)
The K-Y tensors play an important role in models for relativistic spin-1 particles
having in mind their anticommuting property. They enter as square roots in the
structure of several second rank S-K tensors that generate conserved quantities
in classical mechanics or conserved operators which commute with the standard
Dirac operator Ds = 7,JV/J where V^ denotes the canonical covariant derivative for
spinors.
*This research has been partially supported by NUCLEU Program NC/06-35-01-01, MEdC, Ro-
2656
2657
The construction of non-standardDir&c operators which commute with the Dirac
operator Ds depends upon the remarkable fact that the (symmetric) S-K tensor K^
involved in the constant of motion quadratic in the four-momentum p^
Z = \K^Pllpv (3)
has a certain square root in terms of K-Y tensors f^:
K»» = Wl • (4)
The general results are applied to the case of the four-dimensional Euclidean
Taub-Newman-Unti-Tamburino (Taub-NUT) space (A.l).
2. Gravitational anomalies
For the classical motions, a S-K tensor K^ generate a quadratic constant of motion
as in Eq. (3). In the case of the geodesic motion of classical scalar particles, the fact
that K^v is a S-K tensor satisfying (1), assures the conservation of (3).
Passing from the classical motion to the hidden symmetries of a quantized
system, the corresponding quantum operator analog of the quadratic function (3) is:1
K. = D»K»VDV (5)
where D^ is the covariant differential operator on the manifold with the metric
9fj.v Working out the commutator of (5) with the scalar Laplacian 7i — D^D^ we
get that in general the quantum operator IC does not define a genuine quantum
mechanical symmetry.
Using the S-K tensor components of the Runge-Lenz vector for the generalized
Taub-NUT metrics3 we proceeded to the evaluation of the quantum gravitational
anomalies for these metrics.4
3. Dirac equation on a curved background
Carter and McLenaghan showed that in the theory of Dirac ferrnions for any isom-
etry with K vector R^ there is an appropriate operator:5
Xk = -i(i?"VM - \i^R^) (6)
which commutes with the Dirac operator Ds.
Moreover each K-Y tensor f^ produces a non-standard Dirac operator of the
form
Df = -i7"(VV„ - ^YU) (7)
which commutes with the standard Dirac operator Ds.
In the case of the standard Taub-NUT space Dirac-type operators are
constructed from the K-Y tensors of this metric Eq. (A.l).
2658
4. Index formulas and axial anomalies
In4 we computed the index of the Dirac operator on annular domains and on disk,
with the non-local Atiyah, Patodi and Singer (APS)6 boundary condition. For the
generalized Taub-NUT metrics,3 we found that the index is a number-theoretic
quantity which depends on the metrics. In particular, our formula shows that the
index vanishes on balls of sufficient large radius, but can be non-zero for some values
of the parameters c, d (A.2) and of the radius.
We mentioned in4 some open problems in connection with unbounded domains.
The paper7 brings new results in this direction. We showed that the Dirac operator
on M4 with respect to the standard Taub-NUT metric does not have L2 harmonic
spinors.
Appendix A. Generalized Euclidean Taub-NUT spaces
The generalized Taub-NUT manifolds whose metrics are defined on R4 - {0} by the
line element:3
dsNUT2 = f(r)(dr2 + r2d62 + r2 sin2 6 dip2) + g(r)(dX + cos6 dip)2 (A.l)
where the angle variables (6, ip, \) parametrize the sphere S3 with 0 < 9 < 7r,0 <
ip < 2tt, 0 < x < 47r, while the functions
,, , a + br ar + br2
/(r) = , g(r) = — —— , (A.2)
r 1 + cr + drz
depend on the arbitrary real constants a, b, c and d. If one takes the constants
c = —, d = \ the generalized Taub-NUT metric becomes the original Euclidean
Taub-NUT metric up to a constant factor. In the original Taub-NUT geometry
there are four K vectors8,9 . On the other hand in the original Taub-NUT geometry
there arc known to exist four K-Y tensors of valence 2.
The remarkable result of Iwai and Katayama3 is that the generalized Taub-NUT
space admits a hidden symmetry represented by a conserved vector, quadratic in
4-velocities, analogous to the Runge-Lenz vector of the Coulomb/Kepler problem.
The components of the Runge-Lenz vector involve three S-K tensors, but there are
no K-Y tensors for generalized Taub-NUT metrics.
References
1. B. Carter, Phys. Rev. D 16, 3395 (1977).
2. M. Cariglia, Class. Quantum Grav. 21, 1051 (2004).
3. T. Iwai and N. Katayama , J. Geom. Phys. 12, 55 (1993).
4. I. Cotaescu, S. Moroianu and M. Visinescu, J. Phys. A: Math. Gen. 38, 7005 (2005).
5. B. Carter and R. G. McLenaghan, Phys. Rev. D 19, 1093 (1979).
6. M. F. Atiyah, V. K. Patodi and I. M. Singer, Math. Proc. Cambridge Philos. Soc. 77,
43 (1975).
7. S. Moroianu and M. Visinescu, J. Phys. A: Math. Gen. 39, 6575 (2006).
8. G. W. Gibbons and P. J. Ruback , Commun. Math. Phys. 115, 267 (1988).
9. G. W. Gibbons and N. S. Manton, Nucl. Phys. B 274, 183 (1986).
A NEW EXPRESSION FOR THE TRANSITION RATE OF AN
ACCELERATED PARTICLE DETECTOR*
JORMA LOUKOt and ALEJANDRO SATZ*
School of Mathematical Sciences, University of Nottingham,
Nottingham NG1 2RD, UK
We analyse the instantaneous transition rate of an accelerated Unruh-DcWitt
particle detector whose coupling to a quantum field on Minkowski space is regularised by a
finite spatial profile. We show, under mild technical assumptions, that the zero size limit
of the detector response is well defined, independent of the choice of the profile function,
and given by a manifestly finite integral formula that no longer involves epsilon-regulators
or limits. Applications to specific trajectories are discussed, recovering in particular the
thermal result for uniform acceleration. Extensions of the model to de Sitter space are
also considered.
1. Introduction
The Unruh-DeWitt particle detector model12 is a useful tool for probing the physics
of quantum fields. The simplest case to consider is an idealised two-state atom with
a monopole coupling to a massless scalar field in its Minksowski vacuum state. Up
to a detector-dependent proportionality constant, the probability of a transition of
energy uj at proper time t, after "turning on" the interaction at proper time r0, is
given in first-order perturbation theory by the response function
FT(w)= f At' [ dr"e-^(T'-T")VK(T',r"), (1)
where W(t', t") = (O|0(x(t'))0(x(t"))|O) is the Wightman function of the field. The
T-derivative of the response function is the transition rate
FT{u) = 2Rel dse-luJSW(T,T~s). (2)
Jo
Using the conventional ie regularisation prescription, the Wightman function reads
W(x,x')= lim ~ 5 ? , (3)
^' > f^0+ 4tt2 (t - f - ie)2 - |x - xf W
with the limit being taken after integration against smooth functions of x(t') and
x(t"). However, the sharp cutoff assumed in the integrals (1) and (2) implies that
this form of the two-point function is not guaranteed to give unambiguous results.
In fact, Schlicht3 and Langlois4 have shown that this procedure gives Lorentz non-
invariant results for a uniformly accelerated trajectory, instead of the thermal
spectrum expected according to the Unruh effect.
"This research was supported by an EPSRC Dorothy Hodgkin Research Award to the University
of Nottingham
tjorma.louko@nottingham.ac.uk
* pmxas3@nottingham.ac.uk
2659
2660
Schlicht3 has proposed a new regularisation scheme that avoids this problem.
The detector is coupled to a spatially smeared version of the field operator given by
0/(r) = |d3e/e(€)0(x(T,O) , (4)
where fe (£) is a profile function and £ are Fermi-Walker coordinates parametrizing
the simultaneity plane of the detector at time r. The parameter e controls the size of
the detector, recovering the pointlike coupling in the e —> 0 limit. Using a particular
Lorentzian profile function, Schlicht obtained the modified correlation function
We(T,T')= lim -L ^ > (5)
' ^°+47T2(x_x/_le(x + x'))2
and showed that using it in (2) gives the correct Planckian result for the Rindler
motion. But it remained open whether this result depends on the choice of a convenient
profile. This is a motivating question for our research.
2. Results
Our first result is that it is possible to take the explicit e —> 0 limit in Schlicht's
expression for the transition rate, with the outcome
<j 1 [At , fcos(us) 1 \ 1
FT(u) = --- + — ds -^r + ^\ +0 9A , (6)
TV ' 4tt 2tt2 Jo y (Ax)2 s2 J 2tt2At
where (Ax) = (x(r) — x(r — s))2 and At = t — tq.
Formula (6) contains no regulators and is manifestly Lorentz invariant. It
separates the spectrum cleanly into a universal term odd in uj and a trajectory-dependent
term even in u. It is therefore a convenient starting point for concrete calculations
of detector response for generic trajectories (the only condition imposed on x(r) to
derive the result is C9 continuity). Asuming some further but still mild conditions
on the trajectory to control the asymptotic past limit, expression (6) is also valid
when At = +oo. All stationary motions in Minkowski space are covered under these
assumptions, as well as many nonstationary ones (excluded are certain pathological
cases like trajectories that cover an infinite space in a finite proper time). In
particular, for the Rindler motion our formula obtains the Planckian spectrum, and for
certain asymptotically Rindler motions an asymptotically Planckian spectrum. For
the general case with At = +oo the spectrum can also be written as
U! 1 f°° ( 1 1 \
FT(u» = --6(-a,) 4- ^ I dscosM ^ + T2j , (7)
where the first term is the spectrum for inertial motion and the second contains the
effects of acceleration.
Our second result is that the same expression (6) also follows in the zero-size
limit from any profile function fe(£) which has compact support, if a technical
2661
modification is made to the definition of spatial smearing so that the transition rate
is defined by
FT» = f d3£d3£' /e(0/e(€')2Re f \S e~-s W(x(t,£))0(x(t-S,£'))|O> .
(8)
The f —^ 0 limit of (8) can be taken in a general way to obtain (6), assuming the
trajectory to be real analytic. Full details leading to these two main results can be
found in our paper.5
Thirdly, the second result can be easily generalised to de Sitter spacetime (dS).
For a detector moving in dS when the field is in the Euclidean vacuum state, the
zero-size limit of the transition rate for a general compact profile is given by
* , •> u l fAT j ( cos(o;s) 1 \ 1 ,m
*<""=-c+s?y„ dsU,(T))-z(x(T-»))]^j +^' ()
where Z(x) are the Minkowski coordinates corresponding to de Sitter point x in
a five-dimensional space in which dS is embedded as an hyperboloid Z2 = a2.
Expression (9) is manifestly de Sitter invariant and gives the expected Planckian
spectrum for inertial trajectories.
3. Conclusions and outlook
We have calculated the zero-size limit of the transition rate for particle detectors
regularised by a spatial profile. The result, given by expression (6), was applied to
a number of trajectories in Minkowski space and generalises straightforwardly to
de Sitter space. Whether similar expressions hold in more general backgrounds is
an open question. It is worth remarking that when the conventional regularisation
is used and the detector is switched on and off with a smooth function, expression
(6) is also recovered as the approximate transition rate in the fast switching limit.6
This suggests its universal status.
References
1. W. G. Unruh, Phys. Rev. D 14, 870 (1976).
2. B. S. DeWitt, "Quantum gravity, the new synthesis", in General Relativity; an
Einstein centenary survey ed S. W. Hawking and W. Israel (Cambridge University Press,
1979) 680.
3. S. Schlicht, Class. Quantum Grav. 21 4647 (2004). (arXiv:gr-qc/0306022)
4. P. Langlois, Ann. Phys. (N.Y.) 321 2027 (2006). (arXiv:gr-qc/0510049)
5. J. Louko and A. Satz, Class. Quantum Grav. 23 6321 (2006). (arXiv:gr-qc/0606067)
6. A. Satz, (arXiv:gr-qc/0611067)
ON THE GEOMETRIZATION OF THE ELECTRO-MAGNETIC
INTERACTION FOR A SPINNING PARTICLE
FRANCESCO CIANFRANI*, IRENE MILILLO* and GIOVANNI MONTANI*^
*ICRA—International Center for Relativistic Astrophysics
Dipartimento di Fisica (G9), Universita di Roma, "La Sapienza",
Piazzale Aldo Mora 5, 00185 Rome, Italy
^ENEA-C.R. Frascati (U.T.S. Fusione),
via Enrico Fermi 45, 00044 Frascati, Rome, Italy
francesco. cianfrani@icra.it
montani@icra.it
We outline that, in a Kaluza-Klein framework, not only the electro-magnetic field can be
geometrized, but also the dynamics of a charged spinning particle can be inferred from the
motion in a 5-dimensional space-time. This result is achieved by the dimensional splitting
of Papapetrou equations and by proper identifications of 4-dimensional quantities.
Keywords: Kaluza-Klein theories.
After Einstein recognized the gravitational field as the metric of the space-time
manifold, Kaluza and Klein proposed a model in which also the electromagnetic
interaction is a geometrical one.1'2 This result has been obtained by adding a
spatial closed dimension: the new available five degrees of freedom can be recast as a
gauge vector field A^ and a scalar one 0, under a proper restriction of the general
covariance. In particular, the form of the Kaluza-Klein metric tensor is the following
one
■ _(gllv{xP)+e2k2<l>2All{xP)Al/{xP)ek(l>2All{xP)\ ,.
UB ~ \ ek<f>2Av{xe) <f{xP) ) [i)
where Greek letters refer to the standard 4-dimensional space-time coordinates (/i =
0,..., 3), e is the electron charge and k a constant, while g^ is the 4-dimensional
metric tensor.
Hence, by the dimensional reduction of Einstein-Hilbert action, one sees that
the Lagrangian for the vector field is the Maxwell one. For what concern the scalar
field, it determines the size of the extra-dimension; at the same time, it appears in
front of the Maxwell Lagrangian density, so being related to the electro-magnetic
coupling constant. Therefore, the stabilization of the additional space corresponds
to have a constant electric charge, thus standard electrodynamics has to arise.
However, it is not enough in view of the geometrization, since also the interaction
with matter has to be predicted from the same hypothesis.
The simplest case is that of a test particle: it follows a geodesies trajectory in the
5-dimensional space-time. One can easily show3 that the covariant fifth component
of the velocity, «5, is a conserved quantity and that, in a 4-dimensional perspective,
the motion is that of a test particle endowed with a charge proportional q to the
5-momentum mus, i.e. mu^ = q/(2vG).
2662
2663
Moreover, because of the closed nature of the extra-space, the charge is
quantized; by imposing its minimum value to be the electron one, an estimate for the
length L of the fifth dimension comes out asl« 10~31cm. Being its length just
a few order of magnitude greater than Planck's length, we expect to be able to
explain the stabilization of the extra-space in a quantum gravity framework.
A key-point in the derivation is the link between the fifth- and the fourth-
dimensional line elements, i.e.
^ds = ds
i-«!
(2)
which implies |«s| < 1 => -2- < 2\/G) so that the geometrization stands only for
macroscopic objects and not for elementary particles.
The next step is a rotating body: being I1AB the spin tensor, its dynamics is
described by the following system of equations (Papapetrou equations with Pirani
condition)4,5
_D_(5)pA = ^)RbcAd^BC(,)uD
D \^AB
(5)p>
(5)pA(5)uB __ (5)pB(S)u;
(5)m(5)uA_£S^(5)UB
(3)
EAB(5)«B = 0
While the controvariant 4-dimensional components E^" = S^v can be identified
with the 4-spin tensor, the additional components £5^ = S^ determines a vector,
whose physical interpretation is one of the subject of our investigation.
By the dimensional reduction of the system (3), we obtain the following one6
DsJ
P» = ^R^S^u-* + qF^u" + \V»FvPMvp
DS*V
Ds
pnuv __ pvuH + p^ MP"
Fv Mptl
p
Da
(a2P5 + {ekF^S^) = £-sq = 0
(4)
pn = a2P„ + U5mt - ekFpL,UPS^u5 + lekF^SP
Svlluv + S»u5 = 0
where for the quantity M.^" we have
M^ = -ek(S^u5 + u»S" - uvS»).
(5)
Once we think at the quantity q as the electric charge of the system (a strong
indication for this comes from the third equation of the system (4), which tells us it
is conserved during the motion) we find that the first two equations coincide with
2664
those describing the dynamics of a rotating body with an electromagnetic moment
M^v (Dixon-Souriau equations7,8) . Furthermore, it is clear that the additional
components of the spin tensor describe a non-vanishing electric moment, since the
vector S^ enters into the electro-magnetic moment with a term proportional to
the velocity. In fact, in a co-moving frame the spatial part of the electro-magnetic
moment, i.e. Mu, receive no contribution from terms with S^.
Therefore, a rotating body in a Kaluza-Klein background behaves as a charge
rotating particle in a 4-dimensional point of view. A proper feature of such models
is an electric dipole moments, associated with additional components of the spin
tensor.
It arises the question of the possibility to implement this scheme in a quantum
framework, because an electric moment for elementary particles implies the violation
of the parity and of the time-reversal invariance.
We want to emphasize that, while Kaluza-Klein theories preserve both P and T,
definitions of parity and time-reversal on 5-dimensional spinors differ from those on
4-dimensional ones, so violations of the latter do not imply violations of the former.
For example, since in five dimensions the 75 matrix is one of Dirac matrices, an
explicitly 4-parity-violating term appears in the Dirac action, while the
representation of the 5-parity is given by «7°75 and it is conserved.
For what concern the time-reversal, the question is more subtle, however its
violation is not surprising, since an electric dipole moment term arises directly from
spinor connections in five dimensions.9
References
1. T. Kaluza, On the Unity Problem of Physics, Sitzungseber. Press. Akad. Wiss. Phys.
Math. Klasse, (1921), 966
2. O. Klein, Nature 118, (1926), 516
3. F. Cianfrani, A. Marrocco, G. Montani, Int. J. Mod. Phys, D14, 7, (2005), 1195
4. A. Papapetrou, Proc. Roy. Soc. London, A209, (1951), 248.
5. E. Corinaldesi, A. Papapetrou, Proc. Roy. Soc. London, A209, (1951), 259.
6. F. Cianfrani, I. Milillo, G. Montani, Dixon-Souriau equations from a 5-dimensional
spinning particle in a Kaluza-Klein framework, submitted to Mod. Phys. Lett. A.
7. W. G. Dixon, II Nuovo Cimento, A XXXIV, n° 2, (1964), 317.
8. J. M. Souriau, Ann. Inst. H. Poincare, A XX, n° 4, (1974), 22.
9. S. Ichinose, Phys.Rev., D66, (2002), 104015
CAN EPR CORRELATIONS BE DRIVEN BY AN EFFECTIVE
WORMHOLE?
E. SERGIO SANTINI
Centra Brasileiro de Pesquisas Fisicas, Coordenagdo de Cosmologia, Relatividade e Astrofisica
ICRA-BR
Rua Dr. Xavier Sigaud 150, Urea 22290-180, Rio de Janeiro, RJ, Brasil and
Comissdo Nacional de Energia Nuclear
Rua General Severiano 90, Botafogo 22290-901, Rio de Janeiro, RJ, Brasil
santini@cbpf.br
A causal approach to the Einstein-Podolsky-Rosen (EPR) problem, i.e. a two-
particle correlated system, is developed. We attack the problem from the point
of view of quantum field theory considering the two-particle function for a scalar
field and interpreting it according to the Bohm - de Broglie view. In this approach
it is possible to interpret the quantum effects as modifying the geometry in such a
way that the scalar particles see an effective geometry. For a two-dimensional static
EPR model we are able to show that quantum effects introduces singularities in the
metric, a key ingredient of a bridge construction or wormhole. Following a
suggestion by Holland1 this open the possibility of interpreting the EPR correlations as
driven by an effective wormholea . The two-particle wave function of a scalar field,
V'2(xi,X2, t) satisfies(see for example3,4):
E[(5"^ + !^]V'2(xx,X2,t) = 0 (1)
Explicitly we have
vn c m c
[(c^)i + -^]V'2(x1,x2,t) + [(<9^)2 + -p-]V2(x1;x2,t) = 0. (2)
Substituting ip2 = Rexp(iS/H) in Eq. (2) we obtain two equations, one of them for
the real part and the other for the imaginary part. The first equation reads
rf^d^Sd^S + r}'i"*dllaSdVaS = 2M2 (3)
where rf,v is the Minkowski metric and
M2^m\l^^-2) (4)
R
2m26
with Q = ^ »M _ h2 (JH^hR, (4.)
The equation that comes from the imaginary part is
aAn extended version of this talk can be found in.2
2665
2666
V^df,1{R2d„1S)+V^dfi2(RAdU2S) = 0 (5)
which is a continuity equation. Following De Broglie5 we rewrite the Hamilton-
Jacobi equation (3) as
I fl oa o , I fl Cfl a _ o™2„2
Q 0^50^5 + ' Q-^59,25 = 2mV. (6)
U 2m2c2) U 2mVi
We can interpret the quantum effects as realizing a confonnal transformation of
the Minkowski metric ifv in such a way that the effective metric is given by g^ =
(1 — 2rr>2c2)rliJ-v Now, following an approach by Alves (see7), we shall see that, for
the static case, it is possible to obtain a solution as an effective metric which comes
from Eqs. (3) and (5). For the static case these equations are:
VndXlSdXlS + VndX2SdX2S = 2m2c2(l - ^-^) (7)
dXl {R2dXl S) + dX2 {R2dX2 S)=0 (8)
We consider that our two-particle system satisfies the EPR condition8 pi = —p^
which in the BdB interpretation, using the Bohm guidance equation p = dxS, can
be written as dXl S = —dX2 S. Using this condition in Eq. (8) we have dXl (R2dXl S) =
dX2(R2dXlS) and this equation has the solution R2-§§~ = G(x\ + X2) where G is an
arbitrary (well behaved) function of x\ + X2- Substituting in Eq.(7) we have
and using the expression (4') for the quantum potential, the last equation reads
8G2 + h2(dXl {R2)f - h22R2d2XiR2 + h2(dX2(R2))2 - h22R2d2^R2 - 8m2c2R4 = 0.
(10)
A solution of this nonlinear equation is R2 = 2m2c2 (pi sin (^(xi + X2)) + C2)
provided a suitable function G(x% + X2) which can be obtained from (10) by
substituting the solution.
In order to interpret the effect of the quantum potential we can re-write Eq. (7)
using (9) obtaining m2-!lr-2-dx,SdXlS + rn2 \ 2dX2Sdx.2S = 2m2 that we write as
gndXlSdXlS + g11dX2SdX2S = 2m2 (11)
and then we see that the quantum potential was '"absorbed"' in the new metric gn
which is:
— J_ — mi ( G \2
gil — gll — C2m-2\R2)
x 2C2 sin2 (^ (3:i+x2))-Cf cos2 (^ (xl+x2))-2C1C2 sin(^ (x1+x2))
4 (Cisin(^£(cCl+:C2)) + C2)2
(12)
2667
We can see that this metric is singular at the zeroes of the denominator in (12)
and this is characteristic of a two dimensional black hole solution (see6'7). Then our
two-particle system "see" an effective metric with singularities, a fundamental
component of a wormhole.9 This open the possibility, following Holland,1 of interpreting
the EPR correlations of the entangled particles as driven by an effective wormhole,
through which physical signals can propagate. Obviously a more realistic (i.e. four
dimensional) and more sophisticated model (i.e. including the spin of the particles)
must be studied. b
Acknowledgments
I would like to thank Prof. Nelson Pinto-Neto, from ICRA/CBPF, Prof. Sebastiao
Alves Dias, from LAFEX/CBPF, Prof. Marcelo Alves, from IF/UFRJ, and the
'Pequeno Seminario' of ICRA/CBPF for useful discussions. I would also like to
thank Ministerio da Ciencia e Tecnologia/ CNEN and CBPF of Brazil for financial
support.
References
1. P. R. Holland, The Quantum Theory of Motion: An Account of the de Broglie-Bohm
Causal Interpretation of Quantum Mechanics (Cambridge University Press,
Cambridge, 1993).
2. E.S.Santini, Might EPR particles communicate through a wormhole? quant-
ph/0701106.
3. Silvan S. Schweber, An Introduction to Relativistic Quantum Field Theory, (Harper
and Row, 1961).
4. D. V. Long and G. M. Shore, Nuc. Phys. B 530 (1998) 247-278, hep-th/9605004; H.
Nikolic, Found. Phys. Lett. 17 (2004) 363-380, quant-ph/0208185.
5. L. De Broglie, Non Linear Wave Mechanics, (Elsevier, 1960).
6. R. Mann, A. Shiekh and L. Tarasov, Nucl. Phys. B 341 (1990) 134.
7. M. Alves, Mod. Phys. Lett. A 14 No. 31 (1999) 2187-2192.
8. A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47(1935) 777-780.
9. M. Visser, Lorentzian Wormholes: From Einstein to Hawking (AIP Series in
Computational and Applied Mathematical Physics, 1996)
10. S. W. Hawking, Phys. Rev. D37 4 (1988) 904-910.
bIt is interesting to note that a wormhole coming from a (Euclidean ) conformally flat metric with
singularities was shown by Hawking.10 Consider the metric:
ds2 = Q,2dx2 (13)
with
This looks like a metric with a singularity at xq. However, the divergence of the conformal factor
can be thought as the space opening out to another asymptotically flat region connected with the
first one through a wormhole of size 2b.
IS TORSION A FUNDAMENTAL PHYSICAL FIELD?
O. M. LECIAN1'2'", S. MERCURI1-2'6 and G. MONTANI1'2-3^
^ICRA — International Center for Relativistic Astrophysics
2Dipartimento di Fisica, Universita di Roma "La Sapienza",
P.le Aldo Mora 5, 00185 Roma, Italy
3ENEA C.R. Frascati (Dipartimento F.P.N.), Via Enrico Fermi 45,
00044 Frascati, Roma, Italy
a lecian@icra.it
mercuri@icra.it
c montani@icra. it
The local Lorentz group is introduced in flat space-time, where the resulting Dirac and
Yang-Mills equations are found, and then generalized to curved space-time: if matter is
neglected, the Lorentz connection is identified with the contortion field, while, if matter
is taken into account, both the Lorentz connection and the spinor axial current are
illustrated to contribute to the torsion of space-time.
Keywords: Lorentz gauge theory; Torsion.
1. Lorentz gauge theory on flat space-time
Let M4 be a 4-dimensional flat manifold equipped with the metric tensor g^ =
Vap^r^^v = ??Q/3e2e^> where e° are bein vectors, xa are Minkowskian coordinates,
and y*1 are generalized coordinates. Under an infinitesimal generic diffeomorpliism,
xa —-> y/fl = 5£xa + £,fJ'(x~'), and for an infinitesimal local Lorentz transformation
,xa —> x'a = xa + eaJx~')xP, the behavior of a vector field Va —> V^ must be
equivalent: from the comparison of the two transformation laws, the identification
e£ = g^g ' is possible, where the isometry condition dp£a + da£,p = 0 has been
taken into account.
Spinor fields, on the contrary, cannot have the same behavior under the two
transformations, for spinors transform under a spinor representation of the Lorentz
group, while no spinor representation is given for the diffeomorpliism group,1 i.e.
spinor fields must experience the isometric component of the diffeomorpliism as a
local Lorentz transformation. The implementation of a local symmetry requires the
introduction of gauge field, and the space where these gauge transformations live
can be defined by comparing the coordinate transformation that induces vanishing
Christoffel symbols in the point P, yf, = xfb + \ FgJ xthx\h, where tb refers to
the tangent bundle, with the generic diffeomorpliism ya = xa +£,a(x1): the
identification, in the point P,xf> — ""a ' -
xtbxtb ~ £" 1S possible.2 The coordinates
■tb t~ 2
of the tangent bundle are linked point by point to those of the Minkowskian space
through the relation above, and they differ for the presence of the infinitesimal
displacement £. From now on, these coordinates will be referred to as xa.
Let M4 be a 4-dimensional flat Minkowski space-time: the action describing
the dynamics of spin-i fields, S = | J d4x (%l)"fadaTp — daip"/aip), is invariant under
global Lorentz transformations xjj —-> S(A)tp,tp —-> ipS-1 (A), where S (A) = 1 —
|eabEa(, is the infinitesimal global Lorentz transformation, defined as in.3
2668
2669
For local Lorentz transformations, the Lagrangian density will read L =
^e^^D^-D^i,}, DJ, = e^ZV/, = e"a (d^ - \Ab^bc^) being
the pertinent covariant derivatives. The interaction Lagrangian density Lint =
|e"a^{7a,E6c}V< = ~SbcA\C = -i^e^'K' where 3a = ^757> is the
spinor axial current, shows that the gauge field A interacts with the spinor axial
current, which is the source for gauge field of the Lorentz group on flat space-time. After
variation with respect to the adjoint field, and making use of the anti-commutation
properties of Dirac matrices, the Dirac equation e^a [i^d^ + | eabc^757b^4^1 ip = 0
for the spinor ip in an accelerated frame is found: the spinor cannot be considered
as free, because it interacts with a Yang-Mills gauge field. If a Lagrangian density
for the gauge field A is added, i.e. L = — (l/32)tr * F A F, variation with respect
to A leads to the Yang-Mill equation D^F^ = Svab.
2. Lorentz gauge theory on curved space-time
The need to introduce a Lorentz gauge field in curved space-time comes from the fact
that, while spin connections are intended to restore the properties of Dirac matrices
in the physical space-time, gauge connections allow one to recover invariance under
local Lorentz transformations for spinor fields on the tangent bundlea.(For a first
attempt to a gauge theory of the gravitational field, see45).
As a consequence, two different Lorentz-valued 1-forms are required to make
the spinor Lagrangian density invariant: the total connection reads C\ = uab +
Aab, where uj is the usual spin connection of GR, and A is an additional Lorentz
connection and the total action readsb
S (e, uj, A, i>, Ji) =
\ feabcdea A eb A Rcd - ^ jtr *F A F - i j'eabcdea A eb A Jcf A Af^
+ 2 / eabcd ea A eb A ec A
itlnd fd - l-{u + A)) ip - i (d + i (w + A) J VnV
(1)
If fermion matter is absent, variation with respect to the connection gives the
"On curved space-time two different Lorentz transformations can be distinguished, which coincide
in flat space-time. Active Lorentz transformations are due to the action of the Lorentz group on
tensors V1 and spinors tp on the tangent bundle, i.e. V1 —> h.(x)^vV1' and if> —> s(A(x))ifj.Passive
Lorentz transformations are due to isometric diffeomorphisms of the space-time manifold, which
pull back the local basis in the generic point P. While active transformations are defined everywhere
once the matrix A(x) is assigned, passive transformations can be reduced to a Local Lorentz
transformation only in the point P, acting as a pure diffeomorphism on the other points of the
manifold. These two kinds of transformations, indeed, coincide on curved space-time, too: because
of local Lorentz transformations, a tetradic vector transforms as ejf(x') = Aa^(x')e'^(x'), while, for
world transformations, e%(x) —> ejf (x') = e^(x)^^ ss e°(x) + e^(x)J^77. The comparison leads
to the identification e'»(x') = e*(x') + e* (z')<f, where e% = -L>s£a - R%_^, Xaie = Ra~b£ - R-a5b
being the anholonomy coefficients.
bThe interaction term between w and A is added by hand, and will be crucial for the geometrical
interpretation of the Lorentz-group field. We are assuming 8irG = 1.
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structure equation d^ea = AabAeb: pulling back the action to the unique solutionc
uiab = uab + Aab, we get
S(e,A) = ± J eabcd eaAebARcd-^Jtr*FAF
-\j eabcd ea A eb A 2[cf A A™ - ± J eabcd ea A eb A Ac f A A?d, (2)
where tilde denotes Riemannian objects. Variation with respect to the gravitational
field and to the Lorentz connection gives
eabcd eb A Rcd = Ma + e\cd eb A {wc f + Ac f) A A*d, (3)
d(A) ^ pfd = e^[d eaAebA ^cf] + 2Acf] j t (4)
where Ma is the energy-momentum 3-form of the field A, which can be explicitly obtained
variating the Yang-Mills- like action with respect the gravitational 1-form. Since the
solution to the structure equation is analogous to that of the 2 Cartan structure equation,
the Lorentz connection A can be identified with the contortion 1-form, thus implying the
presence of the torsion 2-form Ta = Aab A e . Field equations describe the coupling
between gravitational and Lorentz connections: gravitational spin connections become the
source of torsion.
If fermion matter is present, variation with respect to the connections give the structure
equation d^'e11 = AabAe — \tabcde AecjdAy pulling back the action to its unique solution
uab = u>\ + A\ + \eabcdecjfA), we obtain
S (e, A,V,V) = \ Jtabcdea A eb A Rcd - ^ J tr *F A F
+ jf tabcd ea A eb A ec A
#7d ( d - %- {u> + A) ) V - i ( d + %- (lu + A)) i,~tdi>
-\J tabcd ea A eb A Ac f A A*d - ± J eabcd ea A e» A w[cf A A™ (5)
-^JeaAebAecAA^ab jfA) - A J' dAx i!abja(A)jb{A), (6)
where the last term is the four-fermion interacting term of Einstein-Cartan theory. The
presence of spinor fields in the structure equation means that both the connection A and
the spinor axial current contribute to the torsion of space-time. Variation with respect to
the gravitational field and to the Lorentz connection leads to the generalization of the field
equation obtained in absence of matter.
References
1. F. Hehl, P. von der Heyde, G.D. Kerlick, J. Nester, Rev. Mod. Phys., 48 3 (1976).
2. O.M.Lecian, S.Mercuri, G.Montani, in preparation (2006)
3. F.Mandl and G.Shaw, Quantum Field Theory, Revised edn. (John Wiley and Sons,
2002)
4. R. Utiyama, Phys. Rev. 101, 1597 (1956).
5. T.W.B. Kibble, J. Math. Phys. 2, 212 (1961).
6. A.Ashtekar, J.D.Romano, R.S.Tate, Phys.Rev.D 40, 2572(1989).
cFor a discussion of the reduction of the dynamics, see.6
UNITARY QUANTIZATION OF THE GOWDY T3 COSMOLOGY
ALEJANDRO CORICHI1
JERONIMO CORTEZ2
GUILLERMO A. MENA MARUGAN2
1 Institute) de Matemdticas, UNAM, A. Postal 61-3, Morelia, Michoacdn 58090, Mexico
2 Institute de Estructura de la Materia, CSIC, Serrano 121, 28006 Madrid, Spain
We analyze the quantization of the linearly polarized Gowdy spacetimes with the spatial
topology of a three-torus. The physical, local degrees of freedom of these cosmologies are
described by a scalar field that satisfies a Klein-Gordon type equation in an auxiliary
background. We show that a convenient choice of the basic field renders this background
static. We quantize the Gowdy model by means of a Fock quantization of this scalar
field and prove that the evolution obtained in this way is unitary, in contrast with the
situation found previously in other quantizations. In this sense, our construction provides
the first consistent quantum description of an inhomogeneous cosmological model.
1. Introduction
Symmetry reduced models have received a lot of attention in general relativity as a
suitable arena to study issues that may play a central role in a quantum theory of
gravity. Reductions that keep an infinite number of degrees of freedom are specially
relevant, because they should capture the field complexity of general relativity.
Among this kind of reductions, the simplest model with applications in cosmology
is the family of Gowdy spacetimes1 with linear polarization and the spatial topology
of a three-torus, T3. After gauge fixing, this model is classically equivalent to 2+1
gravity coupled to an axially symmetric scalar field.2 So, by quantizing this field in
the fictitious (2+1) background one obtains a quantum description of the Gowdy
cosmology. It was precisely in this way that Pierri2 introduced a quantization for
the polarized Gowdy model. However, Pierri's quantization has a serious drawback:
the classical dynamics cannot be implemented as a unitary transformation.3'4 In
this work we will propose an alternate quantization which solves this problem.
2. The polarized Gowdy T3 model
After a gauge fixing in which all but a homogenous constraint are removed from
the system, the metric of the linearly polarized Gowdy spacetimes can be written:5
ds2 = ei-My/v (-dt2 + de2) + e-^^fp'da2 + e^^dS2. (1)
Here, da and dg arc the two Killing vector fields of the model, p > 0 is a homogenous
constant of motion, and the field cf> depends on the time coordinate t > 0 and the
angle 6 E S1. The field 7 gets almost fully determined during the gauge fixing
procedure in terms of p. cf> and its canonical momentum P^.5 Only the zero mode
of 7 remains free, containing a degree of freedom Q that is conjugate to P : = hip.
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2672
The degrees of freedom of the reduced system are the canonical pairs (Q, P) and
(4>, Pcf,)- The homogenous constraint that remains on the model generates
translations in S1 and has the form Co := § dQP^dgi^j \f2Tx. The reduced Hamiltonian that
generates the evolution is Hr := § d6[P2 + t2(dg(j>)2]/(2t), which is independent of
Q and P. In the following we will obviate these homogenous non-dynamical degrees
of freedom and concentrate our discussion on the field <fi (and its momentum).
The equation of motion for this field is <920 + (dt(j>/t) — dg(j> = 0, which is that
of a free scalar field with axial symmetry propagating in a 2+1 background with
metric ds^ = ~dt2 + d62 + t2da2. The smooth real field solutions have the form
f = S^L-oo [Anfn(t, 0) + A*/*(f, 6)] where * denotes complex conjugation and
fn{t,9) := Ut)e«", /„(*):= M|!*) n^ 0, f0(t) := 1=^. (2)
Ho is the zeroth-order Hankel function of the second kind. The solutions form a
symplectic vector space with symplectic structure Q(<fii, 1^2) '■= § d6[ip2tdtipi —ipitdt^]-
With this structure, the constants (An, A*) behave like pairs of annihilation and
creation-like variables. In Pierri's quantization, these pairs are promoted to
annihilation and creation variables.2 However, as we have commented, the dynamics
dictated by Hr does not admit a unitary implementation.4 Actually, the problems
with unitarity can be traced back to the appearance of the factor t in the symplectic
structure. At this stage, we note that this factor can be absorbed by scaling the field
by \ft. Moreover, one can check that for large wave numbers |n|, the scaled solutions
y/tfn(t, 9) behave as the standard modes of a free scalar field in a two-dimensional
flat background (trivially equivalent to a three-dimensional formulation with axial
symmetry), scalar field which clearly admits a unitary quantum evolution.
3. New field description of the model
Motivated by our previous discussion, we perform the time dependent canonical
transformation £ = \fi<j> and P5 = (P^ + <j>/2)/\fi. The linear contribution in 0
to Pj is chosen so that the new Hamiltonian, which generates the evolution after
the transformation, does not contain products of the field with its momentum.6
This Hamiltonian is #« := § d6 \p2 + (de£)2 + £2/(4t2)l /2. Thus, H$ is the sum
of the Hamiltonian for a free scalar field in a two-dimensional flat spacetime and
a time dependent potential that vanishes asymptotically for large times. The new
field equation is <92£ - 9|£ + £/(4t2) = 0, whose solutions are obviously of the form
C = En=-oo [Angn{t,6)+A*ng*n{t,8)} with gn(t,0) := Vtfn(t,6).
Let us expand £ = £~=-oo £(n)eine/v^F and Pc = £~=-oo P^n)eme/^ and
In)
use the Fourier coefficients £(„-, and P, ; as coordinates in the canonical phase space.
Alternatively (and disregarding the zero mode)6 we can use as coordinates the pairs
of annihilation and creation-like variables given by bn = \n\£/n\ + zPJ /y/2|n|
and its complex conjugate 6*. For convenience, we will group them in the vectors
Bm := (6m, b*_m, 6_m, b*m)T with m > 0, where T denotes the transpose.
2673
On the other hand, given a fixed section of constant time t = to, we can establish
an isomorphism Ito between the canonical phase space and the covariant phase
space, identified with the space of smooth solutions, so that the values of bn and
6* at t = to can be regarded as initial conditions and adopted as coordinates to
describe the distinct solutions. Furthermore, the evolution from t = to to t = ti
can be understood as the map between initial conditions given by IflIt0 (m the
coordinates Bm). This evolution takes the form Bm(ii) = W(xln)W~1(xc?n)'Bm(to),
where xlm := rati and the matrix W(xlm) provides the relation between Bm(ij) and
the constants Am := (Am,A*_m,A-m,A:^n)T that determine the solutions in the
basis {gn(t,6),gn(t,6)}. This relation, Bm(ij) = W{xim)Am, is given by6'7
c(x) := J — H0(x) - d*(x), d{x) :=
l + ^c)H*0(x)-iH*1(x)
,(4)
where 0 is the zero 2x2 matrix and Hi the first-order Hankel function of the second
kind. Since |c(x)|2 — \d(x)\2 = 1, W(xlm) is in fact a Bogoliubov transformation.
One can quantize the model introducing a Fock representation in which the
variables bn and b^att — t0 are promoted to annihilation and creation variables.6 From
our analysis, the evolution admits a unitary implementation in this quantization if
so does the transformation defined by the matrices W{xlm) (Vm > 0) for all values
of ti > 0. This condition is equivalent to demand that the sequence {d(mti)} be
square summable Vij > 0, namely Ylm=i \d(mti)\2 < oo. Actually, using Hankel's
asymptotic expansions, one can check that \d{mti)\2 is of order 1/m4 for large m.
Therefore the sequence is square summable, so that the evolution is unitary on the
introduced Fock space. Moreover, the result is also valid6 on the physical subspace
determined by the quantum version of the constraint Co that remains on the system.
In conclusion, by means of a time dependent canonical transformation in the
gauge fixed polarized Gowdy model (that amounts to a new field parametrization
of the metric),6 we have been able to attain a Fock quantization in which the classical
dynamics is implemented unitarily, in contrast with the problems found in Pierri's
quantization. In this respect, our construction provides the first consistent quantum
cosmological model with local degrees of freedom obtained in the literature.
References
1. R.H. Gowdy, Ann. Phys. 83, 203 (1974).
2. M. Pierri, Int. J. Mod. Phys. D 11, 135 (2002).
3. C.G. Torre, Phys. Rev. D 66, 084017 (2002).
4. A. Corichi, J. Cortez and H. Quevedo, Int. J. Mod. Phys. D 11, 1451 (2002).
5. J. Cortez and G.A. Mena Marugan, Phys. Rev. D 72, 064020 (2005).
6. A. Corichi, J. Cortez and G.A. Mena Marugan, Phys. Rev. D 73, 084020 (2006).
7. A. Corichi, J. Cortez and G.A. Mena Marugan, Phys. Rev. D 73, 041502 (2006).
ON THE INTERACTION OF THE GRAVITATIONAL FIELD OF A
COSMIC STRING WITH SOME QUANTUM SYSTEMS
GEUSA de A. MARQUES
Departamento de Fisica, Universidade Federal de Campina Grande, Campina Grande, Pb,
Brazil
gmarques@df. ufcg. edu. br
V. B. BEZERRA
Departamento de Fisica, Universidade Federal da Paraiba, Joao Pessoa, Pb, Brazil
valdir@fisica.ufpb.br
1. Introduction
The study concerning the influence of potentially observable effects of gravitational
fields at the atomic level has been an exciting research field. These studies
considered a problem which suggests potentially observable effects of gravitational fields
at atomic level and showed that an atom placed in a gravitational field is
influenced by its interaction with the local curvature as well as with the topology of the
spacetime1 ~.3
The spacetime of a cosmic string is quite remarkable: its geometry is flat
everywhere apart from the symmetry axis. Thus, the external gravitational field due to a
cosmic string4 may be described by a commonly called conical geometry. Therefore,
there is no local gravity in the space surrounding a cosmic string, but its conical
structure can induce several effects like, for example, the shifts in the energy levels
of a hydrogen atom.3
We will investigate the problem concerning the effects of the topology of the
spacetime generated by a cosmic string at the atomic level by considering the
question of how the shifts in the energy spectrum of a particle are when it experiences
different potentials, like the Kratzer and Morse potentials in this spacetime.
2. Kratzer and Morse potentials in the spacetime of a cosmic
string
In order to determine the energy spectrum of a non-relativistic quantum particle
interacting with a potential and in the presence of the gravitational field of a
cosmic string, let us consider the time-independent Schrodinger equation in a curved
spacetime, which reads
-^2LBiP + ViP = EiP (1)
where V|B = g~l/2di [g%3gl/2dj) (i,j = 1,2,3) is the Laplace-Beltrami operator
and g = det {g^).
The line element corresponding to the cosmic string spacetime is given, in
spherical coordinates, by
2674
2675
dsz
-dtz + drA + rAd8A + azrz shr Odtp
(2)
where the parameter a = 1 — AG p. runs in the interval (0,1], with fi. being the linear
mass density of the cosmic string (In this paper we will consider c = 1).
Firstly, let us consider the Schrodinger equation for the Kratzer potential in this
background, which can be separated as
h2 d2u{r)
2/i dr2
-2D
1A?_
2^2"
A
u(r) = Eu(r),
and
1 d f . dQ
— suit' —
smBde \ d6
a2 sin 0
-6-A6 = 0
(3)
(4)
where —2D
is the Kratzer potential, D and A are positive
constants. A is a separation constant and we have used the fact that rp(r,6,ip) =
^p-e(6)eimv; m = 0, ±1, ±2, ±3....
The solution of Eq. (4) is given by a generalized associated Legendre function
Qt ^a) (cos8), in the sense that Z(Q) = I — (1 — ^)|m| and m(Q) = ~ are not necessarily
integers.
The solution of Eq. (3) is given in terms of the confluent hypergeometric function,
M(r), as
1
u(r)=M[- + -y/l + 4P
i) + 24^;/32
where P = l(a){l(a) -r ^ -r ^.-^
This solution diverges, unless
_ JL
2jj,E
h2 ■
, 1 + Vl + 4P; 2/3?
(5)
0»BD\
'2~h2~
'1 + 4 I/(«)(/(«) + 1) + 2^- ) - -^ = -n; n = 0,1, 2....
Then, from this condition we find the energy eigenvalues
E„
2D2A2n
H2
hh11
■Al(a)(l
(q)V'(q)
1
uBD
h2
(6)
(7)
In order to estimate the effect of the presence of the cosmic string on the energy
shifts, let us take a = 0.999999, which corresponds to a GUT cosmic string. In this
case, there is a decrease in the energy spectrum, corresponding the the first two
levels, of about 10_3% as compared to the flat Minkowski spacetime value.
Now, let us take into account the Morse potential, which reads as
V(r) = -D-
-Muj2r2.
(8)
2676
This potential is similar to the one corresponding to the isotropic harmonic oscillator
with frequency lo plus a constant term. For this case, the angular solution is the
same corresponding to the Kratzer potential.
The radial solution of the Schrodinger equation, R(r), can be written as R(r) =
^^, where g{r) satisfies the equation
d2g(r) M2u2r2 . x , (l,a) +1) . . 2M ^ / x n
whose solution is
7(r) = exp f-^M^A r3 + W1+4'(-)('(-)+1)F(r),
(10)
EM = ftu
D, (11)
where
F(r) = m{\-§m. + ^ + yi + Al(a){l(a) + 1), \ + y±l(a){l(a) + 1); ^f)
is the confluent hypergeometric function.
Applying similar condition given by eq. (6), in order to avoid divergence, we get
the following result for the energy spectrum
\ K/l+4Z(a)(Z(a)+l) - 1J + 2nM + |
An estimation of the shift in the energy levels for this case, shows that there is
a decrease in the energies of about 10~5% for GUT cosmic strings as compared to
the flat spacetime corresponding value.
3. Conclusions
The obtained results tell us that the energy spectra are modified as compared to
the flat spacetime Minkowski result and these shifts are connected with the conical
structure of the spacetime generated by a cosmic string. In other words, these shifts
in the energies are due completely to the topological features of this spacetime.
Acknowledgments
We acknowledge CNPq and FAPESQ-PB/CNPq(PRONEX) for partial financial
support.
References
1. J. Audretsch and G. Schaffer, Gen. Rel. Grav. 9, 243 (1978); 9, 489 (1978).
2. L. Parker, Phys. Rev. Lett. 44, 1559 (1980); L. Parker and L. Pimentel, Phys. Rev.
D44, 3180 (1982).
3. Geusa de A. Marques and Valdir B. Bezerra, Phys. Rev. D66, 105011 (2002).
4. A. Vilenkin, Phys. Rev. D23, 852 (1981).
EINSTEIN-ROSEN WAVES COUPLED TO MATTER
J. FERNANDO BARBERO G.
Institute de Estructura de la Materia, CSIC
Serrano 123, 28006 Madrtd, Spain
fbarbero@iem.cfmac.csic.es
INAKI GARAY
Institute de Estructura de la Materia, CSIC
Serrano 123, 28006 Madrid, Spain
igael@iem.cfmac.csic.es
EDUARDO J. S. VILLASENOR
Grupo de Modelizacion y Simulacion Numerica, Universidad Carlos III de Madrid
Avda. de la Universidad 30, 28911 Leganes, Spain and
Institute de Estructura de la Materia, CSIC
Serrano 123, 28006 Madrid, Spain
ejsanche@math.uc3m. es
We discuss some physical applications of a proposed canonical quantization of Einstein-
Rosen waves coupled to a massless scalar field. In particular we will explore how to
use the particle-like modes of the matter field to operationally explore the quantized
geometry of the system. We will do this in several independent but consistent ways: By
using two-point functions, Newton-Wigner states, and one particle states with suitable
wave functions. We show how some features of a space-time equipped with a classical
metric emerge in an appropriate classical limit.
The use of symmetry reductions of General Relativity as toy models for quantum
gravity has a long tradition. In addition to the ininisuperspace reductions (Bianchi
models) with a finite number of degrees of freedom there are other midisuperspace
models, such as Einstein-Rosen waves,1 that arc interesting because they describe
local degrees of freedom. This system has some other appealing features; among
them we would highlight its residual diffeomorphism invariance interesting to
understand the role of this symmetry in quantum gravity-, and the fact that it can be
exactly solved both at the classical and quantum levels.2 It has been recently shown
by the authors3 that it is possible to further enrich the model by adding some matter
fields (specifically massless scalars) that can be included while keeping its
solvability. This gives us the possibility of using their field quanta as quantum test particles
to probe the emergence of classical and quantum geometry in an operational way4
(much in the way light rays and measuring rods are used in relativity).
The key fact that leads to the complete solvability of the system consisting of
Einstein-Rosen waves and a massless scalar field is that, after performing a first
reduction of the system5 in the direction of the translational Killing vector field,
the resulting action corresponds to 2+1 general relativity coupled to two massless,
axially symmetric, scalar fields. One of them encodes the gravitational degrees of
freedom -as in the case where no matter is present- whereas the other describes
matter. The axial symmetry comes from the extra Killing vector field present in
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the model. In order to have a well defined action principle it is necessary to include
appropriate surface terms in the action. In the present case it is also convenient
to include a fiducial Minkowskian metric to define the zero value of the energy.
These surface terms are very important to define the Hamiltonian H, in fact, its
most salient feature3'6 is the fact that it is given by II = 2(1 — e~H°/2) where Hq
denotes the free Hamiltonian for two non-interacting, massless, axially symmetric
scalar fields in 2+1 dimensionsa. Notice that even though the auxiliary Hamiltonian
is free, the fact that the true physical Hamiltonian is a non-linear function of the
former renders the theory interacting (albeit in a non standard way).
Taking advantage of the fact that the Hamiltonian II can be related to a free
Hamiltonian as described above, the quantization of the system can be carried out
by using a Fock space description for the auxiliary free model.7 Here the Hilbert
space of the combined system H = Tg ® Ts is written as a tensor product of two
Fock spaces associated to the gravitational and matter sectors. The vacuum state in
H is written in terms of the corresponding vacua as |0) = |0)s(g)|0)s. By introducing
creation and annihilation operators for scalar particles of gravitational or scalar type
Ag, As, Ag, and A\ it is straightforward to write the quantum Hamiltonian and the
unitary evolution operators as
H = 2
1 - exp (-1- J™ k[A\{k)Ag{k) + Ai(k)Ax(k)}dk)
(1)
U(t,to)=ex.p^-i{t-t0)Hj. (2)
The unitary evolution operator defines the full quantum dynamics of the model so
we are in the position of computing the time evolution of any state vector in the
Hilbert space of the system.
Of particular interest to obtain information about the emergence of classical
spacetime are the two point functions (fi|0s,g(i?2, t2)(f>s,g{Ri,ti)\n). where 0.Sj9
denotes the field operators for the gravitational and matter scalars. Here R.\, R2 are
radial coordinates and t\, and t2 time coordinates. In practice it is convenient to
use the parameters p12 = -j^r and t = 1\qy , introduce the adimensional variable
q = AGk, and write
(fi|^a.g(ii2,i2)^8,!7(ii1,f1)|n>= / Jo(piq)Jo(p2q)ex.p[-iT{l-e-")]dq. (3)
Jo
The two point functions can be used to study the microcausality of the
system (by considering field commutators) or as approximate propagation amplitudes
between different spacetime points. The main result that can be obtained about
microcausality8-10 is the appearance of the characteristic smearing of light cones
expected on general grounds in quantized theories of gravity. Another interesting
feature that can be seen in this analysis is the appearance of distinct spacetime cells
with dimensions defined by the characteristic length scale of the system (reminiscent
iWe use units h = SG = c = 1. G is the effective gravitational constant of the reduced model.
2679
of the Planck scale in full 3+1 dimensional gravity). The image of the propagation
of quanta (both of gravitational or scalar type) obtained from the study of the two-
point function directly supports the conclusions obtained via causality arguments
concerning the special role played by the symmetry axis in the quantization of the
system. This is manifest as an enhanced probability amplitude to find field quanta
there. It is also possible to see that field quanta define approximate classical
trajectories but the impossibility of thinking of the two point functions as normalized wave
functions makes this interpretation a little heuristic. This issue can be addressed
by introducing a suitable base of (radial) position eigenstates. It is straightforward
to define an orthonormal basis of vectors analogous to the Newton-Wigner11 states
introduced in Quantum Field Theory to address the problem of localizability and
the definition of a position space representation. Once these states are introduced it
is possible to consider (radial) position space wave functions and study their space-
time evolution. The results obtained in this study suggest that wave functions with
a wide enough support at a time to provide well defined spacetime trajectories that
do not spread much under time evolution. On the other hand when narrow supports
are considered there is a considerable spreading inside the light cone. One can then
see the emergence of null classical trajectories defined by the time evolution of these
wave functions when the self-gravitational effects due to test field quanta can be
neglected. As we hope to have shown, the system given by Einstein-Rosen waves
coupled to matter provides interesting tools to explore quantum gravity and offers
interesting avenues to understand this difficult problem.
Acknowledgments
We want to thank M. Varadarajan for discussions. I. Garay is supported by a
Spanish Ministry of Science and Education (MEC) under the FPU program. We
acknowledge the support of MEC under the research grant FIS2005-05736-C03-02.
References
1. A. Einstein and N. Rosen, ,/. Franklin. Inst. 223, 43 (1937).
2. K. Kuchaf, Phys. Rev. D4, 955 (1971).
3. J. F. Barbero G., I. Garay, and E. J. S. Villasenor, Phys. Rev. Lett. 95, 050301 (2005).
4. J. F. Barbero G., I. Garay, and E. J. S. Villasenor, Phys. Rev. D74, 044004 (2006).
5. R. Geroch. J. Math. Phys. 12, 918 (1971).
6. A. Ashtekar and M. Varadarajan Phys. Rev. D50, 4944 (1994).
7. A. Ashtekar and M. Pierri, J. Math. Phys. 37, 6250 (1996).
8. J. F. Barbero G., G. A. Mena Marugan, and E. J. S. Villasenor, Phys. Rev. D67,
124006 (2003).
9. J. F. Barbero G., G. A. Mena Marugan, and E. J. S. Villasenor, J. Math. Phys. 45,
3498 (2004).
10. J. F. Barbero G., G. A. Mena Marugan, and E. J. S. Villasenor, J. Math. Phys. 46,
062306 (2005).
11. T. D. Newton and E. P. Wigner, Rev. Mod. Phys. 21, 400 (1949).
ELECTROMAGNETIC RADIATION FROM A CHARGE
ROTATING IN SCHWARZSCHILD SPACETIME
JORGE CASTINEIRAS, LUIS C. B. CRISPINO* and RODRIGO MURTA
Departamento de Fisica,Universidade Federal do Para, 66075-110, Belem, PA, Brazil
* E-mail: crispino@ufpa.br
GEORGE E. A. MATSAS
Instituto de Fisica Teorica, UNESP, Rua Pamplona, 145, 01405-000, Sao Paulo, SP, Brazil
E-mail: matsas@ift.unesp.br
We analyze the radiation emitted by an electric charge rotating around a chargeless
static black hole in the context of quantum field theory in curved spacetimes.
In Schwarzschild spacetime, the Lagrangian density of the electromagnetic field
in the modified Feynman gauge is given by1
1 1
.-F F^v - -C'2
4 "" 2
(1)
with g = r2sin6», G = VM,, + K^A^ and K» = (0, df /dr, 0,0), where / = 1 -
2M/r. The corresponding Euler-Lagrange equations are
XJVF»V + VG - K^G = 0. (2)
The physical modes can be written as
x a«F'-> roil) i ^(r)^ d+Yim) e~ibJt (3)
and
^(IWm) = (Q) Q) r{pU? (r) ^m r(pUn (r) y^ ^t (4)
with I > 1 (since the gauge condition G = 0 is not satisfied for 1 = 0). Ylm and
yjm are scalar and vector spherical harmonics, respectively. The radial part of the
physical modes satisfies the differential equation
(-2 " Vs) [rtpft W] + ffr (/£ W (r)]) = 0, (5)
where A = I, II and Vs = fl(l + l)/r2 is the Schwarzschild scattering potential.
Now let us consider an electric charge with 6 = ir/2, r = Rg and angular velocity
Q = d<p/dt = const > 0 (as defined by asymptotic static observers), in uniform
circular motion around a Schwarzschild black hole, described by the current density
j% (xv) = -j=^5 (r - Rs) 5 (6 - tt/2) S {<p - tot) u". (6)
2680
2681
Here q is the coupling constant and
MM(^,fis)= . 1 = (1,0,0,») (7)
V S> y/f (RS) - R%H2 K ' K '
is the charge's 4-velocity. We note that jg is conserved, V^jg = 0, and thus
/s dT,^ 'jg (xv) = q for any Cauchy surface E.
Next let us minimally couple the charge to the field through the Lagrangian
£■1 = V~~9 Js^fj.- Then the emission amplitude at the tree level of one photon with
polarization A and quantum numbers (n, u) I, m) into the Boulware vacuum is given
by
jXnUm =i dix JZTg ^(Wm) _ (g)
It can be shown that J^n^lm oc 5 (a; — mfl). This implies that only photons with
frequency uq = rutt are emitted once the charge has some fixed il = const.
The total emitted power is
00 ' r + oo
Ws= J2 Y, 5Z5Z / duJUJ \AXn"lm\ /T, (9)
A=I,II n=<-,-> (=1 m-1 ^°
where T = 2ttS (0) is the total time as measured by the asymptotic static observers.
Using now Eqs. (3)-(4) and (6)-(7) we rewrite Eq. (9) as
oo /
Ws = Y, Y.Y, \wlsn"olm + wllnuolm] (10)
n=<—,—> 1=1 m—1
with
wInuolm = 2WmZU? ( _ 2M\
S [1(1 +1)]2 V RsJ
2
d [Rsvln0i(Rs)}
dRs
2
|yim(7r/2,0)|2 (11)
and
Wnnu0lm = 2^q2mnZ [flg ^lln {Rg)f \ylm (7r/2,0)|2 . (12)
According to General Relativity for a stable circular orbit around a
1 /3
Schwarzschild black hole we have Rs = (M/9?) ' . We use this relation to
compute numerically the emitted power given by Eqs. (10)-(12) as a function of fl for
stable circular orbits. The result is plotted as the solid line in Fig. 1. The main
contribution to the total emitted power comes from modes with angular momentum
I = m = 1. As a general rule, (i) the smaller is the I, the larger is the contribution
to the total radiated power, and (ii) for a fixed value of I, the larger is the m, the
larger is the contribution to the total radiated power.
It is interesting to note that the magnitude of the total radiated power in the
electromagnetic case is approximately twice the numerical result found previously
2682
" 2
"a
0.06
Fig. 1. The total power Wg emitted by the electric charge rotating around a Schwarzschild black
hole is plotted as a function of the angular velocity tt as measured by asymptotic static observers.
The solid line represents our numerical result whereas the dashed line represents our analytic result
for low frequencies. The I summation in Eq. (10) is performed up to I = 6. Mtt ranges from 0 up
to 0.068 (associated with the innermost stable circular orbit at Rs = 6iW).
for a scalar source coupled to a massless Klein-Gordon field.2 In principle, this is
not surprising because of the fact that photons have two physical polarizations.
Notwithstanding, it should be emphasized that the two polarizations contribute
quite differently to the emitted power. For our rotating charge, the contribution
from mode A = II is negligible when compared with the one from mode A = I.
Low-frequency approximations for the physical modes can be used to obtain an
analytic approximation for the emitted power.1 The result is plotted as the dashed
line in Fig. 1. We see from it that the numerical and analytical results differ sensibly
as the charge approaches the black hole but coincide asymptotically, since far away
from the hole only low frequency-modes contribute to the emitted power.
Acknowledgments
The authors are grateful to Conselho Nacional de Desenvolvimento Cientifico e
Tecnologico (CNPq) for partial financial support. R. M. and G. M. would like to
acknowledge also partial financial support from Coordenagao de Aperfeigoamento
de Pessoal de Nivel Superior (CAPES) and Fundacao de Amparo a Pesquisa do
Estado de Sao Paulo (FAPESP), respectively.
References
1. J. Castineiras, L. C. B. Crispino, G. E. A. Matsas and R. Murta, Phys. Rev. D71,
104013 (2005).
2. L. C. B. Crispino, A. Higuchi and G. E. A. Matsas, Class. Quant. Grav. 17, 19 (2000).
RECENT DEVELOPMENTS IN QUANTUM ENERGY
INEQUALITIES
CHRISTOPHER J. FEWSTER
Department of Mathematics, University of York,
Heslington, York YO10 5DD, United Kingdom
cjf3@york. ac.uk
Two recent developments in the theory of Quantum Energy Inequalities (QEIs) are
reported: first, an absolute QEI in curved spacetimes; second, the use of local covariance
in combination with QEIs to obtain a priori bounds on the renormalized stress tensor.
Keywords: Quantum field theory in curved spacetime, Quantum energy inequalities
1. Introduction
In General Relativity, the stress tensor Tab is often assumed to obey the Weak
Energy Condition (WEC) that TabUaub should be everywhere nonnegative for all
timelike ua. Although the classical energy conditions are violated by quantum fields,
there are remnants of these conditions, called Quantum Energy Inequalities (QEIs)
[or, more briefly, Quantum Inequalities (QIs)] which apply to suitable averages
(T(f)>w := J(Tab)ufabdvo\
of the expectation value of the renormalized stress-energy tensor in state u>. (See
Refs. 1-3 for recent reviews and references.) There are two types of QEIs: absolute
QEIs (AQEIs), which take the form
(T(f))w > — Q(f) for all (physically reasonable) states ui,
and difference QEIs (DQEIs), which take the form
(T(f))w — (T(f))Wo ^ — Q(fi^o) f°r an (physically reasonable) states u),
where loq is a reference state. As a concrete example, the massless scalar field in
four-dimensional Minkowski space obeys the AQEI
J'(T00(t,0))„\g(t)\2 dt> -j^ J'\g"(t)\2 dt (1)
for all Hadamard states u and all smooth, real-valued functions g vanishing outside
a compact set. A simple consequence4 is that if (Tbo)w(£,0) < £ for 0 < t < t, then
£ > —C/t4, where C = 3.16... in units where h = c = 1. This illustrates the close
links between the QEIs and intuition based on the uncertainty principle.
2. Absolute Quantum Energy Inequalities
In curved spacetimes, the most general results known are difference QEIs (e.g.,
Ref. 5 for the scalar field). This hampers attempts to use QEIs to constrain exotic
spacetimes6^8 because one does not typically have explicit access to a reference
2683
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state uj0. The typical approach is to use the equivalence principle to argue that
Minkowski space QEIs such as Eq. (1) apply on sufficiently small scales. Here, we
describe recent work with C.J. Smith,9 in which the first explicit AQEIs in general
four-dimensional curved spacetimes are obtained.
Consider the quantized minimally coupled Klein-Gordon field with mass m > 0
in four spacetime dimensions. In state ui, the expected renormalized stress tensor is
(T^)M) = [P{nu){K - Hk)} (x,x) - Q{x)^u + C^{x) (2)
when expressed in terms of a tetrad, where AbJ(x, x') = ((j)(x)(j)(x/))u, is the two-point
function, the P^ are differential operators given by
JV = V„ ® V„ - ^V^Q/?V° ® V/3 + 2m2rt^
and Hk(x,x') is the partial Hadamard sum10
A1/2 k
In Eq. (2), the term Q(x) is added to ensure conservation, and Cliu(x) is a conserved,
local curvature term. The definition is independent of k provided it is at least 2.
Our AQEI may now be stated as follows:
Theorem 2.1. Let O be an open region in a globally hyperbolic spacetime such that
the Hk exist on OxO. Let 7 : / —-> O be a proper-time parameterisation of a smooth,
future-directed timelike curve, where I is an open interval o/K, and suppose e° is
a tetrad on O which is invariant under Fermi-Walker transport along ■y, where it
obeys eg|7 = ja. Then the AQEI
I(Tabjaib)^g(r)2 dr > /'(Cab7a7b - Q)5(r)2 dr - - f° Tk{-a, a) da
J-y J7 7T JO
holds for any Hadamard state u, any g £ C^°(I;M.) and any k > 5, where
Fk(T,r') = g(r)g(r') (P0o^)(7(t),7(t'))
and Hk(x,x') = \ [Hk{x, x') + Hk(x',x) + iE(x,x')], with E denoting the advanced-
minus-retarded fundamental bisolution. (We write F{k) = Jdnxelk'xF{x).)
This bound is similar in form to an older DQEI,5 in which the terms involving Cab
and Q are absent and Hk is replaced by the two-point function KbjQ of a reference
state. The proofs differ in that Aw — AWo is smooth, while Aw — Hk is only C*,
necessitating a more refined analysis using Sobolev wave-front sets. Similar results
may be obtained for averages over worldvolumes and other timelike spacetime sub-
manifolds. Note that the AQEI bound is independent of the state uj and is defined
in terms of local geometrically constructed objects such as the Hadamard series
coefficients (the result is independent of the particular choice of eV).
As the support of g shrinks, the a^_ contribution to Hk dominates: the bound
becomes Minkowskian. A more careful analysis of this limit, giving precise estimates,
would justify the arguments used to apply QEIs to constrain exotic spacetimes.
2685
3. QEIs and local covariance
A general DQEI on spacetime M has the schematic form
<Tm(t))w - <Tm(t))W0 > -QM(f,uo)
(using M to denote the underlying manifold, its metric and choices of (time)-
orientation). If an isometry ip embeds a globally hyperbolic spacetime N as a
globally hyperbolic subset of a globally hyperbolic spacetime Ai, then we may pull back
a state u> on M. to a state ^p*us on TV so that {T/v^f)},/,.^ = (Tm(4'*^))ui, where xjjj
is the push-forward of f from M to TV. This relation asserts that the stress-energy
tensor is covariantly defined.11 Certain DQEIs are also covariant, i.e.,
and this permits us to use QEIs on M. to constrain energy densities on TV.4'12,13
As a simple application,4 suppose a stationary spacetime M contains a stationary
timelike geodesic segment 7 of proper duration To, which may be enclosed in a flat
simply connected open globally hyperbolic subset N' of N'. Then N' is isometric to
a globally hyperbolic subset of Minkowksi space, and we may apply the Minkowski
QEIs along 7, to obtain an a priori bound
(Tabiaib)u,0>-^~^ C = 3.16...,
on the energy density on 7 of the ground state ujq of the Klein-Gordon field on J\f.
See Refs. 4,12 for other examples; the same idea can be used to prove the averaged
null energy condition for null geodesies with suitable flat neighborhoods.14
References
1. L. H. Ford, in 100 Years of Relativity - Space-time Structure: Einstein and Beyond
(World Scientific, Singapore, 2006) gr-qc/0504096.
2. T. A. Roman, in Proceedings of the Tenth Marcel Grossmann Meeting on General
Relativity gr-qc/0409090.
3. C. J. Fewster, in XlVth International Congress on Mathematical Physics (World
Scientific, Singapore, 2005); Expanded and updated version: math-ph/0501073.
4. C. J. Fewster and M. J. Pfenning, J. Math. Phys. 47, 082303 (2006).
5. C. J. Fewster, Class. Quantum Grav. 17, 1897 (2000).
6. L. H. Ford and T. A. Roman, Phys. Rev. D 53, 5496 (1996).
7. M. J. Pfenning and L. H. Ford, Class. Quantum Grav. 14, 1743 (1997).
8. C. J. Fewster and T. A. Roman, Phys. Rev. D 72, 044023 (2005).
9. C. J. Fewster and C. J. Smith, in preparation.
10. R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole
Thermodynamics (University of Chicago Press, Chicago, 1994).
11. R. Brunetti, K. Fredenhagen, and R. Verch, Commun. Math. Phys. 237, 31 (2003).
12. P. Marecki, Phys. Rev. D 73, 124009 (2006).
13. C. J. Fewster, math-ph/0611058.
14. C. J. Fewster, K. Olum and M. J. Pfenning, gr-qc/0609007.
BLACK HOLES AS BOUNDARIES
IN 2D DILATON SUPERGRAVITY
LUZI BERGAMIN
ESA Advanced Concepts Team, ESTEC - DG-PI
Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
Luzi. Bergamin@esa. int
DANIEL GRUMILLER
Center for Theoretical Physics, Massachusetts Institute of Technology
77 Massachusetts Ave., Cambridge, MA 02139, USA
grumil@lns.mit.edu
We discuss 2D dilaton supergravity in the presence of boundaries. Generic ones lead
to results different from black hole horizon boundaries. In particular, the respective
numbers of physical degrees of freedom differ, thus generalizing the bosonic results of
hep-th/0512230.
1. Introduction
Frequently it is argued that the microstates responsible for the Bekenstein-Hawking
entropy should arise from some physical degrees of freedom located near or on the
black hole horizon (cf. e.g. Ref. 1 and references therein). Recently we have provided
evidence within the framework of 2D dilaton gravity that instead entropy may
emerge from the conversion of physical degrees of freedom, attached to a generic
boundary, into unobservable gauge degrees of freedom attached to the horizon.2'3 In
this joint proceedings contribution we generalize such considerations to 2D dilaton
supergravity (SUGRA).
We start with the first order 2D dilaton SUGRA action3-
S= f XIdAI + l-PIJAjAAI. (1)
Jm l
IM
We use a notation that is a convenient mixture between the one employed in our
previous paper on the subject2 (consistent with Ref. 5) and our papers on SUGRA.6~9
The graded 1-form fields Ai comprise the (dual) spin-connection cj, the Zweibeine
e±± and the gravitino ij)±. The graded 0-form fields X1 comprise the dilaton 0,
Lagrange-multipliers for torsion X±ziz and the dilatino x± ■ They span a target-
space equipped with a Poisson tensor PIJ, viz., a (graded) Poisson manifold.10
The Poisson tensor is given by Eqs. (2.8), (2.16)-(2.19) in Ref. 9; we refrain from
presenting these formulas here. The action (1) is not consistent with the Gibbons-
Hawking-York prescription used in Ref. 2 but nevertheless a valid (and for various
aThe superspace action by Park and Strominger describes the same theory and has several
advantages over (1). However, the solution of all constraints, the construction of classical solutions
and path integral quantization is much simpler starting with the first order action.
2686
2687
purposes useful) starting point. It is advantageous to define the canonical variables
qi = Au , p1 = X1 , qi=A0I , p1 « 0 . (2)
The indices 0,1 refer to world-sheet coordinates a;0, a;1, where x° plays the role of
time. Evidently, the canonical momenta p1 are primary constraints. We keep all
conventions regarding spinor calculus as defined e.g. in the appendix of Ref. 7. To
keep a simple representation of the constraints we need a Poisson bracket with
{qi,p'J} = {-l)IJ+l{pIJM = Sj5(x - x') , (3)
which can be achieved by the definition
The boundary dAi is supposed to be a hypersurface of constant x1. As in2 it is
considered to be a lower one.
In Section 2 we present results of the constraint analysis and possible choices
for boundary conditions. In Section 3 we discuss the gauge fixing and construct the
reduced phase space. The interpretation of our results is analogous to the bosonic
one in Refs. 2,3, so we focus on issues peculiar to SUGRA.
2. Constraint analysis and boundary conditions
With standard methods we obtain the secondary constraints
G'lv] = Jdx1 (d.p1 +PIJqj)v + pIv\dM « 0. (5)
The constraint algebra including boundary terms reads
{G'[v], GJm = GK[rt]dKPIJ - (pKdK - 1) PIJrt\aM ■ (6)
Notice that all brackets {pJ,GJ} vanish with this choice of the boundary action
in contrast to Ref. 2. Moreover, the boundary term in (6) vanishes whenever the
Poisson tensor is homogeneous of degree one. This is always true for the generators
of local Lorentz transformations, i.e. for the brackets {G^[rj\, GJ[^]}, and the basic
relation defining the supersymmetry algebra {G±[r/], G±[^]} = —2v/2G±±. Among
the purely bosonic models the boundary terms vanish completely for the Jackiw-
Teitelboim model,11 for the Witten black hole12 and for models with an (A)dS2
ground state, as noted in Ref. 13. This characteristic is retained upon supersym-
metrization because the full Poisson tensor is homogeneous of degree one if the
bosonic sector exhibits this property.
Variation of the action (1) yields the boundary conditions
p'SqilaM^O. (7)
As in Ref. 2 we implement them by means of constraints on the phase space with
support at the boundary only. The choices for the three bosonic components are
2688
similar to that work and will be recapitulated briefly below. Here we concentrate
on the fermionic variables, where two different choices of boundary constraints,
B±W\ = (<7± ~ A±)7]\aM or B±[ri]=p±ri\aM , (8)
exist. A mixture of the two for the different components of the spinors is conceivable.
To see how the different choices can affect the result one has to construct the
line element [cf. eqs. (100) and (101) in Ref. 7]. Not surprisingly, all fermionic
contributions to the line element vanish at the boundary if both components of
the dilatino are set to zero there. But even with one component of the dilatino set
to zero the bosonic result for the Killing norm emerges, as the (classical) space of
anti-commuting variables is too small to contribute to a bosonic quantity. If instead
of the dilatino both components of the gravitino are fixed at the boundary, p++p
need no longer be proportional to the Killing norm (this conclusion does not depend
on the value of the gravitino chosen at the boundary.) We do not go into further
details of this question here, but simply stick to the first two choices of boundary
conditions, i.e., we always fix at least one dilatino component at the boundary.
The bulk theory contains only first class constraints. However, due to possible
boundary contributions in (6) and as a consequence of the boundary constraints
enforcing (7), terms are generated in the evaluation of Poisson brackets with support
exclusively at the boundary. They convert some of the constraints into second class.
This feature was observed already in the bosonic case.2'3 We shall discuss now its
extension to SUGRA.
Generic Boundary For a generic boundary to solve the boundary problem (7)
among the bosonic variables the only possible choice is 8qj = 0, which we implement
by means of the constraints
Bi[v] = (qz~Ai)v\dM ■ (9)
The only constraint that remains first class for all possible choices in (8) is the
Lorentz constraint G^. Besides G^ there can remain up to two components of p±
first class depending on the choice in (8). The remaining secondary constraints
become second class due to boundary contributions in (6) and possibly additional
contributions from brackets with B±. Moreover, because of
{Bi[vlPJm=5J71i\dM (10)
the Bj make the primary constraints second class.
Horizon As motivated in Ref. 2 a horizon is best described by
$q<i>\dM = 8q++\dM = p \dM = o . (ii)
Consistency with the equations of motion implies q++\gM = 0 as well. Inspecting
the general solution of the SUGRA model (cf. section 6 of Ref. 7) it appears to be
self-evident to choose p+ = p~ = 0 as boundary conditions of the fermionic sector.
However, it should be noticed that this is not enforced.
2689
Again all secondary constraints except the Lorentz constraint become second
class due to the boundary terms in (6) (some of the contributions vanish weakly
due to the B1 constraints, but this is not sufficient to keep an additional constraint
first class.) Among the primary constraints p~~ and p^ remain first class while
all Bi and B1 become second class. For consistency it is then seen that a linear
combination of the second class constraints actually remains first class (the "Dirac
matrix" has determinant zero.)
In summary a difference between a generic boundary and a horizon is seen at
the level of the constraint algebra similar to the result of Ref. 2: if the boundary
is a horizon more first class constraints are present than in the case of a generic
horizon. Therefore, if the boundary is a horizon there are more gauge degrees of
freedom and fewer physical degrees of freedom.
As mentioned above the boundary terms in (6) vanish for a certain class of
models, in which case more first class constraints are encountered. Notice that some
of the G1 still turn into second class constraints due to the Poisson brackets with
B1 from (8) and/or (11).
3. Gauge fixing and reduced phase space
In order to exhibit explicitly the conversion of physical into gauge degrees of freedom
we now construct the reduced phase space in analogy to Ref. 2. In case of a generic
boundary the gaugeb q++ = —i and qj = 0 for all other I can be used, yielding the
straightforward result:
C++ : p++ = p++(.x°) , G*: p* = p»{x0) + ixlp++ , (12)
G+ : p+=p+(x°) , G" : p- = e^ (p-(*°) + ^£^LW^ , (13)
G— : p-
{r-^-*-\£pg^)- .«)
p++(x°)
Here Q, W and w are all functions of the dilaton p^ = 0, cf. Ref. 7 for their
definitions. At this point it matters which boundary conditions were chosen. Quite
generally each choice of a boundary constraint B1 fixes one of the free functions in
(12)-(14), as the analytic continuation of the bulk solution to the boundary must be
equivalent to the boundary value. In the fermionic sector this means that boundary
degrees of freedom can be present only if we fix the gravitino at the boundary.
This conclusion is independent of the nature of the boundary (generic boundary vs.
horizon.)
To proceed it is important to define the boundary conditions in the fermionic
sector. If we choose p+\om = P~ \om = 0 an fermionic integration constants in (12)-
(14) are removed. The derivation and the results within the bosonic sector are the
b Notice that according to our conventions the light-cone components of a vector are purely imag-
7 9
inary. '
2690
same as in the purely bosonic case, since in all relevant equations explicit fermionic
contributions are set to zero by means of the boundary conditions. Like in Ref. 2
the gauge fixing procedure changes if a generic boundary is replaced by a horizon.
Notice however, that the gauge used in Ref. 2 [cf. eq. (6.10) therein] is not suitable
here, as it would fix the boundary value of the dilaton which remains free in the
current approach. A possible choice is to replace q^ = 0 by p++ = i, which together
with the boundary constraint p~~ removes two bosonic degrees of freedom.
There remains the possibility to fix one component of the gravitino and one of
the dilatino. This turns out to be an especially interesting case as one finds that
the boundary prescription for a horizon
8q<t,\dM = 5q++\dM = 5q+\dM = 0 p \om = P~\dM = 0 (15)
together with the equations of motion implies not just q++\dM = 0 but also
q+\sM = 0. Then it can be checked that this leaves two symmetry parameters
(e and e_ in the notation of Ref. 8) unrestricted at the boundary. The algebra
closes trivially among the unbroken symmetries as all commutators vanish
identically. This implies the necessity of yet another gauge condition. A possible choice
is
q++ = -i, 9— =0, p++ = i, <7+ = 0, p+=0. (16)
This eliminates two bosonic boundary degrees of freedom at the horizon, but only
one fermionic one because one can choose p~~ = 0 as boundary condition in the
generic case as well. Thus, the phenomenon of phase-space reduction through
horizon constraints readily generalizes from the purely bosonic case2,3 to SUGRA.
The existence of unbroken supersymmetries at the boundary is not necessarily
connected to the existence of BPS states. In the present case, however, it is easily
seen that the ground state of a horizon respecting half of the supersymmetries
actually is a BPS state. For solutions with vanishing fermions the only condition
for a BPS state is a vanishing body of the Casimir function (mass)8
M = 2w2 - p++p-~eQ = 0. (17)
A BPS solution therefore requires w{<J))\qm = 0. Due to the quadratic nature of
the first term in (17) it is obvious that the mass attains its minimum in the case
of a BPS state and in this sense the latter is the ground state. Once the gauge
(16) is chosen it is easy to see that all classical solutions have vanishing fermions.
Therefore, in this particular gauge all states with C = 0 actually are ground states.
It is worthwhile pointing out that the boundary conditions (15) are quite
different to the ones in Ref. 14. First we use a different boundary action than therein and
second we choose as boundary a horizon. Even with the alternative prescription a
la Gibbons-Hawking-York it is easy to show that a supersymmetric solution of the
variational principle for a horizon is (again) quite different from the one for a generic
boundary. In the latter case one has to choose a vanishing trace of the extrinsic
curvature, in the former this clearly is not an option as the extrinsic curvature is not
even well defined.
2691
Finally, we would like to comment on the duality presented recently,15 which
connects two different actions (1) leading to the same set of classical solutions for
the line element. It was established at the classical level, without supersymmetry
and in the absence of boundaries, only. It is of interest to check what happens when
boundaries and supersymmetry are included. As the boundary terms are insensitive
to the choice of the potentials an extension of the duality to the case with
boundaries is straightforward. Besides redefining the potentials the duality exchanges the
constant of motion with a dimensionful coupling constant in the action. For bosonic
models allowing a SUGRA extension both of their signs are restricted. The duality
maps the positive coupling/positive mass sector of the original theory to the
negative coupling/negative mass sector of the dual theory. Thus, the physical sector of
the original (dual) model is mapped to the unphysical sector of the dual (original)
model.
Acknowledgments
We are grateful to Wolfgang Kummer and Dimitri Vassilevich for collaboration on
Ref. 2.
This work is supported in part by funds provided by the U.S. Department of
Energy (DOE) under the cooperative research agreement DEFG02-05ER41360. DG
has been supported by the Marie Curie Fellowship MC-OIF 021421 of the
European Commission under the Sixth EU Framework Programme for Research and
Technological Development (FP6).
References
1. S. Carlip, Horizons, constraints, and black hole entropy, hep-th/0601041.
2. L. Bergamin et. al., Class. Quant. Grav. 23, 3075 (2006).
3. L. Bergamin and D. Grumiller, Killing horizons kill horizon degrees, gr-qc/0605148.
4. Y.-C. Park and A. Strominger, Phys. Rev. D47, 1569 (1993).
5. D. Grumiller, W. Kummer and D. V. Vassilevich, Phys. Kept. 369, p. 327 (2002).
6. L. Bergamin and W. Kummer, JEEP 05, p. 074 (2003).
7. L. Bergamin and W. Kummer, Phys. Rev. D68, p. 104005 (2003).
8. L. Bergamin, D. Grumiller and W. Kummer, J. Phys. A37, 3881 (2004).
9. L. Bergamin, D. Grumiller and W. Kummer, JHEP 05, p. 060 (2004).
10. P. Schaller and T. Strobl, Mod. Phys. Lett. A9, 3129 (1994).
11. R. Jackiw and C. Teitelboim, in Quantum theory of gravity: Essays in honor of the
60th birthday of Bryce S.DeWitt, ed. S. Christensen (Hilger, Bristol, 1984).
12. E. Witten, Phys. Rev. D44, 314 (1991).
G. Mandal, A. M. Sengupta and S. R. Wadia, Mod. Phys. Lett. A6, 1685 (1991).
S. Elitzur, A. Forge and E. Rabinovici, Nucl. Phys. B359, 581 (1991).
13. D. Grumiller and R. Meyer, Ramifications of lineland, hep-th/0604049.
14. P. van Nieuwenhuizen and D. V. Vassilevich, Class. Quant. Grav. 22, 5029 (2005).
15. D. Grumiller and R. Jackiw, Phys. Lett. B642, 530 (2006).
QUASINORMAL MODES FOR ARBITRARY SPINS IN THE
SCHWARZSCHILD BACKGROUND*
IOSIF KHRIPLOVICH, GENNADY RUBAN
Budker Institute of Nuclear Physics,
Novosibirsk 630090, Russia
khriplovich@inp.nsk.su, gennady-ru@ngs.ru
The leading term of the asymptotic of quasinormal modes in the Schwarzschild
background, ujn = —in/2, is obtained in two straightforward analytical ways for arbitrary
spins. One of these approaches requires almost no calculations. As simply we demonstrate
that for any odd integer spin, described by the Teukolsky equation, the first correction
to the leading term vanishes. Then, this correction for half-integer spins is obtained in a
slightly more intricate way. At last, we derive analytically the general expression for the
first correction for all spins, described by the Teukolsky equation.
1. Introduction
Quasinormal modes (QNM) are the eigenmodes of the homogeneous wave
equations, describing these perturbations, with the boundary conditions
corresponding to outgoing waves at the spatial infinity and incoming waves at the horizon.
Two boundary conditions make the frequency spectrum u„ of QNMs discrete. The
asymptotic form of this spectrum for scalar and gravitational perturbations of the
Schwarzschild background was found at first numerically in1'2 :
wn= ~\ (n+ M + 0.087424, n -► oo , s = 0, 2. (1)
A curious observation was made in 3 : the real constant in (1) can be presented as
Reun = ^ = TH In3, (2)
where TH is the Hawking temperature3-. The general formula for QNMs of integer
spins s was derived in5 , and in particular results (1), (2) confirmed.
In the present contribution we present results of 6 .
2. Regge — Wheeler Formalism
First, we derive here the following universal truncated wave equation for all spins,
valid in the limit \lo\ —> oo:
dr2
2 2u2 u)2 + 1/4
1 (r-1)2
0. (3)
"This research has been partially supported by the Russian Foundation for Basic Research grant
05/02/16627.
aIt was also conjectured in 3 that the asymptotic value (2) is of crucial importance for the
quantization of gravitational field, fixing the value of the so-called Barbero — Immirzi parameter. In spite
of being very popular, this idea is not in fact dictated by any sound physical arguments; quite the
contrary, it is in conflict with them 4 .
2692
2693
Its solutions are expressed via the Whittaker functions W\ifl, and with two boundary
conditions, at the horizon, r = 1, and at infinity, r = oo, we arrive immediately at
the quantization rule
i
Un = - ^ n > n > 1 > (4)
for all spins.
This result is also derived here in another way. We connect the two singular
points, r = 1 and r = oo, of eq. (3) by a cut in the complex plane r. Then we consider
in this plane a closed contour going around the cut and then following the arc of
radius \r\ —> oo. There is no singularity inside the contour. Therefore the solution
at some point on it, after going around the loop, should come back to its initial
value. It means that the phase of this solution changes by 2iTn} n = 0, ±1, ±2,....
In this way we arrive again at the quantization rule (4).
3. Teukolsky equation
To find the next, subleading correction, of zeroth order in n, to formula (4), we
use the Teukolsky equation which describes in unified way integer and half-integer
spins, at least from s = 0 to s = 2.
In the Schwarzschild background the Teukolsky equation for a massless field is
A^ + (^ s)(2r-l)^+U(r)R = 0, (5)
where
» / n / n r-r, x —r(2r — 3)iuis + r3ui2 , , ,. w.
A(r)=r(r-1), U(r) = ^ -^ Aja , AJS = (j + s)(j~s + 1) .
With the tortoise coordinate z(r) = r + ln(r — 1) and new function xir) =
one obtains the following standard form for this equation:
0+[o;2-l/(r)]X = OI (6)
with the effective potential
Tr. , s2-4 Ajs - s + s2 - 1 Ajs - s + s2 - 3iuj.s 2iuis
^^^TI " -3 + - -2 +~7~- (7)
The complication here is the third singular point, that at r = 0, without an
a priori given boundary condition at it. This singular point generates the second
cut in the complex r plane.
The problem is easily circumvented for odd integer spins. Since r = 0 is a regular
singular point of the Teukolsky equation, the exact general solution of this equation
can be presented as follows:
X(r)=r~s/2
J2akrk+1 +J2bkrk+s+l
lk=0 fc=0
(8)
2694
With an odd integer s, the singularity of this solution at r = 0 is due to the overall
factor r~s/2. Correspondingly, the phase acquired by the solution (8) as a result of
going around the branch point r = 0 is 5(0) = ns. Then, we can use again a closed
contour in the complex plane r, which results here in the quantization rule for any
odd integer spin
un= - -n, s = 1,3,..., (9)
i.e. first subleading correction to leading asymptotic (4) here vanishes.
In the general case, to investigate the singularity at r = 0, we shift z —+ z + in,
so that now z(r) = r + ln(l — r), and in the limit r « 1 we have z(r) = — r2/2.
With new variable p = uz(r) = —ujr2/2, we transform equation (7) in the limit
\uj\ > 1 to
dp
2
,2
3is
2p • 16^j* = °- (10)
Its independent solutions are the Whittaker functions. Though derived for | r| "C 1,
these solutions are valid also for |p = | ur2/2 \ ^> 1, if | uj\ is sufficiently large.
Therefore, they can be compared with the asymptotic form of the exact solution.
For half-integer spins, thus obtained solution is
^4(-V)=r(1^/2)A%,-f(-2^). (11)
It behaves for r —> 0 as j-1-5/2. in this way we arrive at the quantization rule
i ( 1
, 2 ,, „ 1/2,3/2, .... (12)
The general case requires here a proper account for the so-called Stokes
phenomenon and a judicious choice of the cut starting at r = 0. In this way, we obtain
analytically the universal formula
if 1\ 1
„ , ■- , „ , , . ln(l + 2 cos ns), n —+ oo , (13)
2 V 2 J in y '
for eigenmodes of any spin s described by the Teukolsky equation. For even s it
gives
^n= ~\ U+ M + -^ln3, n^oo, (14)
and of course comprises as special cases formulae (9), (12).
References
1. E.W. Leaver, Proc. R. Soc. A402, 285 (1985).
2. H.-P.Nollert, Phys. Rev. D47, 5253 (1993).
3. S. Hod, Phys. Rev. Lett. 81, 4293 (1998); gr-qc/98120072.
4. LB. Khriplovich, Int. J. Mod. Phys. D14. 181 (2005); gr-qc/0407111.
5. L. Motl, A. Neitzke, Adv. Theor. Math. Phys. 7, 307 (2003); hep-th/0301173.
6. LB. Khriplovich, G.Yu. Ruban, Int. J. Mod. Phys. D15, 879 (2006); gr-qc/0407111.
CAN QUANTUM MECHANICS HEAL CLASSICAL
SINGULARITIES?
T.M. HELLIWELL
Department of Physics, Harvey Mudd College, Claremont, CA. 91711
helliwell@HMC.edu
D.A. KONKOWSKI
Department of Mathematics, U.S. Naval Academy, Annapolis, MD. 21402
dak@usna.edu
We study a broad class of spacetimes whose metric coefficients reduce to powers of a
radius r in the limit of small r. We show that a large subset of classically singular
spacetimes is nevertheless nonsingular quantum mechanically, in that the Hamiltonian
operator is essentially self-adjoint so the evolution of quantum wave packets lacks the
ambiguity associated with scattering off singularities.
1. Introduction
This is a summary of an investigation [1] of a broad class of spacetimes that are
classically singular. We show that a large subset of these classically singular space-
times is nevertheless nonsingular quantum mechanically, in that the Klein-Gordon
operator is essentially self-adjoint [4,5], a criterion first developed for relativistic
spacetimes by Horowitz and Marolf [2] building on work by Wald [3], We
implement this criterion by using a physically transparent method due to Weyl [5,6] to
show the associated Schrodinger potential is limit point (LP) not limit circle (LC).
Thus the evolution of quantum wave packets lacks the ambiguity associated with
scattering off singularities. The singularity is "healed."
2. Power-law metrics
We consider the 4-parameter family of spacetimes [1] that take power-law metric
form
ds2 = -radt2 + r^dr2 + -^r~<d62 + rsdz2 (1)
in the limit of small r, where a, (3,7, 5, and C are constants.
Eliminating a by scaling r results in two metric types:
•
Type I:
ds2 = r0{-dt2 + dr2) + j^^dO2 + rsdz2, (2)
if a ^ P + 2, and
Type II:
ds2 = _rf3+2dt2 + r/3dr2 + -Lrld02 + rSdz2^ (3)
2695
2696
if a = (3 + 2.
Generically Type I and Type II spacetimes all have scalar curvature singularities
as r —> 0 if and only if (3 > —2. For more detail on their classical structure (including
the presence of strong curvature singularities) see [7].
3. Limit point-limit circle criteria
For the power-law metrics the Klein-Gordon equation can be separated in the
coordinates t,r,9,z, with only the radial equation left to solve. With changes in both
dependent and independent variables, the radial equation can be written as a one-
dimensional Schrodinger equation. This form allows us to use the Weyl limit point-
limit circle criteria [6] described in Reed and Simon [5] to determine essential self-
adjointness.
4. Essential self-adjointness and the power-law parameters
Our goal is to identify the values of /3,7, 5, C for which the quantum mechanical
operator is essentially self-adjoint. That is, for which parameter values is there a
classical, but no quantum, singularity as r —> 0? The Klein-Gordon equation for a
particle of mass M can be decomposed so the scalar field $ ~ elut^i(r,9,z), with
modes ^{r,0,z) ~ etmeelkzip(r). The radial tp(r) equation can be converted to a
one-dimensional Schrodinger-equation
d2u
— + {E-V{x))u = Q (4)
where E — u2 and
V[x) = C1^-) (^ -1)^2+ m2C2x^ + jfcV-{ + MV. (5)
The LP and LC regimes of Type I geometries for given m, k modes can be displayed
in a three-dimensional Cartesian /3,7,5 parameter space (see Figure 1 in [1]. (The
parameter C is irrelevant for this purpose.) Picture the positive (3 axis rising
vertically at right angles to the 7 and 5 axes. The boundaries of the LP and LC regimes
for given m, k modes are generally defined by five planes in this space. There is a
horizontal "base" plane /3 = — 2 , two vertical planes 7 + S = —2 and 7 + 5 = 6,
and two tilted planes 7 = (3 + 2 and S = (3 + 2. These five planes form a LC "bowl"
with bottom on the (3 = — 2 base plane, and four sides rising infinitely out of the
page. Parameter points within the interior of the bowl correspond to the LC regime,
while points outside the bowl are LP. The description is valid if the particle mass
M =£ 0 (otherwise there is no base plane) and for modes with k ^ 0 (otherwise
the tilted plane S = (3 + 2 is absent) and with m 7^ 0 (otherwise the tilted plane
2697
7 = /3 + 2 is absent). For a discussion of Type I modes with k = 0 or m = 0 see [1].
For Type II metrics, the radial ip(r) equation can be written in Schrodinger form
with E = uj2 - (^)2 and
V{x) = m2C2e^^+2> + k2e^-6+2> + M2e^+2K (6)
Type II metrics are LP for all parameter values.
5. Conclusions
For a broad class of four-parameter metrics, whose metric coefficients behave as
power laws in a radial coordinate r in the limit of small r, there are large regions
of parameter space in which classically singular spacetimes (whose singularities
are indicated by incomplete timelike or null geodesies) are "healed" by quantum
mechanics, in that quantum particle propagation is well-defined throughout the
spacetime.
Acknowledgments
We gratefully acknowledge the very helpful related work of Curtis Vinson, Zachary
Walters, Zoe Boekelheide, Ne-Te Loh, and Andrew Mugler. We also acknowledge a
valuable conversation with Jan Schlemmer.
References
1. Helliwell T M and Konkowski D A 2007 "Quantum healing of classical singularities
in power-law spacetimes" submitted to Class. Quantum Grav. gr-qc/0701149
2. Horowitz G T and Marolf D 1995 Phys. Rev. D 52 5670
3. Wald R M 1980 J. Math Phys. 21 2802
4. von Neumann J 1929 Math. Ann. 102 49
5. Reed M and Simon B 1972 Functional Analysis (New York: Academic Press); Reed
M and Simon B 1972 Fourier Analysis and Self-Adjointness (New York: Academic
Press)
6. Weyl H 1910 Math. Ann. 68 220
7. Lake K 2007 "Scalar Polynomial Singularities in Power-Law Spacetimes" gr-
qc/0702112
QUANTIZING TWO-DIMENSIONAL DILATON GRAVITY WITH
FERMIONS: THE VIENNA WAY
RENE MEYER
Max-Planck Institute for Physics, Werner-Heisenberg Institute,
Fohringer Ring 6, D-80805 Miinchen, Germany
and
Institute for Theoretical Physics, University of Leipzig,
Augustusplatz 10-11, D-04103 Leipzig, Germany
meyer@mppmu.mpg.de
I review recent work on nonperturbative path integral quantization of two-dimensional
dilaton gravity coupled to Dirac fermions, employing the "Vienna school" approach.
Despite much progress in our knowledge of quantum gravity1 during the last
decades, a fully satisfactory quantization of the simplest nontopologicaP gravity
theory, namely general relativity in four dimensions, is still missing. The reasons
are two-fold: On one hand, standard techniques from perturbative quantum field
theory do not apply to arbitrary high energies to perturbatively nonrenormalizable
general relativity. On the other hand, its highly nonlinear dynamics makes
general relativity hard to approach with nonperturbative methods. Adding the lack of
observational data for quantized gravitational effects, it is hard to compare the
suitability of different approaches and methods to quantize gravity. In such a context
it may be useful to consider less complicated situations, where even conservative
methods like standard quantum field theory can be applied to gravity.
Such a situation is given in lower dimensions. In two dimensions, however, pure
Einstein-Hilbert gravity is topological, the action being proportional to the Euler
number. One way of constructing a two-dimensional gravity theory with sensible
dynamics is to add an additional scalar field, henceforth called the dilaton X, to
Einstein-Hilbert gravity, and possible matter. This leads to the vast subject of two-
dimensional dilaton gravity.2 Such models arise from spherical reduction of general
relativity, from string theory as well as as toy models for intrinsic two-dimensional
gravity.
Of the many interesting features of these theories, I focus on the application of
the nonperturbative path integral quantization method, developed by the "Vienna
school" around Wolfgang Kummer,3 to dilaton gravity coupled to Dirac fermions.4~6
This method relies on several crucial points: First, using the spin connection cj =
ijJhldxfJ- and dyad 1-forms ea = e^Ax^ built from the inverse Zweibeine e£, the action
for Generalized Dilaton Theories5,
S(2) = -\ I d2x y^j [XR + U{X) (VI)2 - 2V{X) 1 + S(m> , (1)
2 JM2
aIn the sense that it possesses physical locally propagating degrees of freedom, i.e. gravitons.
hU, V parametrize different models (cf. tab. 1 in5). Notation and conventions are chosen according
to5.
2698
2699
is reformulated as a First Order Gravity action0
va v \ 1
- S{m). (2)
XaTa + Xdu + e ( U{X)^^ + V{X)
5d)
I Ms
If the matter in S't"1) does not couple to the auxiliary fields u and Xa, these fields can
be integrated out s.t. (2) and (1) are equivalent both on the classical and quantum
leveld. This is the case for scalar fields as well as intrinsic two-dimensional Dirac
fermions (a d b = a(db) — (da)b)
I Ms
l-F{X) (*ea) A (X7a^x) " eH(X) (mxx + KXX?) ] , (3)
but not for spherically reduced four-dimensional fermions.7 The functions F, H are
generic dilaton couplings, and the most general self-interaction for fermions in two
dimensions contains at most a quartic term. A second crucial point is the use of
light cone gauge for the local Lorentz frame, e.g. X± = (X° ±Xl)/\/2.
The path integral quantization of (2) and (3) then consists of four steps: 1.
Constraint Analysis The system possesses two diffeomorphisms and the local SO(l, 1)
symmetry They are generated on-shell by three first class constraints Gi, which form
a nonlinear Lie algebra6 {Gj(x), Gj(y)}* = fijk(x)Gh5(x — y) with field-dependent
structure functions fijk(x). (2) and (3) thus behaves like a nonlinear Yang-Mills
theory rather than a gravity theory, in which the constraint algebra would typically
close with derivatives of delta functionsf. 2. BVF Formalism.8Accounting for the
three gauge symmetries, one introduces three (anti)ghosts (ci,pc), i,j = 1, 2, 3. The
BRST charge takes the form as for a Yang-Mills theory, 0, = clGi + \c%c^fijk{x)p%.
With the gauge fixing fermion \I/ = pf,; axial (or Eddington-Finkelstein) gauge
(ti>o,e^~, eg") = (0,1,0) is reached. 3. Nonperturbative Path Integral
Quantization of the Geometric Sector The phase space path integral is then evaluated
follows: 1. Integration over the (anti)ghosts yields the Faddeev-Popov determinant,
solely depending on (X, X±). 2. In the chosen gauge, the action depends linearly on
(uix, ef) s.t. this integration can be carried out directly, yielding delta functionals in
the path integral which contain the classical equations of motion for the (X,X±).
These equations still include (up to that point still off-shell) fermion terms. 3.
Integrating out (X, X±) then sets these fields to their on-shell values, where the fermion
terms are viewed as off-shell external sources. During this step, the Faddeev-Popov
determinant cancels, i.e., as typical for axial gauges, the ghosts decouple. Because
the equations of motion for (X, X±) are solved using classical Green functions, the
cXa are Lagrange multipliers for the torsion Ta = Aea + eaf,u>/\eb (u>ai, = eai,u> in two dimensions.),
e0i, = — e^a, eoi = +1 and e = y/—gd2x denotes the volume 2-form. The Ricci scalar is R = — 2*dw,
where * is the Hodge star operator.
d Cf. e.g. the first paper in3 .
e{f,g}* is the Dirac bracket, taking care of the usual second class constraints in the presence of
fermions.
The classical Virasoro algebra, i.e. the one without central charge, is recovered by field-dependent
linear combinations of the Gi.
2700
asymptotic geometry has to be fixed and thus an asymptotic Fock space can be
constructed. The quantum fields (XjX^ fulfill the classical equations of motion
before integrating out the fermions because no physical locally propagating degrees
of freedom that could yield quantum corrections are present in the geometric sector.
4. Matter Perturbation Theory The effective action obtained so far is
nonlocal in space but local in time, and nonpolynomial in the fermions. Carrying out
the path integration over the fermions perturbatively generically yields effective
nonlocal 2n-point vertices.
Some Results and Outlook Reminiscent of bosonization in two flat
dimensions,10 two of the three four-fermi vertices5 coincide with the two effective four-
boson vertices found in a similar analysis for scalar fields,9 while the third one
vanishes for on-shell external momenta. However, the asymptotic modes for bosons
and fermions differ. In order to investigate bosonization in quantum dilaton
gravity, one thus has to compare observables, e.g. the four-particle S-matrices of the
fermionic and the known bosonic case11 or the specific heat of the Witten black
hole (or CGHS model).12 From the scalar case11 one also expects unitarity, i.e. no
information loss, and CPT invariance of the S-matrix.
The whole quantization procedure is background independent and only uses
standard quantum field theory methods. In order to recover the correct semiclassical
limit one also has to sum over degenerate metrics in the path integral. Another
interesting application would be to reconstruct black holes as macroscopic bound
states of quantum dilaton gravity in a Bethe-Salpeter13 like manner.
References
1. S. Carlip, Kept. Prog. Phys. 64, p. 885 (2001).
2. Reviews: D. Grumiller, W. Kummer and D. V. Vassilevich, Phys. Rept. 369, 327
(2002); D. Grumiller and R. Meyer (2006), hep-th/0604049.
3. W. Kummer, H. Liebl and D. V. Vassilevich, Nucl. Phys. B493, 491 (1997), B513, 723
(1998) and B544, 403 (1999); D. Grumiller, PhD thesis, Technische Universitat Wien
(2001), gr-qc/0105078; L. Bergamin, D. Grumiller and W. Kummer, JEEP 05, p. 060
(2004); L. Bergamin (2004), hep-th/0408229; L. Bergamin, D. Grumiller, W. Kummer
and D. V. Vassilevich, Class. Quant. Grav. 22, 1361 (2005).
4. R. Meyer (2005), hep-th/'0512267.
5. D. Grumiller and R. Meyer, Class. Quant. Grav. 23, 6435 (2006).
6. R. Meyer, Master's thesis, Universitat Leipzig (2006), gr-qc/0607062.
7. H. Balasin, C. G. Boehmer and D. Grumiller, Gen. Rel. Grav. 37, 1435 (2005).
8. E. S. Fradkin and G. A. Vilkovisky, Phys. Lett. B55, p. 224 (1975); I. A. Batalin
and G. A. Vilkovisky, Phys. Lett. B69, 309 (1977); E. S. Fradkin and T. E. Fradkina,
Phys. Lett. B72, p. 343 (1978).
9. D. Grumiller, W. Kummer and D. V. Vassilevich, European Phys. J. C30, 135 (2003).
10. S. R. Coleman, Phys. Rev. Dll, p. 2088 (1975); S. R. Coleman, R. Jackiw and
L. Susskind, Ann. Phys. 93, p. 267 (1975).
11. P. Fischer, D. Grumiller, W. Kummer and D. V. Vassilevich, Phys. Lett. B521, 357
(2001), Erratum ibid. B532 (2002) 373.
12. D. Grumiller, W. Kummer and D. V. Vassilevich, JHEP 07, p. 009 (2003).
13. E. E. Salpeter and H. A. Bethe, Phys. Rev. 84, p. 1232 (1951)
VACUUM POLARIZATION FOR A SPINOR MASSIVE FIELD IN
AN EINSTEIN-MAXWELL SPACETIME
V. B. BEZERRA
Departamento de Fisica, Universidade Federal de da Paraiba, Joao Pessoa, Pb, Brazil
valdir@fisica.ufpb.br
NAIL R. KHUSNUTDINOV
Department of Physics, Tatar State Liberal Pedagogical University,
Mezhlauk 1, Kazan 4-20021, Russia
nail@kazan-spu.ru
1. Introduction
The study quantum fields in the spacetime of static cylindrically solutions has
been considered in different situations. Examples of these studies are the
computation of non-vanishing contribution to the vacuum expectation value of the
energy-momentum tensor of quantum fields, as for example, scalar, spinor and
vector fields1-.7 Particularly in these papers it is emphasized the role played by the
topology of the background gravitational field.
In what follows we find the contribution to the vacuum polarization of a massive
Dirac field in the gravitational field due to a tubular matter source with an axial
interior magnetic field and vanishing exterior magnetic field which is a solution
of the combined Einstein-Maxwell field with cylindrical symmetry (Safko-Witten
spacetime).8
2. Vacuum expectation value in the region outside the source
To start with let us consider the massive Dirac equation in the Euclidean sector,
r = it, in the Safko-Witten spacetime which is described by the following line
element8
ds2 = dr2 + dp2 + ^ V + dz2 (1)
where the parameter v is associated with the interior magnetic field and the mass
of the tube. It is given by v = exp((3), with
1 92 {i + HipD^ + iy
where Hi is the intensity of the interior magnetic field. The quantities p\ and p2
are the interior and exterior radius of the tube and 77 is an arbitrary constant.
This spacetime is locally flat outside the tube of matter which means that the
curvature vanishes everywhere outside the tube of matter. Thus, in this region the
2701
2702
gravitational field generated by this source may be described by a commonly called
conical geometry.
Using the standard representation of the Dirac matrices and an appropriate set
of tetrads,7 we get the following equation for the spinor Green function S
(Yd, - v-^y + m)S{x-x<) = jA{xJgX'\ (3)
where
7r=7(0), 7P = cos<p7(1) +sin</?7(2), 7V = -- sin <^7(1) + - cos<p7(2), 70=7(3),
P P
(4)
Now, let us define the Green function G of the squared Dirac operator by the
relation
S(x-x') = (^D^)_mG(x;x'). (5)
It obeys the following equation
(1«Dll)*G(x;x') = -S4{Xv-X'), (6)
where
(V^)2 = (f^ + ~pdp - ^f- + ^V^, - m* - \R) . (7)
In the coincidence limit, the closed form of the renormalized Green function of
the squared Dirac operator, for the massive case, is given by
Gren(x;x) = -^- V(-l)ntan— K1(2mpsm —) (8)
' n—1
mvcos^- f00 Ki(2mp cosh \) sinh f sinh ^
_2_ / _^_y 2Z 2 2 .^
4tt3/3 Jo cosh | cosh ^j/ — cos ttv
where K\ is the Bessel function of second kind. For mp ^> 1 the above expression
exponentially falls down, according to
G™(x-x)* VR° m2e~2mP (9)
[ >X> 167T5/2 (TO/9)3/2 ' V)
In the opposite case, if mp <C 1, the expression for Gren(x;x) is given by
Grm^ = -^- <10>
To calculate the vacuum expectation value of the energy-momentum tensor we
use the following formula9
{Tliu)rm = -\ Hm lm{tT[^(V^[S + SJen -g^'Vy[S + Sren)nx';x)}}, (11)
4 x'^x '
2703
where Sc is the charge conjugate spinor Green function. For the sake of simplicity
let us consider only the zero-zero component of the energy-momentum tensor. Thus,
straightforward calculations give the following structure of the vacuum expectation
value for this quantity (for simplicity we consider the case v < 2)
v f°° ds 2 /• e~iz^ - ^sin2 §
(T°)ren = -~- / ^e~sm / = a— 2-dz, (12)
For mp 3> 1, the energy-momentum tensor is exponentially small
{ o) ~ S^/2 {mpf* ■ [ '
On the other hand, for mp -C 1, the expression for {T$} is given by
iT0yen _ _\Y VI'" ^^'1 ( u)
.0^en_ (^2-l)(7^2+17)
28807T2/
Therefore the energy is localized very close to the string in a radius smaller then
the Compton length of the spinor particle, p < m~1.
3. Conclusion
There is a gravitational effect on the vacuum polarization for a massive spinor field
outside the source due to the content of matter and the interior magnetic field. As
this spacetime has a conical structure, this means that the local influence that arises
on a spinor field is absent outside the source and that this effect on the vaccuum
polarization is due to the topological features of the Safko-Witten spacetime.
Acknowledgments
We are grateful to CNPq , FAPESQ-PB/CNPq(PRONEX), for partial financial
support.
References
1. Helliwell T.M. and Konkowski D.A., Phys. Rev. D34, 1918 (1986).
2. Dowker J.S., Phys. Rev. D36, 3095 (1987).
3. Frolov V.P. and Serebriany E.M., Phys. Rev. D35 3779 (1987).
4. Guimaraes M.E.X., Class. Quant. Grav. 12, 1705 (1995).
5. Linet B., Phys. Rev. D35, 536 (1987).
6. Harari D. D. and Skarzhinsky V. D., Phys.'Lett. B240, 330 (1990).
7. V. B. Bezerra and N. R. Khusnutdinov, Class. Quantum Grav., 23, 3449 (2006).
8. J. L. Safko and L. Witten, Phys. Rev. D5, 293 (1972); J. Math. Phys. 12, 257 (1971).
9. P. B. Groves, P. R. Anderson, E. D. Carlson, Phys. Rev. D66, 124017 2002).
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Casimir Effect and
Short-Range Gravity
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THE CASIMIR EFFECT IN RELATIVISTIC QUANTUM FIELD
THEORIES*
V. M. MOSTEPANENKOt-t
Center of Theoretical Studies Institute for Theoretical Physics, Leipzig University,
Augustusplatz 10/11, 04109, Leipzig, Germany
■f Vladimir.Mostepanenko@itp.uni-leipzig.de
We review recent developments in the Casimir effect which arises in quantization volumes
restricted by material boundaries and in spaces with non-Euclidean topology. The
starting point of our discussion is the novel exact solution for the electromagnetic Casimir
force in the configuration of a cylinder above a plate. The related work for the scalar
Casimir effect in sphere-plate configuration is also considered, and the application region
of the proximity force theorem is discussed. Next we consider new experiments on the
measurement of the Casimir force between metals and between metal and
semiconductor. The complicated problem connected with the theory of the thermal Casimir force
between real metals is analyzed in detail. The present situation regarding different
theoretical approaches to the resolution of this problem is summarized. We conclude with
new constraints on non-Newtonian gravity obtained using the results of latest Casimir
force measurements and compare them with constraints following from the most recent
gravitational experiments.
Keywords: Casimir effect; exact solutions; Nernst heat theorem; non-Newtonian gravity.
1. Introduction
The Casimir effect1 is a particular type of vacuum polarization which arises in
quantization volumes restricted by material boundaries and in spaces with non-
Euclidean topology due to distortions in the spectrum of zero-point oscillations of
relativistic quantized fields in comparison with the case of free infinite Euclidean
space-time. In case of volumes restricted by material boundaries, the polarization
energy results in the Casimir force acting on these boundaries. In spaces with non-
Euclidean topology, the polarization stress-energy tensor influences the geometry of
space-time through the Einstein equations of gravitational field. In both cases the
applications of the Casimir effect are extraordinary wide and range from condensed
matter physics, atomic physics and nanotechnology to gravitation and cosmology
(see monographs 2-5 and reviews 6-8).
During the last few years the Casimir force was measured with increased
precision in configurations metal-metal9-20 and metal-semiconductor.21-23 The theory
of the Casimir effect was widened to incorporate real material properties7 and more
complicated geometrical configurations.24'25 Much attention was given to the
controversial problem of the thermal Casimir force between real metals (see discussion
in Refs. 26,27) and dielectrics.28~30 The results of precise measurements of the
Casimir force between metal surfaces were used for obtaining stronger constraints
"This research has been partially supported by DFG grant 436 RUS 113/789/0-2.
tOn leave from Noncommercial Partnership "Scientific Instruments", Tverskaya St. 11, Moscow,
103905, Russia.
2707
2708
on the Yukawa-type corrections to Newtonian gravitational law predicted in unified
gauge theories, supersymmetry and supergravity.18-20'31
In the present paper we discuss the above most important achievements in the
Casimir physics for the period after the Xth Marcel Grossmann Meeting which was
hold in July, 2003 at Rio de Janeiro. In our opinion, the theoretical achievement
of major significance during this period is the obtaining the exact solution for the
electromagnetic Casimir force in configuration of a cylinder above a plate24 (see also
further development of this matter in Ref. 25). The related work was done for the
scalar Casimir force in configurations of a sphere or a cylinder above a plate.25,32
In Ref. 33 the scalar Casimir effect in the same configurations was considered
numerically using the worldline algorithms. The combination of the exact analytical
and precise numerical methods permitted to make some conclusions on the validity
limits of the so-called proximity-force theorem (PFT) which is heavily used in the
experimental investigation of the Casimir force. All these results are discussed in
Sec. 2 of the present paper.
Sec. 3 is devoted to new experiments on the measurement of the Casimir
force between metals18-20 and between metal and semiconductor.21-23 The
experiments18 ~20 using a micromechanical torsional oscillator permitted for the first
time to achieve the recordly low total experimental error of about 0.5% within a
wide separation range and reliably decide between different competing approaches
to the theoretical description of the thermal Casimir force. The experiments21-23
using an atomic force microscope opened new prospective opportunities for the
control of the Casimir force in nanodevices by changing the charge carrier density in a
semiconductor test body.
The complicated theoretical problems related to the thermal Casimir force are
discussed in Sec. 4. As underlined in this section, the theoretical approach proposed
by some authors for real metals27 is not only in contradiction to experiment, but
inavoidably results in a violation of the Nernst heat theorem.26'34,35 What is more,
we stress that the same problems, as for real metals, arise for the Casimir force in
configurations of two dielectrics and metal-dielectric if dc conductivity of a dielectric
plate is taken into account.28~30 This suggests that there are serious restrictions in
a literal application of the Lifshitz theory to real materials. Some phenomenological
approaches on how to avoid contradictions with theormodynamics and experiment
proposed in literature are discussed.
In Sec. 5 reader finds the review of new constraints on non-Newtonian gravity
obtained from recent measurements of the Casimir force between metallic test
bodies.18-20'31 These constraints are compared with those obtained from gravitational
measurements.
Sec. 6 contains our conclusions and discussions.
2709
2. New exact solutions in configurations with curved boundaries
It is common knowledge that Casimir1 found the exact expression
, . 7T2 He
FW=-240? (1)
for the fluctuation force of electromagnetic origin per unit area acting between two
plane-parallel ideal metal plates at a separation z. Lifshitz theory36 generalized
Eq. (1) for the case of two parallel plates described by a frequency-dependent
dielectric permittivity e(u>). Experimentally it is hard to maintain the parallelity of
the plates. Because of this, most of experiments were performed using the
configuration of a sphere above a plate. The configuration of a cylinder above a plate also
presents some advantages if to compare with the case of two parallel plates.
Unfortunately, over many years it was not possible to obtain exact expressions for the
Casimir force in these configurations. For this reason, the approximative proximity-
force theorem37 (PFT) was used to compare experiment with theory. According to
the PFT, at short separations (z <C R) the Casimir forces between an ideal metal
cylinder (per unit length) or a sphere and a plate are given by
„ . , 7T3 [R fLC . , 7T3 hcR
FcW = -ii^Vl?» ^ = -360^' <2>
where R is a sphere or a cylinder radius.
Within the PFT it is not possible to control the error of the approximate
expressions in Eq. (2). From dimensional considerations it was evident7 that the relative
error in Eq. (2) should be of order of z/R, but the numerical coefficient near this
ratio remained unknown. In fact, rigorous determination of the error, introduced by
the application of the PFT, requires a comparison of Eq. (2) with the exact
analytical results or with precise numerical computations in respective configurations.
One such result for the electromagnetic Casimir effect was first obtained24 for a
cylinder above a plate using a path-integral representation for the effective action.
Eventually, the Casimir energy is expressed through the functional determinants of
infinite matrices with elements given in terms of Bessel functions.24 The analytic
asymptotic behavior of the exact Casimir energy at short separations was found in
Ref. 25. It results in the following expression for the Casimir force at z <C R:
Fc(z)
3 [20__2_\ z_
5 V372 ~~ 36/ R
(3)
384 y/2
Eq. (3) is of much importance. It demonstrates that the relative error of the
electromagnetic Casimir force between a cylinder and a plate calculated using the PFT is
equal to -0.288618z/ii. Thus, for typical parameters of R = 100/im and z = 100 nm
this error is approximately equal to only 0.03%.
For a sphere above a plate the analytic solution in the electromagnetic case is
not yet obtained. The scalar Casimir energy for a sphere above a plate is found in
Refs. 25 and 32. However, the asymptotic expression at short separations similar to
Eq. (3) is not found. In Ref. 33 the scalar Casimir energies for both a sphere and
2710
a cylinder above a plate are computed numerically using the worldline algorithms.
It was supposed that a scalar field satisfies Dirichlet boundary conditions. As was
noticed in Ref. 33, the Casimir energies for the Dirichlet scalar should not be taken
as an estimate for those in electromagnetic case. In addition, it should be stressed
that the errors of the PFT calculated in Ref. 33 are related to the Casimir energy
and not to the experimentally measured Casimir force. This makes all errors larger.
To illustrate, if we were considering the error of the PFT in application to the
electromagnetic Casimir energy between a plate and a cylinder [instead of the force
considered in Eq. (3)], the value of — 0.48103,z/i? would be obtained as a negative
error of the PFT.25 The magnitude of the latter is by a factor of 1.6667 larger than
the error obtained above for a force.
Eq. (3) confirms that PFT works well at short separations and reproduces the
exact result with a very high precision. This justifies the use of the PFT for the
interpretation of experimental data. In Refs. 38,39 it was claimed, however, that
in the configurations of sinusoidally corrugated plates or a sphere above a plate
the PFT overestimates the lateral Casimir force by up to 30-40%. Comment 40
demonstrates that these claims are not warranted. In Refs. 38,39 metal is described
by the plasma model with a plasma wavelength Xp = 136 nm for Au. Deviations of
the "exact" results obtained in Refs. 38,39 from those given by the PFT in plate-
plate configuration are presented in Fig. 1 of Ref. 38 (Fig. 11 of Ref. 39) in terms of
function p versus k = 2tt/Xc) where Ac is the corrugation wavelength. According
to this figure, the lateral force amplitude is less by 16% than the value given by
the PFT for configuration with Ac = 1.2//m and plate separation z = 200 nm
(it is supposed that corrugation amplitudes are much less than z, Xp and Ac).
This result of Refs. 38,39 is in contradiction with a more fundamental path-integral
theory formulated for ideal metals.42 It is easily seen, that the quantity p, plotted
in the above-mensioned figures as a function of k at different z and with a fixed Ap,
is, in fact, a function of kz. Thus, for corrugated plates with rescaled Ac = 12 pm
and z = 2/iin (but with the same kz) the deviation of the lateral force amplitude
from the PFT value is still 16%. At z = 2/zm, however, the nonideality of a metal
does not play any important role, and Ref. 42 demonstrates the agreement between
the exact result and the result obtained by using the pair-wise summation (PWS)
if z is several times less than Ac- Note, however, that in some cases PWS may lead
to more accurate results than PFT.
In the Reply41 to the Comment40 the authors of Refs. 39,40 claim that the above
arguments raising doubts on their predictions are based on a mistake. This claim is
in error. Reference 41 is right that generally the case of perfectly reflecting mirrors
is recovered in the limit Xp —> 0. In the formalism of Refs. 38,39, however, this
limiting transition is forbidden by the condition that the corrugation amplitudes
are much less than Xp. Thus, for fixed corrugation amplitudes the limiting case of
ideal metal cannot be achieved by decreasing Xp. On the contrary, the formalism of
Refs. 38,39 allows any increase of Ac and z, and this was used in the Comment.40
At separations z 2> Xp (in the Comment z = 2 pin) real metal behaves like ideal
2711
metal and all results should coincide with those for ideal metals as obtained in the
path-integral approach.42'43
For the experimental configuration of a sphere above a plate Refs. 38,39 obtain
the "exact" computational value 0.20 pN for the amplitude of the lateral force at a
separation z = 221 nm between the test bodies with corrugation amplitudes equal to
A1 = 59 nm and A^ = 8nm. According to Refs. 38,39, the linear in the corrugation
amplitudes version of the PFT gives instead 0.28pN, i.e., 40% difference. At this
point Ref. 40 stresses that the amplitudes considered are not small comparing to z
(for instance, Axj z = 0.27) and another assumption Ax, A2 <C Ap used in Refs. 38,
39 is also violated (for instance, Ai/\p = 0.43). It is not surprising, then, that
Refs. 38,39 arrive at a force amplitude of 0.20 pN so far away from the value of
0.33 pN obtained theoretically using the complete PFT and that of 0.32 ±0.077 pN
measured experimentally at 95% confidence in Ref. 17. Thus, the approach used
in Refs. 38,39 is not only in contradiction with a more fundamental path-integral
theory42 but is also excluded by experiment.16
3. New precise measurements of the Casimir force between metal
and semiconductor test bodies
The most important experiments on the Casimir force after the Xth Marcel Gross-
mann Meeting were performed at Purdue University — Indiana State
University18^20 and at the University of California. Riverside.21^23 Two experiments in
Refs. 18-20 are devoted to the determination of the Casimir pressure between two
Au-coated plates using the dynamic techniques based on a micromechanical
torsional oscillator. The improved version of this experiment is described in Refs. 19,20.
The two test bodies of the micromechanical oscillator are a sphere and a plane plate.
Sphere is oscillating with the angular resonant frequency, and the shift of this
frequency under the influence of the Casimir force F acting between a sphere and a
plate was measured as a function of separation z in the region from 160 to 750 nm.
From this shift one can find15'18"20 the force gradient dF/dz and using the PFT
arrive to the equivalent Casimir pressure
( >~ 2nR dz ' [ '
This experiment is characterized by a very low total experimental error which was
determined at a 95% confidence level and varies between 0.55 and 0.60% in a wide
separation region from 170 to 350 nm.
The obtained experimental results were compared with different theoretical
approaches using the Lifshitz theory36 and tabulated optical data for the complex
index of refraction.44 In this comparison all corrections due to surface roughness,
nonzero temperature, sample-to-sample variations of optical data, errors of the PFT,
effects of spatial nonlocality and of patch potentials were carefully analyzed and
taken into account in a conservative way. Specifically, the error of the PFT was
conservatively estimated as equal to z/R, whereas recent results presented in Sec. 2
2712
pth _ pexp (mpa)
500 600 700
z (nm)
PexP (mPa)
200 300
500 600
z (nm)
Fig. 1. Differences of the theoretical and experimental Casimir pressures versus separation (dots)
and the 95% confidence intervals (solid lines). The theoretical pressures Pth (left figure) are
computed using the impedance approach and Pth (right figure) using the Drude model approach.
lead to several times smaller error. It was concluded that data are consistent with
the surface impedance approach to the thermal Casimir force at the laboratory
temperature T = 300 K (see Fig. 1, left, where the differences between theoretical, Pth,
and experimental, _PexP, Casimir pressures are plotted versus separation). The data
were found to be consistent also with the theoretical approach using the plasma
model at T = 300 K, and with the theoretical computations at zero temperature.
At the same time, Fig. 1, right, shows that experimental data exclude theoretical
Casimir pressures, Pth, computed using the Drude model approach at T = 300 K
(discussion of different theoretical approaches is contained in Sec. 4).
The experiment by using a microinechanical torsional oscillator has permitted
also to obtain stronger constraints on non-Newtonian gravity which are considered
in Sec. 5.
Three experiments in Refs. 21-23 are devoted to the measurement of the Casimir
force acting between Au-coated sphere and single-crystal Si plates with different
charge carrier densities using an atomic force microscope. In Ref. 21 B-doped
Si plate with a resistivity p « 0.0035 fl cm and concentration of charge carriers
n w 3 x 1019 cm-3 was used. The measured force-distance relation of the Casimir
force was compared with two theoretical dependences. One of them was computed
for this sample and another one for a sample made of Si with high resistivity equal
to 1000 il cm. It was found that theoretical results computed for the semiconductor
plate used in experiment are consistent with the data. At the same time,
theoretical results computed for high-resistivity Si are experimentally excluded at 70%
confidence. This suggests that the Casimir force is sensitive to the conductivity
properties of semiconductors.
The obtained results were confirmed in the direct measurement of the difference
2713
Fb - Fa (pN)
10 r—^ —
Fig. 2. The differences of the mean measured Casimir forces of the lower and higher resistivity
Si (dots) and respective theoretical difference (solid line) versus separation.
Casimir force acting between Au-coated sphere and two P-doped Si plates of
different charge carrier densities.22 One of the silicon plates (sample a) had the resistivity
pa w 0.43 fl cm and the concentration of charge carriers na w 1.2 x 1016cm-3.
Another one (sample b) had much lower resistivity pb « 6.4 x 10_4S7cm and much
higher concentration of charge carriers rib ~ 3.2 x 1020cm~3. In Fig. 2, taken from
Ref. 22, the difference of experimental mean Casimir forces, acting between Au-
coated sphere and samples b and a, Fb — Fa, versus separation is shown as dots. The
theoretically calculated differences using the Lifshitz formula are shown by the solid
line. Within the separations from 70 to 100 nm the mean difference in the measured
Casimir forces exceeds the experimental error of force difference. This permits a
conclusion that in Ref. 22 the influence of charge carrier density of a semiconductor
on the Casimir force was experimentally measured for the first time.
The third experiment on the measurement of the Casimir force between Au-
coated sphere and single-crystal Si plate demonstrates a new physical phenomenon,
the modulation of the Casimir force with laser light.23 In the absence of light the
used Si plate had a relatively high resistivity p w 10 il cm and relatively low
concentration of charge carriers n w 5 x 1014cm~3. This plate was illuminated with
514 nm pulses, obtained from an Ar laser. In the presence of pulse the
concentration of charge carriers increases up to n w 2 x 1019cm-3. The difference of the
Casimir forces in the presence and in the absence of pulse, AF, was measured using
an atomic force microscope within the separation range from 100 to 500 nm. The
experimental results23 are shown in Fig. 3 as dots versus separation. In the same
figure the solid line is computed using the Lifshitz formula under the assumption
that in the absence of laser light Si possesses a finite static dielectric permittivity
eSl{Q) = 11.66. The dashed line is computed taking into account the dc conductivity
2714
AF (pN)
150 200 250 300 350 400 450 500
z(nm)
Fig. 3. The differences of the mean measured Casimir forces with laser pulse on and off (dots)
versus separation. The respective theoretical differences are computed under the assumption of
finite static dielectric permittivity of Si in the absence of laser light (solid line) and taking dc
conductivity of high-resistivity Si into account (dashed line).
of Si in the absence of laser light at frequencies much below the first Matsubara
frequency. As is seen in Fig. 3, the solid line is in excellent agreement with the
experimental data, whereas the dashed line is in disagreement with data. Physical
consequences following from this observation are discussed in the next section.
The demonstrated dependence of the Casimir force between a metal and a
semiconductor on the density of charge carriers in semiconductor can be applied in nan-
odevices of the next generations such as micromirrors, nanotweezers and nanoscale
actuators. In so doing, the density of charge carriers can be changed either by doping
and/or due to irradiation of a device by laser light leading to respective variations
in the magnitude of the Casimir force.
Since the Xth Marcel Grossmann Meeting in 2003, some other experiments on
the Casimir force have been proposed. One could mention the proposal to measure
the influence of the Casimir energy on the value of the critical magnetic field in
superconductor phase transitions,45 the suggestion to measure the Casimir torques
using the repulsive force due to liquid layers,46 and the proposed Casimir force
measurements at large separations.47-49 Special attention was attracted to new
techniques for the measurement of the Casimir force. Thus, in Ref. 50 the holographic
interferometer was first applied for optical detection of mechanical deformation of a
macroscopic object induced by the Casimir force. All this demonstrates that there
are considerable opportunities in the experimental investigation of the Casimir force
and in applications of the Casimir effect.
2715
4. Problems in the theory of thermal Casimir force between
metals and dielectrics
During all the period between the Xth and Xlth Marcel Grossmann Meetings the
problem of the thermal Casimir force was hotly debated. Until 2005, only the case of
two plates made of real metal was the subject of controversy. In 2005 it was shown,
however, that the case of two dielectric plates leads to problems as well.28
We start from the Lifshitz formula for the free energy of the van der Waals
(Casimir) interaction between two semispaces with a gap of width z in thermal
equilibrium at temperature T:
oo
T{z, T)=kj~Y.{l~ W / k± dk± (5)
l=° V J o
x {\n[l - 4M(^,k±)e^'z] +\n[l ~ r2TE(^,k±)e-2^}} .
Here ks is the Boltzmann constant, £; = 2TrkBTl/h are the Matsubara frequencies,
Qi = Q\i +£f/c2)1/2, k± is the projection of the wave vector on the boundary planes
of semispaces, and ttm,te{£,i, k±) are the reflection coefficients for two independent
polarizations of the electromagnetic field (transverse magnetic and transverse
electric modes).
In the original formulation of the Lifshitz theory the semispace material is
described by using the approximation of dielectric permittivity e(u>) depending only
on the frequency, and the continuity conditions
E\t = E2t, Bu — B2t, Dln — D2n, Bm = B2n (6)
for the electric field, magnetic induction and electric displacement on boundary
planes. Thus, the Lifshitz theory does not take into account the effects of spatial
dispersion. In this model case the reflection coefficients take the form
rTM(t,l,k±)= —, rTE(&,k±) = ■ , (7)
mi +k h + qt
where h = ^Jk\ + e^f/c2 and et = e(i£i).
The central point of the debates is the term of Eq. (5) with I = 0 (the so-called
zero-frequency term). At large separations (high temperatures) it is dominant, and
all terms with I > 1 are negligibly small independently of the specific form of e{<jS).
The case of ideal metal plates is obtained from Eqs. (5), (7) using the so-called
Schwinger prescription,5,51 i.e., that one should take limit e —► oo first and set I — 0
afterwards. Using this prescription, for ideal metal plates one obtains
rTM(0,A:j_) = l, rTE(0,A:±) = l. (8)
The same result follows for ideal metal independently of the Lifshitz formula from
thermal quantum field theory with boundary conditions in the Matsubara
formulation. Thus, at large separations (in fact at separations larger than 6 /an at
2716
T = 300 K) it follows
^(^) = -^C(3), (9)
where ((3) is the Riemann zeta function. Notice that Eq. (9) is in agreement with
the classical limit based on the Kirchhoff's law.52'53
Refs. 54-57 (see also Ref. 27) suggested to calculate the thermal Casimir force
by describing the properties of real metals at low frequencies via the dielectric
permittivity of the Drude model
e«» = i + «ktW (10)
where up is the plasma frequency and -j(T) is the relaxation parameter. Substituting
Eq. (10) in Eq. (7) we obtain
rrM(0,A:j.) = l) rTE(0, k±) = 0. (11)
Eq. (11) is preserved also in the limit of ideal metal plates, and is thus in
contradiction with Eq. (8). From Eqs. (5) and (11) at large separations one arrives at the
result
nz,T) = -^-2C(3) (12)
instead of Eq. (9). This result is in contradiction with the classical limit.
Real metals in the frequency region of infrared optics are well described by the
dielectric permittivity of the plasma model
e(iO = 1 + *p. (13)
If one estrapolates this model to low frequencies, the reflection coefficients
become58,59
■a;2
rTM(0,A;±) = l, rTE(0,k±)= v, . (14)
-a;2
In the limiting case of ideal metal plates it holds ujp —-> oo and Eq. (14) agrees with
Eq. (8) because tte(0, A;j_) —> 1. At large separations the plasma model leads to
Eq. (9) in agreement with the classical limit.
It is notable that the plasma model predicts small thermal corrections to the
Casimir force at short separations in qualitative agreement with the case of ideal
metals (a fraction of a percent at separations below 1/im). Much larger thermal
corrections at short separations are predicted by using the Drude model (19% of
the force at z = 1 /zm).
As was mentioned above, the dielectric permittivity depending on the frequency
provides only an approximative description of metals because it disregards the
effects of spatial dispersion. Another approximative description of metals is provided
2717
by the Leontovich impedance boundary condition
Et = Z(u)[Btxn], (15)
where the index t labels the field components tangential to the plates, n is the unit
vector directed into the medium, and impedance function Z(u) is found from the
solution of kinetic equations.60 It is notable that the Leontovich impedance is well
defined even in some frequency regions (for example, in the region of the anomalous
skin effect in which the spatial dispersion is present) where the description in terms
of e(u>) is not possible. At the same time, the Leontovich impedance is not applicable
at separations z < X.p = 2irc/ujp, where the inequality Z<1 may be violated and
the boundary condition (15) cannot be used. In the frequency regions where both
quantities are well defined it holds Z(uj) = l/y'sito).
In terms of Leontovich impedance, the reflection coefficients in the Lifshitz
formula take the form61,62
r-TM 0, fcjj = —^^, rTE 0, k±) = — — 16
cqi+Z^i £i+cqiZl
where Z\ = Z(i£i). The zero-frequency values of these reflection coefficients depend
on the form of impedance function used. For the impedance function of the normal
and anomalous skin effect,60 one reobtains Eq. (8) obtained previously for ideal
metals. For the impedance function of the infrared optics it follows that
rTM(0, k±) = 1, rTE(0, k±) = Up ~ t^ ■ (17)
Ulp + CK±
In the limit of ideal metal plates up —> oo and Eq. (17) coincides with Eq. (8). The
Leontovich impedance leads to almost the same results for the thermal Casimir force
as the plasma model, i.e., to small thermal corrections to the zero-temperature force
at short separations and to Eq. (9) at large separations.
From the above it is seen, that there are three theoretical approaches using the
Drude model, the plasma model and the Leontovich impedance which lead to
different predictions for the thermal Casimir force. There is also the similarity between
the plasma model approach and the impedance approach which both predict small
thermal effects at short separations and are in agreement with the classical limit at
large separations. This is in opposition to the Drude model approach which predicts
relatively large thermal effect at short separations and is in violation of the classical
limit at large separations.
As was analytically proved in Refs. 63,64 (see also Refs. 26,35), the Drude model
approach leads to a violation of the third law of thermodynamics (the Nernst heat
theorem) in the case of metallic perfect lattices with no defects and impurities. For
such lattices the relaxation parameter 7(T) —> 0 when T —> 0 in accordance with
the Bloch-Griineisen law and the entropy of a fluctuating field at zero temperature
2718
takes a negative value26,35,64
16irz2
s(^°) = T7—2 I ydvin
o
fey- ^z2uj2 + c2y2
\cy + J<iz2u)2 + c2y2
2
< 0, (18)
instead of zero as is demanded by the Nernst heat theorem. At large separations
from Eq. (18) it follows
*<'••» =-is? <"• (19»
i.e., what is called in Refs. 27,55-57 the entropy of a "modified ideal metal" (MIM) at
zero temperature. Recent Refs. 27,57 recognize that their MIM violates the Nernst
heat theorem but argue27 that "the crucial difference between real metals and MIM
is that the former includes relaxation by which there will be no violation of the
third law of thermodynamics". This conclusion is wrong because Eq. (18) proves the
violation of the Nernst heat theorem for Drude metals with dielectric permittivity
(10). These metals have a finite permittivity at all frequencies with exception of
zero frequency and a nonzero relaxation described by the relaxation parameter
7(T). From this it follows that the Drude model approach violates the third law of
thermodynamics for perfect metallic crystal lattices with no impurities but nonzero
relaxation at any nonzero temperature. Thus, theoretically this approach is not
acceptable.
Several attempts were made to avoid this conclusion. In Refs. 56,65 the Drude
model approach was applied to metallic lattices with defects and impurities
possessing some residual relaxation 7(0) 7^ 0. As a result, the equality S(z,0) = 0
was obtained which is in accordance with the Nernst heat theorem. This, however,
does not solve the problem of the thermodynamic inconsistency of the Drude model
approach, because metallic perfect crystal lattice with no impurities has a nonde-
generate dynamic state of lowest energy. Thus, according to quantum statistical
physics, the entropy at T = 0 must be equal to zero for such crystal lattices [a
property violated by the Drude model approach according to Eq. (18)].
Another attempt66 includes spatial dispersion in the calculations of the Casimir
energy. At large separations it arrives at the same Eq. (12) as was obtained by
using the Drude model. At arbitrary separations between the plates computations
in Ref. 66 nearly exactly coincide with earlier computations54 using the Drude
model. In Refs. 30,35,67 it was demonstrated, however, that the results of Ref. 66
are not reliable because the used approximative description of a spatial dispersion
is unjustified. The main mistake in Ref. 66 is that it uses the standard continuity
boundary conditions (6) on the electromagnetic field which are valid only in the
absence of spatial dispersion. If the spatial dispersion is present, one must use
instead the more complicated conditions68
4-7T
Eu = E2t, Bln = B2n, D2n - Dln = 4ira._ [n x (B2 - B1)} = —j, (20)
2719
where the induced charge and current densities are given by
2 2
* =^/div [nx[Dxn]]dZ, j = -J—dl. (21)
i i
In the Reply69 to the Comment67 the author attempts to avoid this conclusion
by introducing the auxiliary fields and by bringing the Maxwell equations to the
form with no induced charge and current densities. This attempt, however, fails
because, as the author himself recognizes, the relations used by him are valid only
in the Fourier space. In the case of temporal dispersion there is no problem in
making the Fourier transform. However, for spatial dispersion in the presence of
boundaries and a macroscopic gap between the two plates, this is not allowed.67
The system under consideration in the Casimir effect is not spatially uniform and
it is not possible to introduce the dielectric permittivity s(q,u>) depending on both
the wave vector and the frequency as is done in Refs. 66,69.
Reply69 denies the note in the Comment67 that the formalism used in the original
work66 involves nonconservation of energy. In support of this denial, it is argued that
the energy leaving a region through an interface is entering the region on the other
side, and, thus, energy is fully conserved. To arrive at this conclusion, the author
admits that the in-plane components of the fields are continuous across the interface.
In the presence of spatial dispersion this assumption is, however, not valid, as was
demonstrated above. We underline that the violation of energy conservation in the
so-called "dielectric approximation" of nonlocal electrodynamics used in Refs. 66,69
has long been rigorously proved70 and discussed in the literature.68
To conclude, presently there is no question that the approach to the thermal
Casimir force using the Drude model is thermodynamically invalid. At the same
time, the plasma model and impedance approaches are consistent with
thermodynamics. In particular, they satisfy the Nernst heat theorem.26'35 In Refs. 28-30 it
was shown that the same problems, as for metals, arise for dielectrics if one
describes their conductivity at zero frequency with the help of the Drude model. This
problem is more detailly discussed in another contribution to these Proceedings.71
Important problem is the comparison of different theoretical approaches to the
thermal Casimir force with experiment. As was already emphasized in Sec. 3, the
computations at zero temperature, and also theoretical approaches using the plasma
model and the Leontovich surface impedance at T = 300 K, are consistent with
experiment (see, for example, Fig. 1, left). At the same time, the theoretical approach
using the Drude model is excluded by experiment at 95% confidence level within
the separation region from 170 to 700 nm. In the separation region from 300 to
500 nm the Drude model approach is excluded by experiment at even higher 99%
confidence level.19'20 For the purposes of comparison between experiment and
theory, the computations of the Casimir pressure were done by using the tabulated
optical data for the complex index of refraction44 extended to lower frequencies. In
fact, a marked difference between approaches arises only when calculating the con-
2720
tribution of the zero-frequency term in the Lifshitz formula which should be found
theoretically because at very low frequencies optical data are not available.
The comparison between experiment and theory in Fig. 1 is quite
transparent. However, in Refs. 27,57 several objections against it were raised. According to
Ref. 57, Purdue group18'19 claims "the extraordinary high precision to be able to
see our effect at distance as small as 100 nm" and "the accuracy is claimed to be
better than 1% at separations down to less than 100nm". These statements are
misleading because the experimental ranges in Refs. 18 and 19 are from 260 to
HOOnm and from 160 to 750 nm, respectively. There are no statements concerning
the separations of 100 nm and below 100 nm in Refs. 18,19. According to another
claim in Refs. 27,57, the determination of the absolute sphere-plate separation with
the absolute error Az = 0.6 nm, as stated in Ref. 19, is difficult because "the
roughness of the surfaces is much larger than the precision stated in the determination
of the separation". This claim is not right because the separations are measured
between zero levels of the surface roughness. These zero levels are uniquely
determined for any value of the roughness amplitude. One more claim57 is that "the
effects of surface plasmons72,73 have not been included". This claim is wrong
because the computations in Refs. 18,19 were performed using the Lifshitz formula
which includes in full the effects of surface plasmons.
As was noticed recently,74 the precise values of the Drude parameters are
important for an accurate calculation of the Casimir force in experimental configurations.
According to Ref. 74, the use of different Drude parameters measured and
calculated for different Au samples may lead to up to 5% variations in the magnitude
of the Casimir force. In the computations of Refs. 18-20 the values u>p — 9.0 eV
and 7(T = 300 K) = 0.035 eV were used which are based on the experimental data
of Ref. 44 and computations of Ref. 75. As was demonstrated above, these values
lead to a very good agreement with traditional approaches to the thermal Casimir
force which predict only small thermal corrections at short separations and exclude
the Drude model approach. If much smaller value for Au plasma frequency were
used in computations (i.e., cup = 6.85 eV or 7.50 eV as suggested in Ref. 74) the
agreement between the traditional theoretical approaches and experimental data
would be worse for a few percent. The same holds for many other experiments on
the Casimir effect.13'14'17-23 It should be particularly emphasized that with smaller
values of ujp the disagreement between the experimental data and the Drude model
approach to the thermal Casimir force becomes much larger than is demonstrated in
Fig. 1 (right). If one uses widely accepted criteria from the statistical theory of the
verification of alternative hypotheses,76 the hypothesis on much smaller magnitude
of lop (than that used in Refs. 18-20) is rejected at high confidence by all already
performed experiments on the Casimir force with Au surfaces. This conclusion was
recently confirmed77 by the determination of the plasma frequency of Au coatings
in the experimental configurations of Refs. 18-20 using the measured temperature
dependence of the films resistivity. The obtained result u>p = 8.9 eV [and a
respective value for j(T = 300 K) = 0.0357eV] is in excellent agreement with Refs. 44,75.
2721
It leads to even better than in Fig. 1 agreement of data with the traditional
approaches to the thermal Casimir force and excludes the Drude model approach at
the impressive 99.9% confidence level within a wide separation range. Thus, to date,
it is beyond question that the Drude model approach is experimentally excluded.
One more important physical phenomenon which sheds light on the problem
of the thermal Casimir force is the modulation of the Casimir force with laser
light discussed in Sec. 3. From Fig. 3 it follows that the experimental data are
consistent with theory if the dc conductivity of high resistivity Si in the absence
of laser light is discounted. On the contrary, the dashed line takes into account dc
conductivity of a Si plate in the absence of laser light described using the Drude
dielectric function. As is seen in Fig. 3, the dashed line is experimentally excluded.
Thus, for both metals and semiconductors the account of actual dielectric response
at very low frequencies leads to contradictions between the Lifshitz theory and the
experiment. To achieve an agreement between experiment and theory, one should
use the dielectric response in the region of characteristic frequency ~ c/(2z) and
extrapolate it to zero frequency. The complete understanding of this problem goes
beyond the scope of the Lifshitz theory.
5. Constraints on new physics beyond the Standard Model
Many extensions of the Standard Model predict a new (so-called "fifth") force
coexisting with the usual Newtonian gravitational force and other conventional
interactions. Such force can arise from the exchange of light elementary particles
(e.g., scalar axions, graviphotons, dilatons, and moduli78'79), and as a consequence
of extra-dimensional theories with low energy compactification scales.80'81 The
interaction potential of the fifth force acting between two point masses raj and m-2
at a distance r is conventionally represented as a Yukawa correction to the usual
Newtonian potential
V(r)=-^i(l + ac-^)J (22)
where G is the gravitational constant, a is a dimensionless constant characterizing
the strength of the Yukawa force, and A is its interaction range.
It has been known18-20'31'82-85 that the best constraints on the parameters
(a, A) in submicron range follow from the measurements of the Casimir force. The
pressure of a hypothetical interaction, Phyp(z), which may act between experimental
test bodies is computed18-20 by the pairwise summation of potentials (22) with a
subsequent negative differentiation with respect to separation. Then constraints
on the hypothetical Yukawa-type pressure are found from the agreement between
measurements and theory at 95% confidence level. According to the experimental
results in Refs. 19,20, no deviations from calculations using the traditional theories
of the Casimir force were observed. Thus, one can conclude that the hypothetical
pressure should be less than or equal to the half-width of the confidence interval
\Phyp(z)\ < Atot [Pth(z) - Pexp(z)] , (23)
2722
log10 \a
-7.5 -7 -6.5 -6 -5.5
log10[A (m)]
Fig. 4. Constraints on the Yukawa interaction constant a versus interaction range A. Line 1 is
obtained from the measurements of the Casimir preesure by use of a micromechanical torsional
oscillator.19,20 Line 2 follows from the isoelectronic differential force measurements.31 Line 3 is
obtained from the measurement of the Casimir force using an atomic force microscope, and line 4
from the torsion-pendulum experiment.9 The strongest constraints following from the gravitational
measurements using a micromachined cantilever86 are indicated by the line 5.
where Atot [Pth(z) - Pexp{z)] is the total absolute error of the quantity Pth(z) -
Pe*p(z). Note that just this error (and negative this error) are plotted in Fig. 1 (left
and right) by the solid lines.
In Fig. 4 we plot the strongest constraints on a for different values of A following
from the measurements of the Casimir force and compare them with the best
gravitational experiments. Each line in Fig. 4 is related to some specific experiment. The
region of (a, A) plane above each line is prohibited from the respective experiment
and below each line is permitted. Constraints shown by line 1 follow from the most
recent measurement of the Casimir pressure using a micromechanical torsional
oscillator.19'20 Line 3 is obtained85 from the Casimir force measurement between an
Au-coated sphere and a plate by means of an atomic force microscope.13 Line 4
follows82 from the measurement of the Casimir force between an Au-coated spherical
lens and a plate by means of torsion pendulum.9 Line 2 presents the constraints
obtained from the first isoelectronic differential force measurement31 between an
Au-coated probe and two Au-coated films, made out of gold and germanium. In
this measurement the Casimir background is experimentally subtracted, thus
avoiding the necessity to model the Casimir force. Finally, line 5 shows the most strong
2723
constraints on Yukawa-type deviations from Newtonian gravity obtained86 from
the gravitational experiment using a micromachined silicon cantilever as the force
sensor at separations of order 25 /im, where the Casimir force is already negligibly
small. Gravitational experiments provide also the strongest constraints on a in the
interaction range A > 10_5m (see Refs. 20,78 and 87 for more details).
As is seen in Fig. 4, at and below a micrometer interaction range there is no
competitors to the Casimir effect in obtaining constraints on non-Newtonian
gravity. During the last few years these constraints were strengthened by up to 104
times basing on the results of different measurements of the Casimir force between
metal surfaces. Nevertheless, existing limits on a below a micrometer are still
relatively weak and should be strengthened by several orders of magnitude to reach
the theoretically predicted regions of strange and gluon modulus, and of gauged
barions.
6. Conclusions and discussion
In the foregoing, we have discussed main achievements in the physics of the Casimir
effect during the period after the Xth Marcel Grossmann Meeting hold in 2003.
In our opinion, the major theoretical breakthrough is the obtaining of the exact
analytical solution for the electromagnetic Casiinir energy in the configuration of
an ideal metal cylinder above a plate. The resolution of this problem opened new
opportunities for the investigation of the Casimir force between curved boundaries
and permitted find first indisputable result on the accuracy of the PFT.
The major experimental breakthroughs during this period are the most precise
measurements of the Casimir pressure between metal surfaces using a microrne-
chanical oscillator (Decca et al.) and the first experiments on the Casimir effect in
configuration of metal sphere and semiconductor plate by means of an atomic force
microscope (Mohideen et al.). Both sets of experiments resulted in far-reaching and
important conclusions. Experiments by Decca et al. have conclusively excluded large
thermal corrections to the Casimir force at short separations as predicted by the
Drude model approach. Experiments by Mohideen et al. demonstrated the
possibility to control the Casiinir force by changing the density of free charge carriers and
led to new knowledge on the applicability of the Lifshitz theory to dielectrics.
Time between Xth and Xlth Marcel Grossmann Meetings was marked by
controversial discussions of different approaches to the theoretical description of the
thermal Casimir force. During these discussions it was clearly demonstrated that
all the proposed approaches are of approximate phenomenological character. None
of them can yet claim to be the final fundamental resolution of the problem. Till the
end of this period it was conclusively demonstrated that the Drude model approach
is in contradiction with the foundations of thermodynamics and is excluded
experimentally at a 99.9% confidence level. Important theoretical problem for future is
the fundamental understanding of the thermal Casimir force and related physical
phenomena caused by vacuum and thermal oscillations of the electromagnetic field
2724
(e.g., atomic friction, radiative heat transfer etc.). The experimental challenge for
near future is the measurement of the thermal effect in the Casimir force which has
not been measured yet.
Both recent theoretical achievements and the performed experiments confirm the
unique potential of the Casimir effect both in fundamental physics for constraining
predictions of new unification physical theories beyond the Standard Model and
in nanotechnology for fabrication, operation and control of a new generation of
microdevices. This confirms the increasing role of the Casimir effect both in modern
physics and in technological applications.
Acknowledgments
The author is grateful for helpful discussions with M. Bordag, R. S. Decca, E. Fis-
chbach, B. Geyer, G. L. Klimchitskaya, D. E. Krause, and U. Mohideen.
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LOCAL AND GLOBAL CASIMIR ENERGIES IN A GREEN'S
FUNCTION APPROACH
K. A. MILTON* and I. CAVERO-PELAEZt
Oklahoma Center for High Energy Physics and H. L. Dodge Department of Physics,
University of Oklahoma, Norman, OK 73019 USA
* milton@nhn. on. edu
t cavero @nhn. ou. edu
K. KIRSTEN
Department of Mathematics, Baylor University,
Waco, TX 76798 USA
Klaus. Kir sten@baylor. edu
The effects of quantum fluctuations in fields confined by background configurations may
be simply and transparently computed using the Green's function approach pioneered
by Schwinger. Not only can total energies and surface forces be computed in this way,
but local energy densities, and in general, all components of the vacuum expectation
value of the energy-momentum tensor may be calculated. For simple geometries this
approach may be carried out exactly, which yields insight into what happens in less
tractable situations. In this talk I will concentrate on the example of a scalar field in a
circular cylindrical delta-function background. This situation is quite similar to that of a
spherical delta-function background. The local energy density in these cases diverges as
the surface of the background is approached, but these divergences are integrable. The
total energy is finite in strong coupling, but in weak coupling a divergence occurs in third
order. This universal feature is shown to reflect a divergence in the energy associated
with the surface, the integrated local energy density within the shell itself, which surface
energy should be removable by a process of renormalization.
Keywords: Casimir energy, divergences, renormalization
1. Casimir Energies for Spheres and Cylinders
The calculation of Casimir self-energies of material objects has become
controversial,1 although these concerns are nearly as old as the subject itself.2-4 Although
it appears possible to extract unique self-energies, they may be overwhelmed by
terms which become divergent for ideal geometries.5,6 Our attitude is that these
terms may be uniquely removed by a process of renormalization, and that even the
divergences revealed by heat-kernel methods7'8 may be unambiguously isolated.
Table 1 summarizes the state of our knowledge concerning total Casimir self-
energies for different simple configurations. The first row of the table refers to the
Casimir energy of a perfectly conducting shell, either spherical or cylindrical,
subject to electromagnetic fluctuations in the exterior and interior regions. The second
row refers to the same results for a scalar field subject to Dirichlet boundary
conditions on the surface. The remaining four rows describe small perturbations: Row
3 describes what happens for electromagnetic fluctuations when the interior of the
sphere or cylinder is a dielectric having a permittivity e differing slightly from the
vacuum value of unity; Row 4 indicates the same when the speed of light is the same
2727
2728
inside and outside the object, where £ = {e' ~ e)/{e' + e) in terms of the permittivity
inside (e) and outside (e1) the object; Row 5 shows the effect for a perfect
conductor of a small ellipticity 8e (± refers to a prolate or oblate spheroid, respectively);
and Row 6 refers to a 5-function potential (semitransparent shell) of strength A,
which will be described in this paper. In these four cases, what is shown in the
table is the coefficient of the second-order term in the relevant small quantity. One
of the ongoing challenges facing quantum field theorists attempting to understand
the quantum vacuum is to understand the pattern of signs and zeroes manifested
in the table.
Table 1. Casimir energy (E) for a sphere and Casimir
energy per unit length (£) for a cylinder, both of radius a. The
signs indicate repulsion or attraction, respectively.
Type ESpherea ^Cylinder"2 References
EM +0.04618 -0.01356 9, 10
D +0.002817 +0.0006148 11, 12
(e - l)2 +0.004767 = j^ 0 13, 14
£2 +0.04974= J-*^ 0 15, 16
^ OZ7T '
Se2 ±0.0009 0 17, 18
A2 +0.009947 = =f- 0 19, 20
OZ7T '
In this talk, we will illustrate the ideas for the interesting case of a circular
cylindrically symmetric annular potential. Most of the calculations will refer to a
^-function potential.
2. Green's Function
We consider a massless scalar field <fr in a ^-cylinder background,
Cint = -±5{r-a)<j>2, (1)
a being the radius of the "semitransparent" cylinder. We recall that the massive
case was earlier considered by Scandurra.21 Note that with this definition, A is
dimensionless. The time-Fourier transform of the Green's function,
G(x,x') = J^e-^-^g(r,r>), (2)
satisfies
-V2-u2 + -5{r-a)
a
Adopting cylindrical coordinates, we write
r dh °° i
<?(ry) = y§V*(*-*> Yl ^e^-^gm(r,r';k), (4)
771= —OO
g(r,r') = S(r-r'). (3)
2729
where the reduced Green's function satisfies
1 d d 2 m2 A
■-—r— +k H — +-d(r - a)
r dr dr rA a
gm(r,r';k) = -S(r-r'),
where
■ u)2. Let us immediately make a Euclidean rotation,
(5)
(6)
where £ is real, so k is always real and positive. Apart from the 5 functions, Eq. (5)
is the modified Bessel equation.
2.1. Reduced Green's function
Because of the Wronskian satisfied by the modified Bessel functions,
Km{x)I'm{x) - K'm(x)Im(x) = -,
(7)
we have the general solution to the Green's function equation (5) as long as r ^ a
to be
gm(r,r';k) = 7m(Kr<)KTO(Kr>) + A(r')Im{nr) + B(r')Km(nr),
(8)
where A and B are arbitrary functions of r'. Now we incorporate the effect of the
5 function at r = a in the Green's function equation. It implies that gm must be
continuous at r = a, while it has a discontinuous derivative,
r=a+
= Xgm(a,r';k), (9)
d
a-rgm(r,r';k)
from which we rather immediately deduce the form of the Green's function inside
and outside the cylinder:
r,r' <a : gm{r,r';k) = 7TO(Kr<)Km(Kr>)
XKl(Ka)
1 + XIm(Ka)Km(na)
r,r' > a: gm(r,r';k) = 7m(Kr<)KTO(Kr>)
Xl^ina)
(/w)/m (/w-0, (10a)
Km(Kr)Km(Kr'). (10b)
1 + XIm(Ka)Km(Ka)'
Notice that in the limit A —> oo we recover the Dirichlet cylinder result, that is,
that gm vanishes at r = a.
3. Pressure and Energy
The easiest way to calculate the total energy is to compute the pressure on the
cylindrical walls due to the quantum fluctuations in the field. This may be computed,
at the one-loop level, from the vacuum expectation value of the stress tensor,
(T""> = ( d»d'v - l-g^dx&^\ -G{x,x')
ttd^d" -g^d2)\G{x,x). (11)
2730
Here we have included the conformal parameter £, which is equal to 1/6 for the
conformal stress tensor. The conformal term does not contribute to the radial-
radial component of the stress tensor, however, because then only transverse and
time derivatives act on G(x,x), which depends only on r. The discontinuity of the
expectation value of the radial-radial component of the stress tensor is the pressure
on the cylindrical wall:
-* = X-'rr/in {-'-rr/out
1 £ fW A«2
167T3 -^ J_00 J_00 1 + XIm{K,a)Km{na)
x [Kl(Ka)C(Ka) ~ ll(m)K£(Ka)]
1 ^ f°° „ f°° ,.k d
^ POO f-OO j
V dk dC-—ln[l + \Im(Ka)Km(Ka)], (12)
__^J-oo J-oo a daa
167T3
m=-ooJ-°° J-°°
where we've again used the Wronskian (7). Regarding ka and ("a as the two Cartesian
components of a two-dimensional vector, with magnitude x = na = \Jk2a2 + (2a2,
we get the stress on the cylinder per unit length to be
i r°° °° d
S = 2iraP = - ^—^ I dxx2 J^ ^ ln I1 + A/™ (x)Km (x)] ,
" rn.= — no
(13)
rn= — oo
implying the Dirichlet limit as A —> oo. By integrating S = —-§^£, we obtain the
energy per unit length
1 f°° °° d
E = ~ W / dx x* 2Z Yx ln [1 + Mm {x)Km (:E)]
(14)
m= —oo
This formal expression will be regulated, and evaluated in weak and strong coupling,
in the following.
3.1. Energy
Alternatively, we may compute the energy directly from the general formula22
£=s/«<r>/£2"2s<">- <i5>
To evaluate the energy in this case, we need the indefinite integrals
dyylUv) = \ [(x2+m2)ll(x)-x2C] , (16a)
i
dyyK2m(y) = -- [(x2+m2)K2m(x)~x2K£\ . (16b)
When we insert the above construction (10) of the Green's function, and perform
the integrals as indicated over the regions interior and exterior to the cylinder, we
obtain
2 °° /-oo /.oo i j
£ =-g^2 E J d^J dk(2-—\n[l + \Im(x)Km(x)}. (17)
o
m— — oo
2731
Again we regard the two integrals as over Cartesian coordinates, and replace the
integral measure by
d£ dkC,2 = tt / dnn3. (18)
-oo J— oo JQ
The result for the energy (14) immediately follows.
4. Weak-coupling Evaluation
Suppose we regard A as a small parameter, so let us expand the energy (14) in
powers of A. The first term is
\ <x> „oo j
m=-oo JV
The addition theorem for the modified Bessel functions is
oo
K0(kP)= J2 em^-^Km(kp)Im(kp'), p>p', (20)
m= — oo
where P = y7' p2 + p'2 — 2pp' cos((j> — </>'). If this is extrapolated to the limit p' = p
we conclude that the sum of the Bessel functions appearing in E^1' is i^o(0), that
is, a constant, so there is no first-order contribution to the energy, £W = 0.
4.1. Regulated numerical evaluation of S^
Given that the above argument evidently formally omits divergent terms, it may
be more satisfactory to offer a regulated numerical evaluation of £^\ We can very
efficiently do so using the uniform asymptotic expansions (m —> oo):
'-w-te'-T + E^]' (21a)
k=i
'1 + »-^).
v fc=l /
^)~#e--(l + »l)^], (21b)
where x = mz, t = I/a/1 + z2, and j^ = ~. The polynomials in t appearing here
are generated by
u0(t) = l, uk(t)=l-t2{l-t2)u'k__1(t) + J ds^—^-Uk-iis). (22)
Thus the asymptotic behavior of the products of Bessel functions appearing in
Eq. (19) is obtained from
^)^(-)-^(l + E^)- (23)
2732
The first three polynomials occurring here are
n(t)
F
(l-6t2 + 5t4),
r2(i) = —(7-148i2
554i4- 708t6 + 295t8),
r3(t) = __(36 - 1666t2 + 1377514 - 44272t6
16
10
67162t8 - 48510ilu + 13475i^)
(24a)
(24b)
(24c)
We regulate the sum and integral by inserting an exponential cutoff, S —> 0+:
f(i)
A f00 d
m=0 ,/U
-x<5
(25)
where the prime on the summation sign means that the m = 0 term is counted with
one-half weight. We break up this expression into five parts,
A
fd)
Sira2
(I + II + III + IV + V).
(26)
The first term is the m = 0 contribution, suitably subtracted to make it convergent
(so the convergence factor may be omitted),
I
dx-2
d
Io(x)K0(x)
jo dx
The second term is the above subtraction,
d 1
2a/T
»"5
dxx
1
26
1,
(27)
(28)
/0 \dx y'T^X2'
as may be verified by breaking the integral in two parts at A, 1 C A < 1/5. The
third term is the sum over the mth Bessel function with the two leading asymptotic
approximants in Eq. (23) subtracted:
+2
1—' /-00
III = 2 ^ / dx x2
m=l J°
d
dx
Im(x)Km(x)
t
2m
t
1 + 3—?(l-6t2 + 5t4)
8mz
0.
(29)
Numerically, each term in the sum seems to be zero to machine accuracy. This is
verified by computing the higher-order terms in that expansion, in terms of the
polynomials in Eq. (24):
Im(x)Km(x)
t
2m
i + ^i-^ + st4)
4m5
r2(*) " ~Arl(t)
8m2
t
Am7
rz(i) -\ri(t)r2(t) + \r\[t)
(30)
which terms are easily seen to integrate to zero. The fourth term is the leading
subtraction which appeared in the third term:
oo „0
IV = J^ m /
m=l J°
dz z
d
dz
t e
(31)
2733
If we first carry out the sum on m we obtain
1 f°° 1 1 111
IV=^7o dZz3(l+z^s^z5/2~-Ti + 25-6> (32)
as verified by breaking up the integral. The final term, due to the sub leading
subtraction, if unregulated, is the form of infinity times zero:
(33)
V = - y - / dzz2 — (t3 - 6t5 + 5t7)e~m^.
Here the sum on m gives
£ e—' = -ln(l-e-**), (34)
m
and so we can write
V = — — I' dull- u)au-2~a(u3/2 - Qu5'2 + W2)
16 da J0 K ' K !
Adding together these five terms we obtain
a=0
(35)
f(1) = 8^ + °< (36)
that is, the 1/5 and constant terms cancel. The remaining divergence may be
interpreted as an irrelevant constant, since 5 = r/a, t being regarded as a point-splitting
parameter. This thus agrees with the result stated at the beginning of this section.
4.2. A2 term
We can proceed the same way to evaluate the second-order contribution to Eq. (14),
**=£#£<** I17"'WA™(I)- (37)
m= — oo
By squaring the sum rule (20), and again taking the formal singular limit p' —> p,
we evaluate the sum over Bessel functions appearing here as
OO „27T J
J2 ll{x)K2m{x) = / ^02(2xsin^/2). (38)
— /n 27T
Then changing the order of integration, we can write the second-order energy as
£(2) =-«T5-2 r -^TdzzKUzl (39)
b4TTzaz JQ sin <£/2 7o
where the Bessel-function integral has the value 1/2. However, the integral over ip
is divergent. We interpret this integral by adopting an analytic regularization based
on the integral (Res > — 1)
Ja \ 2) I' 1 + f)
2734
Taking the right-side of this equation to define the ip integral for all s, we conclude
that the ip integral, and hence the second-order energy £(2\ is zero.
The vanishing of the energy in order A and A2 may be given a quite rigorous
derivation in the zeta-function approach to Casimir energies—See Ref. 20.
4.2.1. Alternative numerical evaluation
Again we provide a numerical approach to bolster our argument. Subtracting and
adding the leading asymptotic behaviors, we now write the second-order energy as
(z = x/m)
5(2)
A2
87TO2
dxx
ll{x)I<l{x)
1
1
oo
lim > 'r
'm " I dz
oo
4(1 + X2)
i-s r2
oo 3
1 + z* J0 4^^m2
-,2k
•^ /-OO J.Z I "
2^/ dxx I2m{x)K2m{x) ~ ^ 1 + E
The successive terms are evaluated as
-,2k
(41)
5(2)
A2
!, , ^ 1, C(2) 7C 4 31C 6
4W ' 4 48 1920 16128
87T02
+0.000864 + 0.000006
A2
87ra-
(0.000000),
(42)
where in the last term in the energy (41) only the m = 1 and 2 terms are significant.
Therefore, we have demonstrated numerically that the energy in order A2 is zero to
an accuracy of better than 10~6.
4.2.2. Exponential regulator
The astute listener will note that we used a standard, but possibly questionable,
analytic regularization in defining the second term in energy above. Alternatively,
as in Sec. 4.1 we could insert there an exponential regulator in each integral of e~xS,
with 5 to be taken to zero at the end of the calculation. For m^Oi becomes mz,
and then the sum on m becomes
oo
-mzS
l
l
Then when we carry out the integral over z we obtain for that term
TV 1
8,5
In 2tt.
(43)
(44)
2735
Thus we obtain the same finite part as above, but in addition an explicitly divergent
term
^ = "64^- (45)
Again, if we think of the cutoff in terms of a vanishing proper time t, 5 = r/a,
this divergent term is proportional to 1/a, so the divergence in the energy goes like
L/a, if L is the (very large) length of the cylinder. This is of the form of the shape
divergence encountered in Ref. 14.
4.3. Divergence in 0(A3)
Although the first two orders in A identically vanish, a divergence in the energy (14)
does occur in 0(A3).
1 °° r°° A 1
(46)
m — -~oo
A3
967ra2s'
That such a divergence does occur generically in third order was proved in Ref. 20,
using heat-kernel techniques. As we shall see, this divergence entirely arises from
the surface energy.
5. Strong Coupling
The strong-coupling limit of the energy (14), that is, the Casimir energy of a Dirich-
let cylinder,
00 »oo
1 f d
£D = -—^ V / dxx2— \nIm(x)Km(x),
8naz Z-J In dx
(47)
m—— 00
was worked out to high accuracy by Gosdzinski and Romeo
12
cD 0.000614794033
6 = ^ • y4H>
It was later redone with less accuracy by Nesterenko and Pirozhenko. For
completeness, let us sketch the evaluation here. Again subtracting and adding the lead-
2736
ing asymptotics, we find for the energy per unit length
£
D
2 I dxx
o
\n(2xI0(x)K0(x))
1 1
8 1+a:2
In (2xIm(x)Km(x)) - In —
xt
ln2x + 2 V /
m
dx x2 — In xt
ax
in(t)
2 m2
n(t)
l l
m—1
4 1 -\-x2
1 oo
-V
oo
dx-
1 +x2
(49)
In the first two terms we have subtracted the leading asymptotic behavior so the
resulting integrals are convergent. Those terms are restored in the fourth, fifth, and
sixth terms. The most divergent part of the Bessel functions are removed by the
insertion of 2x in the corresponding integral, and its removal in the third term.
(Elsewhere, such terms have been referred to as "contact terms.") The terms
involving Bessel functions are evaluated numerically, where it is observed that the
asymptotic value of the summand (for large m) in the second term is l/32m2. The
fourth term is evaluated by writing it as
2 1im V
n2-s
s-»0
m=l
'^^=2C'(-2)--^
(50)
while the same argument, as anticipated, shows that the third "contact" term is
zero.a The sixth term is
■- lim
2 s->o
C(*) +
1
In 2tt.
(51)
The fifth term is elementary. The result then is
£D = (0.010963 - 0.0227032 + 0 + 0.0304485 + 0.21875 - 0.229735)
0.0006146
(52)
which agrees with Eq. (48) to the fourth significant figure.
5.1. Exponential regulator
As in the weak-coupling calculation, it may seem more satisfactory to insert an
exponential regulator rather than use analytic regularization. Now it is the third,
fourth, and sixth terms in the above expression that must be treated. The latter is
aThis argument is a bit suspect, since the analytic continuation that defines the integrals has no
common region of existence. Thus the argument in the following subsection may be preferable.
2737
just the negative of the term (44) encountered in weak coupling. We can combine
the third and fourth terms to give
_i i r dzz3 d2 i
S2 + S2 JQ z2 + S2 dz2 ez - 1' ( '
The latter integral may be evaluated by writing it as an integral along the entire z
axis, and closing the contour in the upper half plane, thereby encircling the poles
at i5 and at 2inir, where n is a positive integer. The residue theorem then gives for
that integral
-*[-M (54)
so once again, comparing with Eq. (50), we obtain the same finite part as in Eq. (52).
In this way of proceeding, then, in addition to the finite part found before in
Eq. (52), we obtain divergent terms
£div= 64^5 + W^ + 4^P' (55)
which, with the previous interpretation for 5, implies terms in the energy
proportional to L/a (shape), L (length), and aL (area), respectively, and are therefore
renormalizable. Had a logarithmic divergence occurred (as does occur in weak
coupling in 0(A3)) such a renormalization would be impossible. However, see below!
6. Local Energy Density
We compute the energy density from the stress tensor (11), or
(T00) = - (d°d0' + V • V) G(x, x') - | V2G(x, x)
2* x'-x l
i r00 r°° ^ r / m2 \
— / dk dto Y, U2 + k2 + — +drdr,\g{r,r')
J-oo J-oo TO=-oo ^ '
16ir3i
1
2£-drrdrg(r,r)
(56)
We omit the free part of the Green's function (10), since that corresponds to the
m enerffV in the a^c'=,l"1'"''=, /~if fVio /-.TrlinrldT A/^/npn T*r<=» incprf fVid rdtnainripr nf f np
's function, we
the cylindrical shell:
pan or me ^reen s mncuon (lv), since mat corresponas to une
vacuum energy in the absence of the cylinder. When we insert the remainder of the
Green's function, we obtain the following expression for the energy density outside
i(r) = -
A
16tt3 /_
r dc r dk v &w
J-oo J-oo ^^l + XImi^K,
.(«a)
2^ + ^ + ^Kl(Kr) + ^K^Kr)
-2Z-^-r^-K2m(Kr)
r dr dr my '
r > a.
(57)
2738
The factor in square brackets can be easily seen to be, from the modified Bessel
equation,
9 9 , N 1 - 4£ 1 d d o
2^K^Kr) + -^---r-K^Kr). (58)
For the interior region, r < a, we have the corresponding expression for the energy
density with Im <-> Km.
6.1. Total and surface energy
We first need to verify that we recover the expression for the energy found before.
So let us integrate the above expression over the region exterior of the cylinder,
and the corresponding interior expression over the inside region. The second term
in Eq. (58) is a total derivative, while the first may be integrated according to the
integrals given in Eq. (16). In fact that term is exactly that evaluated above. The
result is
Hdr)u{r) = -— Y, / dxx2—\n[l + XIm(x)Km(x)}
n ac\ A r°j V^ Im(x)Km(x)
-(l-4£)-—_- I dxx } , r . , ' . ,■ (59)
^ s/47ra270 ^> I + \Im{x)Km(x)
m= — oo
The first term is the total energy (14), but what do we make of the second term?
In strong coupling, it would represent a constant that should have no physical
significance (a contact term—it is independent of a if we revert to the physical
variable k as the integration variable).
In general, however, there is another contribution to the total energy, residing
precisely on the singular surface. This surface energy is given in general by22,24-28
1-4C
2i .is
i dS- VG(x,x')
(60)
which turns out to be the negative of the second term in f(dr) u{r) given in Eq. (59).
This is an example of the general theorem
(dr)u(r) + <B = E, (61)
that is, the total energy E is the sum of the integrated local energy density and the
surface energy. A consequence of this theorem is that the total energy, unlike the
local energy density, is independent of the conformal parameter £.
6.2. Surface divergences
We now turn to an examination of the behavior of the local energy density as r
approaches a from outside the cylinder. To do this we use the uniform asymptotic
2739
expansion (21). Let us begin by considering the strong-coupling limit, a
Dirichlet cylinder. If we stop with only the leading asymptotic behavior, we obtain the
expression (z = nr/m)
1 r°° °°
u{r)~-^L dKK ^
m— — oo
V^ + 2(l-4C)«2
2m
7T 1 "
2mt z2
, (A-
(62)
where
X = -2 [77(2) -77 (2-)
(63)
and we have carried out the "angular" integral as in Eq. (18). Here we ignore the
difference between r and a except in the exponent, and we now replace k by mz/a.
Close to the surface,
X
2r~a
t r
and we carry out the sum over m according to
d3 1 12
2£
m3e-mx
3 iV
4^4
dx3 X XA 4 (r - a)4 '
Then the energy density behaves, as r —> a+,
M(r)
16tt2 (r - a)4
(1 - 6£).
(64)
(65)
(66)
This is the universal surface divergence first discovered by Deutsch and Candelas.2
It therefore occurs, with precisely the same numerical coefficient, near a
Dirichlet plate19 or a Dirichlet sphere.29 It is utterly without physical significance (in
the absence of gravity), and may be eliminated with the coiiformal choice for the
parameter £, £ = 1/6.
6.3. Conformed surface divergence
We will henceforth make this conformal choice. Then the leading divergence depends
upon the curvature. This was also worked out by Deutsch and Candelas;2 for the
case of a cylinder, that result is
1 1
u(r)
720tt2 r{r - af '
a+,
(67)
exactly 1/2 that for a Dirichlet sphere of radius a. To get this result, we keep the
1/m corrections in the uniform asymptotic expansion, and the next term in x:
2 r — a
X ~ 7
t r
r — a
(68)
2740
6.4. Weak coupling
Let us now expand the energy density (57) for small coupling,
1D7T J_00 J-oo -,__—, „—n
V + (1 - 40 K
.1 , m
r
K^(Kr) + (l-4C)KzK'^(Kr)}. (69)
If we again use the leading uniform asymptotic expansions for the Bessel functions
we obtain the expression for the leading behavior of the term of order An,
u(n)(r)~8^v("2 / d**£m3"ne"m**n--1(*2 + 1-80-
^ /Jo m=1
(70)
The sum on m is asymptotic to
£m3-"e-*~(3-n)!(-^-j , r - a+, (71)
so the most singular behavior of the order An term is, as r —► a+,
„(n)(r) „ (_A)n (3-n)!(l-y)
w v ; 967r2r"(r-a)4-'1
(72)
This is exactly the result found for the weak-coupling limit for a <5-sphere and for
a <5-plane,22 so this is a universal result, without physical significance. It may be
made to vanish by choosing the conformal value £ = 1/6.
6.5. Conformal weak coupling
With this conformal choice, once again we must expand to higher order. Besides
the corrections noted in Sec. 6.3, we also need
i=t(za/r) ~ t + {t - tz)T-^^, r->a, (73)
Then a quite simple calculation gives
which is analytically continued from the region 1 < Ren < 3. Remarkably, this
is exactly one-half the result found in the same weak-coupling expansion for the
leading conformal divergence outside a sphere.29 Therefore, like the strong-coupling
result, this limit is universal, depending on the sum of the principal curvatures of the
interface. Note this vanishes for n = 1, so in every case this divergence is integrable.
2741
7. Cylindrical Shell of Finite Thickness
We now regard the shell (annulus) to have a finite thickness 5. We consider the
potential
4„t =-^V(r), . (75)
where
{0, r<a-,
h,a-<r<a+, (76)
0, a+ < r.
Here a± = a±<5/2, and we set hS = 1. In the limit as 5 —> 0 we recover the <5-function
potential. As for the sphere29 it is straightforward to find the Green's function for
this potential. In fact, the result may be obtained from the reduced Green's function
given in Ref. 29 by an evident substitution. Here, we content ourselves by stating
the result for the Green's function in the region of the annulus, a_ < r,r' < a+:
gm(r,r') = Imin'r^Km^'ry) + A/to(kV)/to(kV')
+ B[Im{K'r)Km(K'r') + Km(K'r)Im(K'r')} + CKm(K'r)Km(K'r'),
(77)
where k' = yK2 + Xh/a. The coefficients appearing here are
A = -^[Krm(Ka-)Km(K a-) - k'Im(Ka-.)K'm(k'a-)]
x[KK'm(Ka+)Km(K a+) - K'Km(Ka+)K'm(n'a+)}, (78a)
B = ^[Kl'm(na-)Im(K'a_) - K7TO(Ka_)/^(K'a_)]
x[KK'm(Ka+)Km(Ka+) - KKm(Ka+)K'm(Ka+)], (78b)
C = -^[Kl'm(Ka-)Im(n'a-) - k Im{na^)rm{n'a^)]
x [nK'm(Ka+)Im(K'a+) - k Km(Ka+)l!m(K''a+)], (78c)
where the denominator is
5 = [Kl'm(Ka-)Km(K'a-) - k Im(Ka-)K'm(K'a-)]
x[KK'm(Ka+)Im(K a+) - K,Km(Ka+)l!m(Ka+)]
- [Krm(Ka-)Im(K a-) - k'Im(Ka-)I'm(K'a-)]
x [KK'm(Ka+)Km(Ka+) - K'Km(na+)K'm(Ka+)}. (79)
2742
7.1. Energy within the shell
The general expression for the energy density within the shell is given in terms of
these coefficients by
u(r)
1
8^
dm
Ad d
+ 1-40-^—
r or or
Y, W^'r) + CKl(K'r) + 2BKm(K'r)Im(K'r)}. (80)
m= — oc
7.2. Leading surface divergence
The above expressions are somewhat formidable. Therefore, to isolate the most
divergent structure, we replace the Bessel functions by the leading uniform asymptotic
behavior (21). A simple calculation implies
A
B
C ■
t++t'+
l+ -*'+*- ~ *'-c2m(r,'_-r,'+)
t+ + t'+ i_ + t'_
t -f
t- + t'_
(81a)
(81b)
(81c)
where t+ = t(z+), rj'_ = r)(z'_), z'_ = K'a-/m, etc. If we now insert this
approximation into the form for the energy density, we find
u = <TUU)
1 °° /"OC
YT2 X] m / dz+Z+t'r
a+ m-\ ^°
8tT2
t+ + t'+ t- + t'_
9 9
mzz+
(1 - 80 +
m. a\
(1 - 4£)
i2^2t+ t'+t- t'_^2m{r],__n,+ )
'+t+ + t',t-
t'_
(82)
If we are interested in the surface divergence as r approaches the outer radius
a+ from within the annulus, the dominant term comes from the first exponential
factor only. Because we are considering the limit Xha <C m2, we have
t'+ « t+ ( 1
Xh a
2m2 a
+e
and we have
Xh/a sr^
"327r2a2 2^!
+ m=\
dzzt(l-8S, + t2)e2m^'-'1'+').
(83)
(84)
2743
The sum over m is carried out according to Eq. (71), or
J2 me2m«-'+'
rt'.
2(r - a+)
and the remaining integrals over z are elementary. The result is
Xh 1 - 6£
96n2a (r — a+)2
r —> a
+ ;
(85)
(86)
the expected universal divergence of a scalar field near a surface of discontinuity,30
without significance, which may be eliminated by setting £ = 1/6.
7.3. Surface energy
Now we want to establish that the surface energy € (60) is the same as the integrated
local energy density in the annulus when the limit <5 —> 0 is taken. To examine this
limit, we consider Xh/a s> k2. So we apply the uniform asymptotic expansion for the
Bessel functions of k' only. We must keep the first two terms in powers of k <C k':
l2Im(Ka_)Km(Ka+) . , ,
-k :—: , . smhm(r/_ - rj+)
./,
K K
m
— \\ Tr4(Kfl-)^mK) - — \ 7rIm(Ka-)K'm(Ka+)
-+ V + z- " -
x coshm(ri_ — ?/_).
Because we are now regarding the shell as very thin,
'~atr
where
1
z7
<Xha
using the Wronskian (7) we get the denominator
E ~ - -^ [1 + XIm (Ka)Km (««)]•
cr
Then we immediately find the interior coefficients:
Im(Ka)Km(Ka)
A
B
C
v Xha -™V-/~mv-~/ -2mV
2 1 + XIm (KM,)Km (Ka)
1 nrr— Im(KCl)Km(Ka)
- VXlia —7 ; -——; r ,
2 l + XIm(Ka)Rm(Ka)'
1 ^fyj^ Im{na)K m (Ka) 2mW
2n ' 1 + XIm (K.a)Km (h;a)
(87)
(88)
(89)
(90)
(91a)
(91b)
(91c)
2744
7.4. Identity of shell energy and surface energy
We now insert this in the expression for the energy density (80) and keep only the
largest terms, thereby neglecting k2 relative to Xh/a. This gives a leading term
proportional to h, which when multiplied by the area of the annulus 2ira5 gives for
the energy in the shell
£ann~(l-4£)-j—^ > / dmna y / ^ / 92
which is exactly the form of the surface energy € given by the negative of the second
term in the integrated energy density (59).
7.5. Renormalizability of surface energy
In particular, note that the term in € of order A3 is, for the conformal value £ = 1/6,
exactly equal to that term in the total energy £ in Eq. (46):
£(3)=£(3). (93)
This means that the divergence encountered in the global energy is exactly
accounted for by the divergence in the surface energy, which would seem to provide
strong evidence in favor of the renormalizablity of that divergence.
8. Conclusion
The work reported here and in Refs. 20,29 represents a significant advance in
understanding the divergence structure of Casimir self-energies. We have shown that the
surface energy of a <5-function shell potential is in fact the integrated local energy
density contained within the shell when the shell is given a finite thickness. That
surface energy contains the entire third-order divergence in the total Casimir energy.
The local Casimir energy diverges as the shell is approached, but that divergence is
iiitegrable, so it yields a finite contribution to the total energy. The identification of
the divergent part of the total energy with that associated with the surface strongly
suggests that this divergence can be absorbed in a renormalization of parameters
describing the background potential.
Challenges yet remain. This renormalization procedure needs to be made precise.
Further, we must make more progress in understanding the sign (and for cylindrical
geometries, the vanishing) of the total Casimir self-energy. And, of course, we must
understand the implications of surface divergences on the coupling to gravity. Work
is proeeeding in all these directions.
Acknowledgments
We thank the US National Science Foundation and the US Department of Energy
for partial funding of this research. KAM is grateful to Vladimir Mostepanenko for
inviting him to participate in MG11. We thank S. Fulling, P. Parashar, A. Romeo,
K. Shajesh, and J. Wagner for useful discussions.
2745
References
1. N. Graham, R. Jaffe, V. Khemani, M. Quandt, O. Schroeder and H. Weigel, Nucl.
Phys. B677, 379 (2004) [arXiv:hep-th/0309130].
2. D. Deutsch and P. Candelas, Phys. Rev. D20, 3063 (1979).
3. P. Candelas, Ann. Phys. (N.Y.) 143, 241 (1982).
4. K. A. Milton, Ann. Phys. (N.Y.) 127, 49 (1980).
5. G. Barton, J. Phys. A34, 4083 (2001).
6. G. Barton, J. Phys. A37, 1011 (2004).
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th/9811015].
8. P. B. Gilkey, K. Kirsten and D. V. Vassilevich, Nucl. Phys. B601, 125 (2001).
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10. L. L. DeRaad, Jr. and K. A. Milton, Ann. Phys. (N.Y.) 136, 229 (1981).
11. C. M. Bender and K. A. Milton, Phys. Rev. D50, 6547 (1994) [arXiv:hep-th/9406048].
12. P. Gosdzinsky and A. Romeo, Phys. Lett. B441, 265 (1998) [arXiv:hep-th/9809199].
13. I. Brevik, V. N. Marachevsky and K. A. Milton, Phys. Rev. Lett. 82, 3948 (1999)
[arXiv:hep-th/9810062].
14. I. Cavero-Pelaez and K. A. Milton, Ann. Phys. (N.Y.) 320, 108 (2005) [arXiv:hep-
th/0412135],
15. I. Klich, Phys. Rev. D61, 025004 (2000) [arXiv:hep-th/9908101].
16. K. A. Milton, A. V. Nesterenko and V. V. Nesterenko, Phys. Rev. D59, 105009 (1999)
[arXiv:hep-th/9711168, v3].
17. A. R. Kitson and A. I. Signal, J. Phys. A39, 6473 (2006) [arXiv:hep-th/0511048].
18. A. R. Kitson and A. Romeo, Phys. Rev. D74, 085024 (2006) [arXiv:hep-th/0607206],
19. K. A. Milton, Phys. Rev. D68, 065020 (2003) [arXiv:hep-th/0210081].
20. I. Cavero-Pelaez, K. A. Milton and K. Kirsten (2006) [arXiv:hep-th/0607154],
21. M. Scandurra, J. Phys. A33, 5707 (2000) [arXiv:hep-th/0004051],
22. K. A. Milton, J. Phys. A37, 6391 (2004) [arXiv:hep-th/0401090].
23. V. V. Nesterenko and I. G. Pirozhenko, J. Math. Phys. 41, 4521 (2000) [arXiv:hep-
th/9910097],
24. J. S. Dowker and G. Kennedy, J. Phys. All, 895 (1978).
25. G. Kennedy, R. Critchley and J. S. Dowker, Ann. Phys. (N.Y.) 125, 346 (1980).
26. A. Romeo and A. A. Saharian, J. Phys. A35, 1297 (2002) [arXiv:hep-th/0007242].
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28. S. A. Fulling, J. Phys. A36, 6529 (2003) [arXiv:quant-ph/0302117].
29. I. Cavero-Pelaez, K. A. Milton and J. Wagner, Phys. Rev. D73, 085004 (2006)
[arXiv:hep-th/0508001].
30. M. Bordag and J. Lindig, J. Phys. A29, 4481 (1996).
BOUNDARY INDUCED QUANTUM FLUCTUATION EFFECTS:
FROM MOVING MIRROR TO ELECTRON COHERENCE*
JEN-TSUNG HSIANG and DA-SHIN LEE
Department of Physics, National Dong-Hwa University, Hua-lien, Taiwan 974, R- O. C.
Two distinct, but related issues in quantum fluctuation effects induced by the boundary
are discussed. We first consider a perfectly reflecting mirror moving in a quantum scalar
field. The stochastic behavior of the mirror with the backreaction from the field can be
described by the semiclassical Langevin equation derived from the coarse-grained
effective action with the method of influence functional. Then the backreaction effects by
solving the Langevin equation are discussed. We next exam the influence of
electromagnetic vacuum fluctuations in the presence of the conducting plate on electron coherence
with an interference experiment. The evolution of the reduced density matrix of the
electron is obtained by integrating out electromagnetic fields. We find that the plate
boundary anisotropically modifies vacuum fluctuations that in turn affect the electron
coherence.
1. Perfectly Reflecting Mirror Moving in a Quantum Scalar Field
Consider a perfectly reflecting mirror moving in a quantum field. The boundary
conditions on the quantum field corresponding to perfect reflection result in the
interaction of the mirror with the field. The motion of the mirror, which leads to the
moving boundary, can create quantum radiation that in turn damps out the motion
of the mirror as a result of this motion-induced radiation reaction.1,2 In fact, as
required by Lorentz invariance of the field, the force of radiation reaction vanishes
for a motion with uniform velocity. In a motion of uniform acceleration, the mirror
suffers from the same fluctuations as if it was at rest in a thermal bath due to the
Unruh effects, also leading to the zero dissipative force. Fulling and Davies have
computed this force for a mirror moving under a mass less scalar field in the 1+1
dimensional spacetime. It turns out that the motion induced force is proportional
to the third time derivative of the mirror's position.1 Ford and Vilenkin have
extended the study to the 3+1 dimensions in terms of a first order approximation of
the mirror's displacement. The corresponding force then is given by the fifth time
derivative of the position in the non-relativistic limit.2 However, as we know, all
quantum fields exhibit fluctuations that manifest themselves through fluctuating
forces such as fluctuations of Casimir forces. Thus, through a fluctuation and
dissipation relation, in addition to motion-induced radiation reaction, the mirror must
experience the backreaction dissipation effect arising from force fluctuations.
We here employ the Schwinger-Keldysh formalism to study the moving mirror
problem in a massless scalar field in 3+1 dimensions.3 In the case of the small
mirror's displacement, the coarse-grained effective action is obtained by integrating out
the field with the method of influence functional. In the semiclassical regime, we find
"This work was supported in part by the National Science Council, R. O. C. under grant NSC93-
2112-M-259-007.
2746
2747
that the Langevin equation reveals two levels of backreaction effects: radiation
reaction induced by the motion of the mirror as well as backreaction dissipation arising
from the retarded force correlations. Then, the accompanying noise term with the
Gaussian correlation function consistent with a fluctuation-dissipation relation is
obtained to mimic the stochastic dynamics arising from quantum field fluctuations.
Consider a situation where the mirror is attached to a spring and undergoes
oscillations with a natural frequency. In vacuum, backreaction dissipation is
obtained as the fifth time derivative of the mirror's position with the colored noise.
We find that this backreaction effect results in a long relaxation time such that
a time scale more than 104 oscillations is needed to detect a tiny decrease in the
amplitude of the mirror. The mirror gains energy from vacuum fluctuations by
absorbing fewer than 1CP4 quanta during each oscillation. Thus, the effects of vacuum
fluctuations can hardly be detected. Contrary to the vacuum fluctuations, in the
high-temperature limit, the dominant contribution on dissipation is the term
proportional to the mirror's velocity with the uncorrelated white noise as expected. As
long as the temperature of thermal fields is of order kev, the ratio of the amplitude
fluctuations to the amplitude of the oscillating mirror are of order f 0~8 within the
time scales of 10~2s, leading to detectable effects.
2. Electron Coherence Influenced from Quantum Electromagnetic
Fields in the Presence of Conducting Plates
Quantum coherence entails the existence of the interference effects amongst
alternative histories of the quantum states. These effects are nevertheless not seen at the
classical level. The suppression of quantum coherence can be viewed as the result of
the unavoidable coupling to the environment, and thus leads to the emergence of the
classical behavior in terms of incoherent mixtures. This environment-induced deco-
herence has been studied with the idea of quantum open systems by coarse-graining
the environment where certain statistical measures are introduced.4 Thereby, this
averaged effect appears as decoherence of the system of interest.
The influence of zero-point fluctuations of quantum electromagnetic fields in
the presence of the perfectly conducting plates on electrons is studied. The effects
of modified vacuum fluctuations can be observed through the electron interference
experiment, and are manifested in the form of the amplitude change and phase
shift of the interference fringes.5 The method of influence functional is employed by
tracing out the fields in the Coulomb gauge from which we find the evolution of the
reduced density matrix of the electron with self-consistent backreaction.6
Under the classical approximation with the prescribed electron's trajectory
dictated by an external potential, we find that the exponent of the modulus of the
influence functional describes the extent of the amplitude change of the
interference contrast determined by the Hadamard function of vector potentials, and its
phase results in an overall shift for the interference pattern related to the retarded
Green's function. In addition, it is known that the stochastical Langevin equation of
2748
the particle coupled to a quantum field, involves backreaction dissipation in terms
of the retarded Green's function as well as the accompanying stochastic noise with
its correlation function given by the Hadamard function. These two effects are in
general linked by the fluctuation-dissipation theorem.3 Thus, we may conclude that
reduction of coherence is caused by field fluctuations while the phase shift results
from backreaction dissipation through particle creation that influences the mean
trajectory of the electron.
We evaluate the decoherence functional of the electrons with the boundary on
quantum electromagnetic fields. The boundary conditions can be imposed by the
presence of either a single plate or double parallel plates. In each case, the path
plane on which the electrons travel for the interference experiment can be parallel
or perpendicular to the plate (s). It is found that the effects of coherence reduction
of the electrons by zero-point fluctuations with the boundary are strikingly
deviated from that without the boundary. Thus, the presence of the conducting plate
anisotropically modifies electromagnetic vacuum fluctuations that in turn influence
the decoherence dynamics of the electrons. In particular, as the electrons are close
to the plate, electron coherence is enhanced in the case where the path plane of
the electrons is parallel to the plate. This results from the suppression of zero-point
fluctuations due to the boundary condition in the direction parallel to the plate. On
the other hand, the electron coherence is reduced in the perpendicular configuration
where zero-point fluctuations are boosted along the direction normal to the plate. In
addition, in the presence of an additional parallel plate boundary, zero-point
fluctuations seems to make the electrons more coherent in the parallel configuration, but
less coherent in the perpendicular one, compared with the single-plate boundary.
Thus, the loss of decoherence of the electrons can be understood from zero-point
fluctuations of electromagnetic fields given by the Hadamard function of the vector
potentials. Furthermore, the backreaction dissipation through photon emission can
influence the mean trajectory of the electron, and in turn leads to the phase shift
on the electron inference pattern through the retarded Green's function. We wish in
our future work to address the issue of the relation between the amplitude change
and phase shift of interference fringes via the fluctuation-dissipation theorem, which
might be testable in the interference experiment.
References
1. S. A. Fulling and P. C. W. Davies, Proc. R. Soc. London A 348 (1976); ibid.356, 237
(1977).
2. L. H. Ford and A. Vilenkin, Phys. Rev. D 25, 2569 (1992).
3. C.-H. Wu and D.-S. Lee, Phys. Rev. D 71, 125005 (2005).
4. M. Gell-Mann and J. B. Hartle, Phys. Rev. D 47, 3345 (1993); W. H. Zurek, Phys.
Rev. D 24, 1516 (1981); Phys. Today 44, 36 (1991).
5. L. H. Ford, Phys. Rev. D 47, 5571 (1993); Phys. Rev. A 56, 1812 (1997).
6. J.-T. Hsiang and D.-S. Lee, Phys. Rev. D 73, 065022 (2006).
A THEORY OF ELECTROMAGNETIC FLUCTUATIONS FOR
METALLIC SURFACES AND VAN DER WAALS INTERACTIONS
BETWEEN METALLIC BODIES
GIUSEPPE BIMONTE
Universita degli Studi di Napoli Federico II, Dipartimento di Scienze Fisiche, Complesso
Universitario di Monte S. Angelo, Via Cintia, Edificio N', 80126 Naples, Italy
and
INFN, Sezione di Napoli.
bimonte@na.infn.it
We obtain a new expression for the electromagnetic fluctuations outside a metal surface,
in terms of its surface impedance, providing a generalization to real metals of Lifshitz
theory of van der Waals interactions between dielectric solids. We use the new formulae
to compute the radiative heat transfer between two metal surfaces, separated by an
empty gap. It is shown that an experiment on heat transfer may provide a resolution of
a long-standing controversy about the effect of thermal corrections on the Casimir force
between real metal plates.
Keywords: fluctuations, impedance, Casimir, heat transfer
In recent years much attention has been devoted to the study of
electromagnetic (e.m.) fluctuations, both quantum and thermal, mainly in connection with
current work on dispersion forces, Bose-Einstein condensates, nanotechnology,
radiative heat transfer. In this context, we have recently derived1 a new formula for
the correlation functions of the random e.m. fields that are present outside a metal
surface in thermal equilibrium, as a result of the fluctuating microscopic currents
in the interior of the metal. A key feature is that the correlation functions are
expressed in terms of the surface impedance £, and therefore they apply to the
anomalous region, as well as to superconductors (extreme anomalous effect).
Let the metal occupy the z < 0 half-space. We consider the Fourier
decomposition of the e.m. field outside the metal. For the electric field of TE modes, we
write:
^(te)=?2^w r ^ h2k± °(w'k±) i±ei(*'*~ut)+c-c-' (i)
where k± is the tangential component of the wave-vector k and e±_ is a unit
vector perpendicular to the plane formed by k± and the normal to the metal surface.
The third component of the wave-vector kz = ^/w2/c2 — k\ is defined such that
Re(kz) > 0 and Im(kz) > 0. The e.m. field is therefore a superposition of
propagating waves (p.w.) travelling away from the surface (for k± < oj/c) and of evanescent
waves (e.w.) (for k± > w/c) exponentially dying out away from the surface.
Similarly, we write for the magnetic field of TM modes:
B{TM) = J~y^j cLvJcP^b^k^exe'^-^+cc. (2)
In Ref. 1 we obtained the following expressions for the statistical averages for the
2749
2750
products of amplitudes a(u>, k±) and b(u>, fcj_):
(3)
<^>»>^»™<=*hG&) icrfk^'1^1-^ (4)
with all other correlators vanishing, and k Boltzmann's constant.
Using the above Equations, we have obtained a new derivation of the Casimir
force between two metallic plates.1 Here, however, we shall consider an application of
Eqs.(3), (4) that relates to the presently controversial issue of the modification of the
Casimir force arising from a non zero temperature of the mirrors, when the latter are
treated as real metals. The debate was raised by the findings of Ref. 2, showing that
the combined effect of temperature and finite conductivity leads to large deviations
from the ideal metal case. This result was obtained within the framework of Lifshitz
theory, by using the Drude dielectric function e.o(w) = l~flp/[cu (cu+ij)} to describe
the metal. Finite conductivity is taken into account by allowing a non-vanishing
value for the relaxation frequency 7. The results of Ref. 2 have been criticized by
several authors, and supported by others (see Ref. 3, and Refs. therein).
Recent studies4 shed much light on the problem, showing that the large
deviations from the ideal metal case obtained in Ref. 2 arise from thermal TE e.w. of low
frequencies (cu = 1010 — 1013 rad/s for L = 1/zm). It is also shown there that if
surface impedance b.c. are used, with £ = 1/^/Fd (which is the expression valid in the
domain of the normal skin effect), instead of the large repulsive thermal correction
from TE e.w. found in Ref. 2, one obtains an attractive correction, of much smaller
magnitude, while no appreciable differences are found both in the TM and in the
TE p.w. sectors. The important conclusion is that the present disagreement on the
magnitude of the thermal correction to the Casimir force for real metals, arises from
the fact that different models for the metal lead to largely different predictions for
the magnitude of the thermal TE e.w. correction.
Unfortunately, the present precision of Casimir force experiments does not
permit detection of the thermal effect, and therefore it would be valuable to devise
alternative experiments, to establish which model of the metal better describes physical
effects of thermal TE e.w., in the frequency range that is relevant for the Casimir
effect. A key remark now is that the relevant e.w. are not vacuum fluctuations, as
in the Casimir force at zero temperature, but rather real thermal excitations. Now,
it is known that thermal e.w. give the dominant contribution to the power of heat
transfer S between two closely spaced metal surfaces, at submicron separations.0 It
is therefore very interesting to see what is the prediction of impedance theory for
S. and to compare it with the result from Lifshitz theory, as discussed in Ref. 5.
Our formula for the power S (per unit area) has the form of a difference between
2751
two terms, one for each plate:
Ah [°° o f 1 1 \
S = '^l duJUJ Up^/fcro - i " exP(^/fcr2) -1) R«(Ci)Re(Ca)
x Re fdpp\p\2 |e2^Wc| (J_ + _J_\ (5)
J \ aTE £>TM J
where the contour of integration for p is along the real axis, from one to zero (p.w.),
and then along the imaginary axis from iO to ioo (e.w.). The quantities BTE,TM
are defined as:
Bte = |(1 +Pd)(l + Ka) - (1 -Ki)(l -PC2)exp(2ipLW/C)|2 , (6)
BTm = \(p + Ci)(p + Ca) " (P ~ Ci)(P - Ca) exp(2ipLcu/c)\2 . (7)
In Ref. 6 is it shown that the power of heat transfer, at submicron separations, is
extremely sensitive to the model used to describe the metal, and therefore an
experiment measuring S would provide strong indications of which model is preferable, a
knowledge which could then be used in the evaluation of the thermal Casimir effect.
In conclusion, we have presented new formulae for the correlation functions
of e.m. fluctuations present outside a metal. The formulae involve the surface
impedance of the metal, and are therefore applicable in the anomalous region, as
well as in the extreme anomalous case (superconductors), where Lifshitz theory is
not valid. As an application, we have evaluated the radiative heat transfer between
two metal plates at different temperatures and we have shown that a measurement
of this quantity should provide enough information to settle experimentally recent
controversies about the thermal Casimir effect.
The author acknowledges partial financial support by PRIN SINTESI.
References
1. G. Bimonte, Phys. Rev. Lett. 96, 160401 (2006).
2. M. Bostrom and B.E. Sernelius, Phys. Rev. Lett. 84, 4757 (2000); B.E. Sernelius, ibid.
87, 139102 (2001).
3. G.L. Klimchitskaya and V.M. Mostepanenko, Contemp. Phys. 47, 131 (2006).
4. J.R. Torgerson and S.K. Lamoreaux, Phys. Rev. E 70, 047102 (2004); S.K. Lamoreaux,
Rep. Prog. Phys. 68, 201 (2005); G. Bimonte, Phys. Rev. E 73, 048101 (2006).
5. D. Polder and M. Van Hove, Phys. Rev. B 4, 3303 (1971).
6. G. Bimonte, G.L. Klimchitskaya and V.M. Mostepanenko Thermal correction to the
Casimir force, radiative heat transfer and experiment, submitted.
THEORY OF THE CASIMIR EFFECT BETWEEN DIELECTRIC
AND SEMICONDUCTOR PLATES*
G. L. KLIMCHITSKAYAt>t and B. GEYER§
Center of Theoretical Studies Institute for Theoretical Physics, Leipzig University,
Augustusplatz 10/11, 0^109, Leipzig, Germany
t Galina.Klimchitskaya@itp.uni-leipzig.de
§ Bodo. Geyer@itp.uni-leipzig. de
The theory of the thermal Casimir interaction between two dielectric plates or between
one metallic and one dielectric plate are discussed. It is shown that if the static
dielectric permittivity of a dielectric plate is finite, the Lifshitz theory is in agreement with
the requirements of thermodynamics. The inclusion of the nonzero dc conductivity of
a dielectric plate is shown to lead to a violation of the Nernst heat theorem. The
experimental and theoretical results related to the Casimir interaction between metal and
semiconductor with different charge carrier density are also considered.
Keywords: Casimir effect; entropy; Nernst heat theorem.
In the last few years the Casimir effect between metal plates at nonzero temperature,
T ^ 0, was hotly debated.1,2 It was generally agreed, however, that the case of
dielectric plates is basically clear and free of contradictions. The situation has been
changed after the publication of Ref. 3,4 where the low-temperature behavior of the
free energy, entropy and pressure of the Casimir interaction between two dielectric
plates with dielectric permittivity e(u>) was investigated analytically. It was shown
that this behavior is determined by only the static dielectric permittivity eo = £(0).
As an example, if £q < oo, i.e., the dc conductivity is not taken into account, the
Casimir free energy at r e 4irkBaT/(hc) < 1 (a is the separation between the
plates, Ub is the Boltzmann constant) is given by3,4
he
TDD{a1T) = EDD{a)
C(3)(gp-1)2
2567r2a3 [ Tr2(e0 + 1)
(1)
^(4/2-l)(4 + 4/2-2)r4 + 0(r5)
90
Here Edd{o) is the Casimir energy at T = 0, and £(z) is the Riemann zeta function.
From Eq. (1), it follows that the Casimir entropy Sdd((i,T) at low temperatures
goes to zero as T2, i.e., Sdd(o., 0) = 0 in accordance with the Nernst heat theorem.
By contrast, if the dc conductivity of the dielectric plate is included into the model
of dielectric response (recall that any dielectric possesses some small dc conductivity
at T ^ 0), S]j£,(a, 0) takes the nonzero value3'4
SDD(a,0) = —i^{C(3)-Li3
lbiraz
gp - 1
£0 + l
> 0, (2)
*This research has been partially supported by DFG grant 436 RUS 113/789/0-2.
tOn leave from North-West Technical University, St.Petersburg, Russia.
2752
2753
where ~Lin(z) is the polylogarithm function. This means that the Nernst heat
theorem is violated.
Similar results were obtained recently for the configuration of one metal and one
dielectric plate. In this configuration in some temperature interval the Casimir free
energy is a nonmonotonous function of temperature and the corresponding Casimir
entropy can be negative.5 For an ideal metal and dielectric with constant e the
analytic expressions for the Casimir free energy, entropy and pressure were found
in Refs. 5,6. In Ref. 7 they were generalized for the case of real metal and dielectric
with frequency-dependent dielectric permittivities. Here, metal is described by the
plasma model, eM(uj) = 1 — cu2/cu2, where cup is the plasma frequency. Note that
in the configuration of metal and dielectric plates the problem, on how to correctly
describe the transverse electric part of the zero-frequency term of metal plate,1,2
does not influence the result. This is because the reflection coefficient of the dielectric
plate at zero frequency is equal to zero. Similar to the case of two dielectric plates,
the low-temperature behavior of all physical quantities depends only on e^ = £D(0)-
When ^(O) is finite, the Casimir free energy at r <C 1 takes the form7
T (nT\-F <n\ ^ f C(3)(^ - l)2 _3 ,„,
^g(o,r)-£Mg(o)- D -t (3)
1
45
1
2567r2a3 \ 2tt2(4j + 1)
2 (*?)3/2 + (^)5/2] -4 + ^ (tf - 1) K + 11) ^/ + 0(r5)} .
Here Emd{o) is the Casimir energy between metal and dielectric plates at T = 0,
wc = c/(2a) is the characteristic frequency of the Casimir force (the plasma model is
applicable at ujc <C u>p). From Eq. (3), it follows that the Casimir entropy Smd(o>, T)
is of order T2, and Smd(o>, 0) = 0 as the Nernst heat theorem requires. If the dc
conductivity of the dielectric plate is taken into account, one obtains a nonzero
Casimir entropy at T = 0,6,7
Sjifu(a,0) =
I6ira2
C(3) - Li3
1
^ + 1
> 0, (4)
in violation of the Nernst heat theorem.
From Eqs. (2) and (4) one can conclude that the Lifshitz theory becomes
inconsistent with thermodynamics when the actual dielectric response of dielectric
materials at very low, quasistatic, frequencies is taken into account. This suggests
that the actual low-frequency behavior of the dielectric permittivity is not related
to the physical phenomenon of the Casimir force. To calculate the Casimir force,
one should extrapolate to zero frequency the dielectric response in the region of
the characteristic frequency, u>c, rather than to use the actual dielectric response at
quasistatic frequencies. This is just what was done in obtaining Eqs. (1) and (3)
which are consistent with thermodynamics.
A further difficulty arises when at least one Casimir plate is made of a
semiconductor. Semiconductors possess diverse conductivity properties ranging from
metallic to dielectric. The first measurement of the Casimir force between a gold-coated
2754
sphere and a single crystal silicon plate was performed using an atomic force
microscope.8 Later it was shown9 that the theoretically computed Casimir force using
the optical data for dielectric Si is excluded by the measurements with a plate made
of B-doped Si. The recent experiment using two silicon samples of higher and lower
resistivities differing by several orders of magnitude demonstrated10 the dependence
of the Casimir force on the density of charge carriers. The experimental results and
their comparison with theory suggest an approach on how to correctly account the
conductivity properties of semiconductors in the theory of the Casimir force. If
the dielectric permittivity of lower-resistivity Si, £St(tu), exhibits drastic increase in
comparison to the static dielectric permittivity, efj (0) = 11.67, of dielectric Si in the
region of the characteristic frequency, cuc ~ 1014 — 1015rad/s, and first Matsubara
frequency, £i = 27rfcBT//i, this should be taken into account and substituted into
the Lifshitz formula in order to calculate the Casimir force. At the same time, if for
a higher-resistivity Si plate the inclusion of the dc conductivity leads to the increase
of eSt(u>) in comparison to Sjj(0) only at frequencies less than about 108rad/s, i.e.,
much smaller than loc and £j, this dc conductivity should not be taken into account
and substituted into the Lifshitz formula. The formulated approach finds
experimental confirmation10,11 and will be used in future measurements of Casimir force
between semiconductors.
References
1. I. Brevik, J. B. Aarseth, J. S. H0ye and K. A. Milton, Phys. Rev. E71, 056101 (2005).
2. V. B. Bezerra, R. S. Decca, E. Fischbach, B. Geyer, G. L. Klimchitskaya, D. E. Krause,
D. Lopez, V. M. Mostepanenko and C. Romero, Phys. Rev. E73, 028101 (2006).
3. B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. D72, 085009
(2005).
4. G. L. Klimchitskaya, B. Geyer and V. M. Mostepanenko, J. Phys. A39, 6495 (2006).
5. B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A72, 022111
(2005).
6. B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Int. J. Mod. Phys. A21,
5007 (2006).
7. B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, arXiv:0704.3818; Ann. Phys.
(N.Y.), 2007, to appear.
8. F. Chen, U. Mohideen, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev.
A72, 020101(R) (2005).
9. F. Chen, U. Mohideen, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev.
A74, 022103 (2006).
10. F. Chen, G. L. Klimchitskaya, V. M. Mostepanenko and U. Mohideen, Phys. Rev.
Lett. 97, 170402 (2006).
11. F. Chen, G. L. Klimchitskaya, V. M. Mostepanenko and U. Mohideen, Optics Express
15, 4823 (2007).
A NOVEL EXPERIMENTAL APPROACH FOR THE MEASURE OF
THE CASIMIR EFFECT AT LARGE DISTANCES
P. ANTONINI1, G. BRESSI2, G. CARUGNO1, G. GALEAZZI3, G. MESSINEO4 and
G. RUOSO3
1INFN sez. di Padova, via Marzolo 8, 35131 Padova, Italy
2 IN FN sez. di Pavia, via Bassi 6, 27100 Pavia, Italy
3INFN sez. di Legnaro, viale dell'Universitd 2, 35020 Legnaro (PD), Italy
4 Dipartimento di Fisica, via Marzolo 8, 35131 Padova, Italy
We present an apparatus based on a mechanical resonator that will use a homodyne
detection technique to sense the Casimir force in the plane-parallel configuration at
distances larger than one micron.
Keywords: Casimir Effect; Quantum fluctuations; Force measurements.
In quantum electrodynamics the properties of vacuum are modified by the
variation of the boundary conditions. The presence of conducting surfaces changes the
energy associated with the non-zero ground state of the electromagnetic field. This
corresponds to a net force acting on the surfaces, known as Casimir effect.1_3 The
attractive force between two parallel and perfectly conducting metal plates is given
by
where c is the speed of light, h the reduced Planck constant, S the surface of the
plates, and d their separation.
Eq. (1) is valid at zero temperature. For a finite temperature, the contribution
to the force due to the thermal photons must also be taken into account. This
contribution is expected to increase the force. The ratio between the non zero
temperature force and the zero temperature force increases with the distance between
the two metal plates.
In the last decade we assisted to an increased interest on Casimir effect, with the
appearance of several theoretical and experimental papers. Several measurements of
the Casimir force were made in the plane-sphere configuration, and one was made on
the plate-plate configuration.4 Due to the d~4 dependence of the force, that results
in a very fast decrease of its magnitude when increasing the distance between the
plates, so far the force has only been measured at distances up to 1 /zm. Yet there
is a need for measurements made at larger distances. This is mainly due to the
study of the thermal-induced correction factor to the Casimir force, that at room
temperature starts to be important only at separations between the to plates of the
order of a few microns.5 Since at distances larger than 1 /xm the force is already very
small, the most promising experimental setup for such a measurement seems to be
the plate-plate configuration, where the extension of the surface could compensate
2755
2756
for the large distance. Another way to measure the thermal contribution to the force
is using the Casimir-Polder effect, that is the force between a bulk object and an
atom. In this configuration the thermal contribution to the force has been measured
for a dielectric and a Bose-Einstein condensate of rubidium atoms.6
We present the setup of an experiment that aims to measure the Casimir force
between two parallel metal plates at distances larger than 1 /jm, a more detailed
presentation can be found in Ref. 7.
Fig. 1 and Fig. 2 show a scheme of the setup.
The position of one of the two metal plates (called 'the source') is modulated
at a fixed frequency. The presence of a force between the two plates results in a
movement of the second plate (called 'the resonator'), at the same frequency, that
can be measured using an interferometer.
Electrostatic and interferometric calibrations are used to determine the elastic
constant of the resonator, the distance between the two plates, their relative angle,
and bias voltage between the two plates. The electrostatic voltages are mainly due
to the presence of different material in the electric contact, and of charged dust.
To decrease the magnitude of these potential all the contacts are made of the same
material (Al). The potential is then measured and counter-biased. A measurement
of the Casimir force at large distances is only possible if that potential is controlled
at a level of a mV, which is possible with this setup.
For the measurements we use the homodyne detection technique (see Ref. 7 for
details).
Detection _/f^\
Photodiode-V?-
n
He-Ne Laser
Isolator
Michelson
Interferometer
BS
To LCR meter or
voltage supply
#
4 f\ Positi
C/%i ire a I ' ...
Source/ pzT1
Vacuum chamber
6-axis
tioning stage
PZT2
] Moving
mirror
Passive low-pass mechanical filter
Optical bench
Fig. 1. Top-view of the experimental setup. An He-Ne laser is used for an interferometric
measurement of the movement of the resonator due to modulation of the position of the source through
a piezoelectric actuator (PZT). The two surfaces are in vacuum at 10~6 mbar. The parallelization
of the two surfaces is reached by means of a 6-axis translational stage.
The calibrations performed so far permit us to measure the following parameters
characterizing the setup: The minimum detectable force is F = 10_10N; The built-in
voltage can be controlled at 1 mV. This results in a dmax = 6 /jm, the maximum
distance between the plates at which the setup is sensitive to the Casimir force, for
S = 1 cm2.
2757
"Y-
'!- i , id 'I '..ill.., [
1.1 'i ii- i.i
1 >;"i
rt
Fig. 2, A photo of the two metal plates. In foreground the resonator, that covers the source. The
electric contacts axe provided by aluminum foils: the use of the same material (Al) for the whole
electric contact is to reduce the contact potentials.
It seems thus possible to be able to measure the Casimir force between the
parallel plates at distances of few microns. The major limitation of our setup is the
parallelization of the two surfaces, and their flatness. This is what limited us to a
minimum distance of 7 /zm, which is too large for the measurement. The resonators
used for these calibrations presented a flatness at level of 2 /,un. We will receive soon
new more flat resonators, that should allow us get the two plates closer.
References
1. H. B. G. Casimir, Proc. K. Ned. Akad. Wet. B51, 793 (1948).
2. E. M. Lifshitz, Son. Phys. JETP2, 73 (1956).
3. P. W. Milormi, The Quantum Vacuum (Academic Press, 1994).
4. G. Bressi, G. Carugiio, R. Onofrio and G. Ruoso, Phys. Rev. Lett. 88, 041804 (2002).
5. C. Genet, A. Lambrecht and S. Reynaud, Phys. Rev. A82, 012110 (2000).
6. J. M. Obrecht, R. J. Wild, M. Antezza, L. P. Pitaevskii, S. Stringari and E. A. Cornell,
Phys. Rev. Lett. 98, 063201 (2007).
7. P. Antonini, G. Bressi, G. Carugno, G. Galeazzi, G. Messineo and G. Ruoso, New J.
Phys. 8, 239 (2006).
MEASUREMENT OF THE CASIMIR FORCE IN THE RANGE
ABOVE 5 MICRONS AND DETECTION OF THE FINITE
TEMPERATURE EFFECT
G. I. RAJALAKSHMI, D. SURESH and R. COWSIK
Indian Institute of Astrophysics, Koramangala, Bangalore - 34, India
C. S. UNNIKRISHNAN
Gravitation Group, Tata Institute of Fundamental Research, Mumbai - 400 005, India
* unni@tifr. res. in
www.tifr.res.in
We report on the measurement of the Casimir force between a plate and a lens, both
gold coated, in the range 5-10 microns employing a torsion balance. The results show
deviation from the standard zero temperature Casimir force law indicating the first
detection of the finite temperature correction.
Keywords: Casimir force, torsion balance, finite temperature correction.
The Casimir effect has been studied in a number of remarkable experiments in the
recent past. A variety of techniques ranging from torsion balance to micro-mechanical
devices have been used in these studies.1 The full theory of the Casimir force at the
finite temperature at which the laboratory experiments are done predicts a
significant finite temperature correction. The essential idea is that the conducting plates
confine both the vacuum modes and the real thermal electromagnetic modes, and
the full Planck distribution has to be used for estimating the resultant force between
the conducting plates. The Casimir force per unit area in the high temperature limit
is
Fc~ - ' 3 whenx»l, <(3) = 1,20206. (1)
The finite temperature force is attractive, as in the zero-temperature effect, but
the distance dependence is different. For the plate-plate configuration, the zero-
temperature effect is Fc oc l/d4, whereas the finite temperature effect is -Fb(T) oc
T/d3. For the plate-sphere configuration, these force laws are Fc oc l/d3 and Fqit)
oc T/d2 respectively. The finite temperature correction starts to dominate when
the separation between the surfaces, which determines the 'cut-off wavelength', is
comparable to the thermal wavelength defined by
£ * kT. (2)
For a temperature of about 300 K, this corresponds to about 5 microns. This is the
reason why the previous experiments have not seen evidence for the finite
temperature corrections.
We employed a sensitive torsion balance2,3 for the measurement of the Casimir
effect. An exhaustive calculation of the finite temperature effects for various con-
2758
2759
figurations has been done by Reynaud and Lambrecht and by Mostepanenko.4,5
These calculations also include corrections due to finite conductivity and surface
roughness. In our experiment, in the range 3 microns to 10 microns, the finite
conductivity correction is less than 2% reduction in the force, and the correction due
to surface roughness is unimportant.
The main element of the torsion balance is a gold coated flat BK7 glass disc, 8
cm in diameter. This disc is suspended by a 90 micron x 19 micron beryllium-copper
annealed strip, with a length of 39 cm. The disc also serves as the mirror for an
auto-collimating optical lever. The optical lever has a sensitivity of 10~8 rad/\/Hz,
and a range of about 10~2 rad. With a torsion constant of 0.05 dyne-cm/rad, the
natural period of the pendulum is 406 seconds. The r.m.s thermal noise amplitude is
below 10~6 rad. A spherical surface of radius of curvature 38 cm, diameter 25 mm,
made from a lens coated with gold, is mounted on a motorized translation stage. The
lens can be moved with a resolution of 50 nm repeatably. A schematic diagram of
the experimental setup is in Figure 1. Two capacitors, with a grounded guard ring,
are used to apply small forces on the pendulum either to control its velocity and
for damping, or for locking it to a fixed angular position using a negative feedback
circuit coupled to the optical lever signal.
In our experiment, almost every element in the proximity of the balance is
coated with gold. Yet, the electrostatic forces are seen to be upto 50 times larger at
largest separations, and detailed and specific experimental algorithms are used to
fit and subtract these forces. The entire instrument is housed in a UHV chamber,
at a vacuum below 3 x 10~8 torr. A detailed description of the experimental setup,
experimental procedure and algorithms for the analysis are contained in the thesis
of G. Rajalakshmi.6
The initial distance between the plate and the lens is measured by allowing the
pendulum to come close to the fixed lens, and making a soft touch, controlled by the
Separation(microns) 10
Pig. 1. Left: Schematic of the experimental set up showing the plate and the lens, control
capacitors, and the optical lever. 'CP' is a gravity compensator Al plate. Right: Results from the
experiment along with 'world data'. At distances beyond 5 microns, deviation from the zero
temperature law is detected (shaded band), consistent with the finite temperature correction.
2760
voltages on the capacitor plates. This point is taken as the zero of the separation,
and the absolute error is about 0.3 microns. Then we 'release' the pendulum from
a known distance close to the lens such that the initial velocity of the fall is zero.
We measure the angular position every 160 ms and this data is used to determine
the acceleration by differentiating the angle data twice. The data analysis is done
as follows. The information on time vs. angle measured is converted to angle vs.
acceleration to get the total torque acting on the pendulum, as a function of the
angle relative to the position of the lens. A polynomial of the form appropriate
to include the electrostatic and the Casimir forces are then fitted to the data. For
the finite temperature Casirnir force between the flat plate and the lens, the force
is proportional to 1/d2 which is proportional to I/O2. The electrostatic forces can
contribute with distance dependence of 1/0. Also there could be a constant
background forces (or with very weak distance dependence), mainly from gravitational
effects. For the zero temperature Casimir effect the distance dependence is steeper,
and the force varies as I/O3.
The analysis clearly shows that the residuals are considerably smaller in the case
of the fit with the finite temperature expression compared to the zero temperature
Casimir expression for the Casimir force for the entire data from 10 micron to 2
microns. Figure 2 shows the results from our experiment along with several other
results, in the range of 100 nm to 10 microns, and a force (per unit area) range
covering 106. The slope of the data for individual experiments is typically —3 with
an error of about 10% in the region where the zero temperature Casimir effect is
the dominant force, and our data at large distance has a slope of —2 ± 0.4. Thus,
the combined data shows the change in the Casimir force law for the plate-lens
configuration from 1/d3 to 1/d2 when crossing over the thermal wavelength.
Apart from detecting the finite temperature Casimir effect, we have new
constraints, comparable to the previous best constraints in a limited range, on
modifications to Newtonian gravity at distances 3-10 microns. These results will be
discussed in a more detailed publication.
References
1. S. K. Lamoreaux, Rep. of Prog, in Phys. 68, 201 (2005).
2. C. S. Unnikrishnan, Observability of the Casimir force at macroscopic distances: A
proposal, Tata Institute preprint (unpublished 1995).
3. R. Cowsik, B. P. Das, N. Krishnan, G. Rajalakshmi, D. Suresh and C. S. Unnikrishnan,
MG-8 proceedings, 949 (World Scientific, 1998).
4. C. Genet, A. Lambrecht, and S. Reynaud, Phys. Rev. A62, 012110 (2000).
5. M. Bordag, B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. Lett.
85, 503 (2000).
6. G. Rajalakshmi, Torsion Balance Investigation of the Finite Temperature Casimir
Force, Ph. D thesis, Indian Institute of Astrophysics, Bangalore, unpublished (2004).
SCALAR CASIMIR EFFECT WITH NON-LOCAL BOUNDARY
CONDITIONS
ARAM SAHARIAN
Department of Physics, Yerevan State University, 1 Alex Manoogian Street, 375049 Yerevan,
Armenia
Abdus Salam International Centre for Theoretical Physics, 34014 Trieste, Italy
and
Departamento de Fisica-CCEN, Universidade Federal da Paraiba, 58.059-970, J. Pessoa, PB C.
Postal 5.008, Brazil
saharian@ictp.it
GIAMPIERO ESPOSITO
INFN, Sezione di Napoli, Complesso Universitario di Monte S. Angelo, Via Cintia, Edificio N',
80126 Naples, Italy
and
Universita degli Studi di Napoli Federico II, Dipartimento di Scienze Fisiche, Complesso
Universitario di Monte S. Angelo, Via Cintia, Edificio N', 80126 Naples, Italy
giampiero. esposito@na. infn. it
Non-local boundary conditions have been considered in theoretical high-energy physics
with emphasis on one-loop quantum cosmology, one-loop conformal anomalies, Bose-
Einstein condensation models and spectral branes. We have therefore studied the
Wightman function, the vacuum expectation value of the field square and the energy-
momentum tensor for a massive scalar field satisfying non-local boundary conditions on
a single and two parallel plates. Interestingly, we find that suitable choices of the kernel
function in the non-local boundary conditions lead to forces acting on the plates that
can be repulsive for intermediate distances.
Keywords: non-local boundary conditions; Casimir effect
In our analysis of the Casimir effect for scalar fields,1 motivated by the work
in Refs. 2,3, we have considered the geometry of two parallel plates with non-local
boundary conditions
nfadM*1) + J <M| /j(h " xj|IM*/M) = 0, x = ajt (1)
where we use rectangular coordinates xM = (t^x1,^), with xy = (x2, ...,xD), and
n^-s is the inward-pointing unit normal to the boundary at x = a,j. For the region
between the plates the corresponding eigenvalues are solutions of the equation1
(z2 — C1C2) sin z + (ci + C2) z cos z = 0, (2)
where the coefficients Cj are determined by the Fourier transforms Fj of the kernel
functions fj(x\\) in the boundary conditions, i.e.
c3 = (-ly-'aFj^) = (-ly-'aJd^f^e^K (3)
The non-local boundary conditions (1) state that the normal derivative at a given
point depends on the values of the field at other points on the boundary. The
2761
2762
properties of the boundary are expressed by the kernel function fj. In a sense, this
setting is similar to that in electrodynamics for the spatial dispersion of the dielectric
function e, where e depends on the wave vector by virtue of spatial dispersion.
Similarly, our non-local boundary conditions engender dependence of the coefficient
Fj in the eigenfunctions on the wave vector k||.
The evaluation of the corresponding Wightman function is based on a variant
of the generalized Abel-Plana summation formula below:1
£ T
z—\„ ,iyi
h(z)
h(0)
+ cos(z + 2a\) sin z/z
21
dt
o
ir9(Cj)
2cj
dz h(z)
/ifte"/2) - /ifte-"/2)
(t-ci)(t-c2) at _ i
(t+ci)(t+c2)e
g,(c^i/2) + g,(c,e-^)}, (4)
where g}- = (z2 + cj)h(z). The application of this formula has made it possible
for us to extract from the VEVs the parts resulting from the single plate and to
present the part induced from the second plate in terms of integrals exponentially
convergent for points away from the boundary. The Wightman function turns out
to be given by1
<0|^(.x'>(.T"i)|0) = {Osl^M^IOs)
>
dt
^)D J
dklleik»-(x»-xii)
3 ' (27F)i
cosh(iXj + 3j) cosh(to'- + Qj)
a\Jk\
+rnz
(t-ci)(t-c2) ?t
(t+Cl)(t+c2)e
1
cosh
(t - t')Jt2/a2-k2^m2
t2 — k2a2 — m2a2
(5)
having denned u=\ log((t — Cj)/(t + Cj)).
Moreover, the vacuum stress in the direction orthogonal to the plates is uniform.
This stress determines the vacuum forces acting on the plates, and the corresponding
effective pressure reads as1
-(orr/io)
2ffp-1
" (2ir)D
duu
D-2
dtt2
Vu2+rn2 \n?-
(t-F1(M))(t + F2(M)) 2at
(t + F1(u))(t-F2(u))
(6)
We have evaluated numerically the vacuum forces acting on the plates in the
case of the kernel functions1
fj(x) = f0je-<»*.
(7)
2763
The corresponding Fourier transforms -Fj(fcii) are given by the formulae
F'(k') = (T^(kW- <8)
where the parameters F^' are defined by
F1(j)=2^-17rT-ir(JD/2)%. ' (9)
r\u
We find that, for the values F\ ' < —1.08, the vacuum pressure is negative for
all interplate distances and the corresponding vacuum forces are attractive. For
the values f|1} > -1.08 there are two values of the distance between the plates for
which the vacuum forces vanish. These values correspond to equilibrium positions of
the plates. Moreover, for values of the distance in the region between these positions
the vacuum forces acting on plates are repulsive. Thus, the left equilibrium position
is unstable and the right one is locally stable.1
It might be interesting to investigate the relation, if any, with the findings in Ref.
4, where the authors obtain a repulsive Casimir force among parallel plates under
the assumption of a suitable ultraviolet cut-off such that the regularized zero-point
energy of the vacuum can be the source of non-vanishing cosmological constant
driving the acceleration of the Universe. It is also important to understand whether
the non-local boundary conditions (1) admit a generalization to scalar or spinor
electrodynamics.
Acknowledgments
The work of A. Saharian has been supported by the INFN, by ANSEF Grant No.
05-PS-hepth-89-70, and in part by the Armenian Ministry of Education and Science,
Grant No. 0124. The work of G. Esposito has been partially supported by PRIN
SINTESI.
References
1. A.A. Saharian and G. Esposito, J. Phys. A39, 5233 (2006).
2. M. Schroder, Rep. Math. Phys. 27, 259 (1989).
3. G. Esposito, Class. Quantum Grav. 16, 1113 (1999).
4. G. Mahajan, S. Sarkar and T. Padmanabhan, Phys. Lett. B641, 6 (2006).
SAMPLE DEPENDENCE OF THE CASIMIR FORCE*
I. PIROZHENKO, A. LAMBRECHT
Laboratoire Kastler Brossel, ENS, CNRS, UPMC,
4, place Jussieu, Case 74, 75252 Paris Cedex 05, France
E-mail: Irina.Pirozhenko@spectro.jussieu.fr
Astrid.Lambrecht@spectro.jussieu.fr
V. B. SVETOVOY
MESA+ Research Institute, University of Twente,
P.O. 217, 7500 AE Enschede, The Netherlands
E-mail: V.B.Svetovoy@el.utwente.nl
We have analyzed available optical data for Au in the mid-infrared range which is
important for a trust-worthy prediction of the Casimir force. Significant variation of the
data demonstrates genuine sample dependence of the dielectric function. We show that
the Casimir force is largely determined by the material properties in the low frequency
domain. To have a reliable prediction of the force with a precision of 1%, one has to
study the optical properties of metallic films used for the force measurement.
Keywords: Casimir force; Dielectric function; Drude model; Kramers-Kronig relation.
1. Introduction
With the development of micro-technologies, the Casimir force1 has now become
a subject of systematic experimental investigation in different configurations and
using various materials.2 To describe the mirrors of arbitrary material theoretically,
the original expression for the perfect Casimir force1 Fcas is replaced by;3-5
d2k [°° AC, rM[<,k]2 e~2KL
^ = 2E/Ti/ £fi* MLK' J a . (1)
^y47r2y0 2tt l-r^Ckfe"2^ KJ
where L is the mirror separation, n = -\/k2 + C2/c2, and rM denotes the reflection
amplitude for a given polarization \i = TE, TM. The material properties enter
these formulas via the dielectric function e (iQ at angular imaginary frequencies
u> = i£, which is related to the physical quantity e" (cu) = Im (e (cu)) through the
Kramers-Kronig dispersion relation.5 The change in the Casimir force from Fcas to
F can suitably be represented by the reduction factor r\ = F/Fcas-3
The Casimir force is often calculated using the optical data taken from Palik's
Handbook.6 For frequencies lower than the lowest tabulated frequency, ujc, the data
has to be extrapolated. This is typically done with a Drude dielectric function
e(Lu)=e'(Lu) + ie"(tu)
9 9
g/H = i- 2 p 2, g"M= , 2P 2,, (2)
LO2 +W2 LO(i02 +L02) W
*Part of this work was funded by the European Contract STRP 12142 NANOCASE.
2764
2765
to [eVJ to [eVJ
Fig. 1. |e'| as a function of u> for bulk gold. Dots are the experimental data .The solid line is
the prediction according to Kramers-Kronig relation with the given Drude parameters. Left panel:
handbook data.6 Right panel: Weaver data.10
which is determined by the plasma frequency wp and the relaxation frequency lot .
The upper limit for the plasma frequency is cu2 = Ne2/(eoml), with N the
number of conduction electrons per unit volume, e the charge and m* the effective
mass of electron. For gold it gives u>p = 9.0 eV allowing to estimate u>T = 0.035 eV
from the optical data.3 Both parameters may be also extracted, form the optical
data at the lowest accessible frequencies.7'8
Here we analyze the optical data for Au from several available sources to
establish the variation range of the Drude parameters and calculate the uncertainty of
the Casimir force due to the variation of existing optical data. A complete discussion
and the bibliography can be found in Ref. 9.
2. Optical data for gold and evaluation of the Drude parameters
In our analysis we employed four sets of optical data for gold.6'10^12 We extrapolated
the dielectric function from the mid-infrared domain to low frequencies using the
Drude model (2). The parameters of the model were found from the fit of the
available low-frequency data. The results are collected in the Table.
We have also retrieved the Drude parameters employing the Kramers-Kronig
relation between e' and e". First, we extrapolated only the imaginary part of the
dielectric function to low frequencies to < luc using the Drude model. Then the
real part of the dielectric function e'{to) was predicted as a function of the Drude
parameters cup and cuT, which were chosen so as to minimize the difference between
the observed and calculated values of s'(uj). In Fig.l we present the results for the
handbook optical data6 and data from Ref. 10 together with the retrieved Drude
parameters. For more examples see Ref. 9.
For all sets of the experimental data, that we have analyzed, the present
procedure gives for the Drude parameters the values close to the ones, that we obtained
before. Experimental curves for e'{tu) are in good agreement with the calculated
2766
ones at low frequencies. At high frequencies the agreement is not so good.
3. Uncertainty in the Casimir force due to variation of optical
properties
The Table gives the reduction factor 77 at different plate separations. The variation
of the optical data and the associated Drude parameters leads to a variation in
the Casimir force ranging from 5.5% at short distances (100 nm) to 1.5% at long
distances (3 //m). This is an indication of the genuine sample dependence of the
Casimir force. For this reason it is necessary to study the optical properties of
the plates used in the Casimir force measurement if a precision of < 1% in the
comparison between experimental values and theoretical predictions is aimed at.
ujp, u>T(eV) \L(/J,m)
ujp = 7.50, lot = 0.0616
wp = 8.41, lot =0.0210
wp = 8.84, tuT = 0.042211
wp = 6.85, ujt = 0.035712
wp = 9.00, lot = 0.0353
0.1
0.43
0.45
0.46
0.42
0.47
0.3
0.66
0.69
0.69
0.65
0.71
0.5
0.75
0.79
0.78
0.75
0.79
1.0
0.85
0.88
0.87
0.84
0.88
3.0
0.93
0.95
0.94
0.93
0.95
References
1. H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51 793 (1948).
2. S. K. Lamoreaux, Phys. Rev. Lett. 78, 5 (1997), 81, 5475 (1998); A. Roy, C.-Y. Lin,
U. Mohideen, Phys. Rev. D60, 111101(R) (1999); H. B. Chan, V. A. Aksyuk, R. N.
Kleiman, D. J. Bishop, and F. Capasso, Science 291, 1941 (2001); R. S. Decca, E.
Fischbach, G. L. Klimchitskaya, D. E. Krause, D. Lopez, V. M. Mostepanenko, Phys.
Rev. D68, 116003 (2003); M. Lisanti, D. Iannuzzi, F. Capasso, Proc. National Acad.
Sci. USA 102, 11989 (2005).
3. A. Lambrecht and S. Reynaud, Eur. Phys. J. D8, 309 (2000)
4. E. M. Lifshitz, Zh. Eksp. Teor. Fiz. 29, 94 (1956) [Sov. Phys. JETP 2, 73 (1956)].
5. E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 2 (Pergamon Press,
Oxford, 1980).
6. E. D. Palik (ed), Handbook of Optical Constants of Solids (N.-Y.: Acad. Press, 1995).
7. V. B. Svetovoy and M. V. Lokhanin, Mod. Phys. Lett. A15, 1437, A15 1013, (2000).
8. M. Bostrom and B. E. Sernelius, Phys. Rev. A61, 046101 (2000).
9. I. Pirozhenko, A. Lambrecht and V. B. Svetovoy, New J. Phys. 8, 238 (2006).
10. J. H. Weaver, C. Krafka, D. W. Lynch, E. E. Koch, Optical Properties of Metals, Part
II, Physics Data No. 18-2 (Fachinformationszentrum Energie, Physik, Mathematik,
Karsruhe, 1981).
11. G. P. Motulevich and A. A. Shubin, Soviet Phys. JETP 20, 560 (1965).
12. V. G. Padalka and I. N. Shklyarevskii, Opt. Spectr. (USSR) 11, 285 (1961).
CASIMIR INTERACTION
BETWEEN ABSORBING AND META MATERIALS
FRANCESCO INTRAVAIA* and CARSTEN HENKEL
Institut fur Physik, Universitat Potsdam, m69 Potsdam, Germany
francesco.intravaia@physik.uni-potsdam.de
We investigate the Casimir energy between two dissipative mirrors in term of a sum over
mode formula which can be interpreted by analogy to a quantum dissipative oscillator.
We also show that metamaterials engineered at scales between the nanometer and the
micron seem a promising way to achieve a repulsive force.
Keywords: Casimir effect; surface plasmon; dissipative materials; meta material; negative
index.
The Casimir force is one of the most accessible experimental consequences of
vacuum fluctuations in the macroscopic world. It is the most significant force between
neutral, non-magnetic objects at distances on the micrometer scale and below. For
many experiments searching for novel short-range forces predicted by unification
models,1_3 theoretical calculations of the Casimir force are crucial and have to be
done at the same level of precision as the experiments.4
In this context, it is essential to account for the differences between the ideal
Casimir case and real-world experiments, for example non-perfect reflectors made
from absorbing material. This problem, in the plate-plate geometry, was solved by
Lifshitz5
oo oo _ i
F(L)=2Mm/^coth(W^ T (-^ - iV , (1)
where L is the distance between the plates, [3 = h/ksT, kz = (cu2/c2 — fc2)1/2,
and rxE.a, ?~TM,a are the reflection coefficients at plate a = 1, 2 for the two
principal polarizations of the electromagnetic field. This formula allows to calculate the
Casimir force (per unit area) in terms of the optical properties of the plates, with
any non-ideal behaviour (finite permittivity, dissipation) taken into account by
suitable models for the reflection coefficients. For example, with a dissipative medium
one uses a complex dielectric function provided it is compatible with causality
constraints.6'7
Lifshitz' approach rather differs from the one used by Casimir. In fact, Casimir
summed the zero-point energies of the electromagnetic modes inside a cavity of
perfectly reflecting mirrors (Dirichlet boundary conditions), renormalizing this sum
by removing the free vacuum energy. The Casimir energy for a cavity with real
mirrors can also be obtained in this way, the modes being here the ones of the
real cavity.8 Adopting a dissipation-less model for the dielectric function, the final
expression coincides with Lifshitz' formula (1). Lifshitz theory has, however, a wider
'Supported by QUDEDIS (ESF program) and FASTNet (European Research Training Network).
2767
2768
range of applicability because dissipative mirrors can also be described. We have
shown that, also in this case, the Casimir effect can be expressed as a sum over
modes.9 A calculation along lines similar to Ref.10 allows to transform Eq.(l) into
(zero temperature, identical mirrors)
L
oo
F{L)=dhfdkk
^nx(k) - 2i ^ log
(2)
where loc is an arbitrary cutoff frequency and the discrete index n labels the
different modes that exist for a given A;-vector and polarization. The frequencies ujn\(k)
are the complex solutions of e2lk"L/r2x —1 = 0. Their imaginary parts obey a
specific sum rule that removes the dependence on the cutoff ujc. The result (2) can be
understood by analogy to the quantized oscillator coupled to a bath, establishing a
bridge between the quantum field theory and the theory of open systems. At zero
temperature, the oscillator's zero-point energy is shifted because the bath quantum
fluctuations couple to the oscillator observables.11 The logarithmic term arises
because the ground state of the uncoupled oscillator is no longer an eigenstate for the
whole system, therefore its energy shows fluctuations.
In the non-dissipative case and at short distance, it is well known that the
Casimir energy can be understood from the interaction between surface plasmon
resonances on the two (metallic) mirrors.12 This holds also in the dissipative case.
Adopting the lossy Drude model (e = 1 — tu2/(cu2 +i^yuj)), one can show that Eq.(2)
reduces to
T?(T\~(auJP 15C(3)7^ hx2
F{L)X{^-—^-)240L-3 (« = 1.193...), (3)
where we have taken the leading order correction in 7 of Eq.(2) and kept in the sum
only two modes, u>± = -ti/w2(l i e kL) ~ 72/2 — 17/2, which are the dissipative
counterparts of the coupled surface plasmons.
Lifshitz theory also allows to consider materials with engineered properties.
Natural materials have a magnetic permeability which actually can be set always equal
to one in the range of frequencies relevant for the Casimir effect. Nothing forbids,
however, to consider artificial materials (also called metamaterials) which show a
strong modification of their magnetic properties, say, in the visible-light range. We
have recently investigated the simple case of a local magneto-dielectric material
where both permittivity and permeability are given by lossy Drude models.13 More
precisely, the permeability is
MM = 1 + 2 ft ■ , 0 < / < 1. (4)
CUq — LOz — 1KCU
Response functions of this kind have been used previously to describe the response
of a metamaterial to electromagnetic waves. The material contains a regular lattice
of sub-wavelength units (wires and rings) with a size much smaller than the
incident wavelength and filling factor /. With a suitable spatial averaging procedure
(effective medium description),14 one finds the permeability (4).
2769
The calculation of the Casimir force requires response functions at imaginary
frequencies. We have used the Kramers-Kronig relation
MiO
1
dto
wlm/j(w)
(5)
and focused on the limit of weak absorption where Im /j,(u>) collapses to a S-function.
The resulting expression features a "magnetic plasma frequency" u>p = cuo\/J.
As shown in Fig.l, the Casimir interaction becomes repulsive for a 'mixed' pair
of mirrors, one mainly dielectric, the other mainly permeable. This previously
discussed phenomenon15 survives in some range of distances at sufficiently low
temperatures even for dispersive materials. The corresponding parameter window is the
wider, the higher the magnetic plasma frequency.
6
4
2
o
-2
-4
■6x10'5'
0.01 0.1. .. 1
L/A
Fig. 1. Casimir pressure as a function of
distance L between two different metama-
terial plates. Positive values correspond to
an attractive interaction. The force (per
unit area) is normalized to hQ/L3, the
distance to A e 2ttc/D. where O is a typical
plasma or resonance frequency in Eq.(4).
Plate 1 is purely dielectric, plate 2 mainly
magnetic. The temperature takes the
values kBT = (a) 0, (b) 0.03, (c) 0.1, (d)
0.3 Ml Adapted from Fig.2 of Ref.13 where
the parameters can be found.
References
1.
2.
3.
4.
5.
6.
9.
10.
11.
12.
13.
14.
15.
M. Bordag, U. Mohideen and V. Mostepanenko, Phys. Rep. 353, 1 (2001).
S. K. Lamoreaux, Rep. Progr. Phys. 68, 20 (2005).
R. Onofrio, New J. Phys. 8, 237 (2006).
A. Lambrecht and S. Reynaud, Eur. Phys. J. D8, 309 (2000).
E. Lifshitz, Sov. Phys. JETP 2, 73 (1956).
L. Landau, E. Lifshitz and L. Pitaevskii, Electrodynamics of Continuous Media (Perg-
amon Press, Oxford, 1980).
J. Jackson, Classical Electrodynamics (Wiley & Sons, New York, 1975).
K. Schram, Phys. Lett. A43, 282 (1973).
F. Intravaia and C. Henkel, in preparation.
F. Intravaia and A. Lambrecht, Phys. Rev. Lett. 94, 110404 (2005).
K. E. Nagaev and M. Buttiker, Europhys. Lett. 58, 475 (2002).
N. G. V. Kampen, B. R. A. Nijboer and K. Schram, Phys. Lett. A26, 307 (1968).
E. Gerlach, Phys. Rev. B4 (1971) 393; C. Henkel, K. Joulain, J.-P. Mulet and J.-J.
Greffet, Phys. Rev. A69, 023808 (2004).
C. Henkel and K. Joulain, Europhys. Lett. 72, 929 (2005).
S. A. Ramakrishna, Rep. Prog. Phys. 68, 449 (2005).
T. H. Boyer, Phys. Rev. A9, 2078 (1974).
CASIMIR ENERGY AND A COSMOLOGICAL BOUNCE*
CARLOS A.R. HERDEIRO
Departamento de Fisica e Centro de Fisica do Porto ,
Faculdade de Ciencias da Universidade do Porto,
Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
crherdei@fc.up.pt
We revisit the computation of the renormalised energy-momentum tensor for a quantised
scalar field in an Einstein static universe. We show that for a range of couplings to the
Ricci scalar and masses, the renormalised energy momentum tensor violates the strong
energy condition. We discuss the back-reaction problem and in particular the possibility
that this Casimir energy could source both a short inflationary epoch and avoid the big
bang singularity through a bounce.
Keywords: Casimir effect; cosmological singularity.
1. Introduction
The Casimir effect2 is a macroscopic manifestation of the vacuum fluctuations of
a quantum field. It was first considered in systems with boundaries. The effect is
highly sensitive to the geometry, size and topology of such boundaries. In particular
it may change from attractive to repulsive when these parameters are changed.3
In a compact space, on the other hand, there are no boundaries, but the non-
trivial topology imposes periodicity conditions which resemble boundary conditions,
originating a Casimir force as well.
If our universe is compact, every quantum field living on it will give rise to a
Casimir type force. Could this force: 1) originate primordial or present day inflation?
2) avoid the Big Bang singularity? These two questions can be rephrased as the
following one: Could the vev of the renormalised energy momentum tensor of a
quantum field in our universe violate the strong energy condition?
2. Quantum Scalar Field in an Einstein Static Universe
As a first approximation we consider an FRW model with a(t) = 0. Spatial
homogeneity and isotropy mean that the energy density p and pressure p are constant.
Energy conservation means that we can compute the pressure as p = —dE/dV,
where the total energy is pV, and V is the volume of spatial sections of the
Universe. Note that p = p(V). Thus, knowing the energy density (unrenormaUsed or
renormalised) we can easily compute the pressure (unrenormaUsed or renormalised).
"This communication is based on work in collaboration with M. Sampaio.1 The author
was supported by Fundacao Calouste Gulbenkian through Programa de Estimulo a In-
vestigacao and by the FCT grants SFRH/BPD/5544/2001, POCTI/FNU/38004/2001 and
POCTI/FNU/50161/2003. Centro de Fisica do Porto is partially funded by FCT through the
POCTI programme.
2770
2771
In an Einstein Static Universe with radius R, the unrenormalised energy density
for a scalar field with mass p? and coupling £ to the Ricci scalar is
, +00 2
P°=yJ2^n2+a2R2 > a2i?2=M2i?2+6C-l. (1)
Depending on a2!?2 this expression can be renormalised using different techniques.1
• If a2R2 = 0 (eg. massless, conformally coupled scalar field) a simple use of the
Riemann zeta function gives (first obtained by a damping function technique4)
1 1 r+°° t3 n
Pren ~ 4807r2i?4 " 2^ J0 eM-l ' Pren ~ ~Y~ ■ ( '
• If a2R2 > 0, one can apply a damping function technique (with or without
using also the Abel Plana formula) or an Epstein-Hurwitz zeta function to obtain
equivalent expressions1 of which the simplest one is
1 [+°° t2Vt2 - a2R2 Jm
which was first obtained for a conformal coupling.5 The renormalised pressure is
1 p + 00 j.4
Pren = &^Ri J]a]R ^t-x)^w^mdt ■ (4)
• If a2R2 < 0, a damping function technique, together with a use of the Abel-
Plana formula gives1
1 / r+oc i2\h2 - n2R2
11 V andt+^-j t'W-t2-a2 R2 cot TTtdt] , (5)
1 2
1 r\°-\R , \
- / t2\/-t2 -a2 R2 cot ntdt \
, 1 fMR t4cotTrt \ ,n.
dt+- -====dt • (6)
67r2i?4 yj0 {e2^ - 1)V*2 - a2R2 2 J0 v'-t2 - a2R2
Using these expressions we have plotted pren and pren as a function of mass, for
different values of £, in figure 1. The most noticeable feature is that both the
renormalised energy density and pressure may become negative for a range of values.
Clearly this leads to violations of the strong energy condition, that is to p + 3p < 0.
3. Discussion
One can illustrate the cosmological consequences of these violations of the strong
energy condition with a simple toy model. Integrating the Friedinann and Ray-
chaudhuri equations for an FRW model with dust, a cosmological constant and the
renormalised energy momentum tensor of a massless scalar field with a sufficiently
small coupling to the Ricci scalar,1 one find the behaviour exhibited in figures 2.
Indeed, the Casimir effect can lead to an inflationary era in the early universe, which
generically seems to be too short to solve the usual big bang model problems. More
interestingly it can lead to a cosmological bounce and avoid the big bang singularity.
2772
0,0 0.25 05 0.75 1.0 1.25 1.5
00 '-H^rfBaSR^gSj^i i i i i I i i i i I
Fig. 1. Renormalised total vacuum energy 27r2pren and pressure 27r2pren for R = 1, as a function
of/i £ [0,1.5], for six different couplings £ e [0,1/6]. As £ increases the colour of the line in the
corresponding graph becomes darker. £ = 0 corresponds to the most negative curve in both graphs.
..tf°:
1.5 1 1.5
0.06 0.08
Fig. 2. Left: scale factor for a universe with a cosmological constant, dust and the quantum fluid
of a massless scalar field; Right: Detail near t = 0 showing clearly the bounce structure.
References
1. C. Herdeiro and M. Sampaio, Class. Quantum Grav. 23, 473 (2006); To appear.
2. H. Casimir, Proc. Ron. Nederl. Akad. Wet. 51, 793 (1948).
3. M. Bordag, U. Mohideen and V. Mostepanenko, Phys. Rep. 353, 1 (2001).
4. L. H. Ford, Phys. Rev. Dll, 3370 (1975).
5. S. G. Mamayev, V. M. Mostepanenko and A. A. Starobinsky, Sov. Phys. - JETP, 43,
823 (1976).
PHOTON GENERATION FROM THE VACUUM: AN
EXPERIMENT TO DETECT THE DCE
CATERINA BRAGGIO
INFN, via Marzolo 8, 35100 Padova, Italy and
University of Ferrara, via Saragat, J^J^lOO Ferrara, Italy
braggio@pd. infn. it
G. BRESSI
INFN, Sezione di Pavia - Via Bassi 6, 27100 Pavia, Italy
G. CARUGNO
INFN, Sezione di Padova - Via Marzolo 8, 35100 Padova, Italy
G. RUOSO
INFN, LNL - Viale dell'Universita 4, 35020 Legnaro, Italy
D. ZANELLO
INFN, Sezione di Roma - P.le A. Moro 2, 00185 Roma, Italy
We describe our experiment to detect the generation of photons in the laboratory.
1. Introduction
The experiment presented falls on the general framework of the study of quantum
vacuum, a subject that has gained importance in the last decade following precise
experimental results in the measurement of the Casimir effect.1 The Casimir effect
studies the modification of the vacuum energy due to fixed boundaries. A more
general issue is the study of the quantum vacuum with moving boundary conditions,
allowing investigation of unsolved problems in quantum electrodynamics, cosmology
and general relativity. The so-called dynamical Casimir effect should occur when
the motion of the boundaries is performed with non-constant acceleration, giving
rise to dissipative phenomena, i. e. to photon production from the vacuum.
In principle the effect is possible also for a single mirror oscillating in the sea of
vacuum fluctuations, but the predicted number of photons created is immeasurably
small for nonrelativistic mirror trajectories. Nonetheless there is an experimental
configuration which should allow production of an observable number of photons:
the mirror becomes the wall of a cavity and it oscillates at a frequency which is
double of the resonance frequency of the cavity itself (parametric resonance
condition). Through this mechanism, the the number of produced photons should grow
exponentially inside the cavity.
2. The MIR experimental scheme
As huge accelerating mirrors are technologically difficult to achieve, some theoretical
papers2'3 have proposed to modulate the dielectric constant of a semiconductor in
2773
2774
the cavity, instead of physically move the cavity wall. A novelty of our approach is
the periodical modulation of the dielectric constant of a semiconductor slab covering
the wall of a superconducting cavity. At cryogenic temperatures the semiconductor
is an insulator and the cavity is long L, but as soon as a laser pulse shines uniformly
on it, a plasma of carriers is produced on its surface. The quasi-metallic wall that is
formed in position L — D (see Fig. 1), where D is the thickness of the slab, causes
a shift of the cavity fundamental frequency vq.
L train of
j laser pulses
j :■-■-■-.-.—.r -i'"lJS|l
I ■ ontir fibie
| ,;
Nhsjpi!ici)"ci.n: "j
cavity
semiconductor
layer
(a) (b)
Fig. 1. (a) Mirror effective motion: a composite mirror changes its reflection properties
under laser illumination, and the microwave reflecting surface switches its position between
L and L~ I) accordingly, (b) Arrangement of the composite mirror in a microwave resonant
cavity.
The laser pulses are guided into the cavity via an optical fibre. The total energy
per train of pulses is limited, so must be the number of available pulses, which will
be between 103 and 104 pulses for each train.
Using a, train of laser pulses with repetition rate twice the resonance frequency of
the cavity, the parametric resonance condition is satisfied.
3. The frequency shift problem
After a preliminary feasibility study, in which it has been demonstrated that the
idea at the basis of the detection scheme is feasible,4 more recent experimental work
has been devoted also to the study of another critical point of the experiment, that
is the frequency shift problem. One has in fact to demonstrate that the appearance
of the plasma, on a semiconductor slab placed over the wall of a resonant cavity
produces a shift of the frequency of resonance vq. The relative frequency change
^jf = ~~ is connected to the relative cavity length change ^r, where D is the
thickness of the semiconductor slab, L length of the cavity.
The problem has been studied experimentally and it was demonstrated that a
conductive film can be a good mirror in the sense of the frequency shift even if its
thickness is smaller than the calculated skin depth.
To study experimentally the problem of the frequency shift a few plexiglas slabs
2775
with evaporated copper films on one side have been used; the film thicknesses were
chosen in such a, way to be smaller than the calculated skin depth for copper. The
slab was set over the 71 x 22 mm2 cavity wall of a copper cavity, which had the same
dimensions of the niobium cavity of the experiment (see Figure 2). The expected
DG
coppe! cavity
evaporated film
plexiglass slab
Fig. 2. Cavity measurements of the frequency shift with evaporated films on plexiglass slabs.
frequency shift of 28 MHz has been obtained with a minimum layer of thickness
G = 75 uni, which is about 20 times smaller than the calculated skin depth at the
resonance frequency vq (at the resonance frequency of the used cavity it is S ~ 1.3
fan.).
Provided that the film is displaced by the distance D, which is much bigger than
the skin depth 5 and much smaller than the length of the cavity, a thin film is an
ideal mirror even if G <C 8 when a parameter A, defined in the chosen geometry as
A = 8G/p,5 is bigger than unity, p (in /iO-cm) is the resistivity of deposited metal
and G is the film thickness(in nanometers).
4. Conclusions
In the proposed experiment we will use 800 nm light impinging on GaAs at 5 K;
the corresponding absorption length is 1 fim that can be considered the equivalent
thickness of the plasma. For a laser pulse energy of 100 pj, a mobility of /* = 104
cm2/V-s, we can calculate a resistivity of a, few mficm and the condition A > 1 is
still satisfied.
References
1. G. Rressi, G. Caruguo, G. Ruoso, R. Onofrio. Phys. Rev. Lett. 88 499 504 (2002).
2. Y. E. Lozovik, V. G. Tsvetus, and E. A. Vinogradov. Physica Scripta, 52:184-190,
1995.
3. E. Yablonovitch. Phys. Rev. Lett, 62, 1989.
4. C. Braggio, G. Bressi, G. Caruguo, C. D. Noce, G. Galeazzi, A. Lombardi, A. Palmieri,
G. Ruoso, and D. Zanello. Europhysic? Letters, 70:754, 2005.
5. Braggio C , Bressi G , Carugno G , Dodonov A V, Dodonov V V , Galeazzi G, Ruoso
G and Zanello D. Phys. Lett. A (2006), doi:10.1016/j.physleta.2006.11.071.
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Loop Quantum Gravity,
Quantum Geometry, Spin
Foams
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THE EMERGENCE OF AdS2 FROM QUANTUM FLUCTUATIONS
J. AMBJ0RNa'c, R. JANIK6, W. WESTRAc and S. ZOHRENd
a The Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100 Copenhagen 0,
Denmark
b Institute of Physics, Jagellonian University, ul. Reymonta 4, 30-059 Krakow, Poland
c Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, NL-3584 CE Utrecht, The
Netherlands
d Blackett Laboratory, Imperial College, London SW7 2AZ, United Kingdom
ambjorn@nbi.dk, ufrjanik@if.uj.edu.pl, w.westra@phys.uu.nl, stefan.zohren@imperial.ac.uk
We have shown how the quantization of two-dimensional quantum gravity with an
action which contains only a positive cosmological constant and boundary cosmological
constants leads to the emergence of a spacetime which can be described as a constant
negative curvature spacetime with superimposed quantum fluctuations.
1. Introduction
The causal dynamical triangulations approach to quantum gravity (CDT) is an
attempt to define the gravitational path integral in a background independent and
nonperturbative manner. As in the case of Euclidean dynamical triangulation
approaches (DT), CDT provides a regularization of the path integral through a sum
over piecewise linear geometries where the edge length of the individual building
blocks serves as an ultraviolet cutoff. However, in contrast to DT a global time
foliation is imposed on each individual history in the path integral.
The method of CDT was first applied to two-dimensional quantum gravity where
the model was shown to be analytically solvable.1 Although two-dimensional
quantum gravity does not have any propagating degrees of freedom it is a fertile
playground to study certain aspects of diffeomorphism invariant theories. Among the
issues that have been addressed within the two-dimensional framework are the
inclusion of a sum over topologies2 and the emergence of a background geometry
purely from quantum fluctuations3 where the latter is the subject of this article.
One of the most natural quantities to study in CDT is the loop-loop-propagator
which is the amplitude for a transition from a spacelike loop with boundary
cosmological constant X to a loop with boundary cosmological constant Y in time
T,
GA(X,Y;T) = jv[g]e-s^\ S[g] =aJ d2x^/g(x, t) + X j> dh + Y j
d/2) (1)
where the action only includes bulk and boundary cosmological constants, since the
curvature term in the Einstein-Hilbert action is trivial in 2D.
Evaluating the path integral using the CDT regularization and taking the
continuum limit yields the following1
CJXY-T) *2(r'*)~A l (2)
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2780
where X(T,X) is the solution of
_ = _(X2-A), X(0,X)=X. (3)
2. The emergence of AdS2
To determine the background geometry of the 1+1 dimensional universe we calculate
the average spatial length at proper time £ £ [0,T]
1 f°°
W))x,y,t= Ga{XY-T)1 dLG^X^L^)LG^,Y;T-t). (4)
Evaluating the average length at the boundary t = T and taking the limit T —> oo
gives
lim {L(T))XtY,T = TT^Tt' (5)
Interestingly, one observes that there is a special value Y = —\/A of the boundary
cosmological constant for which the boundary length diverges and the geometry
becomes non-compact. Using this critical value for the boundary cosmological constant
Y one can obtain the boundary length for finite T
LC(T) = (L(T))X Y„ /^ _ = -\= J= • (6)
V, J/X,Y—VA,T ^AcothVAT-1
Instead of using boundary cosmological constants one can also fix the spatial length
of the boundaries. Using the Laplace transformed propagator G\(Li,L2',T) we can
evaluate the average spatial length {-L(£))z,i,z,2,r for fixed lengths L\ and L2 of the
boundary loops.
In the following we want to investigate the quantum geometry in the case where
it becomes non-compact. Therefore we set the boundary length at t = T to the
critical value LC(T) as defined in Eq. (6) and for simplicity we shrink the spatial
geometry at t = 0 to a point. In the limit T —► 00 one obtains the average length of
the spatial geometry at proper time t £ [0,T]
(L(t)) = rlim)<L(t))il=o,i2=ic(r),r = ^ sinh(2VAt). (7)
Due to the fact that L and T are denned from the continuum limit of a simplicial
geometry there is a relative constant of proportionality that can only be fixed by
comparing with continuum calculations4 yielding Lcont(t) = w{L(t)). From this
result the metric for the background geometry is readily obtained,
ds2 = dt2 ^ d02 = dt2 antyyxt) df)2
4tt2 4A v '
This is nothing but the metric of the Poincare disc which can be seen as a Wick
rotated version of AdS2 with constant negative curvature R = —8A.
2781
To better understand the quantum nature of the geometry it is useful to compute
the fluctuations of the spatial length. From expressions analogous to Eq. (4) one
can determine the relative fluctuations
AL(t) = ySFW P-v/At (q)
W)) {L(t)) ■ { ]
Surprisingly, the fluctuations of the spatial geometry become exponentially small for
t S> A-1/2. Concluding from Eqs. (8) and (9), one can view the quantum geometry
as a version of Wick rotated AdS2 dressed with small quantum fluctuations.
3. Discussion
We have shown that in 2D quantum gravity defined through CDT there is a
transition from compact geometry to non-compact AdS^-like geometry for a special value
of the boundary cosmological constant. This phenomenon is similar to the Euclidean
case where non-compact ZZ-branes appear in a transition from compact 2D
geometries in Liouville quantum gravity.5 A surprising feature of the CDT result is that the
fluctuations become exponentially small which enables us to interpret the emerging
AdS2 spacetime as a genuine semiclassical background. It is interesting that similar
results have been reported in four-dimensional CDT where numerical simulations
indicate the emergence of a semi-classical background from a nonperturbative and
background-independent path integral.6
Acknowledgments
All authors acknowledge support by ENRAGE (European Network on Random
Geometry), a Marie Curie Research Training Network in the European Community's
Sixth Framework Programme, network contract MRTN-CT-2004-005616.
References
1. J. Ambj0rn and R. Loll, Nucl. Phys. B 536, 407 (1998).
2. R. Loll, W. Westra and S. Zohren, Nucl. Phys. B 751, 419 (2006).
3. J. Ambj0rn, R. Janik, W. Westra and S. Zohren, Phys. Lett. B 641, 94 (2006).
4. R. Nakayama, Phys. Lett. B 325, 347 (1994).
5. J. Ambj0rn, S. Arianos, J. A. Gesser and S. Kawamoto, Phys. Lett. B 599, 306 (2004).
6. J. Ambj0rn, J. Jurkiewicz and R. Loll, Phys. Rev. Lett. 93, 131301 (2004), Phys. Lett.
B 607, 205 (2005), Phys. Rev. D72, p. 064014 (2005).
THE PONZANO-REGGE MODEL AND REIDEMEISTER TORSION
JOHN W. BARRETT
ILEANA NAISH-GUZMAN
School of Mathematical Sciences, University of Nottingham
University Park, Nottingham, NG7 2RD, UK
1. Introduction
The Ponzano-Regge model of quantum gravity1 on a triangulated 3-dimensional
manifold was originally presented in terms of a state-sum over representations of
SU(2). It is well-known that the analogous model for a finite group can be
reformulated in terms of a sum over group elements located on triangles (or dual edges).
It is commonly assumed that this is still possible with SU(2). We note that there
are subtle questions both about the convergence of the state-sum and also about
the fermionic character of the SU(2) representations. To avoid these questions, we
present the definition of the Ponzano-Regge model in terms of integrals over
elements of SU(2) assigned to the triangles of the triangulation.
There are several different candidates for observables in this model. We define
observables specified by giving a conjugacy class in SU(2) to each edge of a graph in
the manifold. In general there is still a question about whether the resulting integral
for the partition function, or 'expectation value', of an observable is well-defined.
Our first result provides an answer to this question: the criterion for the formula
to make sense is that the second twisted cohomology group should vanish at each
point of the integration. Our second result says that if this criterion is satisfied, then
the resulting expression can be written in terms of the Reidemeister torsion. This
proves the independence of the partition function on both the regularisation used
in its definition and the triangulation of the manifold. We discuss the particular
features of both planar graphs and knots. For a good treatment of the cohomology
theory involved, the reader is referred to Dubois.2
2. Definition of the partition function
Let M be a closed 3-manifold with a specified triangulation. The triangulation
will have a finite number of simplexes. To specify an observable, we need a graph
embedded in il/, and some data on each edge of the graph. More precisely, let V
be a connected subcomplex consisting of edges and vertices of M. For each edge
e of this graph, choose a conjugacy class 9e of the group SU(2). The conjugacy
class is specified by an angle 0e e [0,27r], the angle of the corresponding rotation
in Euclidean space. The idea of the Ponzano-Regge model is that it calculates a
number Z which is the 'expectation value' of this observable. The number Z is
often called the partition function, due to the analogy with statistical mechanics.
Let Ai (Ti) be the set of edges of the triangulation (graph). To define Z, it is
necessary to pick a regularising subset of edges T C Ai \ I\ satisfying the following
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conditions:
• Each connected component of the graph formed by T is a tree (i.e. contains
no loops) and is attached to V at exactly one vertex
• T is maximal, i.e. visits each vertex of M not contained in I\
The definition of the partition function is as follows. We use the dual cell
decomposition of M in which there is one dual fc-cell for each 3 — fc-cell of M. On
each dual edge / of M, with an arbitrary choice of orientation, there is a variable
<?/ G SU(2) (and gjl is assigned to the opposite orientation). This set of variables
is called a connection, and given a path consisting of a sequence of oriented dual
edges 7 = (/i, /2, ■ ■ ■, /iv), there is a holonomy element
H(J) = g% g% ... 9y»
where e/t = ±1 according as fi is traversed in a positive/negative sense (with respect
to its orientation).
On each oriented dual face e, there is then the holonomy he = H{j) given by the
sequence 7 of dual edges around its boundary. This is well-defined up to conjugation.
Finally, the definition uses some delta-functions on SU(2). The first of these is the
delta-function at the identity element i, defined by
5(g)F(g)dg = F(i),
SU(2)
for any function F, where / dg = 1. The second is the delta-function at a conjugacy
class </>, given by an ordinary delta-function S((f> — 0(g)). Here, 0(g) denotes the
conjugacy class of g G SU(2).
The partition function is obtained by integrating over these variables.
Z(M,T0)= f H dgf n 6(9(he)~6e) H S(he) (1)
J /eA2 eeri eeAi\(riUT)
Similar definitions appears in previous works.3'4
The roles of the various factors in (1) are as follows. The delta-functions for the
edges on T force the holonomy of the connection around that edge of the graph to lie
in the conjugacy class #e; the delta-functions at the identity force the g variables to
give a flat SU(2) connection on the complement of T; the set of edges T eliminates
excess delta-functions, which would otherwise reduce to integrating S2 in one of the
variables.
3. Existence criterion
Theorem 3.1. The partition function (1) exists for a region 1Z of the space of
parameters {(&!, 62, ■ ■ •)} as a distribution if and only if the second twisted cohomology
group H2(L,p) of the graph exterior L is trivial for each flat connection p whose
conjugacy classes (9\, 02, ■ ■ ■) lie in 71.
2784
The proof of theorem 3.1 and further results below will be given in our forthcoming
paper. In the special case of a planar graph, the existence criterion is always satisfied
and so its partition function is always well-defined. It is interesting to consider, in
light of our result, the formula for the tetrahedron graph calculated by Freidel and
Louapre.5 For certain values of the parameters, it yields an infinite answer, calling
into question the well-definition of this observable. Theorem 3.1 tells us that the
correct interpretation of the result is as a distribution. It is the distributional nature
of graphs in general that requires the statement of theorem 3.1 in terms of a region
of parameters.
4. Invariance of the partition function
Theorem 4.1. If the existence criterion is satisfied then the partition function
(1) can be expressed as an integral over the space of flat connections on the graph
exterior L with measure given by the Reidemeister torsion, tor(L).
The Reidemeister torsion is known to be a homeomorphism and simple homotopy
invariant, and so we have the following
Corollary 4.1. The partition function (1) is independent of the choices of trian-
gulation and regularising set T.
If the graph is a knot K, then the partition function vanishes unless all conjugacy
classes are equal, so we may, without loss of generality, take the knot to have a
single edge (and a single vertex). If 9 is the associated conjugacy class, then the
existence criterion is satisfied for 9 less than a critical angle, 9C(K), depending on
the knot K. For 9 in this range, the partition function is simply a constant times the
Reidemeister torsion tor(L). The simple homotopy invariance of the Reidemeister
torsion means we may calculate tor(L) using the cell complex for L coming from
the Wirtinger presentation of IIi(L). Doing so, one obtains
. 2/1
Z(S3,K0)=const.^^ ? 0<9<9C(K)
|AK(e'e)|2
where AK is the Alexander polynomial of K. This generalises Barrett's result for
the trefoil knot.6
References
1. G. Ponzano and T. Regge, Semiclassical limit of Racah coefficients. In: Spectroscopic
and group theoretical methods in Physics, ed. F. Bloch, North-Holland (1968), pp 1-58.
2. J. Dubois, math.GT/0403304.
3. L. Freidel and E. Livine, Class. Quant. Grav. 23, 2021 (2006).
4. L. Freidel and D. Louapre, gr-qc/0410141.
5. L. Freidel and D. Louapre, Class. Quant. Grav. 20, 1267 (2003).
6. J. W. Barrett, Mod. Phys. Lett. A 20, 1271 (2005).
THE PROCA FIELD IN LOOP QUANTUM GRAVITY
GABOR HELESFAI
Eotvos Lorand University (ELTE-TTK),
Pazmany Peter setany 1/a,
Budapest 1117, Hungary
heles@manna. elte.hu
In this paper we investigate the Proca field in the framework of Loop Quantum Gravity.
By introducing an auxiliary (non-Higgs) scalar field, we arrive at a theory with first class
constraints. This makes possible a rigorous, consistent, non-perturbative quantization of
the Proca field.
1. Classical theory
The action of the Proca field coupled to gravity has the form
S = J d*x [V^gR + V^~99ac9bd ( ^ \FibFid - im2ffa5A^)], (1)
where gab is the metric-tensor, g is its determinant, R is the scalar curvature of g, A^
is a U{1) connection with curvature F^b. After the 3+1 decomposition one obtains
a non gauge invariant Hamiltonian with second class constraint algebra. It would be
desirable to have a gauge invariant Hamiltonian with a first class constraint algebra
since then it is much easier to apply the tools developed in loop quantum gravity.
There is an elegant way of curing both problems4'5'6 , and that is to introduce an
auxiliary scalar-field and modify the Lagrangian to have the following form:
-AF_tbF4cd - \vr?gab{A\ + d^)(Ad + dfo)] (2)
ac bd
Pm = V~gR + y-99
The Hamiltonian of the above system is
Hm= f {NH + NaHa + Aid + A0O) (3)
H = -±-tr(2[Ka, Kb] - Fab)[Ea, Eb] + ^(EaEb + BaBb)
1 ^ ' "rqm\a\Aa+da4>)(Ab+db4>) (4)
2v/gm2
na = F3abE) + eabcEbBc + (Aa + dacfr)ii (5)
G = VaEa-ir, G{=VaE?, (6)
where ir is the conjugate momenta for </>. The quantities H, TCa, Gi,G are referred to
as the scalar, diffeomorphism, gravitational Gauss and Maxwell Gauss constraints.
It is easy to verify that 1) the above system is first class 2) diffeomorphism and
"This research has been supported by the Hungarian and Polish academies. The full article can
be found at gr-qc/0605048.
2785
2786
gauge symmetries are independent of m 3) the scalar field and the Yang-Mills field
is only coupled to each other in the scalar constraint and only through a derivative
term and no scalar mass-term required, which means that if we will quantize this
system the scalar field will have a totally different role than the one introduced via
symmetry breaking.
2. Quantization and results
Since wc have a covariant Hamiltonian and first class constraints, we can directly
apply the tools developed in loop quantum gravity to quantize the Proca field. Using
the results for quantizing general gauge systems (8~13) and the scalar field (17,18),
we obtain the Hamiltonian operator Hm of the mass term:
Hm = HP + HM (7)
v K v(A)=v(A') = v
l ~ ■ f - i
x Q'Sl(a')(". \)QZ^)(v, \)Q:n{a'}(v, \) (8)
it \ '
v(A)=v(A')=v
x [U(l,Sn(A)) - U(l, v) + hSn{A) - 1][U(1, sr(A')) - U(l,v) + hSr{A,} - 1]
x Qt(A)(«, l)Qs9(A)(v, f )Qi.(A.)(". -4)QTtm(v, -4) (9)
where Qke(v,r) = tr(rfc/ie[/i-1,F(t;)r]) and E(v) = »("-iK"-2) ^ n standing for the
valance of the vertex v.
The most important results are the following:
f) Since the structure of this Hamiltonian is similar to the pure gravitational
Hamiltonian (it contains operators that either add additional vertices/edges/loops or do
not change it), it is possible to construct a solution - introduced by Thiemann -
with a recursive method (see details in1).
2) We did not get any constraint on the mass, so it has to be given either from
experiments or from additional physical input. Actually mass acts as a coupling
constant.
3) The scalar field used in this formalism is an auxiliary field without mass term
and nonlinearity, and no Higgs field was required.
4) To arrive to the original formalism (1) one has to use a gauge fixing. This can
be done by introducing additional constraints
Ca = da4> = 0 (10)
C = n-V-qm2A°~^aAa=0. (11)
2787
In this case one has to introduce Dirac-brackets to be consistent, since now we have a
second class constraint algebra. In the case of field theories, it is done in the following
way (see2 for details): first one calculates the matrix Mij(x,y) := {Bi(x), Bj(y)},
where Bi(x) are the second class constraints in the theory. After that one calculates
the inverse of Mij(x,y) in the following sense (since Mij(x,y) is a distribution):
j dzzMlk{x,z)(M^)k3(z,y) = 8i35{x-y) (12)
After this the Dirac-bracket is defined as
{f,9}D:={f,g}- j cPxtPyiftBiWUMl-^afayXB^g} (13)
After this the quantization procedure is the same as before except that the Poisson-
bracket should be replaced with the Dirac-bracket (see1 for a detailed analysis).
References
1. Helesfai G 2006 Preprint gr-qc/0605048
2. Weinberg S 1995 The Quantum Theory of Fields (Cambridge University Press,
Cambridge, United Kingdom)
3. Ashtekar A 1987 Phys. Rev. D 36 1587
4. Henneaux M and Teitelboim 1992 Quantization of Gauge Systems (Princeton
University Press, Princeton, New Jersey)
5. Hong S, Kim Y, Park Y and Rothe K D 2002 Mod. Phys. Lett. A17 4335
6. Banerjee R and Barcelos-Netol J 1997 Nucl.Phys. B 499 453
7. Ashtekar A, Romano J D and Tate R S 1989 Phys. Rev. D 40 2572
8. Thiemann T 1998 Class. Quant. Grav. 15 839
9. Thiemann T 1998 Class. Quant. Grav. 15 875
10. Thiemann T 1998 Class. Quant. Grav. 15 1281
11. Thiemann T 2000 Preprint gr-qc/0110034
12. Ashtekar A and Lewandowski J 2004 Class. Quant. Grav. 21 R53
13. Ashtekar A, Lewandowski J, Marolf D, Mourao J and Thiemann T 1995 J. Math.
Phys. 36 6456
14. Alfaro J, Morales-Tecotl H A and Urrutia L F Loop quantum gravity and light
propagation 2002 Phys. Rev. D 65 103509
15. Varadarajan M 2000 Phys. Rev. D 61 104001
16. Varadarajan M 2001 Phys. Rev. D 64 104003
17. Thiemann T 1997 Preprint HUTMP-97/B-364
18. Ashtekar A, Lewandowski J and Sahlmann H 2003 Class. Quant. Grav. 20 Lll-1
19. Kaminski W, Lewandowski J and Bobienski M 2005 Preprint gr-qc/0508091
20. Pons J M 1996 Int. J. Mod. Phys. All 975
21. Gambini R and Pullin J 2005 Phys. Rev. Lett. 94 101302
AMBIGUITY OF BLACK HOLE ENTROPY IN LOOP
QUANTUM GRAVITY
TAKASHI TAMAKI and HIDEFUMI NOMURA
Department of Physics, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan
tamaki@gravity.phys.waseda.ac.jp
nomura@gravity.phys.waseda.ac.jp
We reexmine some proposals of black hole entropy in loop quantum gravity and consider
a new possible choice of the Immirzi parameter.
1. Introduction
Loop quantum gravity (LQG) has attracted much attention because of its
background independent formulation, account for microscopic origin of black hole
entropy,1 etc. The spin network has played a key role in this theory.2 Using this,
expressions for the spectrum of the area can be derived as3 A = 8^7 J2 \/ji(Ji + 1),
where 7 is the Immirzi parameter. The sum is added up all intersections between
a surface and edges. The number of states that determines the black hole entropy
was first estimated as1
s= Aln(2jmin + 1)
87T7\/imin(imin + 1) '
where A and jm;n are the horizon area and the lowest nontrivial representation
usually taken to be 1/2 because of SU(2), respectively. In this case, the Immirzi
parameter is determined as 7 = ln2/(7r\/3) to produce S = A/4.
However, (1) was corrected as4,5 S = -L^-, where jm is the solution of
00
1 = J2 2exp(-27r7MVj(i + l)) , (2)
j=Z/2
where j takes all the positive half-integer. In this case, 7m — 0.23753 • • •. Another
possibility has also been argued. It is to determine 7m as the solution of6,7
00
1 = E W +!) gM-^im VJtiTrj). (3)
In this case, 7m = 0.27398 • • •. These provide us with the following question: which
is the best choice for the Immirzi parameter? Therefore, we reanalyze these
possibilities. For details, see.8
2. Summary of the ABCK framework
First, we introduce the isolated horizon (IH) where we can reduce the SU(2)
connection to the U(l) connection. Next, we imagine that spin network pierces the
IH. By eliminating the edge tangential to the isolated horizon, we can decompose
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2789
the Hilbert space as the tensor product of that at the IH Hjh and that in the
bulk Hj:, i.e., Hjh ® H-^. If we specify the points that are intersections of edges
having spin (j'1,,72, •"" >jn) and the IH, we can write H^ as the orthogonal sum
H-£ = 0 . Hji'm% where m, takes the value —ji, —ji + 1, • • •, jV This is related
to the flux operator eigenvalue e™1 := 8irjmi that is normal to the IH (s' is the
part of the IH that have only one intersection between the edge with spin ji.). Since
we eliminate the edge tangential to the IH, we have m; ^ 0. The horizon Hilbert
space can be written as the orthogonal by eigenstates "t^ of the holonomy operator
hi, i.e., hi^b = e^i^^b.
Next, we consider the constraints at the IH. At the IH, we do not consider
the scalar constraint since the lapse function disappears. If we require that the
horizon should be invariant under the diffeomorphism and the U(l) gauge
transformation, The horizon area A is fixed as A = Air^k, where k is natural number
and it is the level of the Chern-Simons theory. In addition, it is required that we
should fix an ordering (bi, 62, • • • , bn). The area operator eigenvalue Aj should
satisfy (i) Aj = 8^7 y^ vjiiji + 1) < A-- From the quantum Gauss-Bonnet theorem,
(ii) 5Z™=1 bi = 0. From the boundary condition between the IH and the bulk,
(Hi) bi = —2m,i modk. All we need to consider in number counting are (i)(ii)(iii).
3. Number counting
If we use (ii) and (iii), we obtain (ii)' 5Z™=1 rrii = n'|. In,5 it was shown that
this condition is irrelevant in number counting. Thus, we perform number counting
only concentrating on (i) below. For this purpose, there are two different points
of view. The one adopted in the original paper1,4'5 counts the surface freedom
(b\, &2) •"" ) bn). The second counts the freedom for both j and b.6'7
We first consider the second possibility since (we suppose) it is easier to
understand. To simplify the problem, we first consider the set M^ by following,4
Mk := j (Ji, • • • , jn)\0 ^Jie~,J2ji<^\ ■ (4)
Let Nk be the number of elements of Mk plus 1. Certainly, N(a) < Nk, where
N(a) (a := -^-) is the number of states which account for the entropy. Note that
if (ji,--- Jn) € Mk-i, then (j
l)"'" j in) \) ^ Mk- In the same way, for natural
0 < s < k,
(jir ■ ■ ,Jn) e Mk-s => (j\, ■ ■ ■ ,jn, |) e Mk ■ (5)
Then, if we consider all 0 < s < k and all the sequence (ji,--- ,jn) € Mk-S,
we found that (ji,--- ,Jn,f) form the entire set Mk- Moreover, for s ^ s',
(ji, ■ ■ ■ ,jn, |) ^ (ji, ■ ■ ■ ,jn, y) e Mk- The important point to remember is that
we should include the condition m, ^ 0 (or equivalently bi ^ 0). Thus, each ji has
freedom 2jt for the ji integer and the 2ji + 1 way for the jt half-integer. They are
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summarized as 2[-J^] where [• • • ] is the integer parts. The recursion relation is
iVfc = £2[^](iVfc_s-l) + l. (6)
s=l
This is the point which has not been examined out so far.
As a stright forward extension of this, we can consider N(a), which is
N(a) := j (ji, • • • , j„)|0 ^ ji G |, £ ^JiC/i + l) < \ = a |
(7)
In this case, we obtain the recursion relation
N(a) = 2N(a - y/3/2) + 2N(a - V2) + ■ ■ ■ +
r 2 J + 1 n
2[-^-]N{a - ^3%{j% + !)) + ■■■ + W4a? + 1 - 1] . (8)
If we notice that the solution of \/ji(ji + 1) = a is j, = (Via2 + 1 — l)/2, meaning
of [\/4a2 + 1 — 1] is obvious. If we use the relation N(a) = Ce A~< , where C is a
constant, that was obtained in,5 we obtain
1= E 2[^^]exp(-27r7Mv/i(i + l)), (9)
J = Z/2
by taking the limit A —> oo. Then if we require S1 = A/4, we have 7 = 7m- In this
case, 7m = 0.26196
For the case that counts only the surface freedom, we have (2).
4. Conclusions
We have considered two possibilities for the number of states of black holes in the
ABCK framework. One of them gives a new value for the Immirzi parameter.
References
1. A. Ashtekar, J. Baez, A. Corichi, and K. Krasnov, Phys. Rev. Lett. 80, 904 (1998);
A. Ashtekar, J. Baez, and K. Krasnov, Adv. Theor. Math. Phys. 4, 1 (2000).
2. C. Rovelli and L. Smolin, Phys. Rev. D 52, 5743 (1995).
3. C. Rovelli and L. Smolin, Nucl. Phys. B 442, 593 (1995); Erratum, ibid., 456, 753
(1995).
4. M. Domagala and J. Lewandowski, Class. Quant. Grav. 21, 5233 (2004).
5. K. A. Meissner, Class. Quant. Grav. 21, 5245 (2004).
6. I.B. Khrlplovich, gr-qc/0409031; gr-qc/0411109.
7. A. Ghosh and P. Mitra, Phys. Lett. B 616, 114 (2005); ibid., gr-qc/0603029, hep-
th/0605125.
8. T. Tamaki and H. Nomura, Phys. Rev. D 72, 107501 (2005).
EXPLORING THE DIFFEOMORPHISM INVARIANT HILBERT
SPACE OF A SCALAR FIELD
HANNO SAHLMANN
Spinoza Institute/ITF, Utrecht University, Postbus 80.195, 3508 TD Utrecht
h. sahlmann @phys. uu.nl
As a toy model for the implementation of the diffeomorphism constraint, the
interpretation of the resulting states, and the treatment of ordering ambiguities in loop quantum
gravity, we consider the Hilbert space of spatially diffeomorphism invariant states for a
scalar field. We give a very explicit formula for the scalar product on this space, and
discuss its structure. Then we turn to the quantization of a certain class of diffeomorphism
invariant quantities on that space, and discuss in detail the ordering issues involved.
1. Introduction
The space of spatially diffeomorphism invariant states, Hdiff, is important in Loop
Quantum Gravity (LQG): It may be home to the physical states of the theory,
and it is the space on which the Hamiltonian constraint, arguably the most
important operator of the theory, is defined. We think however that Wdiff is not very
well understood. For example elements of TL^m are obtained by a group averaging
procedure that is quite subtle.2 Also the physical meaning of the states is rather
unclear. Finally, there are only few quantities that can be quantized on that space
without substantial ambiguities. Resolution of those ambiguities is important, for
example in the case of the Hamilton constraint.
Here we present resultsa on a toy model in which the above-mentioned points can
be studied with relative ease. We study a scalar field in the polymer representation.4'
The basic field quantities derived from the canonical pair (</>, ir) on a spatial slice E
that are subject to quantization are
Tx,a = exp[ia<p(x)}, tt(/) = / 7r(y)/(y), a el.
We consider two cases, X = Z, R.b We discuss the construction of Hdis for the field
in analogy to that of LQG, and we study the quantization of
La = ir(x)ex.p[ict(f)(x)] a EX. (1)
They form an algebra under Poisson brackets
{La, La>} = i(a -a')La+a,, La = L-a. (2)
There is thus a simple and thorough test of the quantization: Is (2) reproduced
(in appropriate commutation- and adjointness relations)? For X = Z one recognizes
aFor more details as well as all the proofs, see1 .
bAs will become apparent when we introduce the representation for these quantities, it is
mathematically more appropriate to describe X as the Pontryagin dual of U(l), and of Mb (the Bohr
compactification of M), respectively.
2791
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the Witt algebra. Direct calculation confirms that in this case the La generate
diffeomorphisms of the target U(l). The ordering problems that can be expected
for the quantization of (1) are analogous to (though much simpler than) those
encountered for the "FEE" term in the Hamiltonian constraint.3
2. The diffeomorphism invariant Hilbert space
The polymer representation for a scalar field is given on a Hilbert space a basis of
which can be labeled by functions A., A',... from the spatial slice E to X that are
non-zero at most in a finite number of points. The scalar product is
(x.\x'.)=l[s(xx,K),
and the representation is given by
f,,A|A.) = |A. + X6?), 7f(f)\X.) = J2 Ax/(s)|A.>.
To define Tidis, we use a rigging map r>, in analogy with LQG2 :
(r>*7)(*)= J2 F(lGS7l) E <vi * ^ * *71*)
VJiSDiff/Diff^ »J2£GS7
with F(n) a strictly positive function0 on N. GS7 is the group of graph symmetries
and * is the action of diffeomorphisms on states. The structure of Has depends on
the group of diffeomorphisms (analytic, smooth, semianalytic etc.) and on dim(E).
Instead of a case-by-case analysis, we make the following
Assumption 2.1. For any two ordered sets (pi,... ,pn), {Pi, ■ ■ ■ ,p'n)> °f n points
of M there is ip G Diff such that <p(pi) = p'iti = 1,... n.
Under this assumption, Ha\s can be described as follows: Let T* denote I\ 0, and
Af the set of functions N. : I* —> N, zero on all but finitely many elements of
X*. Consider the free vector space over such functions and equip it with the inner
product
(TV. | TV'.) = Yl NXIS(NX,N'X).
Aei*
We define annihilation operators by aa\N.) = Na\N. — Sa). We find
laL al>] = K, aa>] = 0, K, al,] = S(a, a')id.
We also define the number operators Na = a^cia and TV = J2a£i* Na- Vectors |TV.)
can be identified with diffeomorphism invariant elements of Cyl* via
JV#) = J2 5(\x,\'), (TV.|(|A.)) = (N.\N{y))
Proposition 2.1. Provided Assumption 2.1 holds, the rigging map is given by
TF\\.) = (TV.(A)|F(TV!). The resulting scalar product is (• | -)F = (• 11/F(TV!) •)■
cIn LQG F(ri) = 1/n, but in the present case this is less clear, so we chose to keep it general.
2793
3. Quantization
Quantization of (1) presents obvious ordering issues. We choose to implement
symmetric ordering, i.e. we will first define an operator Sa with an ordering of ir to
the right, and then symmetrize by setting La == (Sa + S'lQ,)/2. Starting on the
kinematical Hilbert space we define
Sa\X.) = J2^\^+a5x).
A short calculation shows that indeed [Sa,Sa>] = (a' — a)Sa+a>. However,
symmetric ordering runs into severe problems: No element of Cyl is in the domain of
definition of S^ for a ^ 0. Quantization on Wdiff fares better: Since Sa commutes
with diffeomorphisms, it gives rise by duality to an operator Sa on the diffeomor-
phism invariant elements of Cyl*. We find
Sa = ^2{\-a)a\_aax.
A
where a$ = a'0 = id. It turns out that the adjoints (for any F) are densely defined.
We set La = (S^a + <S^)/2. Now L^a = L-a by construction. However now the
commutation relations are anomalous:
Proposition 3.1. Setting A(n) = F((n + 1)!)/F((n + 2)!) - F(n\)/F((n + 1)!) + 1,
one finds
[L'a,L'a>\ = («' " ")%+«' + ?j-(<&A(N)a-al - alA(N)a-a)
A detailed analysis of the equation A(n) = 0 can be found in1 . Suffice it to say
that there is none with strictly positive F.
4. Conclusions
Under Assumption 2.1 we gave an explicit description of Wdiff f°r a scalar field.
We showed that it carries a Fock structure. This may be interesting in its own
right when quantizing diffeomorphism invariant scalars. We used these results to
study the ordering problem for the diff invariant quantities (1), which turned out
to be surprisingly difficult. We understand this as a cautionary example regarding
quantization of the (much more complicated) Hamiltonian constraint.
References
1. H. Sahlmann, arXiv:gr-qc/0609032.
2. A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourao and T. Thiemann, J. Math. Phys.
36 (1995) 6456.
3. T. Thiemann, Class. Quant. Grav. 15 (1998) 839.
T. Thiemann, Class. Quant. Grav. 15 (1998) 1281.
4. T. Thiemann, Class. Quant. Grav. 15 (1998) 1487.
A. Starodubtsev, arXiv:gr-qc/0201089.
A. Ashtekar, J. Lewandowski and H. Sahlmann, Class. Quant. Grav. 20 (2003) Lll.
NIEH-YAN INVARIANT AND FERMIONS IN
ASHTEKAR-BARBERO-IMMIRZI FORMALISM
SIMONE MERCURI
Dipartimento di Fisica, Universita di Roma "La Sapienza", Piazzale Aldo Moro 5, 1-00185,
Rome, Italy
ICRA — International Center for Relativistic Astrophysics
In order to introduce an interaction between gravity and fermions in the Ashtekar-
Barbero-Immirzi formalism without affecting classical dynamics a non-minimal term
is necessary. The non-minimal term together with the Hoist modification to the Hilbert-
Palatini action reconstruct the Nieh-Yan invariant. As a consequence the Immirzi
parameter, differently from the minimal coupling approach, does not affect the classical
dynamics, which is described by the Einstein-Cartan action.
The introduction by Ashtekar of self-dual SL(2, C) connections,1 which reduces the
phase space of General Relativity to that of a Yang-Mills gauge theory, has given
a boost to the program of a background independent quantum theory of gravity
and has finally led to the formulation of the so called Loop Quantum Gravity.2,3
The use of the complex Ashtekar connections simplifies remarkably the Hamiltonian
constraints of the theory, which are reduced to a polynomial form, but, on the other
hand, in order to assure that the evolution be real, a reality condition is necessary.
Implementing the reality condition at the quantum level is a very difficult task, so
the real Barbero connections4 are in general preferred, even though the Hamiltonian
scalar constraint results more complicate and non-polynomial. The relation existing
between the complex Ashtekar connections and the Barbero's real ones was
clarified by Immirzi,5 with the introduction of the so called Immirzi parameter (3, in the
definition of the new connections. Being introduced via a canonical transformation
the Immirzi parameter does not affect the classical dynamics, but it has important
effects in the quantum non-perturbative regimes as explained in.6 This double role
of the parameter (3 suggests an analogy with the parameter 9 in QCD.7 In fact the
analogy exists, because both the parameters results to be multiplicative factors in
front of topological terms(a), as the Hoist covariant approach clearly shows.8
Basically the Hoist action contains a modification with respect to the Hilbert-Palatini
action, which vanishes once the torsionless second Cartan structure equation is
satisfied, if torsion is present things could change. As a consequence spinor fields could
affect this picture. In fact, as well known, the presence of spinors in the
dynamics generates a non-vanishing torsion 2-form, which modifies the Cartan structure
equation and, in the usual Einstein-Cartan theory, yields a Fermi-like four spinors
aIt is worth noting that the adjective topological is generally referred to objects like the integrals
of Pontryagin or Chern classes, which, if the space is compact, depend only on the topological
characteristics of the manifold, but it is often, even though improperly, used referring to the
object multiplying the Immirzi parameter, which does not belong neither to the Chern nor to the
Pontryagin classes and is defined on a pseudo-Riemannian manifold.
2794
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interaction term; the questions we want to address in this brief paper are: Does the
Hoist modification to the Hilbert-Palatini action affect the Einstein-Cartan picture?
And then: If it is the case, is it possible to postulate a non-minimal coupling in
order the resulting effective theory is the Einstein-Cartan one? Does the non-minimal
coupling any geometrical meaning?
The answer to the first question is addressed in a couple of papers and confirms
what initially expected, in fact, minimally coupling spinors to the gravitational
field described by the Hoist action and variating the total action with respect to
the Lorentz valued connection, one finds a non-vanishing right side in the Cartan
structure equation. After having extracted the expression of the right-hand side
2-form:
Ta= -\lT^(£ab^ + ]pS["S«) ■W'Ae", (1)
(where J?As = i[)jaj5ip) one immediately realizes it differs from the Torsion tensor
coming out in the Einstein-Cartan theory, both for the presence of an additional
term and for the dependence on the Immirzi parameter (obviously as soon as the
limit (3 —> oo is calculated the 2-forin above reduces to the torsion of the Einstein-
Cartan theory): as a consequence also the effective action depends on the Immirzi
parameter.9,10 It is worth noting that the 2-form in line (1) cannot be associated
with the torsion of space-time, even though it represents the right hand side of a
dynamical equation analogue to the structure equation of the Einstein-Cartan
theory. The point is that the 2-form (1) contains a pseudo-vectorial term, which cannot
be traced back to anyone of the irreducible components of the torsion tensor11 (b).
The resulting modification to the Einstein-Cartan effective action and the
classical role the Immirzi parameter would play in this framework, suggest to search for
a different formulation of the interaction between gravitational and spinor fields. In
particular, we found that using the following non-minimal action
S (e, tu, V,?) = \ j (\ eabcd ea A eb A Rcd - ~ ea A eb A Rab)
* ea A
i>la ( 1 " ^75 j Z>V -Thpfl- ^75 ) 7>
(2)
we can describe the interaction between the gravitational field and spinor matter
without affecting the effective limit and leading to a natural generalization of the
Hoist approach.12 In fact, the above action reduces to the usual Einstein-Cartan
effective action once the second Cartan structure equation is satisfied and
generates consistent dynamical equations for every value of the Immirzi parameter(c),
bWe stress that, even though the resulting connection contains two parts with different
transformation properties under the sector P of the Lorentz group, the effective theory does not violate
the parity discrete symmetry.
cIt is worth noting that the minimal approach previously described applies only to real values of
the Immirzi parameter.
2796
generalizing the Ashtekar-Romano-Tate one.12 The non-minimal spinor coupling
term together with the Hoist modification reconstruct, once the Cartan structure
equation is satisfied, the so called Nieh-Yan invariant.13 In other words we have(d)
Yq j [ea A eb A Rab + * ea A (^757aXty - Whalv[>)} = -^ I d (Ta A ea). (3)
Moreover the non-minimal spinor action (2) can be, unexpectedly, separated in
two independent actions with different weights depending on the Immirzi
parameter, where the respective interaction terms contain the self-dual and anti-self-dual
Ashtekar connections; this suggests to search for a similar separation in the Hoist
action, in order to rewrite the total action as the sum of two actions describing
independently the self-dual and anti-self-dual sector of the complete theory. This
separation is in fact possible and, as noted by Alexandrov in,14 referring to the
pure gravitational case, both the constraints and the reality condition simplify
using the self-dual and anti self-dual Ashtekar connections as separate variables. On
the other hand, once one realizes that the real Barbero connections can be written
as a weighted sum of self-dual and anti-self-dual connections with weights
depending on the Immirzi parameter, the calculation of the Hamiltonian constraints for
the real connections can be performed starting, directly, from the separated action.
References
1. A. Ashtekar, Phys. Rev. Lett. 57, 2244, (1986) and Phys. Rev. D36, 1587, (1987).
2. C. Rovelli, Quantum Gravity, Cambridge University Press, (2004).
3. A. Ashtekar, J. Lewandowski, Class. Quant. Grav. 21, R53, (2004), gr-qc/0404018.
4. F. Barbero, Phys. Rev. D51, 5498, (1995) and Phys. Rev. D51, 5507, (1995).
5. G. Immirzi, Nucl. Phys. Proc. Suppl. 57, 65, (1997), gr-qc/9701052.
6. C. Rovelli, T. Thiemann, Phys. Rev. D57, 1009, (1998).
7. R. Gambini, O. Obregon, J. Pullin, Phys. Rev. D59, 047505, (1999), gr-qc/9801055.
8. S. Hoist, Phys. Rev. D53, 5966, (1996).
9. A. Perez, C. Rovelli, Phys. Rev. D73, 044013, (2006), gr-qc/0505081.
10. L. Freidel, D. Minic, T. Takeuchi, Phys. Rev. D72, 104002, (2005), hep-th/0507253.
11. C. Rovelli, private communication, Marseilles, (2007).
12. S. Mercuri, Phys. Rev. D73, 084016, (2006), gr-qc/0601013.
13. H.T. Nieh, M.L. Yan, J. Math. Phys. 23, 373, (1982).
14. S. Alexandrov, Class. Quant. Grav. 23, 1837, (2006).
dFor the details of the demonstration and a brief discussion of the Nieh-Yan topological term we
address the reader to.12
A GENERALIZED SCHRODINGER EQUATION
FOR LOOP QUANTUM COSMOLOGY
D. C. SALISBURY* and A. SCHMITZ
DEPARTMENT OF PHYSICS, AUSTIN COLLEGE,
Sherman, TX 75090, USA
* dsalisbury@austincollege. edu
www.austincollege.edu
A temporally discrete Schroedinger time evolution equation is proposed for isotropic
quantum cosmology coupled to a massless scalar source. The approach employs
dynamically determined intrinsic time and produces the correct semiclassical limit.
Keywords: constrained dynamics, loop quantum gravity, quantum cosmology
1. Introduction
Popular approaches to loop quantum cosmology recover a notion of time within a
" frozen time" formalism through requiring that the Hamiltonian constraint
annihilate physical states.1 It is claimed that the resulting states encode unique
correlations between dynamical observables and intrinsically defined time. In contrast we
present here a simple isotropic cosmological model with a massless scalar source in
which we argue that it is possible to formulate a unique quantum time evolution.
Furthermore, we demonstrate explicitly that this evolution produces the correct
semi-classical limit. The program utilizes intrinsically defined time, and is
motivated by the recognition that classical cosmological variables expressed in terms of
intrinsic time can be shown to be invariant under the canonically realized group of
time coordinate transformations.2'3 This work is based on an improved
understanding of the nature of the diffeomorphism-induced canonical symmetry group in which
it is recognized that lapse functions must be retained as canonical variables.4 We
will first discuss the classical implications of this group from four different but
equivalent perspectives, and then we will propose a generalized intrinsic-time-dependent
Schrodinger equation.
2. Classical intrinsic time and canonical reparameterization
invariance
We will consider an isotropic cosmological model with expansion factor a(t)£p,
massless scalar source <j){t)y/mp/tp and lapse function N(t)£p, where for later
convenience we express all fields in Planckian units (so that a(t), <j>(t), and N(t), as well as
the time coordinate t are all dimensionless). The reduced Lagrangian takes the form
L = h (-gf + ^Py The resulting Hamiltonian is H = f (-%& + ^r) + ApN
where A is an arbitrary positive-definite function oft, and the factors multiplying N
and A are primary constraints. The Lagrangian model is covariant under
infinitesimal reparameterizations in time of the form t' = t — 7V_1£(£), and corresponding
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variations in the canonical variables are faithfully generated by the phase space
generator G^ := | f — ^£- + J^r J + £p./v- We shall choose as our intrinsic time T the
square of the expansion factor, thus in terms of the general solution of the equations
/ / /' \2/3
of motion T(t) = a2(t) = (N0t + /0 dt' /0 dt"\(t") + ag 1 . The naught subscript
signifies variables evaluated at time t = 0. There are now four equivalent ways to
construct reparameterization invariants:
(1) Perform the time reparameteriztion T(t). Thus the invariant variables
are <j>(T) = cj>(t(T)) = ^o ± ^ (f logT - 31oga0) = cj>{t) ±
\ 6rT (11°S^ — 3 log a(t)) where it is significant in the final expression that
the initial values may be replaced by the full coordinate time dependence; the
invariants are constants of motion in the sense that they are independent of t.
Also, N{T) = N(t(T))-§, = IT1'2.
(2) Dynamical variables may be gauge transformed through the use of the finite
canonical generator V^(s,t) = exp(s{ —, G^(t)}). In particular, setting for s — 1
the gauge transformed expansion factor a? equal to t, one can solve for the
required dynamical-variable-dependent finite descriptor £. Employing this
descriptor in the gauge transformation of the remaining variables we obtain the
same invariant variables </>(£) and N(t) as above.
(3) Impose the gauge condition t = a2(t). Preservation of this condition under time
evolution leads to a new condition, 7V = — jjp-. The Dirac-Bergmann
procedure then yields the gauge-fixed Hamiltonian HGF = - (—^ + J^) j^
fljvj-T?— with the equations of motion N = —5—^—, a = ^-, pn = — t?%,
^Jv Sfra^pa ^ 8-jra2pa ' 2a' va 2a2'
<p = — i7r^} , and p,/, = 0. The general solution is of course the same intrinsic
time solution as above.
(4) Simply solve the constraints, taking a(t) = t1'2 and N(t) = 3t1/2/2, leaving only
<j> as a dynamical variable. The corresponding reduced Lagrangian is ht<p2/3,
o 2
with Hamiltonian H(t) = -^. To recover the correct classical solutions one
must in addition impose the condition that pi = ^.
3. Generalized time-dependent Schrodinger equation
Bojowald has shown that the expansion factor a2 in the loop quantum gravitational
approach has the discrete eigenvalues tk = fc/6, where k is a nonnegative integer.5
We propose to employ the Hamiltonian obtained above to implement discrete time
stepping. Thus we posit that
Mt*+i)>= (i-j.&H(tk+1))wk)>= (i-2h^+l)pi)Wk)>, (i)
We will work in a </> representation for which the operator P<t> = ^-§z- The classical
field </> can range from minus infinity to plus infinity. Our Hilbert space is thus L2(3?).
2799
can
The minimum uncertainty state ip(<j>,t0) = (2tt(72)-1/4 exp - ^ J2o) + i^
easily be shown to display the correct semi-classical behavior. We assume that the
initial time to = ^frf, f°r large k. (j>o is the expectation value of <j> at time to, while
po is the expectation value of p^. One finds that
|V(<Mo + At)|2<W «&) + ;|£-At, (2)
i.e., the expectation value satisfies the classical evolution equation. In addition,
the expectation value of p^ is constant in time. The classical correspondence limit
requirement that p1, = g^- can be imposed only as an expectation value. This
supplementary condition would be removed in a more realistic massive scalar field model
with a potential.
4. Discussion and conclusions
We have employed the reparameterization in time symmetry to argue that the
imposition of an intrinsic time gauge condition produces reparameterization invariants.
These invariants enjoy an evolution that can be modeled at the classical level, and
promoted to a discrete quantum evolution. The lapse function itself undergoes a
corresponding unique evolution; every choice of intrinsic time yields a fixed
evolution in the lapse. Although in this model the lapse operator is merely a c-nuinber
function, in general it will be a non-trivial operator. Consequently it will generally
undergo fluctuations. One might well question the legitimacy of this approach in
which the intrinsic time, being itself a physical variable, does not itself seem to be
subject to fluctuation.
The only physical criterion employed in this construction is that the model yield
the correct semiclassical limit. In this regard it is permissible to avoid the initial
quantum singularity in the simple manner we have proposed; there is no time zero.
The smallest time is tp/6. The Bojowald difference equations that result from the
imposition of the Hamiltonian constraint do not permit this choice.5 A detailed
discussion of the relation between our construction, Bojowald's semiclassical limit,
and the Wheeler-DeWitt equation will appear elsewhere.
References
1. M. Bojowald and F. Hinterleitner, Phys. Rev. D66, 104003 (2002) [gr-qc/0207038]
2. J. M. Pons and D. C. Salisbury, Phys. Rev. D71, 12402 (2005) [gr-qc/0503013]
3. D. C. Salisbury, J. Helpert, and A. Schmitz, to appear [gr-qc/0503014]
4. J. M. Pons, D. C. Salisbury and L. C. Shepley, Phys. Rev. D55, 658-668 (1997) [gr-
qc/9612037].
5. M. Bojowald, Class. Quant. Grav. 19, 2712 (2002) [gr-qc/0202077]
SPECTRAL ANALYSIS OF THE VOLUME OPERATOR IN LOOP
QUANTUM GRAVITY
J. BRUNNEMANN* and D. RIDEOUTt
* Hamburg University, Mathematics Department, Bundesstrasse 55, 201^6 Hamburg, Germany
t Blackett Laboratory, Imperial College, London SW7 2AZ, United Kingdom
johannes.brunnemann@math.uni-hamburg.de, d.rideout@imperial.ac.uk
We describe preliminary results of a detailed numerical analysis of the volume operator
as formulated by Ashtekar and Lewandowski.2 Due to a simplified explicit expression for
its matrix elements,3 it is possible for the first time to treat generic vertices of valence
greater than four. It is found that the vertex geometry characterizes the volume spectrum.
1. Introduction
Loop Quantum Gravity1 is an attempt to apply canonical quantization to General
Relativity (GR). For this four dimensional spacetime M is foliated into an
ensemble of three dimensional spatial hypersurfaces E. GR can then be rewritten as an
SU(2) gauge theory with the canonical variables being densitized triads E?(x), and
connections AJb(y), encoding information on the induced metric q on E. Here (x, y)
are points in E. a,b = 1,2,3 are tensor indices, and i,j = 1,2,3 are ST/(2)-indices.
In this treatment the theory is subject to constraints: three vector and one scalar
constraint ensuring invariance under diffeomorphisms within E and deformations
of E within M. respectively, and three Gauss constraints G, which ensure
invariance under SU(2) gauge transformations. In the quantum theory one considers
the integral of A3b(y) over one dimensional edges e C Et, that is the holonomies
he(A) = JeA, and fluxes Ei(S) = fs*Ei resulting from the integration of the dual
of Ef(x) over two dimensional surfaces ScS(. Finite collections of edges are called
a graph 7. The edges mutually intersect at their beginning and end points, which
are called the vertices {v}\~( of 7. The canonical pair (he,Ej,(S)) can then be
represented as multiplication and derivation operators respectively, on the space spanned
by spin network functions (SNF) T^AH(hei (A),..., heN (A)) = UeCi [nje (he)} ^
formulated with respect to a particular 7. Each of the edges (ei,..., e^v) of 7
carries a matrix element function \jtjp (he)] of an irreducible S£/(2 ^representation
of weight (j\,... ,Jn) ='■ j with matrix elements denoted by (mi,..., tun) ='■ rn,
(ni,..., tin) =: n. There is for each copy of ST/(2) attached to an edge e C 7 a one
to one correspondence between the action Ei(S) [ttj(-)] (') and the action of the
usual angular momentum operator J; on an angular momentum state | j m ; n )
with spin (J2i=i JiJi)\ j rn ;n) = j(j + l)\ j m ;n) and J^\ j rn ;n ) = m\ j m ; n ),
and an additional quantum number n which is not affected by the action of J{.
2. The Volume Operator
As the theory is formulated classically in terms of the geometric objects (A,E), it
is possible to formulate a quantum version of the classical expression for the volume
2800
2801
Fig. 1. (a) Overall 2048 bin histograms for the gauge invariant 5-vertex (Z = £p = 1) up to
jmax = 4^ (top curve, below it are histograms for smaller jmax). There are 4.8 X 1012 eigenvalues
in all, of which 4.5 x 1011 are zero (and are excluded), (b) Portion of histogram for A^ < 9.
V{R) of a spatial region EcS given by f„ ^/detq d3x = J„ y | det E\ (fix, where
the classical identity | deti?| = detg is used (we assume detg > 0). Upon quantiza-
tion one obtains2-3 V{W^) = 4£Wl,n* V^E/J^^^^WO-
Here £P is the Planck length, Z is a constant and quk '•= 4eyfeJf J|J J|K is a
polynomial of operators, J^1 denoting the i-component of angular momentum
acting on the SU'(2)-copy attached to the edge ej. In the action of V(R) the classical
integration fR is replaced by a sum ^2iv\\ over vertices v of 7 contained in R, so
volume is concentrated at vertices only. At each vertex v of 7 one obtains a matrix
Qijk for each triple ej Hej Hex = v of edges incident at v. These matrices are added
with prefactors e(IJK) := sgn (det (ei(v),ij(v),eK(v))) = 0, ±1, which carry
spatial diffeomorphism invariant information on the orientation of the triple of edge
tangent vectors e,i(v) := -^ei(s)\v for each edge ex at v, with curve parameter s.
If the tangents are coplanar then e(IJK) = 0. Taking the matrix sum we obtain
a purely imaginary antisymmetric matrix with real eigenvalues Xq (which come in
pairs ±|Aq| or are 0) and eigenstates T\q (linear combinations of the T jAfl(')),
V(R) then has T\^ = T\q as eigenstates with according eigenvalues A-^ = y|Ag|.
3. Spectral Analysis
The action of V(R) on an arbitrary SNF decays into a sum over single vertices,
so it is sufficient to compute its spectrum for a single vertex only. We have
implemented the matrices qjjK for a single SU(2)-g&uge invariant JV„-valent vertex v on
2802
a supercomputer. Here techniques from recoupling theory of angular momenta for
the construction of a gauge invariant SNF as linear combinations of the T ^-(-)
are heavily used: The gauge invariant subspace contained in the span of the SNF
is computed by considering all ways to recouple the angular momenta of the edges
incident at v to a resulting trivial representation of SU(2). The second task is to
examine which edge triple sign combinations e : = {e(IJK)} are realizable in an
embedding of Nv edges. There are (^") triples e(IJK) — 0, ±1 resulting in 3^ z>
possibilities. However for valences > 4 not all of these possibilities can be realized.
We have computed the set of realizable sign combinations e by a Monte Carlo
random sprinkling of Nv points on a unit sphere, where each point is regarded as the
end point of a vector emanating from the origin. The according e(IJ K)-iactors can
then be computed. We exclude coplanar edge triples e(IJK) = 0 from our
analysis, as such configurations will never arise via sprinkling. For a 5-vertex with 10
triples we find that only 384 out of 210 possibilities can be realized. For valences
Nv = 4, 5, 6, 7 we have computed the eigenvalues A^. for the matrices V for all sets
of spins ji,..., jnv < jmax and all realizable esign configurations. Here jmax is
an upper cutoff. The A^. can then be sorted into histograms to obtain a notion
of spectral density. We find that the spectral properties of V depend strongly on
the e. In particular one can choose e such that the smallest non-zero eigenvalues
either increase, decrease or stay constant as jmax is increased. There are also e-
configurations for which all A^. — 0 independently of the spins, as a consequence
of gauge invariance. Figure 1 shows the resulting overall histogram for the gauge
invariant 5-vertex where all Xy for all 384 e configurations are collected. For large
eigenvalues (> 10) we obtain a rapidly increasing eigenvalue density which can be
fitted by an exponential. For smaller eigenvalues (~ 3) the density becomes minimal
and then increases again close to zero. This suggests that zero is an accumulation
point of the volume spectrum. This property is shared by 6 and 7-valent vertices.
The complete results can be found in a forthcoming paper.4
Acknowledgments We thank Thomas Thiemann for encouraging discussions as
well as the Numerical Relativity group of the Albert Einstein Institute Potsdam.
J.B. thanks the Gottlieb Daimler- and Karl-Benz-foundation for financial support.
The work of D.R. was supported by the European Network on Random Geometry,
ENRAGE (MRTN-CT-2004-005616).
References
1. T. Thiemann, "Introduction to Modern Canonical Quantum General Relativity",
Cambridge University Press, Cambridge 2006, [arXiv: gr-qc/0110034].
2. A. Ashtekar and J. Lewandowski, "Quantum theory of geometry. II: Volume operators,"
Adv. Theor.-Math. Phys. 1, 388 (1998) [arXiv:gr-qc/9711031].
3. J. Brunnemann and T. Thiemann, "Simplification of the spectral analysis of the volume
operator in loop quantum gravity," Class. Quant. Grav. 23 (2006) 1289, [arXiv:gr-
qc/0405060].
4. J. Brunnemann and D. Rideout, "Properties of the Volume Operator in Loop Quantum
Gravity", to appear.
COUNTING ENTROPY IN CAUSAL SET QUANTUM GRAVITY
D. RIDEOUT* and S. ZOHRENt
Blackett Laboratory, Imperial College, London SW7 2AZ, United Kingdom
* d.rideout@imperial.ac.uk, t stefan.zohren@imperial.ac.uk
The finiteness of black hole entropy suggest that spacetime is fundamentally discrete, and
hints at an underlying relationship between geometry and "information". The foundation
of this relationship is yet to be uncovered, but should manifest itself in a theory of
quantum gravity. We review recent attempts to define a microscopic measure for black
hole entropy and for the maximum entropy of spherically symmetric spacelike regions,
within the causal set approach to quantum gravity.
1. Introduction
The various entropy bounds that exist in the literature (see for a review1) suggest
that an underlying theory of quantum gravity should predict these bounds from
a counting of microstates and should clarify which are the fundamental degrees of
freedom one is actually counting. This verification of the thermodynamic laws is an
important consistency check for any approach to quantum gravity.
In what follows we review an earlier work by Dou and Sorkin2 defining a
microscopic measure for black hole entropy together with our recent proposal3 for
measuring the maximum entropy contained in a spherically symmetric spacelike
region, within the causal set approach to quantum gravity.
2. Causal set quantum gravity
Causal set theory is an approach to fundamentally discrete quantum gravity (see
for a recent review4). Besides taking fundamental discreteness as a first principle,
the primacy of causal structure is the main observation underlying causal sets.
Mathematically a causal set is a locally finite partially ordered set, or in other
words a set C endowed with a binary relation 'precedes' -<, which satisfies: (i)
transitivity: if x -< y and y < z then x -< z, (ii) irreflexivity: x -/< x, (Hi) local
finiteness: for any pair of elements x and z of C, the set of elements lying between
x and z is finite, \{y\x -< y -< z}\ < oo. Some useful definitions are the past of an
element past(x) = {y £ C\y ~< x] and its future future(a;) = {y S C\x -< y}.
Further, a relation x ~< y is called a link iff future(a;) (~l past(y) = {x, y}. Elements
of the causal set whose future (past) is empty are called maximal (minimal).
The hypothesis of causal set theory is that spacetime at short scales such as
the Planck length is fundamentally discrete, and is better described by a causal set
than a differentiable manifold. The notion of continuum Lorentzian spacetime M.
at larger scales is recovered as an approximation of the causal set. This occurs when
the causal set can be faithfully embedded into M., where faithfully means that the
embedding respects not only the causal relations, but also a correspondence between
cardinality and spacetime volume.
2803
2804
Fig. 1. (a) Schwarzschild spacetime and null hypersurface L. (b) Spherically symmetric spacelike
region S, its future domain of dependence £>+(S) and future Cauchy horizon #+(£).
3. Black hole entropy
In an earlier work, Dou and Sorkin considered the four-dimensional Schwarzschild
black hole in its dimensionally reduced form, ds2 = —4a3/r e~r'adudv, where a is
the Schwarzschild radius of the black hole and u and v are the Kruskal coordinates.2
Assuming that this spacetime arises as an approximation to a causal set which
can be faithfully embedded into it, they propose to count the number of causal
links from causal set elements x € TZ\ = J~(H) n J~(L) to elements y € 7?-2 =
J+ (H)PiJ+ (L) (see fig. (a)). The motivation for counting links comes from regarding
the black hole entropy as arising from quantum entanglement across the horizon
H evaluated at a null hypersurface L, and noting that the links are effectively
irreducible elements of potential information flow in a causal set. The number of
such links is given by (n) = Jn Jn e~v^x^dVxdVy, where V(x,y) denotes the
volume of J+(x) D J~(y). (The dimensional reduction is necessary to make feasible
the computation of such regions.) To suppress certain unphysical nonlocal links
one further has to impose that the elements y are minimal in J+{H). Evaluating
the above integral at scales much larger than the discreteness scale then yields
(n) = 7r2/6 + • • • (where the • • • represent higher order terms in the ratio of the
discreteness scale to the macroscopic scale a). Unfortunately, when one considers
the angular dimensions, it now seems clear that the expected number of links will
diverge, essentially because the intersection of the future light cone of a candidate
element x with H has an infinite extent. However, it seems likely that a minor
variation, such as counting triples of elements rather than pairs, will lead to a
convergent integral in the full four-dimensional case.
4. The spherical entropy bound
We now discuss our recently proposed microscopic evidence for the spherical entropy
bound arising from causal set theory. Susskind's spherical entropy bound5 states
that the entropy of the matter content of a spherically symmetric spacelike region
2805
E (of finite volume) is bounded by a quarter of the area of the boundary of E in
Planck units, S < A/(4Zp), where lp is the Planck length.
In the case of black holes the counting of links is computationally difficult in
the full four-dimensional geometry, because of the complicated causal structure in
the angular coordinates. For the simpler case of the spherically symmetric region
E let us now propose the following measure of entropy. Note that the entropy of
the matter contained in E must eventually "flow out" of the region by passing
over the boundary of its future domain of dependence D+(E), the future Cauchy
horizon iJ+(E) (see fig. (b)). But because spacetime is fundamentally discrete, the
amount of such entropy flux is bounded above by the number of discrete elements
comprising this boundary. These elements can be seen as just the maximal elements
of the causal set faithfully embedded into the future domain of dependence D+(Ti).
This is similar to the case of the black hole, where the links started at the elements
x which were maximal in TZ\ (by definition of being linked to y). Hence we define
the maximal entropy contained in E as the number of maximal elements in Z)+(E),
Smax = (n) = ./*£>+(£) e~v(x*)dVx, where V(x) is the volume of future(x) n D+(E).
The claim is that if the fundamental discreteness scale is fixed at a dimension-
dependent value this proposal leads to Susskind's spherical entropy bound in the
continuum approximation, Smax=A/(AlP), where A is the area of the boundary of
E.
For the case where E is a three dimensional-ball in four-dimensional Minkowski
spacetime, (n) can be evaluated analytically yielding, at scales much larger than
the discreteness scale, (n) = y/6A/(^lP) + • • •. This shows that indeed the result is
proportional to the area of the boundary of E. If we fix the fundamental discreteness
scale to lj = \/&lp, we arrive at the desired result Smax = A/(4lp). Further, we
could numerically show that one obtains the same result in the case of different
spherically symmetric spacelike regions in four-dimensional Minkowski spacetime
as well as for different dimensions, where the value of the fundamental discreteness
scale changed with the dimension. Work in progress indicates that this result is also
true in the case of conformally flat Friedmann-Robertson-Walker spacetime.
Acknowledgments
The authors acknowledge support by the European Network on Random Geometry,
ENRAGE (MRTN-CT-2004-005616). Further, we would like to thank F. Dowker for
enjoyable discussions, comments, and critical proof reading of the manuscript.
References
1. R. Bousso, Rev. Mod. Phys. 74, 825 (2002).
2. D. Dou and R. D. Sorkin, Found. Phys. 33, 279 (2003).
3. D. Rideout and S. Zohren, Class. Quant. Grav. 23, 6195 (2006).
4. J. Henson in Approaches to Quantum Gravity: Towards a New Understanding of Space
and Time, ed. D. Oriti, Cambridge University Press, (2006).
5. L. Susskind, J. Math. Phys. 36, 6377 (1995).
ALGEBRAIC APPROACH TO 'QUANTUM SPACETIME
GEOMETRY'
IOANNIS RAPTIS
Algebra and Geometry Section, Department of Mathematics, University of Athens,
Panepistimioupolis, Athens 157 84, Greece
and
Theoretical Physics Group, Imperial College London, Prince Consort Road, South Kensington,
London SW7 2BZ, UK
i. raptis iSic.ac.uk
PETROS WALLDEN
Raman Research Institute, Theoretical Physics Group, Sadashivanagar, Bangalore - 560 080,
India
petros@rri.res.in, petros.wallden@gmail.com
ROMAN R. ZAPATRIN
Department of Information Science, The State Russian Museum, Inzenernaya 4, 191186,
St.Petersburg, Russia
Roman.Zapatrin@gmail.com
In General Relativity, the topology of spacetime is an entity which is given once and
forever. From the operationalistic, quantum mechanical point of view this deprives the
topology the status of an observable quantity. Recently a mathematical formalism for
treating spacetime topology (in particular, the description of spacetime foam in algebraic
terms) as a quantum observable was provided by the authors. The suggested formalism
lacked in operationalistic treatise as no binding it with at least thought experiment was
provided. For that, the histories approach to Quantum Mechanics was drawn in order to
pass from description in terms of vectors in Hilbert spaces to more realistic issues like
records of experimental events.
1. Motivation
In the standard formulation of relativity theory, the spacetime topology is a priori
fixed by the theorist to that of a continuous manifold; hence, it is not an observable
entity. Only the metric structure is traditionally supposed to be dynamically
variable. But even in General Relativity, where no variable is supposed to be quantum,
we need histories to actually define the topology of spacetime. This is because the
concept of neighborhood turns out to be something which an observer, located at
some point in spacetime, deduces for regions that belong to her causal past. The
key point, is the existence of an upper bound in the speed of transfer of matter and
information. Due to this, the set of possible events (P) has the extra structure of
a partially ordered set (with respect to the causality relation). This property, and
provided we can have access to the set of possible events V by some measurements,
allows us to recover some proximity relation between spacelike points and therefore
deduce the topology.
2806
2807
2. Algebraic Description of Spacetime Foams
Let us sketch out the basic ingredients of our previously proposed algebraic
formalism for spacetime foam description.1 Each observer of quantum causality creates
her own picture of the dynamics of quantum causality; as it may, she creates her
own 'time-gauge'. We formulate first how quantum spacetime topology can possibly
move (ie, its kinematical structure), and then entertain ideas of how it actually
moves (ie, its dynamics). For that, we employ non-*-algebras, and provide the
algebraic machinery, which endows their irreducible representations (treated as points)
by non-trivial topologies. These representations—referred to subalgebras—are
considered as quantum states, on one hand, and as finitary substitutes of spacetime,
on the other.
Two issues are worth mentioning. Firstly, the non-*-algebras are often claimed
to be unphysical. This is because the usual algebras of observables of relativistic
matter quanta using quantum field theory on Minkowski spacetime are *-algcbras.
The latter, are theories intrinsically time-reversible. Our stand point, is related to
the discussion above about kinematics versus dynamics and in particular, Penrose's2
suggestion that "the true quantum gravity is a time asymmetrical theory". We
therefore expect our theory to address the problem of the quantum arrow of time
at the kinematic level. The second point we stress, is that at this early stage of
the construction of the theory it seems more natural to us to sacrifice unitarity for
fmiteness. One reason for this choice, is because the former is usually perceived as a
non-local conception (since, in non-relativistic quantum mechanics it conventionally
involves an integration over all space), while our algebraic approach is fundamentally
local.
3. Histories and Records
The basic ideas of our next work3'4 can be summarized as follows. To make
propositions about spacetime topology we apply the decoherent histories approach: an
alternative formulation of Quantum Theory design to deal with closed systems, and
that has as main objects of interest whole histories of the system rather than the
one time propositions of the Copenhagen interpretation. We can assign
probabilities to histories when the set of histories decoheres. This is given by considering the
decoherence functional which is a function that effectively measures the interference
between two histories. Note that, decoherence is closely related with the existence
of records, in particular, we have decoherence if and only if there exist records of
these histories somewhere in the universe5 (a record is a set of projection operators
at the final time that is perfectly correlated with a particular history).
In our operational approach we use exactly this property as starting point and do
the inverse, ie deduce the topology of the underlying effective spacetime, given a set
of records and certain assumptions about them.3'4 This set of records corresponds
to outcomes of actual experiments, thus remaining true to operationalism.
The assumptions about the records, in order to recover the effective topology, are
2808
the following. The records capture the spatiotemporal properties of the system. This
means that the record of each history will correspond to a coarse grained trajectory.
We will also assume that each of these records is composed from sub-records that
correspond to the coarse grained events. We therefore end up with a set of
(coarsegrained) events V and a collection of subsets C, corresponding to each causal chain.
The causal order of the events within each chain, is not given.
This order can be reconstructed up to some ambiguities that are also classified
in Ref. 4. We therefore end up with an effective causal set (discrete version of a
manifold). From there we recover the topology of this causal set following similar
methods with Ref. 6.
4. Conclusions
Our approach can be summarized as follows. We attempted to have the topology as
a quantum variable. First we reviewed some algebraic considerations, presented in
Ref. 1 where discrete spacetime topologies are associated with appropriate subspaces
of the state space, endowed with an extra structure of associative (non-*)-algebra.
We then considered a more 'realistic' situation using the concept of record from the
decoherent histories.3,4 From the set of unordered causal chains we recover the full
causal order and thus recover a causal set. From the causal set we, in turn, derive
the topology of a spacelike surface following Ref. 6.
Acknowledgments
RRZ is grateful to the Organizing Committee of the Eleventh Marcel Grossmann
Meeting on General Relativity for hospitality and financial support.
References
1. I.Raptis, R.R.Zapatrin, Classical and Quantum Gravity, 18, 4187 (2001), gr-
qc/0102048.
2. Penrose, R., Newton, Quantum Theory and Reality, in 300 Years of Gravitation, Eds.
Hawking, S. W. and Israel, W., Cambridge University Press, Cambridge (1987).
3. Ioannis Raptis, Petros Wallden and Roman R. Zapatrin, International Journal of
Theoretical Physics 45, 1589 (2006), gr-qc/0506088.
4. Ioannis Raptis, Petros Wallden and Roman R. Zapatrin, International Journal of
Theoretical Physics 45, 2199 (2006), gr-qc/0510053.
5. M. Gell-Mann and J. Hartle, Phys. Rev. D 47, 3345 (1993).
6. S. Major, D. Rideout and S. Surya, Classical and Quantum Gravity, 23, 4743 (2006),
gr-qc/0506133.
NONCOMMUTATIVE TRANSLATIONS AND ^-PRODUCT
FORMALISM* *
MARCIN DASZKIEWICZ, JERZY LUKIERSKI and MARIUSZ WORONOWICZ
Institute of Theoretical Physics
Wroclaw University pi. Maxa Borna 9, 50-206 Wroclaw, Poland
We consider the noncommutative space-times with Lie-algebraic noncommutativity (e.g.
re-deformed Minkowski space). In the framework with classical fields we extend the it-
product in order to represent the noncommutative translations in terms of commutative
ones. We show the translational invariance of noncommutative bilinear action with
local product of noncommutative fields. The quadratic noncommutativity is also briefly
discussed.
In noncommutative space-time, in general case, the translations are also
noncommutative. The aim of this note is to study the translational invariance of local
noncommutative actions.
The noncommutative Minkowski space
i
[x^Xv] = —0(kx) , (1)
where we choose (x = kx)
9(x) = 9$xxx+e$xpxxxp, (2)
is invariant under the translations
X^i > X„ — X^ -\- Vn , \0)
if
[v^v, } = -0$xvx + iO$XpvxvP , (4)
K
[xt,,v„} = ^0(ix2Jxp{xxvP+xpvx) . (5)
If the relation (3) describes a coproduct from the relation (5) follows that for
quadratic deformations such a coproduct is a braided one (see also1'2). Contrary to
the recent proposal3 , in Lie-algebraic case the formula (3) implies that the
noncommutative translations are represented by standard Hopf-algebraic coproduct. It
should be recalled that such standard coproduct describes the translation sector of
quantum K-Poincare group4 .
Let us choose firstly in (1-5) 8$ ^ 0 and 9\j.Jp = 0 (Lie-algebraic case). In
such a case the relations (1) and (4-5) describe two commuting copies of Lie algebra
with the structure constant 8\iJ .
* Supported by KBN grant 1P03B01828.
tPresented by J. Lukierski, e-mail: lukier@ift.uni.wroc.pl
2809
2810
Using CBH formula for the multiplication of the group elements of the corresponding
Lie group (see e.g.5)
eia''£Mei/3'J£M _ &i^{a,fi)xIL /g\
where
y>, p) = a» + p + -eWorp" + —^Ml^K" V + /T/3 V) + • • • , (7)
one can introduce the following ^-product of the classical exponentials
For two arbitrary classical field the formula (8) generates the following *-
multiplication
(j)(x)-kx(x) = \im^{y) exp(ix^( —, — jjx(z)
dixidiX2K(x; xi, x2)4>(xi)x(x2) . (9)
where 7^(0:,/?) = 7M(a,/?) — aM — [3^ and the nonlocal kernel K(x;y,z) describes
the bidifferential operator of infinite order.
The product of two noncommutative fields (j>(x, v)x(x, v) is represented as the
product of two commuting •-products (9)
,dy
d_ d_
•.dy' dz
For <f)(x,v) = <f)(x + v) and x(x,v) = x(x + v) one can put on r.h.s. of (10) <j){y,u)
_a_
dy
x, v) * x(x, v) = lim lim 4>(y,u) explix^f— ,—)-
y,z^x u,w^v \ \C)y CJZJ
4>{y + u) and x(z, w) = x(z + w). Using -§- = £,£ = £; and (9), one gets
(x + v)-k x(x + v) ■= \m\^ <)){y + v) exp {{(x^+v^^i—,— ] x(z + v)
dixidix2K(x + v;xi>x2)(f>(x1)x(x2) ■ (H)
We introduce the noncommutative integration satisfying the relation
dAxF{x) = [ dixn(x)F(x) , (12)
where n{x) is adjusted by the cyclic property of the noncommutative integral when
F(x) = (p(x)x(x) (see e.g.6). The translational invariance of standard integration
and the formula (11) implies that
<Tx(j)(x + v)x(x + v)= d*x<j>(x)x(x) . (13)
The formula (13) describes explicitly the translational invariance of bilinear action
under noncommutative coordinate shifts (3).
2811
The star product (8) describes the multiplication of nonordered noncomutative
plane waves. In particular case of Lie-algebraic deformation, it is useful to consider
the noncommutative plane waves ordered in particular way. For example, if we
assume that the commutator (1) describes K-deformed Minkowski space4'7
[x0,Xi] = -Xi , [xi,Xj]=0, (14)
K
one can introduce the normally ordered exponentials7'8
Using the relation
(15)
(16)
gip* X^ _ gip XOglp Xi
where
P°=P° , Pl = ~(l-e-^p\ (17)
one can translate the CBH star product (8) into the standard star product, used in
K-deformed field theory8,9 , which is homomorphic to the multiplication of normally
ordered exponentials. In fact, there is an infinite number of ways to define the star
product, homomorphic to noncommutative multiplication rule, which is related by
various nonlinear transformations of the four-momentum variable (see e.g.8'10).
Finally, let us consider quadratic deformations of Minkowski space. If we choose
in (1-5) #}t„ = 0 and ff^v ^ 0, the star product representing the
noncommutative translations has to take into consideration the braiding (relation (5)), i.e.
contrary to the formula (10), it does not factorize into the product of two identical
•-products. If we correctly, however, introduce one "big" star product
representing the noncommutativity given by (1), (4) and (5), it is possible to represent the
noncommutative quadratic translations by the classical ones. In order to show the
translational invariance of corresponding noncommutative local field theory, one has
to find for quadratic deformations the counterpart of the relation (12), which is less
obvious than in the case of Lie-algebraic space-time commutation relations.
References
1. S.Majid, Journ. Math. Phys. 34, 2045 (1993)
2. C.Chryssomalakos and B.Zumino, Salamfest 1993 Proa, p.327 (1994)
3. A.Agostini, G.Amelino-Camelia, M.Arzano, A.Marciano and R.A.Tacchi, hep-
th/0607221
4. S.Zakrzewski, Journ. of Phys. A 27, 2075 (1994)
5. V.Kathotia, math.qa/9811174
6. M. Dimitrijevic, L. Jonke, L. Moller, T. Tsouchnika, J. Wess and M. Wohlgenannt,
hep-th/0307149.
7. S.Majid, H.Ruegg, Phys. Lett. B 334, 348 (1994)
8. P.Kosinski, J.Lukierski, P.Maslanka and A.Sitarz, Czech. J. Phys. 48, 1407-1414 (1998)
9. P.Kosinski, J.Lukierski, P.Maslanka, Phys. Rev. D 62 (2000) 025004
10. L.Freidel, J.Kowalski-Glikman and S.Nowak, hep-th/0612170
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Brane Worlds and String
Motivated Cosmology
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BLACK HOLES ON COSMOLOGICAL BRANES *
LASZLO A. GERGELY
Departments of Theoretical and Experimental Physics, University of Szeged,
Dora ter 9, H-6720 Szeged, Hungary
gergely@physx.u-szeged.hu
While in general relativity black holes can be freely embedded into a cosmological
background, the same problem in brane-worlds is much more cumbersome. We present here
the results obtained so far in the explicit constructions of such space-times. We also
discuss gravitational collapse in this context.
Keywords: brane-worlds, cosmology with inhomogeneities, gravitational collapse
Although almost perfectly homogeneous and isotropic at very large scales, as probed
by the measurements of the cosmic microwave background, our universe contains
local inhomogeneities in the form of galaxies and their clusters. Therefore the
cosmological model of Friedmann-Lemaitre-Robertson-Walker (FLRW) geometry with
flat spatial sections, considered valid on large scales, has to be modified on lower
scales. The simplest way to do it in general relativity is to cut out spheres of
constant comoving radius from the FLRW space-time and fill them with Schwarzschild
vacua, modeling stars, black holes or even galaxies with a spherical distribution of
dark matter. Such a model was worked out by Einstein and Straus..1 In the
framework of this, so-called Swiss-cheese model, it was shown that (a) cosmic expansion
has no influence on planetary orbits and (b) the luminosity-redshift relation receives
corrections.2 The Einstein-Straus model however is unstable against perturbations.3
Brane-world models4"8 with our universe as a 4-dimensional hypersurface (the
brane) with tension A embedded in a 5-dimensional bulk is also confronted with the
challenge of introducing local inhomogeneities in the cosmological background. The
basic dynamic equation in these models is the effective Einstein equation,9,10
Gab = ~^gab + K2Tab + 7i4Sab-£ab +Lab +Pab ■ (1)
On the right hand side we find the unconventional source terms Sab = V~TacTbc +
TTab/3 - gab(-TcdTcd + T2/3)/2]/4, quadratic in the energy-momentum tensor Tab
(modifying early cosmology11); the average taken over the two sides of the brane
of the electric part £ab = Cabcdnbnd of the bulk Weyl tensor Cabcd (in a
cosmological context £ab is known as dark radiation with magnitude limited by Big Bang
Nucleosynthesis (BBN) arguments11,12); the asymmetry source term Lab which is
the trace-free part of the tensor Lab = KabK — KacKb — gab(K — KabK )/2
* Research supported by OTKA grants no. T046939, TS044665 and the Janos Bolyai Fellowships
of the Hungarian Academy of Sciences. The author wishes to thank the organizers of the 11th
Marcel Grossmann Meeting for support.
2815
2816
(with Kai, the mean extrinsic curvature); and the pull-back to the brane Vab =
(21? /?>)(gagfli.cd)TF oi the bulk energy momentum tensor Ila6 (with k2 and 7? the
brane and bulk coupling constants and gab the induced metric on the brane). The
function A = (£2/2)(A —ncndHcd—L/4) contains the possibly varying normal
projection of the bulk energy-momentum tensor and the trace of the embedding function
Lab. Under special circumstances A becomes the brane cosmological constant. Here
we consider this simpler case; also £ai, = 0 = Vab-
For a perfect fluid with energy density p and pressure p the non-linear source
term Sab scales as p/X as compared to Tai,. Due to the huge value of the brane
tension, this ratio is in general infinitesimal, excepting the very early universe and the
final stages of gravitational collapse. The strongest bound on A was derived by
combining the results of table-top experiments on possible deviations from Newton's law,
probing gravity at sub-millimeter scales13 with the known value of the 4-dimensional
Planck constant. In the 2-brane model6 this gives14 A > 138.59 TeV4.Much milder
limits arise from BBN constraints15 (A > 1 MeV4) and astrophysical considerations
on brane neutron stars16 (A > 5 x 108 MeV4). Nevertheless, even when small, the
presence of the source terms Sab implies that the pressure of the perfect fluid at the
junction surface with a vacuum region does not vanish.17
The junction conditions between FLRW and Schwarzschild regions on the
brane18 imply a Swiss-cheese model that forever expands and forever decelerates.
The energy density and pressure of the fluid tend to the general relativistic values at
late times (on the physical branch; there is also an unphysical branch never allowing
for positive values of p). At early times however p is smaller than in the Einstein-
Straus model and p takes large negative values.19 When we allow for a cosmological
constant in the FLRW regions, the deviation from the Einstein-Straus model is
present at late-times as well. As the universe expands, first p turns positive, then
eventually p turns negative. Moreover, for A overpassing a threshold value Am;n a
pressure singularity accompanied by regular cosmological evolution appears.
Such a Swiss-cheese model may be interpreted as a cosmological brane
penetrated by a collection of bulk black strings.19 When the brane is embedded asym-
metrically, with different left and right bulk regions, the source term Lab slightly
modifies this scenario.20 The evolution of the cosmological fluid is further
degenerated, proceeding along four possible branches, two of them being physical. The
future pressure singularity becomes generic, it appears even below the threshold for
A, due to the difference in the bulk cosmological constants. For any A < Am;n there
is a critical value of a suitably defined asymmetry parameter which separates Swiss-
cheese cosmologies with and without pressure singularities.20
The mathematically similar problem of the gravitational collapse on the brane
has been also studied. If the pressure of the collapsing fluid is set to zero, we
recover the analogue of the general relativistic Oppenheimer-Snyder collapse.21 But
in contrast with general relativity, the exterior space-time is either characterized
(beside the mass) by a tidal charge22 (and the collapse possibly leads to a bounce, a
2817
black hole or a naked singularity), or is non-static,16'23 infiltrated by radiation,24'25
or by a Hawking flux.26 An effective Schwarzschild solution on the brane can be
found when phantom bulk radiation is absorbed on the brane.27
By allowing for non-vanishing pressure in the collapsing star, the exterior can
be again static.28'29 In this case the collapsing fluid is described by the FLRW
metric, which fills spheres of constant comoving radius cut out from the Schwarzschild
space-time. The modified gravitational dynamics (1) again gives two branches. On
the physical branch the fluid is near dust-like at the beginning of the collapse: it
has an infinitesimal negative pressure (tension) p = wp with w « —p/2\,
arising from the interaction of the fluid with the brane. The tension vanishes in the
general relativistic limit, but as the collapse proceeds and p increases, it becomes
more important. For astrophysical brane black holes the tension stays small even at
horizon forming. However well below the horizon, at the final stages of the collapse
w « —1/2 and the condition for dark energy p + 3p < 0 is obeyed. This however has
little repulsive effect, as at such high energy densities the source term Sab (which is
always positive) dominates, and the singularity inevitably forms.
References
1. A.Einstein andE.G.Straus,Rev.Mod.Phys. 17,120(1945), errata,ibid. 18,148(1946).
2. R. Kantowski, Astrophys.J. 155,89(1969).
3. A. Krasiriski, Inhomogeneous Cosmological Models, Cambridge University Press (1997).
4. N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Lett. B 429, 263 (1998).
5. N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Rev. D 59, 086004 (1999).
6. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999).
7. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999).
8. R. Maartens R, Living Rev. Rel. 7, 1 (2004).
9. T. Shiromizu T, K. Maeda, and M. Sasaki, Phys. Rev. D 62, 024012 (2000).
10. L. A. Gergely, Phys. Rev. D 68, 124011 (2003).
11. P. Binetruy, C. Deffayet, U. Ellwanger, and D. Langlois, Phys.Lett. B 477, 285 (2000).
12. K.Ichiki,M.Yahiro,T.Kajino,M.Orito,andG.J.Mathews,Phys.Rev.D66,043521 (2002).
13. J. C. Long, et al., Nature 421, 922 (2003).
14. L. A. Gergely and Z. Keresztes, JCAP 06(01), 022 (2006).
15. R.Maartens,D.Wands,B.A.Bassett,andI.P.C.Heard, Phys.Rev.D62,041301(R)(2000).
16. C. Germani and R. Maartens, Phys. Rev. D 64, 124010 (2001).
17. N. Deruelle, gr-qc/0111065 (2001).
18. L. A. Gergely, Phys. Rev. D 71, 084017 (2005), erratum, ibid. 72, 069902 (2005).
19. L. A. Gergely, Phys. Rev. D 74, 024002 (2006).
20. L. A. Gergely, I. Kepfro, hep-th/0608195 (2006).
21. J. R. Oppenheimer and H. Snyder, Phys. Rev. 56, 455 (1939).
22. N.Dadhich,R.Maartens,P.Papadopoulos, andV.Rezania, Phys.Lett.B487,1 (2000).
23. M. Bruni, C. Germani, and R. Maartens, Phys. Rev. Lett. 87, 231302 (2001).
24. N. Dadhich N and S. G. Ghosh, Phys. Lett. B 518, 1 (2001).
25. N. Dadhich and S. G. Ghost, Phys. Lett. B 538, 233 (2002).
26. R. Casadio and G. Germani, Prog. Theor. Phys. 114, 23 (2005).
27. S Pal, Phys. Rev. D 74 124019 (2006).
28. L. A. Gergely, hep-th/0603254, JCAP 07(02), 027 (2007).
29. L. A. Gergely, gr-qc/0606073, to appear in Int. J. Mod. Phys. D (2006).
GENERALIZED COSMOLOGICAL EQUATIONS FOR
A THICK BRANE
SAMAD KHAKSHOURNIA
Nuclear Science and Technology Research Institute (NSTRI),
Atomic Energy Organization of Iran, Tehran, Iran
skhakshour@aeoi. org. ir
We obtain the generalized cosmological equations for a thick brane immersed in a five-
dimensional Schwarzschild Anti-de Sitter spacetime. It turns out that, at late times,
one can naturally recover the standard cosmological evolution on the core of the thick
brane without the need for splitting the brane energy-momentum tensor into a constant
background part called the brane tension and a time dependent matter contribution.
Particularly our results show that an accelerating brane cosmology emerges at late times
provided there is either a negative transverse pressure component in the brane energy-
momentum tensor or a positive effective cosmological constant.
1. Introduction
Recently we have developed a formalism based on the gluing of a thick wall
considered as a regular manifold to two different manifolds on both sides of it.1 Such
a matching of three different manifolds has envisaged of having many applications
in general relativity and cosmology. It enables one to have any topology and any
spacetime on each side of the thick wall or brane. One may apply it to the dynamics
of galaxy clusters and their halos or to a brane in any spacetime dimension with any
symmetry on each side of it. By construction such a matching is regular and there
is no singular surface in this formulation. Therefore Darmois junction conditions for
the extrinsic curvature tensor on the thick wall boundaries with the two embedding
spacetimes can be applied. Using an expansion scheme in the proper thickness of the
wall we have then been able to obtain an approximate equation of motion for the
thick wall. Our formalism is valid for the wall whose thickness is small compared to
its curvature radius. Very recently we have applied our formalism for a codimension
one brane of finite thickness to study its cosmological evolution. In this note we
give a summary of that work done by the author together with S. Ghassemi and R.
Mansouri.2
We use A for the five-dimensional cosmological constant and k for its
gravitational constant. The core of the thick brane is denoted by So- The symbol |s0 means
"evaluated on the core of the thick brane". For any quantity S let So denote S\s0.
Latin indices range over the intrinsic coordinates of So denoted by £g> and Greek
indices over the coordinates of the 5-manifolds.
2. Thick Brane Cosmological Equations
Our main equation being written on the core of the brane up to the first order of
the brane proper thickness 2w is given by1
2818
2819
Kn
~Kab
So
+w (KacKcb - R^v^aevbrfn^)
(1)
+{KacKcb - R^xe^n17^) -2{KacKcb - R^xe^bn°nA)
So
0,
where the superscripts +,—, and w refer to two slices of the outside spacetime,
and the spacetime within the wall respectively, nM is the normal vector field to
the brane, Kab extrinsic curvature tensor, R^ava the five-dimensional Riemann
curvature tensor and e^ = ^- are the four basis vectors tangent to the brane. The
Schwarzschild Anti-de Sitter bulk spacetime is given by
where f(r) = k
A»,2
ds2
c
-f(r)dT2
dT2
fir)
r2dQ2,
(2)
, the constant C is identified with the mass of a black
hole located at r = 0, and dVi2k is the metric of the 3D hypersurfaces E of constant
curvature that is parameterized by k = 0, ±1. We then take the following ansatz for
the metric of the thick brane
ds2 = -n2{t, y)dt2 + dy2 + a2(t, y)dQ2k. (3)
The energy-momentum tensor of the matter content in the brane is written as
T? = (-p,PL,PL,PL,Pr), (4)
where the energy density p, the longitudinal pressure Pl, and the transverse pressure
Pt are functions of t and y.
Putting all this together, we see that the angular component of the equation (1)
takes the following form2
m
k 8ttG
36
A4
3
C
(5)
where the effective four-dimensional energy density q associated to the five-
dimensional energy density p has been defined as
pdy ~ 2wp0 + 0(w2),
and the following identifications have been made
A w2A2 . „ k2w(-A)
A4
3
6
9
8ttG
(6)
(7)
It is easy to show that in the limit of a vanishing brane thickness Eq. (5) reduces
to the unconventional Friedmann equation of thin brane cosmology.3
Defining
PL =ulq,
pT = coTg,
(8)
2820
with constants u>i and lot- The time component of the equation (1) turns out to
be2
a0 a0 3 \ 2 1-3ujlJ 6 \ 2
6 V 6 ) a0 [at <
,2
It follows the constraints for possibility of accelerating universe at late times are
3wr, Slot „ . — A .„„.
l + ^-Wr+I-^-<0, A4>-. (10)
Particularly in the case of wj, = 0 for dust matter we get
ujt < --. (11)
3. Concluding Remarks
Our main results can be summarized as follows:
(1) The generalized cosmological equation (5) shows a linear in addition to a
quadratic term in the density. Therefore, the late time behavior is the same as the
standard cosmology without introducing an ad hoc brane tension into the energy-
momentum tensor of the brane.
(2) An accelerating brane cosmology may emerge at late times provided there is
either a negative transverse pressure component in the brane energy-momentum
tensor or the effective brane cosmological constant is positive.
References
1. Sh. Khosravi, S. Khakshournia, and R. Mansouri, Class. Quantum Grav. 23, 5927
(2006).
2. S. Ghassemi, S. Khakshournia, and R. Mansouri, JHEP 08, 019 (2006).
3. D. Ida, JHEP 0009, 014 (2000); gr-qc/9912002.
CERENKOV RADIATION FROM COLLISIONS OF STRAIGHT
COSMIC (SUPER)STRINGS
ELENA MELKUMOVA, DMITRI V. GAL'TSOV and KARIM SALEHI
Department of Physics,
Moscow State University, Moscow, 119899, Russia
elenamelk@srd. sinp.msu. ru
We consider Cerenkov radiation which must arise when randomly oriented straight
cosmic (super)strings move with relativistic velocities without intercommutation. String
interactions via dilaton, two-form and gravity (gravity being the dominant force in
the ultra-relativistic regime) leads to formation of superluminal sources which generate
Cerenkov radiation of dilatons and axions. Though the effect is of the second order in the
couplings of strings to these fields, its total efficiency is increased by high dependence of
the radiation rate on the Lorentz-factor of the collision.
1. Introduction
Recently the early universe models involving strings and branes moving in higher-
dimensional space-times received a renewed attention1-.4 In particular, the problem
of the dimensionality of space-time can be explored within the brane gas scenario1-.3
Another new suggestion is the possibility of cosmic superstrings with lower tension
than those in the field-theoretical GUT strings.3 Superstrings as cosmic strings
candidates revive the idea of the defect origin of cosmic structures and stimulate
reconsideration of the cosmic string evolution with account for new features such as
existence of the dilaton and antisymmetric form fields and extra dimensions. The
main role in this evolution is played by radiation processes. The radiation mechanism
which has been mostly studied in the past consists in formation of the excited closed
loops which subsequently loose their excitation energy emitting gravitons5 axions6
and dilatons7-.10
In this paper we consider the bremsstrahlung mechanism of string radiation11
which works for initially unexcited strings undergoing a collision. We develop a
classical perturbation scheme for two endless unexcited long strings which move
one with respect to another in two parallel planes being inclined at an angle.
It was shown earlier that in four space-time dimensions there is no gravitational
bremsstrahlung under collision of straight strings.11 This can be traced to absence
of gravitons in 1+2 gravity. It is not a coincidence that in four dimensions there
is no gravitational renormalization of the string tension either.13 But there is no
such dimensional argument in the case of the axion field there such dimensional
argument and it was demonstrated that string bremsstrahlung takes place indeed12
within the model in flat space. Here we extend this result to the full gravitating case
including also the dilaton field. Strings interacts via the dilaton, axion and graviton
exchange. Radiation arises in the second order approximation in the coupling
constants provided the (projected) intersection point moves with superluminal velocity.
2821
2822
Thus, the string bremsstrahlung can be viewed as manifestation of the Cherenkov
effect.
2. String interactions
Consider a pair of relativistic strings x^ = x^(cr^), /i = 0,l,2, 3, aa = (t, a), a =
0,1, where n = 1, 2 is the index labelling the two strings. The 4-dimensional space-
time metric signature +, and (+, —) for the string world-sheets metric
signature. Strings interact via the gravitational g^v = rj^ + h^, dilatonic (j>(x) and
axion (Kalb-Ramond) field B^(x):
S=-V / ^daxZdbx»ng^1ab^exp2a^+2irfdax%dbx»eabB^}d2a
^ / l2l
+ J fa^gT + \H»„pH^e-^ - ^|
-gd4x. (1)
Here \xn are the (bare) string tension parameters, a and / are the corresponding
coupling parameters, e01 = 1, jab is the induced metric on the world-sheets. In
what follows, we linearize the dilaton exponent as e2a* ~ 1 + 2acj>.
The totally antisymmetric axion field strength is defined as H^\ = d^Bv\ +
dvBx^ + dxB^. Variation of the action (1) over x^ leads to the equations of motion
for strings
da (/iS6.<ff^7a6v^e2Q* + 4irfdbx»eabB^)
-I
Variation with respect to field variables </>, Bpl/ and gpl, leads to the dilaton equation:
-MaM&4ffa/J7°6\/=7e2a^ - ^dax^dbx^abV^ie2^dfigap = 0. (2)
9,
(g^d^V^g) + ^H2e~4^ + ^J dax^dbx^gtlulabe2aH\x - xn(<jn))d2a = 0,
(3)
the axion equation:
d^ (H^xe-4a*^) + 2irf f dax»ndbxxneal'54(x - xn(an))d2a = 0 (4)
<j> B st
and the Einstein equations: R^ — ^g^R = 8ttG(T^+ T^+ TfW),
7> = 4 (d^d^ - y^(\70)2) , T^ = {H^pHfP - \E2g^) e'4^.
Our calculation follows the approach of11-12 and consists in constructing
solutions of the string equations of motion and dilaton, axion and graviton iteratively
using the coupling constants a, /, G as expansion parameters.
The total dilaton, axion and graviton fields are the sums due to contributions
of two strings: <j> = <j>i + <fo, B^ = B{v + B%v, h^v = h^v + h%v. Since in the
i l
zero order the strings are moving freely , the first order dilaton </> , axion B ^ and
2823
i
graviton variables hn" do not contain radiative components. Substituting them into
the Eq. (2) we then obtain the first order deformations of the world-sheets x^, which
are naturally split into contributions due to dilaton, axion and graviton exchange:
2 2
Radiation arises in the second order field terms (j>n and B^u which are generated
by the first order currents J(<i>),j{J'g) in the dilaton and axion field equations ((3),(4)).
Note that gravitational radiation in four dimensions is absent,11 so we do not
consider the second order graviton equation. The dilaton and axion radiation power
can be computed as the reaction work given by the half sum of the retarded and
advanced fields upon the sources.12 The final formula for the dilaton and axion
bremsstrahlung from the collision of two global strings can be obtained analytically
in the case of the ultrarelativistic collision with the Lorentz factor 7 = (1— v2)~1/2 S>
1. We assume the BPS condition for the coupling constants13 a\i = 2\/2nf. The
main contribution to radiation turns out to come from the graviton exchange terms.
The spectrum has an infrared divergence due to the logarithmic dependence of the
string interaction potential on distance, so a cutoff length A has to be introduced:
PW = ^gVm^/^ + I/^)), p(*> = l^G2^LfK\l{y)-Hv)),
(5)
where L-length of the string, y = —^, k = 7 cos a, a is the strings inclination
angle, d is the impact parameter and f(y) = 12^/^2-^2 (|, \\ §, §; —y) - 3In (4yec) .
h (y) = erfc(^) (|y3 - 30y2 + 114y + ±f) - ^ (|y2 _ SAy + 131) , h{y) =
erfc(x/y) (|y3 + 6y2 - 6y - |) - "—^- (|y2 + fy - 7) , F is the generalized hy-
pergeometric function and C is the Euler's constant.
This work was supported in part by RFBR grant 02-04-16949.
References
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HIGH-ENERGY EFFECTS ON THE SPECTRA OF
COSMOLOGICAL PERTURBATIONS
IN BRANEWORLD COSMOLOGY
TAKASHI HIRAMATSU1, KAZUYA KOYAMA2 and ATSUSHI TARUYA1
1 Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
2 Institute of Cosmology and Gravitation, University of Portsmouth,
Portsmouth POl 2EG, United Kingdom
We study the evolution of scalar curvature perturbations in a braneworld inflation model
in a 5D Anti-de Sitter spacetime. The inflaton perturbations are confined to a 4D brane
but they are coupled to the 5D bulk metric perturbations. We numerically solve full
coupled equations for the inflaton perturbations and the 5D metric perturbations
using Hawkins-Lidsey inflationary model. At an initial time, we assume that the bulk is
unperturbed, while the inflaton field is perturbed. We find that the inflaton
perturbations at high energies are strongly coupled to the bulk metric perturbations even on
sub-horizon scales, leading to the suppression of the amplitude of the comoving
curvature perturbations at a horizon crossing. This indicates that the linear perturbations of
the inflaton field does not obey the usual 4D (linearised) Klein-Gordon equation due to
the coupling to 5D gravitational field on small scales and it is required to quantise the
coupled brane-bulk system in a consistent way in order to calculate the spectrum of the
scalar perturbations in a braneworld inflation.
1. Introduction
We consider a 4D inflaton field confined to the brane in the Randall-Sundrum
single brane model [1,2]. This paper focuses on the classical evolution of inflaton
perturbations coupled to bulk metric perturbations in order to study the effects of
bulk metric perturbations has been initiated in Ref. [3,4]. We assume an inflaton
potential proposed by Hawkins and Lidsey, which realises a power-law inflation on
the brane (ao(t) ~ tl'G ) [5], and take into account the backreaction of inflaton
dynamics consistently. This entails a numerical analysis to solve the coupled system
directly. We investigate whether the inflaton perturbations behave as free massless
fields on small scales.
The basic equations can be found in Ref. [3,4,6]. The perturbed metric in the
Gaussian-normal coordinate with the 5D-longitudinal gauge is given by
ds2 = -n2(l + 2A)dt2 + a2(l + 2K)8ijdxidxj + (1 + 2Ayy)dy2 + nAydydt. (1)
The metric perturbations can be derived from a master variable Q as long as the
master variable fl in the bulk satisfies a wave equation given by [7]
-GH'+0')'+("2+5)?n=»- <2»
where a prime and a dot denote the derivatives with respect to y and t, respectively.
The equation of motion for the inflaton perturbations, 6(j>, confined to the brane
is derived from the conservation law V^rfT^j, = 0. In the present study, we introduce
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a gauge-invariant variable, the Mukhanov-Sasaki variable[8], defined as Q = 8<p —
(4>/H)1Zb, which satisfies
Q + 3HQ + ^Q+\§-2§^--2(§) +v"(<t>)\Q = JM, (3)
where J(f2) represents the contribution from the bulk metric perturbations. In fact,
J vanishes in the 4D limit.
The junction condition imposed on the perturbations is derived from the effective
Einstein equation [9]. We can relate Q and Q as
•Mf")'-^^-™)^"^)),- <4)
which gives a non-local boundary condition for Q.
We solve numerically the evolution equations for Q (2) and Q (3) with the
junction condition (4). The aim of this paper is to check whether we can neglect
J(Q) on small scales or not. If this term could be neglected on the small scales, the
quantity aoQ would behave just as plane waves with a constant amplitude as in the
standard 4D cases. For this purpose, we take the simplest possible initial conditions
for £l(y,t) and Q(t) : £l(y,ti) = 0, Cl(y,ti) = 0, Q(U) — 1, and Q(U) is determined
so that they are consistent with the junction condition (4).
2. Evolution of curvature perturbations
The observable is the comoving curvature perturbation defined by TZC = —(H/(f>)Q.
We focus on the dynamics of this quantity. First we have confirmed that the
curvature perturbations 1ZC becomes constant on super-horizon scales, which has been
shown to be valid even in brane world models [10]. On sub-horizon scales, while
1ZC oc 1/ao in the 4D cosmology, a suppression of the amplitude is observed in the
braneworld model, which is due to the coupling to the bulk metric perturbations.
In Fig. 1, we compare the amplitude of 7ZC obtained in numerical simulations with
the one obtained by neglecting the coupling to the bulk metric perturbations, i.e.
J(Q) = 0 in Eq. (3). While the difference is very small for the long-wavelength
modes (left panel; k/ao(ti)(j, = 296), the suppression becomes significant for the
short-wavelength modes (right panel; k/ao(U)fj, = 2960). Due to this, the spectrum
of 1ZC just before the horizon crossing acquires a scale dependence as is shown in
Fig. 1. Perturbations with larger wavenumber stay on sub-horizon scales for a longer
time than those with smaller wavenumber, so they receive more suppression. The
suppression of the amplitude 7ZC under horizon may be understood as the excitation
of metric perturbations. The suppression of the amplitude of Q is transferred into
the enhancement of the metric perturbations Q in the bulk.
This extra suppression on small scales shown here means that it is impossible
to neglect a coupling to gravity even on small scales due to the coupling to the
higher-dimensional gravity through J(Q). The suppression of the amplitude may be
fw- vlfw*\>/
understood as a loss of energy due to the excitation of the bulk metric perturbations
as we took the initial condition that n(y,t,) = 0 and tl(y,ti) = 0. On super-horizon
scales, the curvature perturbations become constant, which confirms the fact that
the constancy of the curvature perturbations is independent of gravitational theory
for adiabatic perturbations [10].
k/(S(j(ti)\i=296
41.5 42 ;
M'
k/adt^
Fig. t. The curvature perturbations (multiplied by the scale factor) in the inflationary epoch for
a long-wavelength mode (left) and for a short-wavelength mode (centre). The solid lines represent
numerical results and the dashed lines show the 4D predictions obtained by neglecting the coupling
to the bulk metric perturbations, that is, by solving Eq. (3) with J(il) = 0. We set C = 0.1 and
fiti = 40. rigid : The scale dependence of the curvature perturbations evaluated just before the
horizon crossing. The horizontal axis represents the physical scale of perturbations evaluated at
the initial time fjii = 40. We estimated the ratio YRj^P/TZfP] for each wave number where 7?.^D is
the curvature perturbation in the brane world model and TZfP is the one in a 4D model with the
identical background dynamics.
Our result suggests that an usual assumption that the iuflaton perturbations
(the Mnkhanov-Sasaki variable) approach to a free massless field on small scales
cannot be applied in a brane world models on small scales at high energies. For the
detail discussions about this point, see Ref. [6]. Furthermore, a quantum mechanical
analysis for the present issue can be seen in our recent paper [11].
References
1. L. Randall and R, Sundrum, 1999 Phys. Rev. Lett. 83 4690,
2. R. Maartens, D. Wands, B. A. Bassett and T. Heard, 2000 Phys. Ran. D 82 041301.
3. K, Koyama, D. Langlois, R. Maartens and D. Wands, 2004 J CAP 0411 002.
4. K. Koyama, S. Mizuno and D. Wands, 2005 JCAP 0508 009.
5. R. M. Hawkins and J. E. Lidsey, 2001 Phys. Rev. D 83 041301.
6. T. Hiramatsu and K. Koyama, 2006 JCAP 0612 009.
7. S. Mukohyama, 2000 Phys. Rev. D 82 084015.
8. M. Sasaki, 1986 Prog. Theor. Phys. 78 1036; V. F. Muklianov, 1988 Zh. Eksp. Tear.
Fiz. 94N7 1
9. T. Shironrizu, K. i. Maeda and M. Sasaki, 2000 Phys. Rev. D 82 024012.
10. D. Wands, K. A. Malik, D. H. Lyth and A. R. Liddle, 2000 Phys. Rev. D 82 043527.
11. K. Koyama, A. Mennim, V. A. Rubakov, D. Wands and T. Hiramatsu, 2007 JCAP in
press [arXiv:hep-th/0701241].
BRANEWORLDS AND QUANTUM STATES
OF RELATIVISTIC SHELLS
S. ANSOLDI*
International Center for Relativistic Astrophysics (ICRA), Italy, and
Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Trieste, Italy, and
Dipartimento di Matematica e Informatica, Universita degli Studi di Udine,
via delle Scienze 206, 1-33100 Udine (UD), Italy [Mailing address]
*ansoldi@trieste.infn.it — Web-page: http://www-dft.ts.infn.it/^ansoldi
We review some applications of relativistic shells that are relevant in the context of
quantum gravity/quantum cosmology. Using a recently developed approach, the stationary
states of this general relativistic system can be determined in the semiclassical
approximation. We suggest that this technique might be of phenomenological relevance in the
context of the brane-world scenario and we draw a picture of the general set-up and of
the possible developments.
Keywords: Brane World Scenario; Bohr-Sommerfeld States; Cosmology; General
Relativistic Shells; Junction conditions; WKB Quantization.
Let us consider two (N+l)-dimensional domains of spacetime, (N+1'>M±, which are
parts of two solutions of Einstein equations; let ^N^T,± be isometric parts of their
boundaries. Then ^N^E± can be identified, and f>N+1'>M± can be joined across we.
If (w)£ is also equipped with some matter-energy content and if it is a timelike
hypersurface in (N+^M = (JV+1lM_ U (W)E U (-N+1^M + , then it describes the
evolution of this matter/energy. (N>Y, is traditionally known as a general relativistic shell
or a co-dimension one brane. General relativistic shells have been often used as a
framework for astrophysical and cosmological models (for an extensive bibliography
see Ref. 1). A good reason for this success is certainly the geometrically flavored
description provided by Israel junction conditions,2 thanks to which the classical
dynamics of the system is under control. If we call WKffi (n,v= l,...,N + l) the
extrinsic curvature of (W'E with respect to its embeddings in (N+1^M± and ^S^v
the stress-energy tensor describing the matter-energy content of (iv'E, Israel
junction conditions are, in suitable units,
W4+)_(^(-)=87rM^, MF = (%-(\„(%/2, (1)
where ^g^v is the metric on ^'E and ^S is the trace of ^S^. Soon after the
earliest classical applications of shells, a number of works discussed their semiclassical
quantization (see again Ref. 1 for additional references). Most of them had the goal
to investigate situations where the emergence of singularities was breaking down the
predictive power of general relativity as in the cosmology of the early universe (with
the initial singularity problem) and in gravitational collapse (with its, also singular,
final fate). In the first case we would like to explicitly remember the paper of Far hi
et al.,3 which showed how useful the idea of shell tunnelling can be raising some
interesting (still open) issues.4 About the second aspect, we remember the early works
of Berezin5 and Visser6 (additional bibliography can be found in Ref. 1). In what
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follows we will elaborate on the case in whicha the metrics in (N+1^A4± can be cast in
the form (N+1)ds2± = -h±(a±; {&±})dt2±+da2±/h±(a±; {^±}) + ^N-1UQ2±({^±})a2±
in the coordinates (t±,a±,(... )±), where "(. ..)±" is a set of coordinates for the
maximally symmetric spaces of metric (N~1'd£i±({'&±}). In this setup the junction
conditions (1) can be reduced to just one equation
6+^jA2 + h+(A; {S?+}) - e.^A2 + h_(A; {S?_}) = M(A; {£}), (2)
where A(r) is the value of a± at the brane location as a function of the proper time r
of on observer living on the brane, an overdot denotes a derivative with respect to r
and e± are the signs of the radicals. M(A; {£}) encodes the properties of the matter-
energy source living on the shell. Studies to develop the fully covariant Hamiltonian
formulation started also early7 (see again Ref. 1 for later ones) and represent the
proper framework to correctly interpret effective Lagrangian/Hamiltonian
formulations on which, for simplicity, we will concentrate. Indeed Eq. (2) can be obtained
from an effective, dimensionally reduced Lagrangian/Hamiltonian as a first integral
of the Euler-Lagrange/Hamilton equations. From this Lagrangian/Hamiltonian,
following for instance Ref. 8, the effective momentum P(A,A; {&}, {<%}) conjugated
to A can also be determined. Moreover, from (2), it is possible to solve for A and
substitute this result into P(A,A; {&}, {£}) to obtain P(A; {&}, {£}), i.e. an
expression for the momentum evaluated on a solution of the classical equations of
motion. If the system admits bounded solutions, so that classically A oscillates between
j4min({^'}, {<^}) and Amax({@}, {<o}), we can then evaluate (sometimes analytically
but, otherwise, at least numerically) the value of the action on a classical solution
S(m,{*}) = 2 P{A;{<3},{g})dA. (3)
JAmi„(m,{<?})
When the action 5({Sf}, {<?}) is of the order of the quantum the gravitational system
is in a quantum regime and the Bohr-Sommerfeld quantization condition
S({&},{£})~nh, n=l,2,..., (4)
defines the semidassical states of the system. In this case (4) is a constraint: not
all combinations of values of the parameters are allowed. Let us now further
specialize our discussion to TV = 4 and discuss Robertson-Walker cosmologies in five-
dimensional Schwarzschild anti-de Sitter spacetime9 in the spirit of the Randall-
Sundrum scenario.10 Then h+ (a; {S?+}) = /i_(a;{S?_}) = h(a; {k,l,m}) = k+l2a2 +
2m/a2 and e+ = —e_ = +1; we also choose the coordinates in the maximally
symmetric space as (...)± = (x±,0±,(f>±). Then W<m2±({&±}) = ^dn2±(k) = dX2± +
f2k(X±)(de2±+Sm2e±d<p2±), where fk(y) = (exp(fc1/2|/)-exp(-fc1/2|/))/(2fci/2|/) and
aWe will use the notation {CS±\ to collectively indicate the dependence from the geometry-related
parameters of the model (for example, the Schwarzschild mass, the cosmological constant and so
on) as well as the notation {£} to denote the dependence from the parameters defining the brane
matter-energy content (for example, the surface tension and so on). Later we will also use, with
similar meaning, the shorthand {&} according to the following definition: {^} = {#+} U {#—}.
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k = —1,0, +1 determines if the maximally symmetric space is a 3-sphere, §3, a 3-
Euclidean space, E3, or 3-Hyperbolic space, H3, respectively. We now observe that
in the model we are discussing, M(R; {<o}) contains most of the relevant physical
information, since it describes the matter-energy content of the brane, i.e. of our
universe. It is, then, interesting to choose the set of parameters {£} to describe
the matter component of the universe pm, the radiation component pr, the
cosmological constant pa the dark energy p? and so on; thus {£} = {pm, pr, pa, P?},
whereas {^} = {k, I, m} is fixed by the bulk spacetime structure. We then see that,
already in the very simple and natural semiclassical approach discussed above (of
which the toy model in Ref. 11 is a preliminary test), the quantization condition
S({k,l,m}, {pm,Pr,PA,P?}) ~ nh, n = 1,2,..., provides a constraint among the
cosmological parameters. Phenomenological implications and further refinements of
this approach are currently under investigation and will be reported elsewhere.12
Acknowledgements
I would like to thank Mr. Bernardino Cresseri, Prof. Gianrossano Giannini and
Mr. Enrico Ramot for some administrative and practical arrangements which made
possible my participation to the MG11 meeting. I would also like to gratefully
acknowledge financial support from ICRA (International Center for Relativistic
Astrophysics) and INFN (Istituto Nazionale di Fisica Nucleare, Sezione di Trieste).
References
1. S. Ansoldi, Class. Quantum Gray. 19, 6321 (2002), gr-qc/0310004.
2. W. Israel, Nuovo Cimento B 44, 1 (1966) [Erratum-ibid. 48, 463 (1967)]; C. Barrabes
and W. Israel, Phys. Rev. D 43, 1129 (1991).
3. E. Farhi, A. H. Guth and J. Guven, Nucl. Phys. B 339, 417 (1990).
4. A. Aguirre and M. C. Johnson, Phys. Rev. D 72, 103525 (2005), gr-qc/0508093; ibid.
73, 123529 (2006), gr-qc/0512034; S. Ansoldi, "Gravitational tunnelling of
relativistic shells", in Frontiers of Fundamental and Computational Physics, Springer (2005),
gr-qc/0411042; S. Ansoldi, "Bubbles and Quantum Tunnelling in Inflationary
Cosmology", to appear in the proceedings of the 16th Workshop on General Relativity and
Gravitation (JGRG16), Niigata, November 27th-December 1st, 2006.
5. V. A. Berezin, Phys. Lett. B 241, 194 (1990).
6. M. Visser, Phys. Rev. D 43, 402 (1991).
7. P. Hajicek and J. Bicak, Phys. Rev. D 56, 4706 (1997), gr-qc/9706022; P. Hajicek
and J. Kijowski, Phys. Rev. D 57, 914 (1998) [Erratum-ibid. 61, 129901 (2000)], gr-
qc/9707020; S. Mukohyama, Phys. Rev. D 65, 024028 (2002), gr-qc/0108048.
8. S. Ansoldi, A. Aurilia, R. Balbinot and E. Spallucci, Class. Quantum Grav. 14, 2727
(1997), gr-qc/9706081.
9. D. Ida, JHEP 014, 009 (2000), gr-qc/9912002.
10. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999), hep-ph/9905221; ibid.
83, 4690 (1999), hep-th/9906064.
11. S. Ansoldi, AIP Conf. Proc. 751, 159 (2005), gr-qc/0410080.
12. S. Ansoldi, E. I. Guendelman and H. Ishihara, Semiclassical States in Brane
Cosmology, in preparation.
ROTATING BRANEWORLD BLACK HOLES
ALIKRAM N. ALIEV
Feza Giirsey Institute, P.M. 6 Cengelkoy, 34684 Istanbul, Turkey
aliev@gursey.gov.tr
We present a Kerr-Newman type stationary and axisynimetric solution that describes
rotating black holes with a tidal charge in the Randall-Sundrum braneworld. The tidal
charge appears as an imprint of nonlocal gravitational effects from the bulk space. We
also discuss the physical properties of these black holes and their possible astrophysical
appearance.
1. Introduction
The braneworld idea is a revolutionary idea to relate the properties of higher
dimensional gravity to the observable world by direct probing of TeV-size mini black boles
at high energy colliders . According to this idea our observable Universe is a slice,
a "3-brane" in higher dimensional space.1'2 This in particular gives: (i) An elegant
geometric resolution of the hierarchy problem between the electroweak scale and
the fundamental scale of quantum gravity, (ii) the large size of the extra dimensions
supports the weakness of Newtonian gravity on the brane and makes it possible to
lower the scale of quantum gravity down to the electroweak interaction scale, (iii)
the braneworld model (RS2 model) also supports the properties of four-dimensional
Einstein gravity in low energy limit. In light of all this, it is natural to assume the
formation of black hole in the braneworld due to gravitational collapse of matter
trapped on the brane.
Several strategies have been discussed in the literature to describe the
braneworld black holes. First of all, it has been argued that if the radius of the
horizon of a black hole on the brane is much smaller than the size of the extra
dimensions (r+ <C L), the black hole, to a good enough approximation, can be
described by the usual classical solutions of higher dimensional vacuum Einstein
equations. In the opposite limit when (r+ 3> L), the black hole becomes effectively
four-dimensional with a finite extension along the extra dimensions. The first simple
solution pertinent to the latter case is based on the idea of a usual Schwarzschild
metric on the brane that would look like a Mack string solution from the point
of view of an observer in the bulk.3 However, the black string solution exhibits
curvature singularities at infinite extension along the extra dimension.
We shall discuss another strategy namely, we shall specify the metric form
induced on the 3-brane assuming a Kerr-Schild ansatz for it. With this ansatz the
system of the effective gravitational field equations on the brane4,5 becomes closed
and the solution to this system turns out to be a Kerr-Newman type stationary
axisynimetric black hole which possesses a tidal charge instead of a usual electric
charge.
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2. The metric form on the 3-brane
To describe a rotating black hole in the Randall-Sundrum scenario we shall make a
particular assumption about metric on the brane, taking it to be of the Kerr-Schild
form
ds2=(ds2)flat+H(kdxl)\ (1)
where H is an arbitrary scalar function and U is a null, geodesic vector field in
both the flat and full metrics. Earlier,6 this type of strategy was emoloyed for a
static black hole localized on the brane. With the metric form (1) the effective
gravitational field equations on the brane
Rij = —Eij , (2)
where E^ the traceless "electric part" of the five-dimensional Weyl tensor, and the
associated constraint equation
R = 0, (3)
admit the solution which in the usual Boyer-Lindquist coordinates takes the form7
ds2 = - (l - ™1^) dt2 _ 2a(2Mr-P) ^ ^
+1 dr2 + E d92 + (r2 + a2+ 2-MlzA fl2 sin2 &\ sin2 Q d(j)2 ) (4)
where
A = r2 + a2 - 2Mr + (3 , S = r2 + a2 cos2 9 . (5)
We see that that this metric looks exactly like the Kerr-Newnian solution in general
relativity, in which the square of the electric charge is " superceded" by a tidal charge
parameter (3. The Coulomb-type nature of the tidal charge is verified by calculating
the components of the tensor E^ through equation (2). Therefore one can think
of it as carrying the imprints of nonlocal gravitational effects from the bulk space.
Furthermore, the tidal charge may take on both positive and negative values.
3. Major Features
In complete analogy to the Kerr-Newman solution in general relativity, the metric
(4) possesses two major features: The event horizon structure and the existence of a
static limit surface, the ergosphere. The event horizon is a null surface determined
by the largest root of the equation A = 0. We have
r+ = M + y/M2 - a2 - (3 (6)
The horizon structure depends on the sign of the tidal charge. The event horizon
does exist provided that
M2 >a2+p. (7)
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Thus, for the positive tidal charge we have the same horizon structure as the usual
Kerr-Newman solution. New interesting features arise when the tidal charge is taken
to be negative. For (3 < 0 from equation (6) it follows that the horizon radius
r+ -> (M + y/^p) > M (8)
as a —> M. This is not allowed in the framework of general relativity. From
equations (6) and (7) it follows that for /3 < 0, the extreme horizon r+ = M corresponds
to a black hole with rotation parameter a greater than its mass M . Thus, the bulk
effects on the brane may provide a mechanism for spinning up the black hole so that
its rotation parameter exceeds its mass. Meanwhile, such a mechanism is impossible
in general relativity.
The static limit surface is determined by the equation gtt = 0, the largest root
of which gives the radius of the ergosphere
r0 = M + yjM2 - a2 cos2 9 - (3 . (9)
Clearly, this surface lies outside the event horizon coinciding with it only at angles
0 = 0 and 9 = ir. The negative tidal charge tends to extend the radius of the
ergosphere around the braneworld black hole, while the positive (3 just as the
usual electric charge in the Kerr-Newman solution, plays the opposite role. For the
extreme case, we find the radius of the ergosphere within
M <r <M + sin 9 yjM2 - (3 . (10)
We see that in astrophysical situations, the rotating braneworld black holes with
negative tidal charge are more energetic objects in the sense of the extraction of the
rotational energy from their ergosphere.
References
1. N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B 429, 263 (1998).
2. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999).
3. A. Chamblin, S. W. Hawking and H. S. Reall, Phys. Rev. D 61 065007 (2000).
4. T. Shiromizu, K. Maeda, and M. Sasaki, Phys. Rev. D 62, 024012 (2000).
5. A. N. Aliev and A. E. Gumrukcuoglu, Class. Quant. Grav. 21, 5081 (2004).
6. N. Dadhich et. al., Phys. Lett. B 487, 1 (2000).
7. A. N. Aliev and A. E. Gumrukcuoglu, Phys. Rev. D 71, 104027 (2005).
GENERAL SOLUTION FOR SCALAR PERTURBATIONS IN
BOUNCING COSMOLOGIES
VALERIO BOZZA
Dipartimento di Fisica "E.R. Caianiello", Universitd di Salerno,
Via S.Allende, 1-84081, Baronissi (SA), Italy
and
Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Italy
valboz@sa.infn.it
Bouncing cosmologies, suggested by String/M-theory, may provide an alternative to
standard inflation to account for the origin of inhomogeneities in our universe. The
fundamental question regards the correct way to evolve the scalar perturbations through the
bounce. In this talk, we present the evolution of perturbations and the final spectrum
for an arbitrary (spatially flat) bouncing cosmology, with the only assumption that the
bounce is governed by a single physical scale. In particular, we find the condition for the
pre-bounce growing mode of the Bardeen potential (which is scale-invariant in the Ekpy-
rotic scenario) to survive unaltered in the post-bounce. If some new physics acting at
the bounce satisfies such a condition, then bouncing cosmologies are entitled to become
a real viable alternative to inflation for the generation of the observed inhomogeneities.
1. Introduction
Several theories of quantum gravity suggest that the initial big bang singularity
may be cured by some high energy cut-off, be it the Planck scale, the string scale or
anything else. In such scenarios, the big bang is preceded by a contraction phase, in
which the spacetime curvature grows up to the cutoff value. At this stage high energy
physics comes into play and drives the universe towards the standard decelerated
expansion.
A contraction phase in the early universe may solve the horizon and flatness
problems as efficiently as standard inflation. If it were possible to justify the observed
primordial spectrum of cosmological perturbations, then the so-called bouncing
cosmologies would become a serious alternative to standard inflation. In this spirit, the
string-inspired Ekpyrotic model has proposed that quantum fluctuations during a
very slow contraction before the bounce would generate the correct scale-invariant
spectrum for scalar perturbations. This statement has been criticized by many
authors, while investigations of specific toy models in which perturbations are
explicitly calculable analytically or numerically have provided conflicting results, with no
definite conclusions.
While the fact that during the pre-bounce the Bardeen potential grows with a
scale-invariant spectrum is universally accepted, there are two alternatives in the
post-bounce: either the scale-invariant spectrum is transmitted to a constant mode,
or it is present just in a decaying mode that becomes subdominant with respect to
some constant mode with a blue spectrum (Fig. 1).
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Log a
\s%
/10
/ 7.5
/ 5
/ 2.5
\Ps=1
\ns=5
-4 -2
2 4 6
Log a
Fig. 1. The two alternatives for the Bardeen potential evolution in a bouncing cosmology.
2. General solutions for scalar perturbations in bouncing
cosmologies
In a general investigation of regular bounces with two sources we have shown that
the Bardeen potential growing mode is always entirely converted into a decaying
mode in the post-bounce, clarifying some puzzles emerged in previous studies.1
Afterwards, we have looked for a general solution for the evolution of scalar
perturbations through a cosmological bounce,2 retaining a minimal number of
assumptions:
i) It makes sense to define a 4-dimensional metric tensor at all times. Then we
can always write effective Einstein equations as G^ =TIJV.
ii) The universe is homogeneous and isotropic; thus the background metric is FRW.
iii) The bounce is entirely determined by a unique physical scale.
iv) Before and after the bounce, the universe is characterized by constant w and
c2, with w > —1/3 (no inflation, no deflation).
Assumption (i) allows us to use the Einstein equations throughout the
cosmological evolution, provided that all corrections coming from high energy physics, which
become important during the bounce phase, are encoded in the effective energy-
momentum tensor on the right hand side. Assumption (iv) simply states that the
evolution before and after the bounce is dictated by ordinary matter sources,
excluding inflationary stages, which would spoil the purpose of our investigation.
Perturbing Einstein equations, we can write appropriate first order evolution
equations for the perturbations. As gauge-invariant variables, we choose the Bardeen
potential $, the curvature perturbation on comoving slices £, the energy density and
the pressure on comoving slices 6pv, 5pv and the anisotropic stress £.
5pv is constrained to be proportional to V2$ by the Hamiltonian constraint.
The other equations can be recast in a set of two independent first order equations
for $ and £ with 5pv and £ as sources.
We can put these equations in the form of two integral equations, whose solution
can be formally written as a recursive series. Luckily, since we are interested into
modes that are outside the horizon at the bounce, we can truncate the series to the
first three terms.
2835
In order to close the system, the sources must be expressed as functions of $
and £• In general, we can say that they are linear combinations of the two variables
with operator coefficients, which can be expanded in powers of V2.
At the end, we can write the solution for the pre-bounce and the post-bounce
phase, taking advantage of the fact that any integral of any function covering the
bounce phase can only contain two physical scales: the wave number k and the
bounce scale r\B- Assumption (iii) says that these are the only scales governing the
bounce and thus the integrals can be estimated by simple dimensional arguments.
The solution for the pre-bounce can be matched to the asymptotic vacuum
fluctuations, determining the initial spectrum. The post-bounce solution for the
Bardeen potential contains four modes: a decaying mode (endowed with a scale-
invariant spectrum in the limit of very slow pre-bounce contraction), two blue
constant modes and an additional constant mode with the same spectrum as the
decaying mode. This additional mode is present only if 5pv oc $ rather than V2$.
3. Discussion
Perfect fluids and scalar fields have 5pv oc V2$ and this explains why in bounces
based on these sources there is no constant mode carrying the original Bardeen
potential spectrum. Spatial curvature has 5pv oc <&, but the need to get rid of the
curvature by some accelerated expansion in the post-bounce make bounces with
spatial curvature uninteresting. On the other hand, the transfer condition is fulfilled
by models of bouncing cosmologies with extra-dimensions. This leaves the possibility
open that very slow contraction may represent a real and complete alternative to
standard inflation.
References
1. V. Bozza and G. Veneziano, Phys. Lett. B625, 177 (2005); JCAP 0509, 007 (2005).
2. V. Bozza JCAP 0602, 009 (2006).
CONSTRAINTS ON ACCELERATING BRANE COSMOLOGY
WITH EXCHANGE BETWEEN THE BULK AND BRANE*
GRANT J. MATHEWS
University of Notre Dame, Center for Astrophysics, Notre Dame, IN 46556 USA
gmathews @nd. edu
K. UMEZU,1 T. KAJINO1-2
1 National Astronomical Observatory of Japan, and Graduate University for Advanced Studies,
2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
2Department of Astronomy, Graduate School of Science, University of Tokyo, 7-3-1 Hongo,
Bunkyo-ku, Tokyo 113-0033, Japan
K. ICHIKI
Research Center for the Early Unverse, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo
113-0033, Japan
R. NAKAMURA, M. YAHIRO
Department of Physics, Graduate School of Science, Kyushu University, 6-10-1 Hakozaki,
Higashi-ku, Fukuoka 812-8581, Japan
We have analyzed the observational constraints on a brane-world cosmology in which
the exchange of mass-energy between the bulk and the bane is allowed. We have shown
that it is possible to have a A = 0 cosmology for an observer on the brane which
satisfies standard cosmological constraints including Type la supernovae at high redshift,
the CMB temperature fluctuations, and the matter power spectrum. This model even
accounts for the observed suppression of the CMB power spectrum at low multipoles.
In this paradigm, the cosmic acceleration is attributable to the flow of matter from the
bulk to the brane. An interesting observational consequence of this cosmology is that
the present dark-matter content is significantly larger than that of a standard ACDM
cosmology.
We have analyzed1 a mechanism by which the observed cosmic acceleration can
be produced without the need to invoke dark energy and its associated complexities.
Specifically, the cosmic acceleration is driven2-7 by the flow of dark matter from a
higher dimension (the bulk) into our three-space (the brane).
A thin three-brane embedded in an AdS§ space is a practical model8 for higher
dimensional physics. It has been shown9 that the quasi-normal modes of massive
particles on the brane could be metastable to decay into the bulk dimension. We have
* Work supported in part by the US Department of Energy under Nuclear Theory grant DE-FG02-
95ER40934. N.Q.L. also supported in part by NSF grant PHY 02-16783 for the Joint Institute for
Nuclear Astrophysics (JINA). Work at Lawrence Livermore National Laboratory performed under
the auspices of the U.S. Department of Energy under under contract W-7405-ENG-48 and NSF
grant PHY-9401636. Work supported in part by the Mitsubishi Foundation, the Grants-in-Aid for
Scientific Research (13640313, 14540271) and for Specially Promoted Research (13002001) of the
Ministry of Education, Science, Sports and Culture of Japan. K.I.'s work has been supported by
a Grant-in-Aid for JSPS fellows.
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tested1 models with mass-energy flow from the bulk to the brane and inversely10
by comparing to the observations of Type la supernovae at high redshift, the
temperature fluctuation spectrum of the cosmic microwave background (CMB), and
the matter power spectrum. All of these constraints can be satisfied in this model
even without introducing a cosmological constant on the brane. In fact, this model
provides a natural explanation for the a suppression of the CMB power spectrum
for the lowest multipoles.
The essential physics is that we decompose the dark-matter energy-
momentum tensor, (£>M)j1^; jnto the usual three-density p and pressure p
of dark matter on the brane, plus the bulk components (dm-bulk)^^ _
5(y)d\ag(-p,p,p,p,Q) +(dm-bulk) TAB) wher6; (dm-bulk)T05 „ (p + p)[/5;
represents the matter-energy flow from the bulk to the brane, while (dm-bulk)j<5^ _
(p + p)U5U5 + p, represents a bulk pressure in the limit of vanishing U5. We also
parametrize the EOS for matter in the bulk: p <x (pcr/aq) , where pcr is the present
critical density, and a(t) is the scale factor on the brane, with q = 3(1 + w), where
w = p/p. For the five velocity of matter in the bulk we write , U5 oc —IH, where
/ = [—6M3 / A5]1/2, is the bulk curvature radius.10 We also consider a model with
constant C/5. We thus parameterize the 0-5 component of the bulk dark-matter
energy-momentum tensor as (dm-bulk)j<o^ _ (a/2)(pcr/aq)lH. For the case of
constant U5 we replace H with the present Hubble parameter.
The cosmological equations of motion with brane-bulk energy exchange have
been formulated in Refs.1"7 We have compared various cosmological models with
this modified expansion with the SNIa data.11 Our best fit A = 0 growing cold dark
matter (GCDM) models are nearly indistinguishable from the best fit Standard
A+cold dark matter (SACDM) model. An accelerating cosmology requires that
px + p be nearly constant and that q be small for matter in the bulk.
There are two ways in which growing cold dark matter models alter the CMB
power spectrum. First, there is less dark matter at earlier times leading to a smaller
amplitude of the third acoustic peak. Second, the decay of the gravitational potential
at late times is diminished. This leads to a smaller late integrated Sachs-Wolfe effect
and less power for the smallest multipoles. We explored1 the likelihood in an eight
dimensional parameter space consisting of six WMAP12 standard parameters (fife/i2,
flch2, h, zre, ns, As) plus the two brane-world parameters, a and q. For the combined
SNIa and CMB data we used a seven dimensional parameter space (Vi^ti2, ho, ze,
ns, As, a, q). These data imply a slightly smaller minimum in x2 f°r the GCDM
model.
The optical depth is rather large for the optimum fit to the CMB alone (r =
0.533). In the combined fit with the SNIa data, h is better constrained so that
a smaller value of r = 0.133 results. In all of these fits a large value of Qdr ~
2 — 3 is offset by the negative dark radiation component. The key constraint is that
fiflM + &DR ~ fiflM + ^A-
To fit the galactic matter power spectrum P(fc),13,14 we assume that the dark
matter and dark radiation enter with uniform distributions and then evolve as
2838
normal matter. In a simultaneous fit to the CMB+SNIa+P(fc) data. The best fit
parameters are q = 0.037 and a = 8.33. The power spectrum derived in the best
fit growing dark matter model is almost indistinguishable from a SACDM model
until one gets to the very largest structures. The bias parameter is somewhat larger
b = 2.1 than that deduced in the usual SACDM models, b = 1.05, because the dark
matter potentials are not as deep at early times.
In summary, we have found that GCDM exchange is consistent with
observations including the supernova magnitude-redshift relation, temperature fluctuations
in the CMB, and the matter power-spectrum data. This cosmology is even slightly
preferred as it fits better the suppression of the CMB power spectrum at low mul-
tipoles. We have thus demonstrated that this cosmology represents an alternative
model to the SACDM cosmology for an observer on the 3-brane.
The value of Qdm here is much larger than in the standard cosmology, though its
gravitational effect is canceled by the dark-radiation contribution. This large dark
matter content, suggests new observational tests. Direct terrestrial measurements of
the total density of cold dark-matter particles should indicate a higher density than
expected based upon their mass and gravitation effect. Another test is that there
should be a suppression of the matter power spectrum on the scale of the horizon
compared to a SACDM cosmology. There is also a suppression of the third acoustic
peak in the CMB power spectrum. We also note that this cosmology produces large
oscillations in the CMB polarization power spectrum. A final amusing feature of
this model is that, if the flow were to cease, the universe would become a matter-
dominated VLm ~ 3 cosmology and collapse in about a hubble time.
References
1. K. Umezu, K. Ichiki, T. Kajino, G. J. Mathews, R. Nakamura, Phys.Rev. D73 (2006)
063527,- astro-ph/0507227.
2. E. Kiritsis, G. Kofinas, N. Tetradis, T. N. Tomaras and V. Zarikas, JHEP, 02, 035
(2003).
3. N. Tetradis, Phys. Lett., B569, 1 (2003).
4. P. S. Apostolopoulos and N. Tetradis, Class. Quant. Grav., 21, 4781 (2004).
5. Y.S. Myung, J.Y. Kim, , Class. Quant. Grav., 20, L169 (2003).
6. N. Tetradis, Class. Quant. Grav., 21, 5221 (2004).
7. P. S. Apostolopoulos and N. Tetradis, Phys. Rev., D71, 043506 (2005).
8. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999); 83, 4690 (1999).
9. S. L. Dubovsky, V. A. Rubakov and P. G. Tinyakov, Phys. Rev. D 62, 105011 (2000).
10. K. Ichiki, P. M. Garnavich, T. Kajino, G. J. Mathews, and M. Yahiro, Phys. Rev. D
68, 083518 (2003).
11. A. G. Riess et al. [Supernova Search Team Collaboration], Astrophys. J. 607, 665
(2004) [arXiv.astro-ph/0402512];
12. D. Spergel, et al. ( WMAP Collaboration, Astrophys. J. Suppl., 148, 175 (2003).
13. S. Dodelson, et al. (SDSS Collaboration), Astrophys. J., 572, 140 (2002); M. Tegmark,
A. J. S. Hamilton, and Y. Xu, MNRAS, 335, 887 (2002).
14. W. Percival, et al. (2dF Collaboration), MNRAS, 328, 1039 (2001).
TESTING DGP MODIFIED GRAVITY IN THE SOLAR SYSTEM
LORENZO IORIO*
Viale Unith. di Italia 68, 70125, Bari (BA), Italy
lorenzo .iorio @libero. it
In this talk we review the perspectives of testing the multidimensional Dvali-Gabadadze-
Porrati (DGP) model of modified gravity in the Solar System. The inner planets, contrary
to the giant gaseous ones, yield the most promising scenario for the near future.
1. The DGP picture
In the Dvali-Gabadadze-Porrati (DGP) braneworld scenario1 our Universe is a
(3+1) space-time brane embedded in a five-dimensional Minkowskian bulk. AH the
particles and fields of our experience are constrained to remain on the branc apart
from gravity which is free to explore the empty bulk. Beyond a certain
threshold ro, which is a free-parameter of the theory and is fixed by observations to
~ 5 Gigaparsec, gravity experiences strong modifications with respect to the usual
four-dimensional Newton-Einstein picture: they allow to explain the observed
acceleration of the expansion of the Universe without resorting to the concept of
dark energy. For a recent review of the phenomenology of DGP cosmologies see
Ref. 2. With more details, an intermediate regime is set by the Vainshtein scale
r* = [rgr^)1l'i, where rg = 2GM/c2 is the Schwarzschild radius of a central
object of mass M acting as source of gravitational field; G and c are the Newtonian
gravitational constant and the speed of light in vacuum, respectively. For a Sun-like
star rv amounts to about 100 parsec. In the process of recovering the 4-dimensional
Newton-Einstein gravity for r << r* << r0, DGP predicts small deviations from
it which yield to effects observable at local scales.3 They come from an extra radial
acceleration of the form4-6
aDGP = T (~) f^f. (1)
The minus sign is related to a cosmological phase in which, in absence of cosmo-
logical constant on the branc, the Universe decelerates at late times, the Hubble
parameter H tending to zero as the matter dissolves on the brane: it is called
Friedinann-Lemaitre-Robertson-Walker (FLRW) branch. The plus sign is related
to a cosmological phase in which the Universe undergoes a de Sitter-like
expansion with the Hubble parameter H = c/ro even in absence of matter. This is the
self-accelerated branch, where the accelerated expansion of the Universe is realized
without introducing a cosmological constant on the brane. Thus, there is a very
important connection between local and cosmological features of gravity in the DGP
model.
* Fellow of the Royal Astronomical Society
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2. The testable effects and their measurability
About the local effects, Lue and Starkman in Ref. 5 and Iorio in Ref. 6 derived an
extra-secular precession of the pericentre u> of the orbit of a test particle
of 5 x 10 4 arcseconds per century (" cy 1), while Iorio in Ref. 6 showed that also
the mean anomaly M. is affected by DGP gravity at a larger extent
lie A 39 2
the longitude of the ascending node fl is left unchanged. In (2)-(3) the upper sign is
for the FLRW branch, while the lower sign is for the self-accelerated one. As a result,
the mean longitude A = w + Q + M, which is a widely used orbital parameter for
nearly equatorial and circular orbits as those of the Solar System planets, undergoes
a secular precession of the order of 10~3 " cy-1. Such precessions are independent of
the semi-major axis a of the planetary orbits and depends only on their eccentricities
e via second-order terms. The effects of DGP gravity on the orbital period of a test
particle were worked out by Iorio in Ref. 7; the DGP precession of a spin can be
found in Ref. 8, but it is too small to be detectable in any foreseeable future.
Recent improvements in the accuracy of the data reduction process for the inner
planets of the Solar System,9'10 which can be tracked via radar-ranging, have made
the possibility of testing DGP very thrilling.6'7'11'12 In particular, Iorio in Ref. 12
showed that the recently observed secular increase of the Astronomical Unit13'14
can be explained by the self-accelerated branch of DGP and that the predicted
values of the Lue-Starkman perihelion precessions for the self-accelerated branch
are compatible with the recently determined extra-perihelion advances,10 especially
for Mars, although the errors are still large. Rather surprisingly, it was recently
showed in Ref. 15 that the Kuiper belt objects, if not properly modelled in the
dynamical force models of the data-reduction softwares used to process planetary
data, might affect the dynamics of the Earth and Mars at a non negligible level with
respect to the DGP features of motion. The possibility of using the outer planets
of the Solar System, suggested by Lue in Ref. 2 and, in principle, very appealing
because all the competing Newtonian and Einsteinian orbital effects so far modelled
are smaller than the DGP precessions, is still very far from being viable.16 Finally,
we mention that it was argued17 that the launch of a LAGEOS-likc Earth artificial
satellite would allow to measure the DGP perigee precession, but such a proposal
was proven to be highly unfeasible in Ref. 11.
Acknowledgements
I am grateful to R. Ruffini and H. Kleinert for the grant received to attend
the Eleventh Marcel Grossmann Meeting on General Relativity, 23-29 July, Freie
Universitat Berlin, 2006.
2841
References
1. G. Dvali, G. Gabadadze and M. Porrati Phys. Lett. B 485, 208 (2000).
2. A. Lue Phys. Rep. 423, 1 (2006).
3. G. Dvali, A. Gruzinov and M. Zaldarriaga Phys. Rev. D 68, 024012 (2003).
4. A. Gruzinov New Astron. 10, 311 (2005).
5. A. Lue and G. Starkmann Phys. Rev. D 67, 064002 (2003).
6. L. Iorio Class. Quantum Grav. 22, 5271 (2005a).
7. L. Iorio J. Cosmol. Astropart. Phys. 1, 8 (2006a).
8. L. Iorio Int. J. Mod. Phys. D 15, 469 (2006b).
9. E.V. Pitjeva Sol. Sys. Res. 39, 176 (2005a).
10. E.V. Pitjeva Astron. Lett. 31, 340 (2005b).
11. L. Iorio J. Cosmol. Astropart. Phys. 7, 8 (2005b).
12. L. Iorio J. Cosmol. Astropart. Phys. 9, 6 (2005c).
13. G.A. Krasinsky and V.A. Brumberg Celest. Mech. Dyn. Astron. 90, 267 (2004).
14. E.M. Standish, E.M., The Astronomical Unit now, in Transits of Venus: New Views of
the Solar System and Galaxy, Proceedings IAU Colloquium No. 196, ed. D.W. Kurtz
(Cambridge University Press, Cambridge, 2005).
15. L. Iorio Mon. Not. Roy. Astron. Soc. 375, 1311 (2007).
16. L. Iorio and G. Giudice J. Cosmol. Astropart. Phys. 8, 7 (2006).
17. I. Ciufolini gr-qc/0412001 (2004).
THE DYNAMICS OF SCALAR-TENSOR COSMOLOGY FROM RS
TWO-BRANE MODEL
P. KUUSK*, L. JARV and M. SAAL
Institute of Physics, University of Tartu,
Riia 142, 51014, Tartu, Estonia
* piret.kuusk@ut.ee
We consider Randall-Sundrum two-brane cosmological model in the low energy gradient
expansion approximation by Kanno and Soda. It is a scalar-tensor theory with a
specific coupling function. We find a first integral of equations for the A-brane metric and
estimate constraints for the dark radiation term. We perform a complementary analysis
of the dynamics of the scalar field (radion) using phase space methods and examine
convergence towards the limit of general relativity. We find that it is possible to stabilize
the radion at a finite value with suitable negative matter densities on the B-brane.
Keywords: two-brane cosmology, scalar-tensor theory in the Jordan frame, phase space
methods.
1. Introduction
Following Kanno and Soda,1 we consider the Randall-Sundrum type I cosmological
scenario2 with two branes (A and B) moving in a 5-dimensional bulk. Both branes
are taken to be homogeneous and isotropic and supporting energy-momentum
tensors of a perfect barotropic fluid with barotropic index T (p = (T — l)p) on the
A-brane (identified with our visible Universe) and T on the B-brane; for simplicity
we assume r = V .
The field equations of the effective 4-dimensional theory obtained by Kanno and
Soda1 are the equations of a scalar-tensor theory with one scalar field \P (interpreted
as a radion) with a specific coupling function w{^) = 3\P/(2(1 — <J>)) which describes
the proper distance between the branes. The cosmology of this model was first
studied by Kanno et al.3 and later by us.4,5
2. Field equations and analytic solutions
The field equations1 on the A-brane for the Friedmann-Lemaitre-Robertson-Walker
(FLRW) line element ds2 = -dt2 + a2(t)[(l - kr2)'1 dr2+r2dQ2} and perfect fluid
matter on both branes read {H = a/a)
2__ * 1 ji2 K2V_ tf_p_ /t2(l-ff)2 B k_
* + 4*(i-*) + y* + 3* + y * p "^' ()
2g + 3g2 = -4--(l-*)y + -V-^-2^-g ^ -4, (2)
q,F <5 v I F ^ <f $ $ 4*(1- <J) a2' v ;
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* = -3^ - -oT^ + ^ U - Vpl (l _ *)
2(1-*) 3 V d^ J
2 2
+ y (1 - *) (p ~ 3p) + y (1 - *)V - 3p"), (3)
they reduce to general relativity when * —► 1, * —► 0. The conservation laws as
measured by an A-brane observer, p + 3HTp = 0 , pB + 3HTpB - (3*rpB)/(2(l -
*)) =0 , imply a relation between the energy densities on the A- and B-brane
1-4- \ 2 f p
>">t\j^) UJ- <4)
The dynamical equation for H decouples from the scalar field and B-brane matter4-5
due to the specific form of the coupling function w(*) and its first integral reads
rr2 n2 k2 ( a\-3T k K2C('a\-4'
S 3 \ao/ a 3 \ao/
which is a Friedmann equation with a dark radiation term, C. Comparison with
recent results of light element abundances, BBN, and CMB observations constrain
the dark radiation term5 (po is the radiation energy density),
-0.054 < — < 0.138. (6)
Po
Analytic solutions are found5 for the flat Universe scale factor in the case of
cosmological constant (r = 0), radiation (r = 4/3), dust (r = 1), cosmological
constant and radiation, and for the scalar field in the case of radiation (r = 4/3),
cosmological constant and radiation (to be expressed in terms of elliptic functions).
3. The dynamics of the scalar field
Defining a new time variable6 , dp = hcdt, hc = H + A- , it is possible to derive
dp
3 . /q,n
a decoupled "master equation" for the scalar field (here V = 0, k = 0, -J- = /'l
8(1 - *)-^- " 3(2 - T) / - J - 2((4 - 6*) - (4 - 3r)(l - tt)W(tt)) /-
12(2 - T)(l _*)£-_ 8(4 - 3r)(l - *)2VF(*) = 0, (7)
where
W{9) = POb) °J or W(*)= 2i£, (8)
corresponding to the case when a = 0, aB = 0, p ^ 0, pB ^ 0 and to the case when
a ^ 0, aB ^ 0, p = 0, pB =0, respectively. Phase portraits are found5 depending on
the values of constraints involved (Fig. 1).
2844
(a) " (b)
Fig. 1. Phase portraits (x = *(p), y = *'(p)) (a) for cosmological constants: a = l,oB = —0.5
and (b) for dust: po = 0.5, pj = -1, *o = 0.5.
All trajectories are constrained to be in the physically allowed region of the phase
space determined by Friedmann equation (1). Figure la for a cosmological constant
dominated Universe contains a saddle point (\P = 0, ^' = 0) and a spiralling
attractor corresponding to general relativity (\P = 1, \P' = 0). Figure lb for a dust
dominated Universe with pf < 0 contains two saddle points, (<3> = 0, ^' = 0) and
(<5 = !,$'= 0), and also an attractor (<5 = 1 - pg2(l - ^)3/pl, *' = 0) which can
stabilize the branes in a position that does not correspond to general relativity on
the A-brane.
4. Summary
We have considered a braneworld inspired scalar-tensor cosmology with a specific
coupling function, cosmological constant and perfect fluid matter on both branes.
The first integral of equations for the metric tensor of the A-brane contains the dark
radiation term. Phase portraits of the scalar field reveal fixed points and allow us to
find late time fates for different cosmological models. The solutions may approach
general relativity (\P = 1), or go to brane collision (\P = 0), depending on the initial
conditions. There are also additional fixed points in between the two extremes: a
saddle for cosmological constants and an attractor for dust (Fig. 1(b)).
References
1. S. Kanno and J. Soda, Phys. Rev. D 66, 083506 (2002), [hep-th/0207029].
2. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999), [hep-ph/9905221].
3. S. Kanno, M. Sasaki, and J. Soda, Prog. Theor. Phys. 109, 357 (2003), [hep-
th/0210250].
4. P. Kuusk and M. Saal, Gen. Rel. Grav. 36, 1001 (2004), [gr-qc/0309084].
5. L. Jarv, P. Kuusk, and M. Saal, Phys. Rev D (accepted), [gr-qc/0608109].
6. T. Damour and K. Nordtvedt, Phys Rev. D 48, 3436 (1993).
SELF-T-DUAL BRANE COSMOLOGY
MASSIMILIANO RINALDI *
School of Mathematical Sciences, University College Dublin,
Belfteld, Dublin 4, Ireland,
and
Dipartimento di Fisica and I.N.F.N, Universita di Bologna,
Via Irnerio 46, 40126 Bologna, Italy.
We show how T-duality can be implemented with brane cosmology. As a result, we
obtain a smooth bouncing cosmology with features similar to the ones of the pre-Big
Bang scenario. Also, by allowing T-duality transformations along the time-like direction,
we find a static solution that displays an interesting self tuning property.
1. Introduction
In the past years, various cosmological models were inspired by different aspects
of string theory. In some cases, these rely upon the fundamental symmetries of
string theory, the most notable example being the pre-Big Bang (PBB) scenario
motivated by T-duality.1 In other cases, models are based on extended objects
such as branes.2 These two approaches are often seen as competing. However, if
our Universe is seen as a brane moving in a higher-dimensional bulk, obtained by
compactification of string theory, it is likely that the effective cosmology inherits
some of the symmetries of the uncompactified theory. A first application of this idea
can be found in the context of type IIA/IIB supergravity. When compactified to
five dimensions, these theories possess static black hole solutions with flat horizon,3
which are directly related by T-duality transformations. By studying a brane moving
in these dual spaces, it was found that these transformations induce the inversion of
the cosmological scale factor on the brane, along the lines of the PBB scenario.4 The
latter, however, is based on a self-T-dual action, with time-dependent background
solutions. In the next section, we show that it is possible to construct a self-T-dual
action, which, instead, has static background solutions. Also, an embedded moving
brane displays an effective cosmological evolution, which smoothly connects a pre-
and a post-big bang phase, through a non-singular bounce, in complete analogy
with some of the PBB models. Finally, in the last section, we will also show how
Self-T-dual brane models can tackle the problem of fine-tuning between the brane
vacuum energy density (tension) and the bulk cosmological constant.
2. Pre-Big Bang on the brane
To recreate a PBB scenario on the brane, we must find first a self-T-dual action,
such that the related equations of motion have static solutions. Let us consider the
* rinaldim@bo.infn.it
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2846
dilaton-gravity action5
Sbuik= f d5x^e-^[TZ + A(VcP)2 + V] , (1)
Jm
where V is an exponential function of the so-called shifted dilaton 0. If we choose
the line element
ds2 = -A2{r) dt2 + B2{r) dr2 + R2{r) 8l3 dx{ dx3 , (2)
then the shifted dilaton is defined as <j>{r) = 4>{r) — § \nR(r). It can be shown that
the action (1) is invariant under the T-duality transformation R(r) —> i?(r)_1,
which leaves the shifted dilaton unchanged. Therefore, to any solution with metric
(2), there exists another with R replaced by 1/R. This property holds if we neglect
the boundary terms springing from variation of the action with respect to the fields.
However, if we want to preserve self-T-duality, these terms must be kept when we
introduce a Z2 symmetric 3-brane, which acts as a boundary. In this way, it turns
out that the full action, obtained by adding Eq. (1) to the brane action
Sbrane = - [ d3xdrVhe'2* [4K + C] , (3)
is still invariant under the transformation R(r) —> i?(r)_1, provided C —» £, i.e.
provided the brane matter Lagrangian is itself T-duality invariant. In the expression
above, h is the determinant of the induced FLRW metric on the brane, ds2 =
—dr2 + R2{r{T))5ij dx% dx3 , K is the trace of the brane extrinsic curvature, and
r is the proper cosmological time (and the parametric position of the brane in the
bulk). It is clear that the duality acting on the bulk metric leads to the inversion
of the scale factor R. Now. let the matter on the brane be a perfect fluid, with
equation of state p = cop. By carefully studying the Israel junction conditions, it
T
can be shown that the self-T-duality of C implies that ui —> — ui, exactly like in the
PBB scenario a.
By studying the bulk equations of motion, one can find black hole solutions with
one regular horizon. A brane moving in such a background encounters a turning
point outside the horizon. By assuming that the T-duality transition occurs at the
bounce, one can construct a non-singular transition between a pre-big bang phase
(with, say, — u> and scale factor l/R(r)) and post-big bang phase (with to and scale
factor R{t)). Finally, it also turns out that the cosmic evolution far away from the
bounce, both in the past and in the future, is always of de Sitter type.
3. Time-like T-duality and the fine tuning problem
Wc now turn our attention to the fine tuning problem of Randall-Suiidrum-likc
models. We consider again the action (1), but now we assume that the bulk metric
It is important to remark that, in this model, u is an function of r.
2847
has the Poincare-invariant form
ds2 = e2aMt]liVdx'1dxu + dz2 , (4)
while the shifted dilaton reads <p = 6 — 2a. Along the lines sketched in the previous
section, one can show6 that the action is invariant under T-duality transformations
along both the time and space coordinates x^, i.e. under a(z) —> —a(z). Of course,
the same holds for the equations of motion, for which the only non-singular solutions
are given by a constant shifted dilaton 0o and a = ±\{z — zq), where A and zq
are integration constants. The positive and negative sign solutions are related by
T-duality, and they simply correspond to two slices of anti-de Sitter space b. By
inserting a ^-symmetric brane on this background, with a perfect fluid as matter,
we can preserve the self-T-duality of the total action provided we impose (ui + 1) —>
— (u> + 1). Interestingly, this duality transformation is identical to the one found in
the context of phantom cosmology c. But the most important result is that, in the
case of a static brane, the (constant) energy density of the brane matter is given by
p2 = W0e^° , (5)
where (3 and Vq are arbitrary constants. Therefore, any value of p can be reached
given any vacuum expectation value of the shifted dilaton.
These encouraging results call for further investigations into self-T-dual brane
cosmology, and the main target is to find some signatures (such as particular CMB
fluctuations or relic gravitons) of this model, which might be tested by observations.
Acknowledgments
I wish to thank P. Watts and O. Corradini for their fundamental contributions to
these results, and Prof. D. Gal'tsov for inviting me to speak at the parallel session.
References
1. M. Gasperini and G. Veneziano, Phys. Rept. 373 1 (2003).
2. P. Horava and E. Witten, Nucl. Phys. B 460 506 (1996); L. Randall and R. Sundrum,
Phys. Rev. Lett. 83 3370 (1999); L. Randall and R. Sundrum, Phys. Rev. Lett. 83
4690 (1999).
3. M. Rinaldi, Phys. Lett. B 547 95 (2002).
4. M. Rinaldi, Phys. Lett. B 582 249 (2004).
5. M. Rinaldi and P. Watts, JCAP 0503 006 (2005).
6. O. Corradini and M. Rinaldi, JCAP 0601 020 (2006).
7. M. P. Dabrowski, T. Stachowiak and M. Szydlowski, Phys. Rev. D 68 103519 (2003).
bThis is consistent with time-like T-duality, which requires the time-like direction to be compact.
cThanks to Prof. M. P. Dabrowski for pointing out this similarity.7
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Brane Worlds
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CATCHING PHOTONS FROM EXTRA DIMENSIONS
A. DOBADO1 and A.L. MAROTO2
Departamento de Fisica Tedrica,
Universidad Complutense de Madrid, 28040 Madrid, Spain
1 dobado@fts.ucm.es
2 maroto@fts.ucm.es
J.A.R. CEMBRANOS
Department of Physics and Astronomy,
University of California, Irvine, CA 92697, USA
jruizcem@uci. edu
In extra-dimensional brane-world models with low tension, brane excitations provide a
natural WIMP candidate for dark matter. Taking into account the various constraints
coming from colliders, precision observables and direct search, we explore the
possibilities for indirect search of the galactic halo branons through their photon producing
annihilations in experiments such as EGRET, HESS or AMS2.
Keywords: Brane-worlds: branons; dark matter.
1. Low-tension braneworld phenomenology
According to recent suggestions our universe could be a 3-dimensional brane, where
the SM fields live, embedded in a D-dimensional space-time1 (D = 4 + N). The
most important parameters of this setup being the fundamental scale of gravity
in D dimensions Md (which is no longer the Planck scale Mp) and the brane
tension r = /4. Besides the SM fields, other new excitations appear on the brane:
Kaluza-Klein gravitons2 and brane fluctuations na, a = 1,2, ...TV, where /V is smaller
or equal than the number of extra dimensions.3 These branons arc the Goldstorie
Bosons associated to the spontaneous breaking of the translational invariance in the
extra dimensions induced by the presence of the brane. However, in the general case,
translational invariance is not an exact symmetry of the bulk space, i.e: branons
acquire a mass M. For / <C MD (low tension), KK gravitons decouple from the SM
particles. Consequently, at low energies the only relevant degrees of freedom are the
SM particles and the branons whose interactions can be described by the effective
Lagrangian:
CBr = Ig^d^dvir" - ImW + ~(4d^ac%ira - A^>fVV)7^
(1)
in which, one can see that branons interact by pairs with the SM and with a coupling
controlled by the brane tension scale /. For simplicity, we assume that all branons
have the same mass, Map = M6ap. Therefore branons are a kind of new scalar
fields, whose properties (stability, weak couplings and masses) coincide with those
expected for a WIMP (Weakly Interacting Massive Particle).4
2851
2852
From the above effective Lagrangian it is possible to obtain the branon
production cross sections for different colliders,5 the typical signature being missing
energy and missing Pt, and thus to find bounds on the / and M parameters for
different values of N. Other constraints can also be obtained by computing the
effect of virtual branons on various precision observables6 including the muon g — 2
measurements. Taking all this into account, one can calculate the rate for direct
detection of branons in the current and future experiments designed for WIMP
detection. Remarkably these particles can be well accommodated within all these
bounds and still they offer definite predictions for future direct search experiments.4
In addition WIMPs are expected to annihilate in the galactic halo producing
photons in different ways. Such photons could be caught by detectors on Earth or in
space, thus providing a new indirect way to detect their presence which could nicely
complement the above mentioned more direct searches. In the following we analyze
the potential detection of these photons coming from the galactic halo branons.
2. Gamma rays from branon annihilation
The photon flux in the direction of the galactic center coming from dark matter
annihilations can be written as:7'8
Y,Mjw <2>
dQdE7 NM2 ^ ' dE7
where Jo is the integral of the dark matter mass density profile, p(r), along the path
between the galactic center and the gamma ray detector:
■*> = -/-/ P2dl, (3)
47r J path
N is the number of dark matter species with mass M and (aiv) is the thermal
average of the annihilation cross section of two dark matter particles into another two
particles. On the other hand, the continuum photon spectrum from the subsequent
decay of particles species i presents a simple description in terms of the photon
energy normalized to the dark matter mass, x = E1jM. Thus, for each channel i,
we have:
diV< dN; a' _btx
= M = e (4)
dx dE7 x3/2 [)
where a* and 6, are constants. In the case of heavy branons, if we neglect three body
decays and direct production of two photons, the main contribution to the photon
flux comes from branon annihilation into ZZ and W+W~. The contribution from
heavy fermions, i.e. annihilation in top-antitop can be shown to be subdominant.
The concrete values for the above constants in those channels are: azz = aw w =
0.73 and bzz = bw±wT = 7.8.7>8
2853
On the other hand, the thermal averaged cross-section (<jz,Wv) which enters in
eq. (2) has been calculated in4 and in the non-relativistic limit is given by:
M2;/1 - %^(4M4 - 4M2m|;H/ + 3m|^)
(az.wv) = ^^ (5)
The produced high-energy gamma photons could be in the range (30 GeV-
10 TeV), detectable by Atmospheric Cerenkov Telescopes (ACTs) such as HESS,
VERITAS or MAGIC. On the contrary, if M < mz,w, the annihilation into W
or Z bosons is kinematically forbidden and it is necessary to take into account
the rest of channels, mainly annihilation into the heaviest possible quarks.9 In this
case, the photon fluxes would be in the range detectable by space-based gamma ray
observatories10 such as EGRET, GLAST or AMS, with better sensitivities around
30 MeV-300 GeV.
Acknowledgments: This work has been partially supported by DGICYT (Spain)
under project numbers FPA 2004-02602 and FPA 2005-02327, by NSF CAREER
grant No. PHY-0239817, NASA Grant No. NNG05GG44G, the Alfred P. Sloan
Foundation and Fulbright-MEC award.
References
1. N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B429, 263 (1998); Phys.
Rev. D59, 086004 (1999); I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G.
Dvali, Phys. Lett. B436 257 (1998)
2. J. Hewett and M. Spiropulu, Ann. Rev. Nucl. Part. Sci. 52: 397-424, (2002)
3. R. Sundrum, Phys. Rev. D59, 085009 (1999); A. Dobado and A.L. Maroto Nucl. Phys.
B592, 203 (2001); J.A.R. Cembranos, A. Dobado and A.L. Maroto, Phys.Rev. D65,
026005 (2002)and hep-ph/0611024
4. J.A.R. Cembranos, A. Dobado and A.L. Maroto, Phys. Rev. Lett. 90, 241301
(2003); Phys. Rev. D68, 103505 (2003); Int. J. Mod. Phys. D13: 2275, (2004); hep-
ph/0307015; hep-ph/0402142; and hep-ph/0406076; A.L. Maroto, Phys. Rev. D69,
043509 (2004) and Phys. Rev. D69, 101304 (2004)
5. J. Alcaraz et al., Phys. Rev. D67, 075010 (2003); J.A.R. Cembranos, A. Dobado and
A.L. Maroto, Phys.Rev. D70, 096001 (2004)
6. J.A.R. Cembranos, A. Dobado and A.L. Maroto, Phys. Rev. D73, 035008 (2006);
Phys. Rev. D73, 057303 (2006)
7. L. Bergstrom, P. Ullio and J.H Buckley, Astropart. Phys. 9, 137 (1998)
8. J.L. Feng, K.T. Matchev, F. Wilczek, Phys. Rev. D63, 045024 (2001)
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10. AMS Collaboration, AMS Internal Note 2003-08-02; J.A.R. Cembranos, A. Dobado
and A.L. Maroto, work in progress.
LORENTZ INVARIANCE VIOLATION
IN BRANEWORLD MODELS
P. A. KOROTEEV
Institute for Nuclear Research, Moscow, 117312, Russia
koroteev &ms2. inr. ac.ru
www.inr.ac.ru
Lorentz invariance (LI) violation in brane world scenario is considered. The family of
Lorentz violating static solutions of bulk Einstein equations are found with ideal rela-
tivistic fluid in the bulk. The no go theorem about null energy condition (NEC) in the
bulk and for matter on the brane is proved in general case and it is established that
for static bulk solution one cannot satisfy them both for Universes of finite volume. We
derive the graviton spectrum in the obtained background and show that Newtonian
gravity on the brane is restored up to small corrections at short distances. It is remarkable
that in the framework under consideration we have graviton zero mode. Localization
of fermions is performed by means of bulk Dirac mass. Thus LI violation provides us
with interesting technics of localizing of fields on topological defects. In the end we
discuss various Lorentz violating backgrounds and analyze their physics. All results will be
published in.1
Keywords: General Relativity; Cosmology; Lorentz Invariance; Field Theory.
In resent years it has been speculated that Lorentz invariance in our world can be
violated. Different scenarios were developed in!0>13.14>18>19 (and references therein). In
models with large extra dimensions2-9 we have a 3+1 - dimensional brane embedded
into bulk of higher dimension. The simplest toy-model one can imagine is the one
with one extra dimension. Under appropriate choice of coordinates (t,xi,X2,X3,z)
the brane is a 3+1 (t,Xi,X2,xs) hypersurface located at z = 0 with z-axis being
orthogonal to it. To violate LI means to peak up 5D metric coefficients in such a way
that no smooth transformation could map it into the one conformal equivalent to
Minkowskian metric. To be in consistence with experimental observations Lorentz
invariance should be conserved at the location of the brane.
1. Null Energy Condition and No Go Theorem
For physical applications one needs null-energy condition (NEC) to be valid on the
brane. In terms of the stress-energy tensor it reads Tab£A£B > 0, gABS,A£B = 0
for arbitrary null vector £A. Thus to figure out wether NEC is satisfied or not one
should minimize the bilinear form T over the surface gAB£A£B = 0. For simple
equation of state p = wp it reads w > — 1.
Here we will consider special 2-parametric 5*0(3) x Z2 ansatz respecting spatial
brane rotations and bulk reflections ds2^ c) = e~2k^z^dt2 - e~2fec|z|dx2 - dz2. One
can see that the case £ = ( corresponds to AdSs bulk and Randall-Sundrum Lorentz
invariant scenario. It appears that what is important for physics on the brane is
ratio of £ and (.
2854
2855
The solution of Einstein equations with ideal relativistic fluid with density p and
anisotropic pressure p\ = wp in £1,2:2, £3 directions and ps = up in the direction
of extra dimension reads
C* + Ott-0, w = _1 + fc«^0 m
p=-A-6rC, w = -l + Ar — ^^ ^, w = -l + fc
where A is a bulk cosmological constant. We put domain wall with the following
energy-momentum tensor Tgbrane = diag(pb + a -pb + a -pb + u - pb + a 0) S(z)
where a is the brane cosmological constant. One can write down Israel junction
conditions to obtain pb = 6(k — a and Ub = — 1 -I ^ . It is remarkable that the
Pb
same constraint £ > £ arises here for the null energy condition to be satisfied. One
can see that one cannot make NEC to be valid both in the bulk and on the brane.
It paper12 no-go theorem was proved. We generalize it1 to generic static Lorentz
violating background. The theorem reads: Let the spatial curvature of the brane
be equal to zero. Then one cannot screen bulk naked singularity from the brane by
means of horizon if NEC on the brane and in the bulk are satisfied.
2. Newtonian Gravity on the Brane
Investigation of scalar field perturbations is reasonable since traceless transverse
(TT) perturbations of metric tensor satisfy the same equation. This fact was used by
authors in Lorentz invariant case in16 to calculate effective gravitational potential on
the brane but one can show by straightforward calculations that this correspondence
also takes place in LI violating case.
In many papers10,16"19 metrics with £ > (" were considered. We focus here
on another case where £ = 0, C = 1- Let us put scalar field in the bulk S =
f d5XyfgdA(f>dA(f>. General L2-class solution of the field equation above equation
contains McDonald function 0(z) = const e"3/2fe|z|ii'y(f efe|z|). Matching at z = 0
yields the condition from which the spectrum of excitation can be derived.
pKv+l{l) _3
where v = yj - %-■ Modified Newtonian law for gravitational potential between
two masses m\ and m.2
y(r).«(1 + _L) (3)
r v nkr/
In the main paper1 we investigate spectrum of spin 0 and 1/2 perturbations and
conclude that its properties (continuity or discreteness) are entirely determined by
ratio of £ and C parameters. If £ > C we have continuous spectrum and quasilocalized
modes (particles can escape10'11 into extra dimension), if £ < £ than we have discrete
spectrum and localized modes.1
2856
References
1. P. Koroteev and M. Libanov. Lorentz Invariance Violation in Braneworld Models, in
preparation.
2. J. Polchinski, [ArXiv:hep-th/9611050],
3. A. Lukas, B. Ovrut, K. Stelle and D. Waldram, Phys. Rev. D 59, 086001 (1999)
[arXiv:hep-th/9803235],
4. P. Bowcock, C. Charmousis and R. Gregory, Class. Quant. Grav. 17, 4745 (2000)
[arXiv.hep-th/0007177],
5. G. Dvali and M. Shifman, Phys. Lett. B 396, 64 (1997) [arXiv:hep-th/9612128],
6. G. Dvali and G. Gabadadze. Phys. Rev. D 63, 065007 (2001) [arXiv:hep-th/0008054].
7. N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B 429, 263 (1998)
[arXiv:hep-ph/9803315],
8. Z. Kakushadze and S. Tye, Nucl. Phys. B 548, 180 (1999) [arXiv:hep-th/9809147],
9. A. Perez-Lorenzana, AIP Conf. Proc. 562, 53 (2001) [arXiv:hep-ph/0008333].
10. S. Dubovsky. JHEP 0201:012 (2002) [ArXiv:hep-th/0103205],
11. S. Dubovsky, V. Rubakov and P. Tinyakov. [ArXiv:hep-th/0006046].
12. J. Cline and H. Firouzjahi. Phys.Rev. D65 (2002) 043501 [ArXiv:hep-th/0107198],
13. D. Gorbunov and S. Sibiryakov. JHEP 0509:082 (2005) [ArXiv:hep-th/0506067]
14. C. Csaki, J. Erlich and C. Grojean. Nucl.Phys.B604:312-342 (2001) [ArXiv:hep-
th/0012143]
15. N. Arkani-Hamed, S. Dimopolus and G. Dvali, Phys Lett. B 429 (1998) 263
[ArXiv:hep-ph/9803135]
16. L. Randall and R. Sundrum. Phys.Rev.Lett.83:3370-3373 (1999) [ArXiv:hep-
ph/9905221]
17. L. Randall and R. Sundrum. An Alternative to Compactification.
Phys.Rev.Lett.83:4690-4693 (1999) [ArXiv:hep-th/9906064]
18. M. Libanov and V. Rubakov. JCAP 0509, 005 (2005) [arXiv:astro-ph/0504249]
19. M. Libanov and V. Rubakov. Phys.Rev.D72:123503 (2005) [ArXiv:hep-ph/0509148]
20. P. Binetruy, C. Deffayet, U. EUwanger and D. Langlois. Phys.Lett.B477:285-291 (2000)
[ArXiv:hep-th/9910219]
21. A. Karch and L. Randall. JHEP 0105 (2001) 008 [ArXiv:hep-th/0011156]
22. T. Gherghetta. [arXiv:hep-th/0601213]
23. Y. Grossman and M. Neubert. Phys.Lett. B474 (2000) 361-371 [ArXiv:hep-
ph/9912408]
24. R. Jackiw and C. Rebbi. Phys. Rev. D13, 3398 (1976)
25. R. Contino and A. Pomarol. JHEP 0411:058 (2004) [ArXiv:hep-ph/9912408]
THE BAZANSKI APPROACH IN BRANE WORLDS:
A BRIEF INTRODUCTION
M.E. KAHIL
The American University in Cairo, Cairo 11511, Egypt kahil@aucegyp.edu
Paths of test particles, rotating and charged objects in brane-worlds using a modified
Bazanski Lagrangian are derived. We also discuss the transition to their corresponding
equations in four dimensions. We then make a comparison between the given equations
in brane-worlds (BW) and their analog in space-time-matter (STM) theory.
Keywords: Style file; I^T^X; Proceedings; World Scientific Publishing.
1. The Bazanski Approach in 5D
Motion of test particles in higher dimensions is obtained by using the usual
Bazanski Lagrangian [1] which has the advantage that we obtain path and path
deviation equations from the same Lagragian:
L->»UA-DS (1)
where A = 1,2, 3, 4, 5. By taking the variation with respect to the deviation vector
^c and the tangent vector Uc, we obtain the well known geodesic and geodesic
deviation equations respectively:
DS2 =R\BDUAU^U (3)
Recently, the Bazanski Lagrangian has been modified in order to describe motion
of charged particles and rotating objects in 5-dimensions whether they be compact
or noncompact spaces [2]:
In Compact Spaces
The process to unify electromagnetism (gauge fields) and gravity depends on extra
component(s) of the metric. Using the cylinder condition, a charged particle whose
behavior is described by the Lorentz equation in 4D behaves as a test particle
moving on a geodesic in 5D. At the same time, its deviation equation becomes like the
well known geodesic deviation equation [2]. This result is obtained from applying
the usual Bazanski method in 5D.
In Non-Compact Spaces
the path equation has two main defects:
(i) it is not gauge invariant ,
(ii) the additional extra force from an extra dimension is parallel to the four vector
velocity.
2857
2858
Some authors [4] and [5] have introduced different types of transformations in
order overcome the above mentioned problems. These are expressed like the ususal
geodesic equation (2). Applying the Bazanski approach we can obtain equation (2)
and its corresponding geodesic deviation equation, satisfying the Campbell-Magaard
theorem [3]. Thus (2) becomes :
-53*--0- <4»
2. The Bazanski Approach in Brane World Models
In the Brane world scenario our universe can be described in terms of 4+N
dimensions with N > 1 and the 4D space-time part of it is embedded in 4+N.
manifold [6]. Accordingly, the bulk geodesic motion is observed by a four
dimensional observer to reproduce the physics of 4D space-time [7]. Consequently, the
importance of the equation of motion for a test particle in the bulk space-time of
brane worlds is to describe the apparant motion in 4D space-time. Applying the
Bazanski approach, we can obtain the motion for a test particle on a brane using
the followinng Lagrangian :
L = g^(x»,y)U»^ + f^, (5)
where gpu(xp,y) is the induced metric and/M = \U'pUa'-f22- ~^UM describes a parallel
force due to the effect of non-compactified extra dimension to give [8]:
^ + ( »Xu«u? = (Vcr - gp-^QLu*. (6)
ds \a(3) v2 ' dy ds K '
As in Brane world models, one can express | -Sf2- in terms of the extrinsic curvature
Qpa i.e. Q,ap = \-jfSL [9]. Thus, the path equation for a test particle in brane world
models becomes:
dU" , f /' \uauP = 2(I[/P[7- - gp°)npcJ^U». (7)
ds {apj v2 * ' p° ds
Also,for a rotating object the Papapetrou equation [10] in 4 dimensions becomes :
?- + ( ^V0f//3 = -n?Ah]Ps^us + 2{l-upu° -gp°)npa^u». (8)
ds {oifjj m n ' 2 ds
The above equation is derived from the following Lagrangian
Ds 2m ' 2 ay ds
L = giw(xp,y)u»-— + ( — R^paS»°Uv + -^u<>U°U^)*», (9)
with an additional factor related to Guass-Codassi equation (cf.[ll]) and taking
into consideration the Campbell-Magaard theorem, the four dimensional curvature
becomes i.e. Rap7s = 20o[7f2(5]/3. Similarly, the usual equation of motion for a
charged object [12] can be described in the presence of brane world models:
ds
( ^XuaUP = ?-F%UP + 2(\upU° - gpa)Slpa^-U»,: (10)
[afj) m '* 2 ds
2859
which is derived from the following Lagrangian:
l - S,^',v)u^ + £f„w + i^wu-u^' (ii)
3. Discussion and Concluding Remarks
In Brane world models, it can be easily found that matter in 4D is regarded as the
effect of curvature of the extra dimension in a 5D bulk. While in STM, the bulk
is obtained due to the solution of 5D Einstein equations in vacuum [13]. This may
show the equivalence between Brane world models and Space-time-matter theory
as the first is embedding physics of 4D in order to describe the geometry of the bulk
in 5D. While, in STM the process is based on projecting the geometry of a 5D flat
curvature onto a 4D space to unify matter with geometry. From this perspective, it
is worth mentioning that equations of motions of a test paricle defined in STM (2),
after pojecting the 5D equations onto 4D are equivalent to using their counter-part
in brane worlds (6). Each of them can be derived from a different Lagrangian using
the Bazanski approach.
Also, in this work we have developed the equations of motion for rotating and
for charged objects while taking into account the effect of the extrinsic curvature on
these sets of equations. Then equations (8) and (10) reduce to their ususal rotating
(Papapetrou)[10] and charged (Lorentz) equations (cf.[12]) respectively when the
effect of the extra dimension is dropped.
Finally, it remains an important point should be discussed in future work:
Is the Riemannian Geometry in higher dimensions (compact/non-compact)
sufficient for defining the physics of our cosmos? This apprach might allow brane
world models to be described using some non-symmetric geometries admitting non-
vanishing curvature and torsion simulatenously. Perhaps these could lead a
unification of all fields within the context of Brane world models.
References
[1
[2:
[3:
[4:
[e:
[7
[8j
[9
[10
[11
[12
[13
Bazanski, S.L. (1989) J. Math. Phys., 30, 1018.
Kahil, M.E. (2006), J. Math. Physics 47,052501.
Dahia, E., Monte,M. and Romaro,C. (2003), Mod. Phys. Lett.A18,1773;
gr-qc/0303044
Ponce de Leon, J. (2002) Grav. Cosmology, 8, 272; gr-qc/ 0104008
Seahra, S.,(2002) Phys Rev. D. 65, 124004 gr-qc/0204032
Liu, H. and Mashhoon, B. (2000), Phys. Lett. A, 272,26 ; gr-qc/0005079
Youm, D. (2000) Phys.Rev.D62, 084002; hep-th/0004144
Ponce de Leon, J. (2001) Phys Lett B, 523 ;gr-qc/0110063
Dick, R. Class. Quant. Grav.(2001), 18, Rl.
Papapetrou, A.(1951), Proc. Roy. Soc. Lond A 209, 248.
Maartens, R. (2004) Living Rev.Rel.7; gr-qc/0312059
Sen, D.K. , Fields/Particles, The Ryerson Press Toronto (1968).
Ponce de Leon, J. (2001) Mod. Phys. Lett. A 16, 2291 ;gr-qc/0111011.
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M-Theory and Dualities
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M-THEORY AND DUALITIES
OISIN A. P. Mac CONAMHNA
Theoretical Physics Group, Blackett Laboratory,
Imperial College, London SW7 2AZ, United Kingdom
and
The Institute for Mathematical Sciences,
Imperial College, London SW7 2PG, United Kingdom
1. Introduction
In this article, we review the contributions to the M-theory and dualities parallel
session at MGll. A broad range of topics was discussed, reflecting the diversity of
current research in string and M-theory.
One of the major themes for the session, reflecting its central importance in
modern research, was the AdS/CFT correspondence,1 with five talks (Klebanov,
Landsteiner, Mac Conamhna, Plefka and Stefanski) addressing various aspects of
gauge/gravity duality. The AdS/CFT correspondence, in its simplest form, states
that string or M-theory on an AdS background is dual to a non-gravitational
quantum field theory. This relationship, since its original proposal by Maldacena, has
lead to great insights into both the nature of quantum gravity with a negative cos-
mological constant, and also the physics of gauge theories. The gravity duals of
a family of "cascading" gauge theories were discussed by Klebanov; Landsteiner
discussed the problem of the non-decoupling of Kaluza-Klein states on the gravity
side of the correspondence which do not belong to the gauge theory spectrum; Mac
Conamhna reviewed progress in mapping out the space of supersymmetric AdS
geometries in M-theory; and Plefka and Stefanski discussed various aspects of the
quantisation of superstrings in the AdS 5 x S5 background of type IIB.
Another theme for the session was the physics of string and M-thcory com-
pactifications to lower dimensions. This is of course a very important question for
the construction of phenomenologically viable models of our universe within the
stringy paradigm. However even in contexts that are not phenomenologically
motivated, it can often be a very useful way to probe the theoretical structure of
string or M-theory. The contribution of Lust discussed orientifold compactification
of IIB string theory to four dimensions, and the issues of moduli stabilisation in the
scenario proposed by KKLT.2 Stelle discussed the topological considerations and
flux quantistion issues which much be addressed when M-theory is compactified on
Calabi-Yau five folds.
Herdeiro discussed the issue of chronology protection in string theory, and
proposed that the dynamical chronology protection agent is the condensation of light
winding strings near closed null curves. The talk of Mohaupt was concerned with
the status of electro-magnetic duality in the context of the OSV proposal,3 which
conjectures that the partition function of a supersymmetric black hole in a string
compactification is closely related to the topological string partition function. Har-
2863
2864
tong spoke on the global issues which arise in constructing D7 brane solutions of
IIB with sixteen supersymmetries.
2. The AdS/CFT correspondence: Klebanov, Landsteiner, Mac
Conamhna, Plefka and Stefanski
The AdS/CFT correspondence has provided an arena in which ideas about quantum
gravity may be tested in a rigorous setting. It has also led to greatly enhanced
understanding of supersymmetric gauge theories, and it is hoped that some day it
may be possible to extend these ideas to QCD. Much the best understood example
of the duality is that between IIB string theory on AdS5 x S5, and N = 4 super
Yang-Mills.
This is not surprising, as both sides of this duality have so much symmetry;
AdS5 x S5 is one of the three maximally supersymmetric solutions of IIB super-
gravity (the others are flat space and the plane wave). However while string
quantisation is understood, and the spectrum computable, on the other two backgrounds,
it is not for AdSs x S*5, and because of the motivation provided by the AdS/CFT
correspondence, this question has recently received much attention. This interest
has been further stimulated by the realisation that in planar perturbation theory
in the gauge theory, the dilation operator is isomorphic to the Hamiltonian of an
integrable quantum spin chain.4 This implies the existence of a Bethe ansatz - a
means of reformulating the problem of determining the quantum spectrum into
that of solving a set of non-linear algebraic equations, called the Bethe equations. A
conjecture has been made regarding the structure of these equations to all loops.5
On the string theory side, the classical AdSs x S5 string sigma-model is integrable,
and it is believed that this integrability should persist to the quantum level. So
far the spectrum can only be computed perturbatively around certain solvable
limits, for example the plane-wave limit of BMN.6 However, a conjecture for a set of
Bethe equations for the full quantum string has been made.7 This is the context
for the contributions of Plefka8 and Stefanski.9 In his talk, Plefka described a novel
gauged fixed description of strings on AdSs x S5, in uniform light cone gauge. This
choice of gauge allows for the exact determination of the gauge-fixed Lagrangian
and light-cone Hamiltonian. Using this description, perturbative quantisation in
the near plane-wave limit was performed. These results motivated the proposal of a
new set of light-cone Bethe equations, which were tested and verified in the spinning
string and flat space limits.
The contribution of Stefanski gave a brief review of the role of the Large Charge
Limit in gauge/string dualities. In particular, it was shown how certain Landau
Lifshitz sigma models emerge on the one side as non-relativistic limits of string
sigma models, and, on the other side, as continuum limits of certain integrable spin
chains which have been used to compute anomalous dimensions in the dual gauge
theory. A definition of the general Landau-Lifshitz sigma model on super-cosets was
also given.
2865
These contributions indicate some of the significant progress which has been
made in understanding the AdS5 x S5/J\f = 4 SYM correspondence, though much
remains to be done in exploring the full implications of integrability in this
context. However, since Maldacena's original proposal, the correspondence has been
extended to other AdS backgrounds in string theory, and the dual field theories
identified. A particularly well-studied case is for AdS5 spacetimes dual to J\f = 1 SCFTs
in four dimensions in IIB; the gravity backgrounds in this case are AdS5 x M5,
where M5 is a Sasaki-Einstein five-manifold. These backgrounds arise from the near-
horizon limit of D3 branes at the tip of a Calabi-Yau cone; the base of the cone is, by
definition, Sasaki-Einstein. One important recent devopment was the construction
of the doubly-countably infinite family of Sasaki-Einstein five-manifolds Yp-q ,10 and
the identification of the associated dual quiver gauge theroies.11 A particular case
of the Yp'q spaces is the conifold, T1'1. This has of course long been known; its dual
field theory12 has been intensively studied. The contribution of Klebanov13 pursued
this investigation. For p D3 branes at the tip of the T1'1 cone, the dual theory is
an SU(p) x SU(p) gauge theory. T1'1, in common with all members of the Yp'q
family, has a two-cycle on which D5 branes can wrap. Including M such branes, the
conformal invariance of the field theory is broken (inducing a logarithmic running
of the gauge couplings); the gauge group is deformed to to SU(M+p) x SU(p), and
of course, the AdS isometries of the gravity dual are also broken. The field theory
undergoes a cascade of Seiberg dualities14 along the RG flow, each of which reduces
p by M units; the gravity background is dual to the RG flow.15 In the work reviewed
by Klebanov in his talk,12 a complete analysis of the quantum structure of the
moduli space of this field theory, and its D-brane interpretation, is given. In the IR, the
theory undergoes confinement and chiral symmetry breaking, while in the UV the
couplings run logarithmically and it exhibits a duality cascade. The supergravity
dual is referred to as the "warped deformed conifold" ; the duals of the entire bary-
onic branch of confining vacua - the "resolved warped deformed conifolds"- have
been evaluated numerically. Also, in these backgrounds, it is possible to obtain a
small potential for D3 branes, depending on the radial coordinate of the gravity
solution. This suggests a possible embedding of an inflationary model in this scenario,
where the branes roll slowly to values of lower radius, somewhat along the lines of
those proposed by KKLMMT.16
The J\f = 1 AdS^/CFTA correspondence also provided motivation for the
material discussed by Landsteiner.17 In recent work, Nunez and Gursoy18 studied the
field theory for D5 branes wraped on an S2 in a Calabi-Yau, preserving J\f = 1.
They observed that the scale of the KK masses was of the order of the scale of
the gauge theory - thus, the KK states could not be disentangled from the gauge
theory dynamics. They also observed that this could be improved by looking at a
dipole deformed D5 brane theory. This involved turning on the B-field in the gravity
dual, with one leg along the S2 and one leg transverse to the brane worldvolume,
which reduced the relative size of the compactification S2. Turning on a B-field in
the supergravity background in this fashion leads to a non-commutative field the-
2866
ory on the brane worldvolume, the so-called dipole deformed theory.19 This work
motivated the study reviewed by Landsteiner, investigating in detail the issue of
KK masses in dipole deformed theories from a purely field theoretic perspective. As
part of the investigation, it was found that dipole scalar field theories might allow
for the spontaneous breaking of translation symmetry.
In addition to fully working out the AdS/CFT dictionary for examples where
both the field theory and the gravity backgrounds are known explicitly, considerable
effort has been devoted to finding new examples of the duality. General techniques
for classifying the local geometrical properties of all supersymmetric spacetimes
admitting any desired number of arbitrary Killing spinors have been developed,20
making use of the notion of a G-structure. The existence of a set of Killing spinors
implies the preferred local reduction of the frame bundle of a manifold to a sub-
bundle, and the G-structure thus defined allows one to encode the necessary and
sufficient conditions for the existence of the Killing spinors in a way which makes
manifest their geometrical content. An obvious target for these techniques is the
AdS spacetimes of string and M-theory. Indeed, much work has already been done
in this direction,21 and the Yv,q spaces were found directly from the G-structure
classification of AdS 5 spaces in M-theory. The supersymmetry conditions which
arise as a result of a G-structure classification are first order partial differential
equations (resulting as they do from a repackaging of the information contained in
the Killing spinor equation) which must of course be solved to find explicit new
metrics. However the geometrical insight provided by the G-structure formalism is
often sufficient to allow for the integration of the supersymmetry conditions (and
any other field equations/ Bianchi identities which must be imposed). The
contribution of Mac Conamhna reviewed further progress in the application of these
techniques to the problem of mapping out the AdS landscape of M-theory. The
objective of this project is ultimately to give a set of first order equations, expressed as
algebraic conditions on the intrinsic torsion of an appropriate G-structure, which,
for any given dimensionality and number of preserved supersymmetries, are
satisfied by every AdS spacetime in M-theory. One of the major motivations in doing
so is that the results of the classification can then be used to construct new explicit
AdS solutions, and if the field theories can then be identified, new examples of
AdS/CFT duals. Because all AdS spaces should ultimately arise as the decoupling
limit of some brane configuration, the procedure reviewed by Mac Conamhna22
involved first classifying the geometry of various wrapped brane configurations, and
then taking the AdS limit. This procedure has led to the construction of many new
explicit infinite families of supersymmetric AdS^ solutions of M-theory and type
IIB supergravity,23 reviewed in the talk. These new solutions are dual to N = (2, 0)
two-dimensional superconformal field theories. In M-theory, the AdS spaces arise as
the near-horizon limit of M fivebranes wrapped on a Kahler four-cycle in a Calabi-
Yau fourfold, with membranes extended in the directions transverse to the fourfold
and intersecting the fivebranes in a string.
In summary, the results presented in relation to the AdS/CFT correspondence
2867
during the session reflected, and were made possible by, the great advances which
have been made in recent years in understanding quantum gravity with a
negative cosmological constant, and the corresponding advances in our understanding
of supersymmetric gauge theories. They also highlighted some of the outstanding
challenges and difficulties which need to be addressed in the future; fully exploring
the consequences of integrabilty for AdS^ x S5; completing the AdS/CFT
dictionary for other known examples; and finding new examples of the duality to study.
And of course, there are very many other questions which can be asked to deepen
our understanding of the AdS/CFT correspondence, which are likely to ensure its
continued status as a centrally important topic of modern research in string and
M-theory.
3. The physics of compactification: Lust and Stelle
The most pressing problem faced by string theory, regarded as a branch of physics
as opposed to pure mathematics, is the issue of making contact with, and
predictions for, experimental data. This question is made particularly urgent as the LHC
is nearing completion. Two main approaches to this problem have been proposed.
The first is the brane-world scenario,25 which proposes that our universe may be
described as the worldvolume of a brane embedded in a higher-dimensional bulk
space. This scenario was not discussed in the M-theory parallel session, though it
received much attention elsewhere during the conference. The second main proposal
for connecting string theory with observation is the idea of compactification, that
the extra six (or in M-theory, seven) dimensions are tightly rolled up in a compact
space, on a scale small enough to have avoided detection hitherto. This idea has
been around for a long time, and there exists an enormous literature on the subject;
an up-to-date review has recently appeared.26 Since the discovery of the first models
of universes with accelerating cosmological expansion by KKLT,2 it has been
realised that a vast number of (apparently) anthropically viable universes are allowed
within the stringy paradigm. This has been dubbed the "landscape" of string
theory by Susskind,27 and its existence has stimulated much debate in the community.
In the abscence of any recognised principle which allows for the selection of one
phenomenologically viable model in favour of another, one seems to be forced to
accept a great reduction in the power of string theory to explain why the universe is
as we observe it to be. Of course the hope remains that the universe we observe will
ultimately prove to be completely describable within a string theory framework. In
the current state of the art, in order to explain why any particular model should in
fact be realised by our universe, statistical28 and anthropic29 arguments are being
proposed.
The debate inspired by the discovery of the landscape and its implications for the
predictive power of string theory has not prevented continued progress in
constructing and understanding specific concrete stringy cosmological models. The main
challenge overcome by the KKLT scenario was the problem of fixing all the moduli of
2868
specific compactifications. The moduli appear as massless scalar fields in the low-
energy effective description, and fixing them involves generating a potential, by
including fluxes, branes, anti-branes and non-perturbative effects in the model. The
contribution of Lust to the session30 reviewed work in which the KKLT proposal
could be implemented (or ruled out) for a variety of specific orientifold
compactifications of IIB string theory.
The intensive study of Calabi-Yau manifolds over many years, inspired by string
theory compactification, means that Calabi-Yau spaces provide controlled
theoretical laboratories in which less phenomenologically motivated issues may be
addressed. The understanding of topological strings on Calabi-Yau manifolds is an
essential piece of input into the OSV proposal for computing black hole partition
functions, which forms the arena for the material presented during the session by
Mohaupt, and which is reviewed in more detail below. The talk of Stelle31 was
motivated by the desire to study quantum corrections to d = 11 supergravity, in
the context of compactification on Calabi-Yau fivefolds. The fact that Calabi-Yau
manifolds are so well understood allows the effect of these corrections to be
studied in detail, in a controlled fashion. The starting point for Stelle's talk was the
purely gravitational solution of d — 11 supergravity, given by the direct product of
a timelike line with a Calabi-Yau fivefold, with vanishing four-form flux. To obtain
a solution of M-theory, various considerations, beyond those required in
supergravity, must be taken into account. One of these is a Dirac quantisation condition for
the four-form flux, which is needed to ensure single-valuedness of M-brane wave-
functions, and to ensure invariance under large three-form gauge transformations.32
Another is the presence of a Ci?4 term, required to cancel anomalies on the world-
volumes of M5 branes. This leads to quantum corrections to the Killing spinor
and field equations of the supergravity. These correction terms have received much
study, and have been calculated using various approaches33 in M- and string theory.
However, the effect of these terms on the geometry of M-theory solutions has not
been widely studied, and filling this gap is the primary motivation for the work
described by Stelle. In summary, it is found that the four-form is sourced gravita-
tionally when the corrections are included, and that flux quantisation then imposes
a topological constraint on the ten-manifold. The ten-manifold is itself deformed
away from SU(5) holonomy, but while preserving SU(5) structure, and a warping
for the timelike direction is also induced.
4. Other topics: Hardeiro, Mohaupt and Hartong
The remaining talks of the session gave a flavour of some other current research
topics. One of these is the pressing issue of understanding chronology protection,
and more generally, time-dependent geometries in string theory. Generally speaking,
these are much less well-understood than static (and especially, supersymmetric)
spacetimes; and clearly a better understanding of the issues involved is required for
the successful application of string-theoretic ideas to our universe. Much work has
2869
been done on this topic. From a theoretical point of view, one of the most blatant
issues which must be resolved is that of chronology protection; spacetimes with closed
timelike curves abound in string theory. Many of these are Godel-like, and can even
be supersymmetric;24 other examples can be found from over-rotating black holes
where the chronology-violating region is outside the horizon34 or over-rotating su-
pertubes.35 The talk of Herdeiro36 addressed the issue of chronology protection in
string theory, and gave a proposal for a dynamical chronology protection agent.
The famous chronology protection conjecture of Hawking37 proposed long ago that
chronology is protected by UV quantum field theoretic effects. Hawking's argument
was that the one-loop energy momentum tensor of a quantum field grows without
bound in the vicinity of a closed null curve, backreacting on the geometry to
either form a singularity or prevent the chronology violating region from forming.
In contrast, the string-theoretic chronology protection agent proposed by Herdeiro
is a manifestation of IR physics, and involves the condensation of certain string
states in a chronology-violating region. The idea is that winding string states
become light just before they wrap a closed null curve, and that a phase transition
occurs when their proper length is of the order of the string scale. The winding
states condense in the chronologically pathological region of the spacetime, and it
was conjectured that the end-point is a chronologically well-behaved target space
geometry. The phase transition studied is closely analagous to the Hagedorn phase
transition, where winding string states become light for a compactification of the
order of the string scale.38 This proposal for chronology protection in string theory
was examined in detail in the talk in the context of a toy model, the O-plane orb-
ifold.39 There the proposed condensation of winding modes is observed to happen,
and the mechanism was conjectured to occur generally.
Another major theme of current research addressed at the conference was the
issue of black hole entropy. The celebrated semiclassical Beckenstein-Hawking
relationship between the entropy of a black hole and the area of its event horizon has
long posed a challenge to quantum gravity theories to provide an account of the
black hole microstates. This was first achieved in the context of string theory, at
least for certain supersymmetric black holes, by the work of Strominger and Vafa.40
This was later understood to be a special case of the AdS/CFT correspondence,
which has since provided much more insight in the microscopic origin of black hole
entropy.41 Recently, a new proposal has been made3 relating the black hole
partition function in the context of Calabi-Yau compactifications of string theory to
the topological string partition function. Important elements in this proposal are
Wald's definition of black hole entropy42 based on Nother charge, and the attrac-
tor mechanism, whereby supersymmetry enhancement drives the compactification
moduli to fixed values, determined by the black hole charges, at the event horizon.
The attractor mechanism persists in the presence of higher order corrections.43 The
contribution of Mohaupt44 first gave a brief review of these ideas. It was then
discussed how the macroscopic entropy and attractor equations for supersymmetric
black holes in J\f = 2 supergravity theories can be derived from a variational prin-
2870
ciple for a certain "entropy function", which was computed in the presence of R2
and non-holomorphic corrections to the supergravity. The intimately-related issue
of the covariance of the OSV proposal under electric-magnetic duality was discussed
in detail. This generalisation of the OSV proposal was tested for the cases where the
microscopic degeneracies can be computed in string theory. For "large" black holes,
where the horizon scale is much greater than the string scale, precise agreement was
found at the semiclassical level. For "small" black holes, with horizons of the
order of the string scale, the results were inconclusive, due to the difficulties involved
in performing reliable calculations. This has continued to be a very active area of
research,45 and recently a review of the relationship between black hole entropy,
topological strings and the attractor mechanism has appeared.46
The contribution of Hartong47 was concerned with the study of half-BPS D7
brane solutions of IIB supergravity. These configurations were first studied in lower-
dimensional supergravity48 and subsequently in ten dimensions.49 D7 branes have
since been used (as part of the D3-D7 system) as ingredients in phenomenological
model-building in string theory, both for particle physics50 and cosmology.51 Su-
persymmetric D7 brane solutions have also been studied in the twelve-dimensional
context of F-theory.52 The main motivation of the work discussed by Hartong was
to re-examine half-BPS D7 brane configurations directly in IIB, without relying on
a higher-dimensional F-theory picture. A careful analysis of the conditions implied
by the existence of a globally-defined Killing spinor was presented, in the presence
of SL(2,Z) invariant source terms added to the equations of motion. New super-
symmetric configurations were found, in particular, some containing objects whose
monodromies are not related to the monodromy of a D7 brane by an SL(2, Z)
transformation. Hartong concluded by speculating on the nature of these objects, and
their possible relationship with 07 planes, by analogy with what is observed for the
D8-08 system in IIA.53
5. Conclusions
In any review of this kind, it is only possible to give a snapshot description of the
current state of the vast and dynamic field of M- and string theory. Nonetheless,
the proceedings of the M-theory and dualities parallel session reflected many of
the major trends in the subject at present. The AdS/CFT correspondence
continues to play a central role, and the whole body of work generated as a result
of this idea probably represents the greatest success of the subject over the past
decade. The greatest challenge is provided by the pressing need to connect the
theory with observation, and the discovery of the landscape suggests serious limitations
for what string theory might ultimately hope to achieve from a phenomenological
point of view. However, the fact that time-dependent (and causally pathological)
backgrounds are starting to be properly understood, offers hope of progress in this
direction. The understanding of the entropy of some black holes in string theory
constitutes another major success of the field, and the relationship between black
2871
hole and topological string partition functions is another direction in which this
success may be pursued. It will be interesting to track the development of all the
ideas discussed during the session in the future.
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33. E. Witten, "On Flux Quantization In M-Theory And The Effective Action",
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AdS SPACETIMES IN M-THEORY
JEROME P. GAUNTLETT, OISIN A. P. MAC CONAMHNA, TONI MATEOS and
DANIEL WALDRAM
Theoretical Physics Group, Blackett Laboratory,
Imperial College, London SW7 2AZ, United Kingdom
and
The Institute for Mathematical Sciences,
Imperial College, London SW7 2PG, United Kingdom
The AdS/CFT correspondence1 has given us many insights into the properties of
quantum gravity with a negative cosmological constant. In this contibution, we will
describe ongoing progress in the classification of supersymmetric AdS solutions of
M-theory,2 together with the construction of many new infinite familes of explicit
AdSz solutions, dual to TV" = (2, 0) superconformal field theories in two
dimensions.3'4 We have two major motivations in performing this classification, the first
being to explore the general geometrical properties of all AdS spacetimes of a given
dimensionality and supersymmetry, and so map out the space of supergravity duals
of CFT ground states in M-theory. Secondly, the geometrical insight this provides
is of much value in the construction of explicit new solutions, which we have been
able to exploit.
In performing the classification, we have exploited the relationship between the
supergravity description of wrapped brane spacetimes with that of their AdS
limits. There are many ways in which branes in M-theory can wrap supersymmetric
cycles, and so admit supersymmetric AdS near-horizon limits. In keeping with the
general philosophy of AdS/CFT, one would expect that (together with
configurations involving only space-filling branes) all AdS spacetimes in M-theory may be
obtained in this fashion - in other words, for every supersymmetric AdS spacetime
there exists a dual field theory associated to a brane configuration admitting a
supergravity description. The cases we have studied so far are tabulated below. We
Cycle
Kahler 4-cycle
Co-associative
Kahler 4-cycle
Associative
Special Lagrangian
Kahler 2-cycle
Kahler 2-cycle
Holonomy
SU(A)
G2
SU(3)
G2
SU(3)
SU(3)
SU(2)
World-volume
R1-1
R1*1
R1-1
R1-2
R1*2
R1-3
R1*3
SUSY
A/" =(2,0)
AT =(2,0)
Af =(4,0)
Af = l
M = 2
M = l
N = 2
derive the AdS supersymmetry conditions in a somewhat indirect fashion. First we
derive the supersymmetry conditions for the wrapped brane spacetimes, using G-
structure techniques. This derivation is technically easier than performing a direct
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analysis of the Killing spinor equation for the AdS limits. The form of the wrapped
brane metrics determined by supersymmetry is in each case
ds2 = L~1ds2(R1'r) + ds2(Mi0-r-s) + ds2(Rs), (1)
where the warp-factor L and the metric on .Mio_r_s are independent of the
Minkowski coordinates, Rx'r represents the unwrapped brane worldvolume, and
Aiio-r-s admits an appropriate G-structure. Then, by taking an AdS limit of the
wrapped brane metric, flux and supersymmetry conditions, we derive the AdS
supersymmetry conditions. The limiting procedure involves picking out an AdS radial
direction from the space transeverse to the Minkowki factor, imposing vanishing of
flux components along this direction, and imposing suitable dependence of the warp
factor on the AdS radial coordinate. Full details of this procedure are to be found
in2 . There the supersymmetry conditions for M fivebranes wrapping supersymmet-
ric cycles in manifolds of G-2, SU(3) or SU{2) holonomy, together with those of
their AdS limits, are derived and discussed in detail.
The new supersymmetric AdS% solutions of string and M-theory we have found
arise in M-theory as the near-horizon limit of M5 branes wrapped on Kahler four-
cycles in Calabi-Yau four-folds, with membranes extended in the directions
transverse to the Calabi-Yau and intersecting the fivebranes in a string. These AdS
solutions are dual to N = (2, 0) two dimensional CFTs. The solutions containing a
T2 factor admit a reduction to IIB. In IIB, the only non-zero flux is the five-form
(and the dilaton is constant) so in IIB we interpret these solutions as coming from
the near-horizon limit of D3 branes wrapped on Kahler two-cycles in Calabi-Yau
four-folds. The M-theory solutions are discussed in detail in3 , while global
properties and flux quantisation for eight doubly countably infinite families of these
solutions in IIB are studied in4 .
Our M-theory ansatz for these solutions is as follows. We look for warped AdS3
solutions of the form
ds2 = to2 [ds2(AdS3) + ds2{M8)] , (2)
where Ms is is an S2 bundle over a base manifold B6 which is itself either a Kahler-
Einstein six-manifold KEq or the product of Kahler-Einstein manifolds KEA x
KE-2 or KE2 x KE2 x KE2. This ansatz was motivated by that which led to the
construction of the Yp'q spaces5 . These arise, in M-theory, as the near-horizon limit
of M5s wrapped on Kahler two-cycles in Calabi-Yau three-folds, and the geometry
is very similar; in both cases, of the five directions transverse to the M5s, four arc
tangent to the Calabi-Yau. More motivation of this ansatz is given in3 .
Given our ansatz, we find new non-singular AdS solutions when Be is one of
KE+ x KE+ x H2, KE+ x H2, KE^ x KE^ x S2, KEl x S2, KE+ x KE+ x T2,
or KE^ x T2. Of particular interest are the solutions with a T2 factor, as these
may be reduced and dualised to IIB. In IIB, the solutions with Be = KE% x T2
have metric
ds2 = \ [ds2(AdS3) + ds2{M7)] , (3)
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where the metric onjVly is given by
with Dip = dip + P, Dz = dz - g(y)Dip, and
9{V) = 2{y*a~2y + a)> ^v) = ^ ~ ^+ ?>ay - a2 . (5)
Here a is a constant and dP = J, the Kahler form of the positive scalar curvature
Kahler-Einstein four-manifold. There exist eight choices for the KE^: CP2, S2 xS2,
or a del Pezzo surface dPk, k = 3,..., 8. An analysis of the global regularity conditions
for these local solutions shows that there exist eight regular doubly countably infinite
compact familes, labelled by comprime integers (p,q). Quantising the periods of the
five-form flux over any five-cycle D <E R^Mt.TL) quantises the AdS radius. The
central charges of the field theory duals may be computed according to c = 31/2G^,
where I is the AdS length and G^ is the three-dimensional Newton constant, and
we find that
9pq2(p + mq) Mq 2
3p2 + 3mpq + m2q2 m2h2
where for S2 x S2 we have m = 2, M = 8; for CP2 we have m = 3, M = 9; and
for the del-Pezzos dPk, we have m = 1, M = 9 — k. Finally, h = hcf{M/m2,q},
and n is an arbitrary integer counting the number of copies of the minimal D3
brane configuration. It would be very interesting to construct the families of field
theory duals with these central charges, and also to extend the classification to other
wrapped brane configurations, in the hope of finding more explicit AdS solutions.
References
1. J. M. Maldacena, "The Large N Limit of Superconformal Field Theories and Super-
gravity", Adv.Theor.Math.Phys. 2 (1998) 231-252; Int.J.Theor.Phys. 38 (1999) 1113-
1133, hep-th/9711200.
2. J. P. Gauntlett, O. A. P. Mac Conamhna, T. Mateos and D. Waldram, "AdS space-
times from wrapped M5 branes", JHEP 0611 (2006) 053, hep-th/0605146.
3. J. P. Gauntlett, O. A. P. Mac Conamhna, T. Mateos and D. Waldram, "New super-
symmetric AdS3 solutions", Phys.Rev. D74 (2006) 106007, hep-th/0608055.
4. J. P. Gauntlett, O. A. P. Mac Conamhna, T. Mateos and D. Waldram, "Supersym-
metric AdS3 solutions of type IIB supergravity", Phys.Rev.Lett 97 (2006) 171601,
hep-th/0606221.
5. J. P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, "Supersymmetric AdS5
solutions of M-theory", Class.Quant.Grav. 21 (2004) 4335-4366, hep-th/0402153.
GLOBAL ASPECTS OF SEVEN-BRANE CONFIGURATIONS
ERIC A. BERGSHOEFF* and JELLE HARTONG"
Centre for Theoretical Physics, University of Groningen,
Nijenborgh 4, 9747 AG Groningen, The Netherlands
*E.A. Bergshoeff&rug. nl
** J .Hartong@rug.nl
TOMAS ORTIN
Instituto de Fi'sica Tedrica UAM/CSIC,
Facultad de Ciencias C-XVI, C.U.,
Cantoblanco, E-28049-Madrid, Spain
Tomas. Ortin&cern. ch
DIEDERIK ROEST
Departament Estructura i Constituents de la Materia,
Facultat de Fisica, Universitat de Barcelona,,
Diagonal, 647, 08028 Barcelona, Spain
droest@ecm.ub. es
In order to construct globally well-defined 7-brane solutions we postulate the existence
of a new type of 7-brane. We show that these new 7-branes play an important role in
understanding both the existing F—theory 7-brane configurations as well as more general
7-brane configurations.
Keywords: Branes; F-theory; Supersymmetry.
1. Introduction
A single 7-brane forms an inconsistent background. The simplest consistent super-
gravity 7-brane solution which has a perturbative string theory interpretation is
obtained by applying two T-duality transformations to type I string theory. This
background can be interpreted as the following orientifold of type IIB supergravity:
Mink1]7 x T2/Z2- The orbifold T2/Z2 has four fixed points and each corresponds
to a coincident set of four D7-branes plus one 07-plane.1 The situation in which
the four D7-branes are no longer coincident is described by F-theory2 on K3. It is
known1'3 that when the four D7-branes are separated from each other the orientifold
plane splits into two non-perturbative parts, each with an SL(2,Z) monodromy
Mi^TM^l for some 5L(2,Z) matrix M1>2, where Tt = t + 1 with r the complex
axidilaton field. One of the purposes of Ref. 3 was to show that this F-theory
solution can be interpreted as type IIB supergravity in the presence of a new type of
7-brane, which we refer to as the "det Q > 0 7-brane", for reasons that will become
clear soon. In Ref. 3 it is shown that the F-theory 7-brane configurations form a
subset of a much wider set of solutions.
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2. Seven—branes and supersymmetry
The Einstein metric and Killing spinor e for the most general 7-brane solution are
given by4~6
ds2 = -dt2 + dx72 + (IxnT)\f\2dzdz , e = (///)1/4 e0 , (1)
where z = x8 + ix9 with x8,x9 the coordinates transverse to the 7-branes and
where eo is a constant spinor which satisfies 7z*eo = 0. The functions r and / are
holomorphic functions of z and are defined on the Riemann sphere. They transform
under SL(2, Z) as follows
— ArS^, f^(cr + d)f, A=(^)e5L(2,Z). (2)
In Ref. 3 source termsa are introduced with charges p, q, r. The local solutions
to the sourced equations of motion are characterized by the monodromy r —> eQr
where Q is a charge matrix defined by
The D7-brane is an element of the set det Q = 0. We assume that 7-branes for which
the monodromy eP has trace less than 2, i.e. det Q > 0, also exist.
In order to construct finite energy solutions we need to divide out type IIB
supergravity by S*L(2,Z). The moduli space of this theory is given by the orbifold
{r upper half-plane}/PS'L(2, Z). Within this moduli space there are three special
points (orbifold points) which are fixed points of eQ; these are zoo, p = ( —l+i'v/3)/2,
and i. With each fixed point of the monodromy e^ we associate a 7-brane. The D7-
branes is associated to r = ioo, and with r = p, i we associate branes with some
positive value of detQ- Any 7-brane configuration can be considered as a certain
mapping of these three orbifold points to the transverse space.
3. F—theory 7-brane configurations
The 7-brane configurations of F-theory have the property that the monodromy of r
close to the points Zj, zp (defined by r(zi, zp) = i, p) is the identity in PSL{2, Z) and
T around ,zloo. Further it is required that the function / has no zeros. To construct
such solutions one must take coincident detQ > 0 branes of opposite masses. In
this case the det Q > 0 branes are not noticeable from any local analysis, but they
do have a non-trivial effect on the global positioning of the branch cuts, see figure
1 for an example of an F-theory solution with six non-trivial T-monodromies. The
splitting of the 07-plane, mentioned in the introduction, can be understood from
the global properties of the branch cuts ending on the det Q > 0 branes.
aFor reasons explained in Ref. 7 the coupling of det Q > 0 branes to type IIB supergravity is not
straightforward in the (r, f) parametrization of the coset manifold 51/(2, R)/50(2); it requires a
different parameterization. In Ref. 3 a trick is used to circumvent this difficulty.
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Fig. 1. F-theory solution with six non-trivial T-monodromies. The filled (dashed) lines are T
(S) branch cuts.
Acknowledgments
We would like to thank D. Sorokin for useful discussions. T.O. and D.R. would like
to thank the University of Groningen for hospitality, while J.H. and D.R. would like
to thank the Universidad Autonoma in Madrid for hospitality. E.B. and T.O. are
supported by the European Commission FP6 program MRTN-CT-2004-005104 in
which E.B. is associated to Utrecht university and T.O. is associated to the IFT-
UAM/CSIC in Madrid. The work of E.B. and T.O. is partially supported by the
Spanish grant BFM2003-01090. The work of T.O. has been partially supported by
the Comunidad de Madrid grant HEPHACOS P-ESP-00346. Part of this work was
completed while D.R. was a post-doc at King's College London, for which he would
like to acknowledge the PPARC grant PPA/G/O/2002/00475. In addition, he is
presently supported by the European EC-RTN project MRTN-CT-2004-005104,
MCYT FPA 2004-04582-C02-01 and CIRIT GC 2005SGR-00564. J.H. is supported
by a Breedte Strategic grant of the University of Groningen.
References
1. A. Sen, Nucl. Phys. B 475, 562 (1996).
2. C. Vafa, Nucl. Phys. B 469, 403 (1996).
3. E. A. Bergshoeff, J. Hartong, T. Ortin and D. Roest, arXiv:hep-th/0612072.
4. B. R. Greene, A. D. Shapere, C. Vafa and S. T. Yau, Nucl. Phys. B 337, 1 (1990).
5. G. W. Gibbons, M. B. Green and M. J. Perry, Phys. Lett. B 370, 37 (1996).
6. E. Bergshoeff, U. Gran and D. Roest, Class. Quant. Grav. 19 (2002) 4207.
7. E. A. Bergshoeff, J. Hartong and D. Sorokin, work in progress.
DUALITY AND BLACK HOLE PARTITION FUNCTIONS*
THOMAS MOHAUPT
Theoretical Physics Division, Department of Mathematical Sciences,
University of Liverpool, Peach Street,
Liverpool L69 7ZL, United Kingdom
Thomas. Mohaupt&liv. ac. uk
Supersymmetric black holes provide an excellent theoretical laboratory to test ideas
about quantum gravity in general and black hole entropy in particular. When four-
dimensional supergravity is interpreted as the low-energy approximation of ten-
dimensional string theory or eleven-dimensional M-theory, one has a microscopic
description of the black hole which allows one to count microstates and to compare the
result to the macroscopic (geometrical) black hole entropy. Recently it has been
conjectured that there is a very direct connection between the partition function of the
topological string and a partition for supersymmetric black holes. We review this idea
and propose a modification which makes it compatible with electric-magnetic duality.
Our setup for constructing supersymmetric black hole solutions is A^ = 2
supergravity couled to n vector multiplets. This arises^ as the effective field theory of heterotic
string compactifications on K3 x T2 and of type-II string theory on Calabi-Yau
threefolds. The field equations are invariant under Sp(2n + 2,IR) rotations, which
generalize the electric-magnetic duality rotations of Einstein-Maxwell theory.^ As
a consequence, all vector multiplet couplings are encoded in a single holomorphic
function called the prepotential F. This function must be homogenous of degree 2
in its variables Y1, which provide homogenous coordinates on the scalar manifold
Mvm- The Kahler potential for the metric on Mvm can be expressed in terms of
the holomorphic prepotential. The resulting geometry is known as special (Kahler)
geometry.1'2 It is possible to include a certain class of higher derivative terms
involving the square of the Riemann tensor and arbitrary powers of gauge field strengths,
by giving the prepotential an explicit dependence on the so-called Weyl multiplet.
Associated to these terms is an infinite series of field-dependent couplings. In type-II
compactifications these couplings can be computed in terms of the free energy of
topologically twisted string theory.3'4
As long as we neglect the higher derivative terms, we are dealing with a
generalized Einstein-Maxwell theory with several abelian gauge fields and field-dependent
couplings, plus a scalar sigma-model. The supersymmetric black hole solutions of
such a theory are natural generalizations of the extremal Reissner-Nordstrom black
hole. Besides that the black hole now carries several electric and magnetic charges,
the new feature is that we have scalar fields which vary non-trivially as a function
"This article is based on results obtained in collaboration with Bernard de Wit, Gabriel Lopes
Cardoso and Jiirg Kappeli.
ttogether with further matter multiplets which are irrelevant for our purposes.
*If the supergravity action is the low energy effective action of a string compactification, then
string dualities, such as S-duality and T-duality, are embedded into the symplectic group.
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of the radial variable.§ At infinity, the solutions are asymptotically flat and the
scalars can take arbitrary values in Mvm • The behaviour at the horizon is radically
different: the scalars cannot take arbitrary values but must take fixed point
values which are determined by the electric and magnetic charges of the black hole.
This is the so-called black hole attractor mechanism,5 which generalizes to the case
where higher derivative terms are included.6,7 Since both metric and gauge fields
are determined by the scalar fields through supersymmetry, it follows that the area
of the event horizon is a function of the electric and magnetic charges, and does not
depend on the values of the scalar fields at infinity. Once higher curvature terms
are included in the action, the black hole entropy is no longer given by one quarter
of the area of the event horizon^ but is given by the surface charge of the Killing
vector field which becomes null on the horizon.8 When evaluating the surface charge
for supersymmetric black holes in N = 2 supergravity, one sees that the entropy is
given by the sum of two symplectic functions of the charges.6 While the first term is
the area of the horizon divided by 4, the second term depends only on the couplings
of the higher derivative terms. Therefore the black hole entropy is modified in two
ways: first through the modification of the area itself, second by the deviation from
the area law. The microscopic state degeneracy9,10 agrees with black hole entropy
if and only if both corrections are taken into account.6
If one performs a partial Legendre transformation of the black hole entropy,
which replaces the electric charges by the associated electrostatic potentials, one
obtains the imaginary part of the 'generalized prepotentiar.11 This is a power series
in the Weyl multiplet which has as its coefficients the prepotential (determining the
two-derivative couplings) and the coupling functions of the higher derivative terms.
By the relation between couplings in the effective action and the topological string,
this function is proportional to the real part of the (holomorphic) free energy of
the topologically twisted type-II string. This suggests to interprete the imaginary
part of the generalized prepotential as the free energy of the black hole, and one
obtains the 'OSV-relation'11 Zbh = \Ztop\2-. which relates the black hole partition
function (exponential of the free energy) to the partition function of the topological
string. However, many details of this proposal need to be made more precise. One
is whether the relation is meant to be an exact statement (strong version) or as an
asymptotic statement in the limit of large charges, which corresponds to the semi-
classical limit (weak version). Before reviewing the evidence supporting the weak
version, we need to address another point. By definition, the black hole free energy
is a function of the magnetic charges and of the electrostatic potential. Thermody-
namically this corresponds to a mixed ensemble, where the magnetic charges have
been fixed, while electric charges fluctuate and the corresponding chemical
potentials are fixed.11 This implies that a fundamental property, namely covariance with
respect to symplectic transformations is not manifest. As a consequence, it is not
^We only consider spherically symmetric solutions here,
are using Planckian units.
2883
clear whether the proposal is compatible with string dualities. In fact, discrepancies
between the actual microscopic state degeneracy and the state degeneracy predicted
by the OSV conjecture show that the OSV-relation must be modified.12'13 A natural
way of deriving the modification is based on the observation that the full Legendre
transformation of the black hole entropy, where both electric and magnetic charges
are replaced by the corresponding potentials has a natural meaning: the resulting
function is a Hesse potential for the metric on the scalar manifold.13 Moreover,
the relations between entropy, free energy (mixed ensemble), Hesse potential and
attractor equations can be formulated in terms of a variational principle.13'14 This
suggests to interprete the Hesse potential as the free energy of the black hole, but
now with respect to a canonical instead of a mixed ensemble. One can show that
this proposal leads to a specific correction factor in the OSV-relation. Explicit tests
can be performed in compactifications with A^ = 4 supersymmetry, which can be
treated within the N = 2 formalism explained in this article.15 Subleading
corrections to the state degeneracy have been computed16^18 and the result agrees with
the canonical black hole partition function proposed in13 in the semi-classical limit.
The agreement is impressive as it involves an infinite series of non-perturbative
corrections to the effective action." The precise relation between the canonical black
hole partition function and the topological string remains to be clarified.
References
1. B. de Wit and A. Van Proeyen, Nucl. Phys. B245, p. 89 (1984).
2. B. de Wit, P. G. Lauwers and A. Van Proeyen, Nucl. Phys. B255, p. 569 (1985).
3. M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Commun. Math. Phys. 165, 311
(1994).
4. .1. Antoniadis, E. Gava, K. S. Narain and T. R. Taylor, Nucl. Phys. B413, 162 (1994).
5. S. Ferrara, R. Kallosh and A. Strominger, Phys. Rev. D52, 5412 (1995).
6. G. Lopes Cardoso, B. de Wit and T. Mohaupt, Phys. Lett. B451, 309 (1999).
7. G. Lopes Cardoso, B. de Wit, J. Kappeli and T. Mohaupt, JHEP 12, p. 019 (2000).
8. R. M. Wald, Phys. Rev. D48, 3427 (1993).
9. J. M. Maldacena, A. Strominger and E. Witten, JHEP 12, p. 002 (1997).
10. C. Vafa, Adv. Theor. Math. Phys. 2, 207 (1998).
11. H. Ooguri, A. Strominger and C. Vafa, Phys. Rev. D70, p. 106007 (2004).
12. D. Shih and X. Yin, JHEP 04, p. 034 (2006).
13. G. Lopes Cardoso, B. de Wit, J. Kappeli and T. Mohaupt, JHEP 03, p. 074 (2006).
14. K. Behrndt et al., Nucl. Phys. B488, 236 (1997).
15. G. Lopes Cardoso, B. de Wit and T. Mohaupt, Nucl. Phys. B567, 87 (2000).
16. R. Dijkgraaf, E. P. Verlinde and H. L. Verlinde, Nucl. Phys. B484, 543 (1997).
17. G. Lopes Cardoso, B. de Wit, J. Kappeli and T. Mohaupt, JHEP 12, p. 075 (2004).
18. D. P. Jatkar and A. Sen, JHEP 04, p. 018 (2006).
"These corrections are world-sheet instantons from the point of view of the type-II string but
space-time instantons for the dual heterotic string.
M-THEORY ON CALABI-YAU FIVEFOLDS*
A.S. HAUPT and K.S. STELLE*
Institute for Mathematical Sciences,
Imperial College, London SW7 2PG, U.K. and
Theoretical Physics Group, Blackett Laboratory,
Imperial College, London SW7 2AZ, U.K.
a.haupt@imperial.ac.uk, k.stelle@imperial.ac.uk
It is important to test M-theory in regions of the moduli space that cannot be reached
by string theory and thus to probe M-theory's intrinsic structure. One such test is
the compactification of M-theory on manifolds with SU(5) holonomy, which require ten
Euclidean-signature dimensions and hence probe beyond anything that can be discussed
in perturbative string theory. We present some preliminary results of ongoing work that
is focused on studying the resulting one-dimensional effective action.
1. Topological considerations
At order a'3, the low energy effective action of M-theory contains a Green-Schwarz
term, A /\X%, coming from the M5-brane anomaly cancellation condition.1 Its
presence leads to a correction to the equation of motion for the 3-form gauge potential
d * G + ]-G A G + (27r)4/3X8 = 0, (1)
where G = dA and (3 := (2n)2a/3. Consider examining this equation for an M-
theory background M =lxX, where X is a compact ten-dimensional Ricci-flat
Kahler manifold, i.e. a Calabi-Yau fivefold (or CY5, for short).
The first term in Eq. (1) is exact and hence the other two are cohomologically
equivalent. Xg, which generally depends on the first two Pontrjagin classes of A4,
is now proportional to the fourth Chern class of X. Equation (1) thus implies a
topological constraint of the form:
ci(X) = 12\g]A\g], (2)
where g := G/((2n)2y/j3)■ This is compatible with the g-flux quantization
condition,3 which for the case A4 = M.x X, reads as follows:
b] + ^etf4(X,Z). (3)
That is, g-flux is quantized in integer or half-integer units depending on the second
Chern class of X.
For compact smooth complete intersection CY5, we find C4(X) > 0, which forces
g-flux to be turned on at order \f]3. Vanishing of cn(X) for non-complete intersection
*Work in progress in collaboration with A.B. Barrett and A. Lukas.
'Research supported in part by the EU under MRTN contract MRTN-CT-2004-005104 and by
PPARC under rolling grant PP/D0744X/1.
aA subtle sign issue in this equation is discussed in Ref. 2, whose conventions are adopted here.
2884
2885
CY5 can be achieved by abandoning smoothness or compactness or by considering
orbifold constructions like (CY3 x T2 x T2)/(Z2 x Z2). In those cases, g-Hux is
turned on at order (3.
Since what we have in mind here is a dimensional reduction on a background
M. =KxI, one may ask how the resulting one-dimensional theory "knows about"
the topological constraint. The answer turns out to be in form of a "pure gauge
term, A Aw, that can be added to any general ansatz for the 3-form gauge potential
A. Here, A is a 1-form on R and a; is a harmonic (1, l)-form on X, implying that
A A a; is closed and hence pure gauge. After the reduction, A appears as a Lagrange
multiplier in the one-dimensional action and its variation reproduces Eq. (2).
2. Lowest order dimensional reduction on M. = R X X
A background M. = R10-2™*1 x CY„, n > 2, generically preserves 26~™ supersym-
metries with the effective lower-dimensional theory being supergravity coupled to a
non-linear a-model (NLerM) for the moduli of CY„.
For M. = R x X, we thus expect to find such a model with J\f = 2 supersym-
metry in one dimension, except for the additional peculiarity that supergravity in
dimensions < 3 has no propagating degrees of freedom. However, it does play a role
when coupled to matter (here, moduli fields), in that it imposes the vanishing of
the Hamiltonian and the siipercurrcnt as a constraint, thereby removing degrees of
freedom from the matter Lagrangian. In that respect, supergravity in dimensions
< 3 may be assigned negative degrees of freedom.
A convenient yet general zero-mode compactification ansatz for the eleven-
dimensional metric and 3-form turns out to be
ds2 = -N'2{T)V'2dT1 + 2g^{X)dz»dzz') A = ^[t)vp + c.c, (4)
where g^ is the Ricci-flat Kahler metric on X, V is the volume of X, {up} is a
basis of Harrr/2,1'^), £p are lS2'l\X) complex scalar fields and N is the einbein (or
lapse function) of one-dimensional gravity. The indices /i and v range over 1,..., 5
and 1,... 5, respectively.
As is typical for Calabi-Yau compactifications, the (lowest order) dynamics of
the internal CY5 metric is governed by Kahler and complex structure deformations,
which correspond to harmonic (1,1)- and (4, l)-forms and hence will appear
respectively as hM^^X) real and hl^A'1\X) complex scalar fields, denoted tl and Za, in
the one-dimensional action.b
The full bosonic NLerM that we find from the reduction is given by
/= ^ J' drN-1 {G[]'1\t)iV+AG^1\t,Z,Z)e^ + 2G^1\z,Z)ZaZh] , (5)
where the mass parameter m is the inverse square of the eleven-dimensional Newton
constant, i.e. m := k^2 and the dot means differentiation with respect to time r.
bComplex structure deformations of Calabi-Yau ra-folds correspond to harmonic (n — 1, l)-forms.
2886
The moduli space metrics appearing in Eq. (5) can be expressed purely in terms of
geometrical quantities such as intersection numbers.
We have also performed the full fermionic reduction up to the (fermi)2 level.
The full one-dimensional action has a wealth of symmetries. There is a global
GL(/i(1'1),R) x GL(/i(2-1),C) x GL(/i(4'x),C) target space symmetry which
corresponds to a change of basis of the harmonic (1,1)-, (2,1)- and (4, l)-forms,
respectively.
A remnant of eleven dimensional gauge invariance A —> A + dA is the fact that
the £p, unlike t% and Za, only appear through £p in the action. They thus enjoy
a continuous Peccei-Quinn shift symmetry £p —> £p + cp, for arbitrary complex
numbers cp, and are identified as axions.
The action is also invariant under wordline reparametrizations r —> t'(t) and
local N — 2 worldline supersymmetry. In ongoing work,4 we endeavour to find the
correct superspace version of the action thereby making the local J\f = 2 worldline
supersymmetry manifest.
3. Corrections to M. = K x X
The effects of order (3 corrections to the background M. = R x X for non-compact
X with ca{X) = 0 have been studied in Ref. 5. In this situation, one should allow for
non-vanishing g-fiux and a warp factor (with a "0-brane" structure) in the metric
ansatz of Eq. (4).
The modified Killing spinor equation, deduced from requiring the unbroken su-
persymmetries of the original M = K x X to persist in the face of the order /3
corrections, deforms X into a manifold that is not only non-Ricci-flat but also non-
Kahler but is still a complex manifold with vanishing first Chern class.5
Even though X no longer has SU(5) holonomy, one may still define a
generalized holonomy for the Killing spinor operator. The generalized transverse structure
group is SL(16, C) and the decomposition of the deformed Killing spinor under the
generalized holonomy still contains singlets, showing that supersymmetry remains
unbroken.
References
1. M. J. Duff, J. T. Liu and R. Minasian, Nucl. Phys. B452, 261 (1995), hep-th/9506126.
2. A. Bilal and S. Metzger, Nucl. Phys. B675, 416 (2003), hep-th/0307152.
3. E. Witten, J. Geom. Phys. 22, 1 (1997), hep-th/9609122.
4. A. B. Barrett, A. S. Haupt, A. Lukas and K. S. Stelle, work in progress.
5. H. Lu, C. N. Pope, K. S. Stelle and P. K. Townsend, JHEP 0507, 075 (2005), hep-
th/0410176.
HAGEDORN TRANSITION
AND
CHRONOLOGY PROTECTION IN STRING THEORY*
CARLOS A.R. HERDEIRO
Departamento de Fisica e Centro de Fisica do Porto ,
Faculdade de Ciencias da Universidade do Porto,
Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
crherdei@fc.up.pt
We conjecture that chronology is protected in string theory due to the condensation of
light winding strings near closed null curves. This condensation triggers a Hagedorn phase
transition, whose end-point target space geometry should be chronological. Contrary to
conventional arguments, chronology is protected by an infrared effect. We support this
conjecture by studying strings in a particular Lorentzian orbifold of Minkowski spacetime,
where we show that some winding string states are unstable and condense in the non-
causal region of spacetime. The one loop partition function has infrared divergences
associated to the condensation of these states.
1. Introduction
Hawking has proposed that the laws of physics do not allow the formation of Closed
Causal Curves (CCCs).2 Hawking's argument was based on the behaviour of
quantum field theory in the presence of closed null curves. More concretely he argued that
the one-loop energy momentum tensor becomes very large near a closed null curve
and hence produces a large backreaction which either creates a spacetime singularity
or prevents deforming the spacetime towards the formation of CCCs. Hawking
supported his conjecture with a toy model based on an orbifold of Minkowski spacetime
called Misner space.
String theory is a candidate to a theory of quantum gravity which has a very
different high-energy behaviour from usual quantum field theory. Thus, one Ccin asK
how would strings behave near closed null curves. It has been suggested that new
"massless" string states would then appear.3 We have shown,1 using a toy model
based on an orbifold of Minkowski spacetime, that light winding strings states will
condense near closed null curves. This condensation was shown to produce a large
back-reaction in the non-causal region of the spacetime, and hence it was conjectured
that it modifies it into a causal region.
2. Strings in the O-plane orbifold
The O-plane orbifold is an orbifold of three dimensional Minkowski spacetime
obtained by identifying along the orbits of a Killing vector field which is a sum of a null
boost plus a null translation. Choosing coordinates adapted to this identification
*This communication is based on work in collaboration with M. Costa, J. Penedones and N.
Sous a.1
2887
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the metric becomes
ds2 = -2dy-dy+ + 2Ey{dy~f + dy2 . (1)
In this coordinates the orbifold identification is simply y~ ~ y~ + 2irR. The circle
along the y~ direction is timelike/null/spacelike for y < 0/y = 0/y > 0. There
are CCCs through any spacetime point. But if one would excise the region y < 0
there would be no CCCs. Therefore wc call y < 0 the (causally) 'bad region'.
The parameter E is the ratio of the null boost parameter to the null translation
parameter R.
The wave functions that describe a string's centre of mass dynamics depend
on four quantum numbers: the light cone energy p+, the Kaluza Klein momentum
p_ = n/R (n G No), the winding number m (m € Z) and the 'classical turning point'
l/o. This latter quantum number arises because the classical dynamics is described
by a one dimensional Hamiltonian system with a linear potential. Generic bosonic
string states require another two quantum numbers: n and n, describing left and
right level. The level matching condition is the usual
n — n = mu> , (2)
whereas the on-shell relation is
2p+~ + 2Ey0(p2+ - (luR)2) = 2(n + h - 2) = X . (3)
There are both stable and unstable on-shell states, the latter having a non-vanishing
imaginary part for p+. The unstable states have |/o smaller than a critical value yc{w)
given by
y^) = ~2E^- (4)
The wave function that describes the centre of mass dynamics for these states
oscillates for y < yo and is exponentially damped for y > yo. Thus, as these states
grow in light cone time, their back reaction becomes non-negligible in the the 'bad
region' of the spacetime.
Computing the string partition function in the canonical formalism
—T^-igio-i) , (5)
where q = e2mT and r = t\ +ir-2, one can first show that divergences arise for large
t-2 (i.e. infrared) whenever p+ has an imaginary part. These are associated to the
unstable modes discussed before. Then, making a series expansion of the Dedekind
eta functions that arise in the integrand of the partition function, one finds, for
large n, that the integrand is dominated by the exponential term
AK^n{2-^j2u)2EyR:2)
so that, for each winding number, the sum in n diverges when
(6)
V<E^' (?)
2889
where y is to be interpreted as the string centre of mass.1 Note that n = n = 0, (4)
reduces to (7).
3. Discussion
This behaviour is quite analogous to the well-known Hagedorn behaviour exhibited
by the bosonic string conipactified on a circle- Therein, new massless (winding)
modes appear in the string spectrum at sufficiently small radius (and unstable
modes beyond that); at the same radius, one verifies the existence of a large n
divergence of the partition function. It is usually accepted that this divergence
is signalling a phase transition that takes place by virtue of the condensation of
the unstable modes.4 In our orbifold, the radius of the compact direction varies
along the spacetime. For sufficiently small radius unstable modes appear. But here,
these unstable modes have a semi-localised profile, since they are described by Airy
functions that oscillate in the bad region and become exponentially damped in
the good region. Thus, the divergence can be traced back to the unstable modes
that grow essentially in the region of spacetime where the identified circle becomes
timelike. The condensation of these unstable modes, which corresponds to the back-
reaction caused by this growth, must eliminate the causally pathological region;
otherwise the instabilities would remain.
One can object that unstable modes with zero winding will have, from (7),
y = +oo. Hence they will condense in the whole spacetime. But these modes are
eliminated by GSO projection in the superstring, and the only unstable modes
remaining (from the NS-NS sector for anti-periodic fermionic boundary conditions)
will have yc < 1/ER2. For the superstring with supersymmetry preserving fermionic
boundary conditions the unstable states will condense exclusively in the bad region
of the spacetime.
As in Hawking's original suggestion, we believe this orbifold presents a good
illustration of our proposal. It remains to be seen explicitly what is the end point
of the condensation and how general the proposal is indeed.
Acknowledgements
The author was supported by Fundagao Calouste Gulbenkian through Programa
de Estimulo a Investigacao and by the FCT grants SFRH/BPD/5544/2001,
POCTI/FNU/38004/2001 and POCTI/FNU/50161/2003. Centra de Fisica do
Porto is partially funded by FCT through the POCTI programme.
References
1. M. S. Costa, C. A.R. Herdeiro, J. Penedones and N. Sousa, Nucl.Phys. B728, 148
(2005).
2. S. W. Hawking, Phys.Rev. D46, 603-611 (1992).
3. D. Brace, C. A. R. Herdeiro, S. Hirano, Phys.Rev. D69, 066010 (2004).
4. J. J. Atick and E. Witten, Nucl. Phys. B310, 291 (1988).
KK-MASSES AND DIPOLE THEORIES*
KARL LANDSTEINERt and SERGIO MONTERO*
Institute de Fisica Teorica
C-XVI Universidad Autonoma de Madrid
28049 Madrid, Spain
t karl. landsteiner@uam. es
t sergio.rnontero@uam.es
We reconsider aspects of non-commutative dipole deformations of field theories. Among
our findings there are hints to new phases with spontaneous breaking of translation
invariance (stripe phases), similar to what happens in Moyal-deformed field theories.
Furthermore, using zeta-function regularization, we calculate quantum corrections to
KK-state masses. The corrections coming from non-planar diagrams show interesting
but non-universal behaviour. Depending on the type of interaction the corrections can
make the KK-states very heavy but also very light or even tachyonic. Finally we point
out that the dipole deformation of QED is not renormalizable.
1. Motivation
Non-commutative field theories have attracted much attention for a long time now;
in the context of string theory they appear on the worldvolume of D-branes in a
B-field background. It turns out that different deformations of the field theory can
be constructed with different polarizations of this field, e.g. a Moyal deformation
corresponds to a B-field with both indices along the directions of the worldvolume
of the D-brane.
It is also possible to arrange the B-field in a different way, with one index along
the brane directions and the other one transverse to them. The deformation in
question is defined by the star-product
M*)*<h(?) ■■= e-^W-wv Mx) Mv)\ = 0i (* - y) fa U + -1
(i)
which was first constructed1 by considering T-duality of Moyal-bracket deformed
theories and its basic field theoretical properties have been first studied in Ref. 2.
As explained there, L± 2 are the so-called dipole lengths of the fields <fii and 02-
Without recourse to string theory, a dipole deformation of a field theory can be
defined by introducing the dipole lengths of the fields according to L^ = ^Q°i,
where Qa, arc U(l) charges of the field <f> and the matrix £% picks out a certain
linear combination.
Supcrgravity duals of confining gauge theories are in general plagued by a rather
unwelcome feature: the scale of the masses of the KK-states coming from the com-
pactified part of the worldvolume is of the same order as the scale of the four
"The research of K. L. was supported by the Ministerio de Ciencia y Tecnologia through a Ramon
y Cajal contract and by the Plan Nacional de Altas Energi'as FPA-2003-02-877. The research of
S. M. was supported by an FPI 01/0728/2004 grant from Comunidad de Madrid and by the Plan
Nacional de Altas Energi'as FPA-2003-02-877.
2890
2891
dimensional gauge theory of interest. Therefore, one cannot disentangle the
interesting strongly coupled gauge theory dynamics from the artefacts of these KK-states.
However, Nunez and Giirsoy pointed out3 that this situation might be improved
if one considers a dipole deformed D5-brane theory using the techniques developed
in Ref. 4. They noted that the volume of the compact internal manifolds in the
deformed background are smaller than in the undeformed one, therefore
indicating a possible disentanglement of the KK-states from the interesting gauge theory
dynamics.a This work motivated us to investigate the issue of KK-state masses in
dipole deformed theories from the purely field theoretical point of view. We study
much simpler examples of dipole deformations of field theories compactified on a
circle. For an expanded discussion with computations see Ref. 7.
2. Results and conclusions
In this section we summarize the results obtained in Ref. 7, namely the appearance of
stripe phases, one-loop corrections to KK-masses in these theories and the analysis
of renormalizability of the QED dipole gauge theory.
2.1. Stripe phases
We begin with a dipole theory for complex scalar fields 0 and ^ with quartic
interactions in D dimensions. The deformed vertex gives rise to a modified dispersion
relation of the form E2 = p2 + 2 9D/2 cos(pL) T (^f^-) L2~D, in the massless limit
and p || L. From this one can define a first critical dipole length Lcl where a
minimum away from the origin develops in momentum space and also a second
critical length Lc2 to be the value where the right hand side of the dispersion
relation becomes negative. See Fig. 2.1 for the D = 3,4 cases. A non-zero momentum
mode condenses for D = 3. The new ground state spontaneously breaks translation
invariance in the direction of the dipole moment.b Our analysis was based on a
simple one-loop computation and it is not clear if the properties of the dispersion
relation allowing for this phase transition persist to higher loops or non-perturbative
corrections, which may be analyzed in the lattice as it was done for the Moyal case.9
2.2. Corrections to KK-masses
The corrections to the masses of KK-states show a very interesting pattern. The
dipole length L together with the radius of compactification R = 1/(2ttT) forms a
dimensionless parameter which we call b = TL. It is remarkable that this parameter
is compact, i.e. takes values only in the interval ( — 1/2,1/2]. The interesting
corrections stem from non-planar graphs, in which the UV-divergences are regulated by
aFurther aspects of KK-states in these supergravity backgrounds have been discussed in Refs. 5,6.
bThis behaviour is reminiscent of the behaviour of Moyal deformed <f>4 theory (see Ref. 8).
2892
Fig. 1. (Left) Dispersion relation for different dipole lengths in D = 3. At Lcl it develops a
minimum away from p = 0 and at LC2 it touches E = 0. (Right) Dispersion relation for different
dipole lengths in D = 4, only small wiggling around E2 = p2 is observed.
the presence b. For 6 —> 0 the regularization becomes less effective, and therefore the
non-planar contribution becomes very large and can even overwhelm the tree-level
contribution. Depending on the form of the tree level interaction the non-planar
graph decrease the value of the square of the KK-mass. For small enough b the
corresponding mode might even become tachyonic.
2.3. Gauge dipole field theory
We also considered a dipole theory where the U(l) used was local and chose a
commutator-like interaction, showing that dipolc-deformed QED with adjoint action
of the gauge group is not renormalizable in a way that would only allow star-product
terms in the tree level Lagrangian. This problem might be cured only in highly
supersymmetric extension like the one based on the N = 4 theory.
Acknowledgements
K. L. would like to thank the organizers of the meeting for a very pleasant
atmosphere and a nice conference.
References
1. A. Bergman and O. J. Ganor, JHEP 0010 (2000) 018.
2. K. Dasgupta and M. M. Sheikh-Jabbari, JHEP 0202 (2002) 002.
3. U. Gursoy and C. Nunez, Nucl. Phys. B 725 (2005) 45.
4. O. Lunin and J. Maldacena, JHEP 0505 (2005) 033.
5. N. P. Bobev, H. Dimov and R. C. Rashkov, JHEP 0602 (2006) 064.
6. S. S. Pal, Phys. Rev. D 72 (2005) 065006.
7. K. Landsteiner and S. Montero, JHEP 0604 (2006) 025.
8. S. S. Gubser and S. L. Sondhi, Nucl. Phys. B 605, 395 (2001).
9. W. Bietenholz, F. Hofheinz and J. Nishimura, JHEP 0406 (2004) 042.
10. N. Sadooghi and M. Soroush, Int. J. Mod. Phys. A 18 (2003) 97.
LIST OF PARTICIPANTS
Abdil'din, Meirkhan
Abel, Paul
Abishev, Medeu
Adamiak, Jaroslaw
Adams, Judith
Ahmedov, Bobomurat
Aksenov, Alexey
Alam, Ujjaini
Albers, Mark
Alekseev, George
Alexeyev, Stanislav
Alic, Daniela
Aliev, Alikram Nuhbalaoglu
Alley, Carroll
Aloy, Miguel-Angel
A man, Jan
Amati, Lorenzo
Amelino-Camelia, Giovanni
Amin A., Omar
Anacleto Arroja, Frederico
Ananda, Kishore
Anderson, Paul
Andersson, Nils
Ando, Masaki
Angelini, Lorella
Anglada-Escude, Guillem
Ansoldi, Stefano
Ansorg, Marcus
Antoci, Salvatore
Antonini, Piergiorgio
Arkhangelskaja, Irene
Aros, Rodrigo
Aschenbach, Bernd
Ashtekar, Abhay
Aulbert, Carsten
Babak, Stanislav
Baiotti, Luca
Bajtlik, Stanislaw
Bakala, Pavel
Ballmer, Stefan
Bambi, Cosimo
Kazakh University
University of Leicester
Al-Farabi Kazakh Nat'l Univ
University of South Africa
Institute of Physics Publishing
Ulugh Beg Astronom. Inst.
Inst, for Theor. and Exp. Physics
ICTP
Institute of Theoretical Physics
Steklov Mathematical Institute
Sternberg Astronomical Institute
University of the Balearic Islands
Gursey Institute
University of Maryland at College Park
Universidad de Valencia
KAZAKHSTAN
UK
KAZAKHSTAN
SOUTH AFRICA
UK
UZBEKISTAN
RUSSIA
ITALY
GERMANY
RUSSIA
RUSSIA
SPAIN
TURKEY
USA
SPAIN
Stockholm University SWEDEN
INAF - IASF Bologna ITALY
University of Rome La Sapienza ITALY
Universidad Autonoma Metropolitana MEXICO
University of Portsmouth UK
Institute of Cosmology and Gravitation UK
Wake Forest University USA
University of Southamtpon UK
University of Tokyo JAPAN
NASA/GSFC USA
University of Barcelona SPAIN
University of Udine ITALY
Albert Einstein Institute GERMANY
University of Pavia ITALY
INFN ITALY
Moscow Eng. Physics Inst. RUSSIA
Universidad Andres Bello CHILE
MPI Extraterrestrische Physik GERMANY
Penn State University USA
Albert-Einstein-Institute GERMANY
Albert Einstein Institute GERMANY
Albert-Einstein-Institut GERMANY
Copernicus Astronomical Centre POLAND
Silesian University in Opava CZECH REPUBLIC
MIT / LIGO USA
University of Ferrara ITALY
2893
2894
Barbero Gonzalez, Jesus
Fernando
Barcelo, Carlos
Barkov, Maxim
Barrau, Aurelien
Barsuglia, Matteo
Bashinsky, Sergei
Bassan, Massimo
Bastiaensen, Benjamin
Basu, Prasad
Battisti, Marco Valerio
Beciu, Mircea
Beesham, Aroonkumar
Beissen, Nurzada
Belinski, Vladimir
Benini, Riccardo
Bergamin, Luzi
Bernardini, Maria Grazia
Berrocal Arellano, Aaron V.
Bertolami, Orfeu
Bertoldi, Frank
Bezerra, Valdir
Bianchi, Eugenio
Bianchi, Massimo
Bianco, Carlo Luciano
Bieli, Roger
Bieri, Lydia
Biermann, Peter
Bimonte, Giuseppe
Bini, Donato
Bishop, Nigel
Bizouard, Marie-Anne
Bjornsson, Gunnlaugur
Blair, David
Blanchet, Luc
Bludman, Sidney
Bluemer, Johannes
Bluhm, Robert
Boccaletti, Dino
Boedecker, Geesche
Bolejko, Krzysztof
Bombaci, Ignazio
Bongs, Kai
CSIC
CSIC
University of Leeds
LPSC Grenoble
CNRS-LAL and EGO
Los Alamos National Laboratory
University of Rome Tor Vergata
Ghent University
Centre for Space Physics
ICRA
Technical University
University of Zululand
Al-Farabi Kazakh Nat'l Univ
INFN and ICRANet
ICRA
ESTEC, EUI-ACT
ICRA and ICRANet
Uuiversidad Autonoma
Instituto Superior Tecnico
University of Bonn
Universidade Federal da Paraba
Scuola Normale Superiore, Pisa
University of Rome "Tor Vergata'
ICRANet, ICRA
Albert Einstein Institute
ETH Zurich
MPI
University of Naples
CNR Applied Mathematics Rome ITALY
University of South Africa SOUTH AFRICA
LAL CNRS/IN2P3 FRANCE
University of Iceland ICELAND
University of Western Australia AUSTRALIA
Institut d'Astrophysique de Paris FRANCE
DESY-T GERMANY
University of Karlsruhe and FZK GERMANY
Colby College USA
University of Rome ITALY
University of Potsdam GERMANY
Copernicus Astronomical Center POLAND
University of Pisa ITALY
University of Hamburg GERMANY
SPAIN
SPAIN
UK
FRANCE
ITALY
USA
ITALY
BELGIUM
INDIA
ITALY
ROMANIA
SOUTH AFRICA
KAZAKHSTAN
ITALY
ITALY
NETHERLANDS
ITALY
MEXICO
PORTUGAL
GERMANY
BRAZIL
ITALY
ITALY
ITALY
GERMANY
SWITZERLAND
GERMANY
ITALY
2895
Boonserm, Petarpa
Bostani, Neda
Bouhmadi-Lopez, Mariam
Boutloukos, Stratos
Bozza, Valerio
Bradley, Michael
Braggio, Caterina
Bregman, Joel
Brill, Dieter
Briscese, Fabio
Brizuela, David
Brodatzki, Katharina Anna
Broekaert, Jan
Bruneton, Jean-Philippe
Buonanno, Alessandra
Burinskii, Alexander
Caito, Letizia
Calchi Novati, Sebastiano
Camacho, Abel
Camarda, Karen
Cano, Andres
Capone, Monica
Capozziello, Salvatore
Caramete, Laurentiu loan
Carminati, John
Case, Gary
Cattoen, Celine
Celerier, Marie-Noelle
Cerda-duran, Pablo
Cerny, Slavomir
Chakrabarti, Sandip
Chardonnet, Pascal
Charters, Tiago
Chen, Chiang-Mei
Chernitskiy, Alexander A.
Cherubini, Christian
Christensen, Nelson
Chu, Yaoquan
Chung, T.J.
Cianfrani, Francesco
Cipko, Alois
Clifton, Timothy
Coley, Alan
Victoria University Wellington NEW ZEALAND
Shiraz University
University of Portsmouth
University of Tuebingen
University of Salerno
Umea University
University of Ferrara and INFN
University of Michigan
University of Maryland
University of Rome "La Sapienza'
IEM (CSIC)
Ruhr University Bochum
Vrije University of Brussels
Institut d'Astrophysique de Paris
University of Maryland
NSI Russian Academy of Sciences
University of Rome "La Sapienza''
University of Salerno
Universidad Autonoma
Washburn University
Inst, de Astrofisica de Andalucia
Polytechnic of Turin
University of Naples
MPI Radioastronomy
Deakin University
Louisiana State University
Victoria Univ. of Wellington
Observatoire de Paris-Meudon
Universidad de Valencia
IRAN
UK
GERMANY
ITALY
SWEDEN
ITALY
USA
USA
ITALY
SPAIN
GERMANY
BELGIUM
FRANCE
USA
RUSSIA
ITALY
ITALY
MEXICO
USA
SPAIN
ITALY
ITALY
GERMANY
AUSTRALIA
USA
NEW ZEALAND
FRANCE
SPAIN
Silesian University at Opava CZECH REPUBLIC
S.N. Bose N'l Centre for Basic Sciences INDIA
University of Savoie FRANCE
University of Lisboa PORTUGAL
National Central University TAIWAN, ROC
State University of Engin. and Econ. RUSSIA
University Campus Biomedico, ICRA ITALY
Carleton College USA
Center for Astrophysics CHINA
University of Alabama in Huntsville USA
ICRA, ICRANET ITALY
Silesian University at Opava CZECH REPUBLIC
Cambridge University UK
Dalhousie University CANADA
2896
Collier, Rainer
Consoli, Maurizio
Contaldi, Carlo
Cotsakis, Spiros
Courty, Stephanie
Craig, David
Crawford, Paulo
Crispino, Luis
Crosta, Maria Teresa
Cumming, Andrew
Cunningham, Liam
Cuoco, Elena
Dabrowski, Mariusz
Dadhich, Naresh
Dafermos, Mihalis
Daghan, Durmus
Dainotti, Maria Giovanna
Damiao Soares, Ivano
Damour, Thibault
Danilishin, Stefan
Darabi, Farhad
Das, Santabrata
De Araujo, Jose Carlos
De Bernardis, Paolo
De Felice, Antonio
De Felice, Fernando
De Laurentis, Mariafelicia
De Luca, Fabiana
De Paolis, Francesco
De Pasquale, Massimiliano
De Pietri, Roberto
Dehne, Christoph
Del Zanna, Luca
Delia Valle, Massimo
Delphenich, David
Delva, Pacome
Demianski, Marek
Denardo, Galieno
Dewangan, Gulab
Di Virgilio, Angela Dora
Dias, Gongalo
Dittus, Hansjoerg
Djorgovski, George
Friedrich-Schiller University
INFN Catania
Imperial College
University of the Aegean
University of Iceland
Le Moyne College
Universidade de Lisboa
Federal University of Para
INAF-Astronomical Obs. Turin
McGill University
University of Glasgow
European Gravitational Observatory
University of Szczecin
Inter-University Center for A&A
University of Cambridge
Istanbul Technical University
University of Rome "La Sapienza"
CBPF
Inst, des Hautes Etudes Scientifiques
Moscow State University
Azarbaijan Univ. Tarbiat Moallem
GERMANY
ITALY
UK
GREECE
ICELAND
USA
PORTUGAL
BRAZIL
ITALY
CANADA
SCOTLAND
ITALY
POLAND
INDIA
UK
TURKEY
ITALY
BRAZIL
i FRANCE
RUSSIA
IRAN
Chungnam National University SOUTH KOREA
INPE
University of Rome "La Sapienza"
University of Sussex
University of Padova
Politecnico di Torino
BRAZIL
ITALY
UK
ITALY
ITALY
University of Zurich SWITZERLAND
University of Lecce
Mullard Space Science Laboratory
Parma University
Leipzig University
University of Florence
INAF-Arcetri Astrophysical Obs.
Bethany College
University Pierre and Marie Curie
University of Warsaw
ICTP
Carnegie Mellon University
INFN-Pisa
Inst. Superior Tecnico - CENTRA
University of Bremen
Caltech
ITALY
UK
ITALY
GERMANY
ITALY
ITALY
USA
FRANCE
POLAND
ITALY
USA
ITALY
PORTUGAL
GERMANY
USA
2897
Dobado, Antonio
Dolan, Sam
Dominik, Martin
Dore, Olivier
Dotani, Tadayasu
Drever, Ronald W.P.
Drexlin, Guido
Duez, Matthew
Duffy, Peter
Dumin, Yurii
Dutan, Ioana
Dyrda, Michal
Ehlers, Jurgen
Eisenstaedt, Jean
Esposito, Giampiero
Everitt, C.W. Francis
Faber, Joshua
Fagnocchi, Serena
Fairhurst, Stephen
Fang, Li-Zhi
Farinelli, Ruben
Faye, Guillaume
Fewster, Christopher
Finn, Lee
Fiore, Fabrizio
Flambaum, Victor
Folomeev, Vladimir
Font, Jose A.
Forte, Luca Antonio
Fortini, Pierluigi
Foulon, Bernard
Fragile, Chris
Frankenhuizen, Walburga
Fraschetti, Federico
Frutos-Alfaro, Francisco
Fuchs, Burkhard
Fujimoto, Masa-Katsu
Fukui, Takao
Fuster, Andrea
Fiizfa, Andre
Fynbo, Johan Fynbo
Gadri, Mohamed
Gair, Jonathan
Universidad Complutense de Madrid SPAIN
Cambridge University UK
University of St Andrews UK
CITA CANADA
Inst, of Space and Astronautical Science JAPAN
California Institute of Technology USA
University of Karlsruhe
Cornell University
University College Dublin
Russian Academy of Sciences
MPI Radioastronomy
Jagellonian University
Albert Einstein Institute
Observatory of Paris
INFN Napoli
Stanford University
GERMANY
USA
IRELAND
RUSSIA
GERMANY
POLAND
GERMANY
FRANCE
ITALY
USA
University of Illinois at Urbana-Champaign USA
Enrico Fermi Centre
University of Wisconsin Milwaukee
University of Arizona
University of Ferrara
Institut d'Astrophysique de Paris
University of York
Penn State University
INAF - Oss. Astronomico di Roma
Univ. of New South Wales Sidney
ITALY
USA
USA
ITALY
FRANCE
UK
USA
ITALY
AUSTRALIA
NAN KR KYRGYZ REPUBLIC
University of Valencia
University of Naples
University of Ferrara
ONERA
College of Charleston
MPI Extraterrestrische Physik
ICRA
University of Costa Rica
Astronomisches Rechen-Institut
NAOJ/TAMA
Dokkyo University
SPAIN
ITALY
ITALY
FRANCE
USA
GERMANY
ITALY
COSTA RICA
GERMANY
JAPAN
JAPAN
NIKHEF NETHERLANDS
University of Paris
DARK Cosmology Centre
Al-Fateh University
University of Cambridge
FRANCE
DENMARK
LIBYAN
UK
2898
Galloway, Duncan
Galtsov, Dmitry
Garattini, Remo
Garecki, Janusz
Gegham, Yegorian
Geralico, Andrea
Gergely, Laszo Arpad
Ghahramanyan, Tigran
Gherson, David
Ghosh, Shubhrangshu
Gilmore, Gerard
Glampedakis, Kostas
Goenner, Hubert
Goklu, Ertan
Goncharenko, Igor
Gonzalez, Guillermo
Gonzalez, Jose
Gonzalez-Diaz, Pedro F.
Gorbonos, Dan
Gorini, Vittorio
Graham, Robert
Grave, Frank
Greiner, Walter
Griffiths, Richard
Grindlay, Josh
Grishchuk, Leonid
Grumiller. Daniel
Gucnther, Uwe
Guida. Roberto
Gurzadyan, Vahe
Guzman Murillo, Francisco S.
Gyula, Fodor
Hadley, Mark
Halat, Milenko
Halliwell, Jonathan
Halzen, Frances
Hammond, Richard
Harada, Tomohiro
Harko, Tiberiu
Harmark, Troels
Harriott, Tina
Hartmann, Bruno
Hartong, Jelle
University of Melbourne AUSTRALIA
Moscow State University RUSSIA
University of Bergamo ITALY
University of Szczecin POLAND
University of Yerevan ARMENIA
ICRA ITALY
University of Szeged HUNGARY
University of Yerevan ARMENIA
Inst, de Physique Nucleaire Lyon FRANCE
MPI Radio Astronomy GERMANY
University of Cambridge UK
University of Southhampton UK
University of Goettingen GERMANY
ZARM - University Bremen GERMANY
Peoples' Friendship Univ. of Russia RUSSIA
Universidad Industrial de Santander COLOMBIA
University of Jena GERMANY
IMAFF, CSIC SPAIN
Hebrew University ISRAEL
Universiy of Insubria ITALY
Universitaet Duisburg/Essen GERMANY
University of Tuebingen GERMANY
Frankfurt IAS GERMANY
Carnegie Mellon University USA
Harvard University USA
Cardiff University, Moscow University UK
University of Leipzig
Research Center Rossendorf
ICRA
ICRANet, Yerevan Physics Inst.
Universidad Michoacana
KFKI Research Institute
University of Warwick
University of Pisa
Imperial College London
University of Wisconsin - Madison
University of North Carolina
Rikkyo University
University of Hong Kong
Niels Bohr Institute
Mount Saint Vincent University
Perimeter Institute
University of Groningen
GERMANY
GERMANY
ITALY
ARMENIA
MEXICO
HUNGARY
UK
ITALY
UK
USA
USA
JAPAN
CHINA
DENMARK
CANADA
CANADA
NETHERLANDS
2899
Hasinger, Gunther
Head, Marilyn
Heifetz, Michael
Heinzle, Mark
Helesfai, Gabor
Hellaby, Charles
Hennig, Jorg
Hentschel, Alexander
Heptonstall, Alastair
Herdeiro, Carlos
Hermann, Nicolai
Herrmann, Frank
Hervik, Sigbjorn
Hestenes, David
Hestroffer, Daniel
Hildebrandt, Sergi
Himemoto, Yoshiaki
Hinterleitner, Franz
Hiramatsu, Takashi
Hirata, Christopher
Hladik, Jan
Hledik, Stanislav
Hoang, Ngoc Long
Holzegel, Gustav
Hough, Jim
Hurley, Kevin
Husa, Sascha
Intravaia, Francesco
Iorio, Lorenzo
Ishihara, Hideki
Itin, Yakov
Jakobsson, Palli
Janke, Wolfhard
Janssen, Michel
Jantzen, Robert
Jaranowski, Piotr
Jetzer, Philippe
Jonker, Peter
Kaaret, Philip
Kagramanova, Valeria
Kahil, Magd Elias
Kahya, Emre
Kamenshchik, Alexander
MPI Extraterrestrische Physik
Radio New Zealand
Stanford University
University of Vienna
Eotvos Lorant University
University of Cape Town
Friedrich Schiller Univ. Jena
Humboldt University Berlin
University of Glasgow
Oporto University
MPI for Gravitational Physics
Pennsylvania State University
Dalhousie University
Arizona State University
GERMANY
NEW ZEALAND
USA
AUSTRIA
HUNGARY
SOUTH AFRICA
GERMANY
GERMANY
UK
PORTUGAL
GERMANY
USA
CANADA
USA
FRANCE
SPAIN
JAPAN
IMCCE/PAris Observatory
Institute de Astrofsica de Canarias
The University of Tokyo
Masaryk University CZECH REPUBLIC
University of Tokyo JAPAN
IAS USA
Silesian University in Opava CZECH REPUBLIC
Silesian University in Opava CZECH REPUBLIC
VAST
DAMTP
University of Glasgow
UC Berkeley
University of Jena
University of Potsdam
Universit di Bari
Osaka City University
Hebrew University of Jerusalem
University of Hertfordshire
Universitaet Leipzig
University of Minnesota
Villanova University
University of Bialystok
University of Zurich
SRON, CfA
University of Iowa
Ulugh Beg Astronomical Inst.
American University in Cairo
University of Florida
University of Bologna
VIETNAM
UK
UK
USA
GERMANY
GERMANY
ITALY
JAPAN
ISRAEL
UK
GERMANY
USA
USA
POLAND
SWITZERLAND
NETHERLANDS
USA
UZBEKISTAN
EGYPT
USA
ITALY
2900
Kaniel, Shmuel
Karimian, Hamidreza
Karthauser, Josef
Kashif, Abdul Rehman
Katanaev, Mikhail
Keeton, Charles
Kellmann, Timo
Kempf, Achim
Kenmoku, Masakatsu
Kerner, Richard
Kerr, Roy
Khakshournia, Samad
Khanna, Ramon
Khriplovich, Iosif
Kidder, Lawrence
Killow, Christian
Kim, Kyung Yee
Kinasiewicz, Bogusz
Klaoudatou, Ifigeneia
Klebanov, Igor
Kleihaus, Burkhard
Kleinert, Hagen
Klimchitskaya, Galina
Klioner, Sergei
Klippert, Renato
Knapp, Johannes
Knox, Lloyd
Knutsen, Henning
Kobayashi, Shiho
Koide, Shinji
Konkowski, Deborah
Konopka, Tomasz
Konoplev, Alexander
Kopeikin, Sergei
Korolyov, Valery
Koroteev, Peter
Korzynski, Mikolaj
Kottanattu, George
Kovzacs, Zoltan
Kovar, Jiri
Kowalski-Glikman, Jerzy
Kramer, Michael
Krasihski, Andrzej
Hebrew University
Gent University
University of Sussex
Nat'l Univ. of Sciences and Tech
Steklov Mathematical Institute
Rutgers University
MPI for Radioastronomy
University of Waterloo
Nara Women's University
University of Paris
University of Canterbury
Sharif University
Springer-Verlag GmbH
Budker Inst, of Nuclear Physics
Cornell University
University of Glasgow
Inje University
Jagellonian University
University of the Aegean
Princeton University
University of Oldenburg
FU Berlin
North-West Technical University
Dresden Technical University
Universidade Federal de Itajuba
University of Leeds
University of California at Davis
Stavanger University
ARI Liverpool JMU
Kumamoto University
U.S. Naval Academy
Perimeter Institute
Moscow State Pedagogical University
University of Missouri Columbia
Moscow State Pedagogical Univ.
Institute for Nuclear Research
Warsaw University
University of Nottingham
University of Szeged
Silesian University in Opava CZECH REPUBLIC
University of Wroclaw POLAND
University of Manchester UK
N. Copernicus Astronomical Center POLAND
ISRAEL
BELGIUM
UK
PAKISTAN
RUSSIA
USA
GERMANY
CANADA
JAPAN
FRANCE
NEW ZEALAND
IRAN
GERMANY
RUSSIA
USA
UK
S. KOREA
POLAND
GREECE
USA
GERMANY
GERMANY
RUSSIA
GERMANY
BRAZIL
UK
USA
NORWAY
UK
JAPAN
USA
CANADA
RUSSIA
USA
RUSSIA
RUSSIA
POLAND
UK
HUNGARY
2901
Krige, Dan
Krimm, Hans
Krishnan, Badri
Kronberg, Philipp
Kuchiev, Michael
Kundt, Wolfgang
Kunz, Jutta
Kiinzle, Hans-peter
Kurita, Yasunari
Kuusk, Piret
Lacquaniti, Valentino
Laemmerzahl, Claus
Lai, Kevin
Landsteiner, Karl
Lange, Benjamin
Lantz, Brian
Larena, Julien
Lasenby, Anthony
Lash, Rachel
Lasky, Paul
Lattanzi, Massimiliano
Le Delliou, Morgan
Le Floc'h, Emeric
Leach, Jannie
Lecian, Orchidea Maria
Lee, Chul Hoon
Lee, Da-Shin
Lee, Hyun Kyu
Lee, Hyung Won
Lee, William
Lee, Wo-Lung
Lee, Wonwoo
Lehnert, Ralf
Lemos, Jose P. S.
Lesame, William
Leubner, Manfred P.
Lewandowski, Jerzy
Li, Zhifeng
Liebscher, Dierck-e.
Linares, Manuel
Linares, Roman
Lipunov, Vladimir
List, Meike
University of KwaZulu-Natal SOUTH AFRICA
USRA / NASA GSFC USA
Albert Einstein Institute GERMANY
Los Alamos National Laboratory USA
University of New South Wales AUSTRALIA
Argelander Institute for Astrophysics GERMANY
Carl-von-Ossietzky Univ. Oldenburg GERMANY
University of Alberta
Osaka City University
University of Tartu
ICRA, Univ. of Rome "RomaTre"
University of Bremen
UNISA
IFT/UAM Madrid
VirtualPBX.Com
Stanford University
Observatoire de Paris-Meudon
Cavendish Laboratory
Yale University
Monash University
ICRA, Univ. de Valencia
CFTC, Lisbon University
University of Arizona
University of Cape Town
ICRA
Hanyang University
National Dong Hwa University
Hanyang University
Inje University
UN AM
National Taiwan Normal Univ.
Sogang University
MIT
Center for Astrophysics Lisbon
University of South Africa
University of Innsbruck
Uniwversity of Warsaw
University of Vienna
Astrophys. Inst. Potsdam
University of Amsterdam
Universidad Autonoma
Sternberg Astronomical Institute
University of Bremen
CANADA
JAPAN
ESTONIA
ITALY
GERMANY
SOUTH AFRICA
SPAIN
USA
USA
FRANCE
UK
USA
AUSTRALIA
SPAIN
PORTUGAL
USA
SOUTH AFRICA
ITALY
S. KOREA
TAIWAN, ROC
KOREA
S. KOREA
MEXICO
TAIWAN, ROC
SOUTH KOREA
USA
PORTUGAL
SOUTH AFRICA
AUSTRIA
POLAND
AUSTRIA
GERMANY
NETHERLANDS
MEXICO
RUSSIA
GERMANY
2902
Lobo, Francisco
Loeffler, Frank
Lora, Fabio
Lorek, Dennis
Lousto, Carlos
Lucchesi, David M.
Luck, Herald
Luck, Tobias
Luest, Dieter
Lukierski, Jerzy
Lusanna, Luca
Mac Conamhna, Oisin
Macias, Alfredo
Madsen, Jes
Maeda, Hideki
Maharaj, Sunil
Majid, Shahn
Majumdar, Archan S.
Malafarina, Daniele
Man-brillet, Catherine
Mancini, Luigi
Mandal, Samir
Mapelli, Michcla
Marcian, Antonino
Marecki, Piotr
Marka, Szabolcs
Marques, Geusa
Marronetti, Pedro
Marshall, Francis
Martin, Iain
Martin-Garcia, Jose M.
Masi, Silvia
Mathews, Grant
Matinyan, Sergei
Mattei, Alvise
Matyjasek, Jerzy
Mauskopf, Philip
Mazur, Pawel O.
Mazzali. Paolo
Mckinney, Jonathan
Mehls, Carsten
Meinel, Reinhard
Melkumova, Elena
University of Lisbon PORTUGAL
SISSA ITALY
Universidad Industrial de Santander COLOMBIA
University of Bremen
Univ. of Texas at Brownsville
IFSI/INAF
University of Hannover
University of Cologne
MPI Physik
University of Wroclaw
INFN
Imperial College
Universidad Autonoma
University of Aarhus
Waseda University
GERMANY
USA
ITALY
GERMANY
GERMANY
GERMANY
POLAND
ITALY
UK
MEXICO
DENMARK
JAPAN
University of KwaZulu-Natal SOUTH AFRICA
Queen Mary Univ. London
Bose Nat'l Centre for Basic Sciences
Politecnico di Milano
CNRS-OCA
Universit di Salerno
Centre for Space Physics
SISSA/ISAS
University of Rome "La Sapienza"
ITP, University of Leipzig
Columbia University
Univ. Federal de Campina Grande
Florida Atlantic University
GSFC/NASA
University of Glasgow
CSIC
University of Rome "La Sapienza"
University of Notre Dame
Yerevan Physics Institute
ICRA and LAPTH
Maria Curie-Sklodowska University
Cardiff University
University of South Carolina
MPI Astrophysics
CfA
Stanford University
University of Jena
Moscow State University
UK
INDIA
ITALY
FRANCE
ITALY
INDIA
ITALY
ITALY
GERMANY
USA
BRAZIL
USA
USA
UK
SPAIN
ITALY
USA
ARMENIA
FRANCE
POLAND
UK
USA
GERMANY
USA
USA
GERMANY
RUSSIA
2903
Mena Marugan, Guillermo A.
Mendez, Mariano
Menon, Govind
Menotti, Pietro
Mercuri, Simone
Mester, John
Meyer, Hinrich
Meyer, Rene
Meylan, Georges
Mielke, Eckehard W.
Mignard, Francois
Mignemi, Salvatore
Milton, Kimball
Milyukov, Vadim
Minkowski, Peter
Mino, Yasushi
Miralles, Juan-Antonio
Miranda, Marco
Miritzis, John
Mishima, Takashi
Misthry, Suryakumari
Mitra, Abhas
Mitskievich, Nikolai V.
Miyamoto, Umpei
Mizuno, Yosuke
Mobed, Nader
Mohaupt, Thomas
Mondal, Souincn
Monnet, Guy
Montanari, Enrico
Montani, Giovanni
Montero, Pedro
Mostepanenko, Vladimir
Mottola, Emil
Moura, Filipe
Mouret, Serge
Mousavi, Sadegh
Mrazova, Kristina
Mukhopadhyay, Banibrata
Muller, Dietrich
Miiller, Jurgen
Munyaneza, Faustin
Mureika, Jonas
CSIC
SRON
Troy University
University of Pisa
Univ. of Rome "La Sapienza",
Stanford University
Univ. Wuppertal and DESY
University of Leipzig
EPFL
Universidad Autonoma
Observatoire de la Cte d'Azur
University of Cagliari
University of Oklahoma
Moscow University
University of Bern
California Institute of Technology
University of Alicante
Inst, of Theoretical Physics
University of the Aegean
Nihon University
Durban University
MPI Kernphysik
Universidad de Guadalajara
Waseda University
NSSTC
University of Regina
University of Liverpool
Bose Nat'l Centre for Basic Sciences
European Southern Observatory
Univ. of Rome "La Sapienza"
ENEA/ICRANet ITALY
University of Valencia SPAIN
Ministry Higher Ed., Science and Tech. RUSSIA
Los Alamos National Laboratory USA
Inst. Theoretische Fysica NETHERLANDS
IMCCE - Paris Observatory FRANCE
Amirkabir University of Technology IRAN
Silesian University at Opava CZECH REPUBLIC
Harvard-Smithsonian CfA USA
University of Chicago USA
University of Hannover GERMANY
MPI Radioastronomy GERMANY
Loyola Marymount University USA
SPAIN
NETHERLANDS
USA
ITALY
ICRA ITALY
USA
GERMANY
GERMANY
SWITZERLAND
MEXICO
FRANCE
ITALY
USA
RUSSIA
SWITZERLAND
USA
SPAIN
SWITZERLAND
GREECE
JAPAN
SOUTH AFRICA
GERMANY
MEXICO
JAPAN
USA
CANADA
UK
INDIA
GERMANY
ITALY
2904
Murphy, Tom
Murray, Peter
Musco, Ilia
Mushotzky, Richard
Mychelkin, Eduard
Myklevoll, Kari
Nadalini, Mario
Nagar, Alessandro
Naish-Guzman, Ileana
Nati, Federico
Natoli, Paolo
Navarro-Lerida, Francisco
Navarro-Salas, Jose
Neemann, Ulrike
Neilsen, David Neilsen
Nelson, Jeanette E.
Nester, James M
Neugebauer, Gemot
Nevsky, Alexander
Ng, Y. Jack
Nieuwenhuizen, Theo M.
Nimtz, Giinter
Nishikawa, Ken-Ichi
Noble, Scott
Nogueira, Flavio
Nolan, Brien
Novello, Mario
Nowak, Michael
Nucita, Achille
Ohashi, Masatake
Okolow, Andrzej
Okuda, Toru
Oliynyk, Todd
Oren, Yonatan
Ortaggio, Marcello
Ortolan, Antonello
Osterbrink, Lutz
Ostermann, Matthias
Ostermann, Peter
Overduin, James
Owen, Benjamin
Ozdemir, Nese
Page, Dany
University of California San Diego USA
University of Glasgow SCOTLAND
Queen Mary University of London UK
NASA/GSFC USA
Fesenkov Astrophysical Institute KAZAKHSTAN
Universitat Oldenburg GERMANY
University of Trento ITALY
Politecnico di Torino ITALY
University of Nottingham UK
University of Rome "La Sapienza" ITALY
University of Rome Tor Vergata ITALY
Carl von Ossietzky Univ. Oldenburg GERMANY
University of Valencia
University of Oldenburg
Brigham Young University
University of Turin
National Central University
University of Jena
University of Diisseldorf
University of North Carolina
Inst, for Theoretical Physics
University of Cologne
NSSTC
SPAIN
GERMANY
USA
ITALY
TAIWAN, ROC
GERMANY
GERMANY
USA
NETHERLANDS
GERMANY
USA
University of Illinois at Urbana-Champaign USA
Free University Berlin
Dublin City University
ICRA-Brasil/CBPF
MIT-CXC
University of Lecce
University of Tokyo
Warsaw University
Hokkaido Univ. of Education
Albert Einstein Institute
Hebrew University
University of Trento
INFN Legnaro
University of York
GERMANY
IRELAND
BRAZIL
USA
ITALY
JAPAN
POLAND
JAPAN
GERMANY
ISRAEL
ITALY
ITALY
UK
Ludwig-Maximilians-Universitaet GERMANY
Independent research
Stanford University
Pennsylvania State University
Istanbul Technical University
UNAM
GERMANY
USA
USA
TURKEY
MEXICO
2905
Page, Don
Pai, Archana
Palomba, Cristiano
Paolino, Armando
Parameswaran, Ajith
Pasic, Vedad
Paumard, Thibaut
Pavon, Diego
Peik, Ekkehard
Pelster, Axel
Pereira, Jose Geraldo
Perez Bergliaffa, Santiago E.
Perlick, Volker
Peters, Achim
Petrasek, Martin
Pfeiffer, Harald
Pfister, Herbert
Phillips, Adam
Pian, Elena
Pidokrajt, Narit
Pietrobon, Davide
Pinto, Innocenzo
Pinto-Neto, Nelson
Pirozhenko, Irina
Pizzella, Guido
Pizzi, Marco
Polarski, David
Polchinski, Joe
Polenta, Gianluca
Pollney, Denis
Polyakov, Alexander
Pompi, Francesca
Popa, Lucia Aurelia
Popov, Serghei Mikhailovich
Porto, Rafael
Possel, Markus
Potting, Robertus
Pravda, Vojtech
Pravdova, Alena
Pretorius, Frans
Prix, Reinhard
Prokhorov, Leonid
University of Alberta CANADA
Albert Einstein Institute GERMANY
INFN Rome ITALY
University of Rome "La Sapienza" ITALY
Albert Einstein Institute GERMANY
University of Bath UK
MPI Extraterrestrial Physics GERMANY
Universidad Autonoma de Barcelona SPAIN
Physikalisch-Technische GERMANY
Bundesanstalt
University of Duisburg-Essen GERMANY
UNESP BRAZIL
Universidade Estadual Rio de Janeiro BRAZIL
TU Berlin GERMANY
Humboldt University GERMANY
Silesian University at Opava CZECH REPUBLIC
California Institute of Technology USA
Universitt Tbingen GERMANY
Institute of Physics Publishing UK
Astronomical Observatory of Trieste ITALY
Stockholm University SWEDEN
University of Rome Tor Vergata ITALY
University of Sannio at Benevento ITALY
CBPF BRAZIL
ENS, CNRS, UMPC FRANCE
University of Rome Tor Vergata ITALY
Icra, University of Rome "La Sapienza" ITALY
Univ. Montpellier
University of Calfornia Santa Barbara
University of Rome "La Sapienza"
Albert Einstein Institut
Princeton University
ICRA
Institute for Space Sciences
Sternberg Astronomical Institute
Carnegie Mellon Univ.
Albert Einstein Institute
Universidade do Algarve
Academy of Sciences
Academy of Sciences
University of Alberta
Albert Einstein Institute
Moscow State University
FRANCE
USA
ITALY
GERMANY
USA
ITALY
ROMANIA
RUSSIA
USA
GERMANY
PORTUGAL
CZECH REPUBLIC
CZECH REPUBLIC
CANADA
GERMANY
RUSSIA
2906
Radicella, Ninfa
Radu, Eugen
Rakic, Aleksandar
Rapetti, David
Polytechnic of Turin
National University of Ireland
University of Bielefeld
Stanford/SL AC
ITALY
IRELAND
GERMANY
USA
Rastkar Ebrahimzadeh, Alireza Azarbaijan Univ. of Tarbiat Moallem IRAN
Rea, Nanda
Reall, Harvey
Reboucas, Marcelo
Rendall, Alan
Renn, Jiirgen
Reynaud, Serge
Rezzolla, Luciano
Ricci, Fulvio
Rideout, David
Riemer-Srensen, Signe
Rinaldi, Massimiliano
Ripamonti, Emanuele
Rogatko, Marek
Roken, Christian
Romero, Carlos
Ror, Nicklas
Rosquist, Kjell
Roszkowski, Krzysztof
Rotondo, Michael
Rowan, Sheila
Rubilar, Guillermo
Ruchayskiy, Oleg
Rudenko, Valentin
Ruder, Hanns
Ruediger, Albrecht
Ruffiiii, Remo
Russell, Neil
Saa, Alberto
Saathoff, Guido
Saclioglu, Cihan
Sahlmann, Hanno
Saifullah, Khalid
Salemi, Francesco
Salisbury, Donald
Salomon, Christophe
Sanchez Villasenor, Eduardo J.
Sandhoefer, Barbara
Santini, Eduardo Sergio
Netherlands Inst. Space Res. NETHERLANDS
University of Nottingham
CBPF
Albert Einstein Institute
MPI Wissenschaftsgeschichte
Laboratoire Kastler Brossel
Albert Einstein Institute
University of Rome "La Sapienza"
Imperial College London
Niels Bohr Institute
University College Dublin
UK
BRAZIL
GERMANY
GERMANY
FRANCE
GERMANY
ITALY
UK
DENMARK
IRELAND
University of Groningen NETHERLANDS
Maria Curie-Sklodowska Univ.
University of Bochum
Universidade Federal da Paraba
Karlstads University
Stockholm University
Jagellonian University
University of Rome "La Sapienza",
University of Glasgow
Universidad Estadual Paulista
IHES Paris
Sternberg Astronomical Institute
University of Tuebingen
Albert Einstein Institute
ICRANET
Northern Michigan University
IMECC - UNICAMP
University of Colorado at Boulder
Sabanci University
POLAND
GERMANY
BRAZIL
SWEDEN
SWEDEN
POLAND
ICRA ITALY
UK
BRAZIL
FRANCE
RUSSIA
GERMANY
GERMANY
ITALY
USA
BRAZIL
USA
TURKEY
Utrecht University NETHERLANDS
Nat'l Univ. of Sciences and Techn.
University of Ferrara
Austin College
Ecole Normale Superieure
Universidad Carlos III de Madrid
University of Cologne
CBPF/ICRA-BR, CNEN
PAKISTAN
ITALY
USA
FRANCE
SPAIN
GERMANY
BRAZIL
2907
Sarioglu, Bahtiyar
Sato, Goro
Satz, Alejandro
Sauer, Tilman
Scarpetta, Gaetano
Schaefer, Gerhard
Schiller, Stephan
Schlemmer, Jan
Schlickeiser, Reinhard
Schmidt, Hans-Jiirgen
Schreier, Ethan
Schubert, Christian
Schulz, Frank
Schulz, Norbert S.
Schunck, Franz
Schutz, Bernard
Scott, Susan M.
Seahra, Sanjeev
Sedlmayr, Erwin
Selig, Hanns
Semiz, Ibrahim
Sepulveda, Alonso
Sereno, Mauro
Serpico, Pasquale Dario
Sfarti, Adrian
Shaposhnikov, Nikolai
Shaposhnikov, Mikhail
Sharif, Muhammad
Shaw, Douglas
Sheth, Ravi
Shima, Kazunari
Shoemaker, David
Sigismondi, Costantino
Silbergleit, Alexander
Silverstein, Eva
Sinai, Yakov
Singh, Dinesh
Sintes, Alicia M
Slagter, Reinoud
Slany, Petr
Slosar, Anze
Snajdr, Martin
Sobouti, Yousef
Middle East Technical Univ.
GSFC/JSPS/USRA
University of Nottingham
Caltech
University of Salerno
Friedrich- Schiller- Universitaet
Universitt Dsseldorf
TURKEY
USA
UK
USA
ITALY
GERMANY
GERMANY
MPI for Mathematics in the Sciences GERMANY
Ruhr-Universitt
Universitt Potsdam
Associated Universities, Inc.
University of Michoacan
Albert Einstein Institute
MIT
University of Cologne
MPI Gravitationsphysik
Australian National University
University of Portsmouth
Technische Universitt Berlin
University of Bremen
Bogazici Univ.
Universidad de Antioquia
University of Zuerich
MPI Physik
UC Berkeley
Goddard Space Flight Center
E. Polytech. Fed. de Lausanne
University of the Punjab
Cambridge University
University of Pennsylvania
Saitama Institute of Technology
MIT
ICRA
Stanford University
Stanford University
Princeton University
University of Regina
Univ. de les Illes Balears
Univ. of Amsterdam
GERMANY
GERMANY
USA
MEXICO
GERMANY
USA
GERMANY
GERMANY
AUSTRALIA
UK
GERMANY
GERMANY
TURKEY
COLOMBIA
SWITZERLAND
GERMANY
USA
USA
SWITZERLAND
PAKISTAN
UK
USA
JAPAN
USA
ITALY
USA
USA
USA
CANADA
SPAIN
NETHERLANDS
Silesian University in Opava CZECH REPUBLIC
University of Ljubljana
University of British Columbia
IAS in Basic Sciences
SLOVENE
CANADA
IRAN
2908
Sokolowski, Leszek
Soria, Roberto
Sorkin, Evgeny
Sotiriou, Thomas
Spergel, David
Sperhake, Ulrich
Speziale, Simone
Spiering, Christian
Springel, Volker
Stanwix, Paul
Starling, Rhaana
Starobinsky, Alexei
Stasielak, Jaroslaw
Steigl, Roman
Steiner, Frank
Stelle, Kellogg
Stephens, Branson
Stergioulas, Nikolaos
Stolin, Oldrich
Stornaiolo, Cosimo
Stuchlik, Zdenek
Suwa, Yudai
Szilagyi, Bela
Szulc, Lukasz
Szybka, Sebastian
Tagliaferri, Gianpiero
Takahashi, Hirotaka
Takiwaki, Tomoya
Tamaki, Takashi
Tanvir, Nial
Tartaglia, Angelo
Tautz, Robert
Tavakol, Reza
Taveras, Victor
Teo, Edward
Tessmer, Manuel
Teyssandier, Pierre
Theisen, Stefan
Thornburg, Jonathan
Tiengo, Andrea
Tillman, Philip
Tino, Guglielmo M.
Titarchuk, Lev
Jagellonian University
Harvard-Smithsonian CfA
University of British Columbia
SISSA
Princeton University
University of Jena
Perimeter Institute
DESY
MPI for Astrophysics
University of Western Australia
University of Amsterdam
POLAND
USA
CANADA
ITALY
USA
GERMANY
CANADA
GERMANY
GERMANY
AUSTRALIA
NETHERLANDS
Landau Inst, for Theoretical Physics RUSSIA
Jagiellonian University POLAND
Masaryk University CZECH REPUBLIC
Ulm University GERMANY
Imperial College London UK
University of Illinois at Urbana Champaign USA
Aristotle Univ. of Thessaloniki GREECE
Silesian University CZECH REPUBLIC
INFN Naples ITALY
Silesian University in Opava CZECH REPUBLIC
University of Tokyo JAPAN
Albert Einstein Institute GERMANY
Warsaw University POLAND
Jagellonian University POLAND
INAF - Oss. Astronomico di Brera ITALY
Albert Einstein Institute GERMANY
University of Tokyo JAPAN
Waseda University JAPAN
University of Hertfordshire UK
Politecnico di Torino ITALY
Ruhr-Universitt Bochum GERMANY
Queen Mary University of London UK
Penn State University USA
Nat'l University of Singapore SINGAPORE
Friedrich-Schiller-Universitt Jena GERMANY
Observatoire de Paris FRANCE
Albert Einstein Institute GERMANY
Albert Einstein Institute GERMANY
INAF-IASF Milan ITALY
University of Pittsburgh USA
University of Florence ITALY
George Mason University and NRL USA
2909
Tobar, Michael
Tomaras, Theodore
Tome, Brigitte
Toporensky, Alexey
Torok, Gabriel
Triay, Roland
Trippe, Sascha
Trotta, Roberto
Truemper, Joachim
Tsokaros, Antonios
Tsubono, Kimio
Tsuda, Motomu
Tsupko, Oleg
Tuiran, Erick
Tyurina, Nataly
Uggla, Claes
Unnikrishnan, C.S.
Urbanec, Martin
Usov, Vladimir
Van Den Broeck, Chris
Vanzo, Luciano
Vargas Auccalla, Teofilo
Vargas Moniz, Paulo
Vasile, Ana
Vassiliev, Dmitri
Vasuth, Matyas
Vecchiato, Alberto
Veitch, John
Veneziani, Marcella
Venter, Liebrecht
Venturi, Giovanni
Verbin, Yosef
Vereshchagin, Gregory
Vigelius, Matthias
Vikman, Alexander
Visinescu, Anca
Visinescu, Mihai
Volkov, Mikhail
Volonteri, Marta
Volovik, Grigory
Vu, Khai
Walsworth, Ronald
Wan, Hao-Yi
University of Western Australia
University of Crete
Universidade do Algarve
Sternberg Astronomical Institute
AUSTRALIA
GREECE
PORTUGAL
RUSSIA
Silesian University in Opava CZECH REPUBLIC
Centre de Physique Thorique
MPE
Oxford University
MPI Extraterrestrische Physik
University of Aegean
University of Tokyo
Saitama Institute of Technology
Space Res. Inst. Russ. Acad. Scienc
Mainz University
Sternberg Astronomical Institute
Karlstads University
Tata Inst. Fundamental Research
FRANCE
GERMANY
UK
GERMANY
GREECE
JAPAN
JAPAN
e RUSSIA
GERMANY
RUSSIA
SWEDEN
INDIA
Silesian University at Opava CZECH REPUBLIC
Weizmann Institute of Science
Cardiff University
University of Trento
Federal University of Itajuba
Universidade da Beira Interior
Institute for Space Sciences
University of Bath
KFKI
Astronomical Obs. of Torino
University of Glasgow
University of Paris
University of South Africa
INFN Bologna
Open University of Israel
ICRA, ICRANet
University of Melbourne
LMU, ASC Munich
Nat'l Inst, for Physics and Nuc. En;
Nat'l Inst, for Physics and Nuc. Enj
University of Tours
University of Cambridge
Helsinki University of Technology
Deakin University
Harvard-Smithsonian CfA
Beijing Normal University
ISRAEL
UK
ITALY
BRAZIL
PORTUGAL
ROMANIA
UK
HUNGARY
ITALY
SCOTLAND
FRANCE
USA
ITALY
ISRAEL
ITALY
AUSTRALIA
GERMANY
;. ROMANIA
;. ROMANIA
FRANCE
UK
FINLAND
AUSTRALIA
USA
P.R.CHINA
2910
Wanas, Mamdouh I.
Wandelt, Benjamin
Wang, Chih-Hung
Watson, Casey
Wells, Alan
Westra, Willem
Whale, Ben
Whelan, John
Whisker, Richard
White, Nicholas
Williams, Floyd
Williams, Jeff
Willke, Benno
Wiseman, Toby
Woodard, Richard
Worrall, Diana
Wu, Yu-Huei
Wylleman, Lode
Xie, Naqing
Xue, She-Sheng
Yajima, Satoshi
Yakovlev, Dmitry
Yazadjiev, Stoytcho
Yilmaz, Huseyin
Yoo, Chul-Moon
York, James
Yoshino, Hirotaka
Yunt, Elif
Zakharov, Alexander
Zamani, Farhad
Zane , Silvia
Zannias, Thomas
Zapatrin, Roman
Zaslavskii, Oleg
Zavattini, Guido
Zayakin, Audrey
Zenginoglu, Anil
Zhu, Xingfen
Zink, Burkhard
Zofka, Martin
Zohren, Stefan
Cairo University EGYPT
University of Illinois at Urbana-Champaign USA
Lancaster University UK
Ohio State University USA
University of Leicester UK
Spinoza Institute NETHERLANDS
Australian National University
Albert Einstein Institute
Durham University
GSFC
University of Massachusetts
Brandon University
Albert Einstein Insitute
Harvard University
University of Florida
University of Bristol
University of Southampton
UGent
Fudan University
ICRANet
Kumamoto University
loffe Physico-Technical Institute
Sofia University
Hamamatsu Photonics K.K.
Osaka City University
Cornell University
Waseda University
Istanbul Technical University
Institute of Theor. and Exp. Physics
IASBS
University College of London
Inst. Fisicas y Matematicas
Russian State Museum
V.N. Karazin Nat'l University
University of Ferrara
Moscow State University
MPI for Gravitation
Center for Astrophysics
MPI Astrophysik
Charles University
Imperial College London
AUSTRALIA
GERMANY
UK
USA
USA
CANADA
GERMANY
USA
USA
UK
UK
BELGIUM
P.R. CHINA
ITALY
JAPAN
RUSSIA
BULGARIA
JAPAN
JAPAN
USA
JAPAN
TURKEY
RUSSIA
IRAN
UK
MEXICO
RUSSIA
UKRAINE
ITALY
RUSSIA
GERMANY
CHINA
GERMANY
CZECH REPUBLIC
UK
AUTHOR INDEX
Abdil'din, Meirkhan M., 2110, 2158
Abishev, Medeu E., 2110, 2158
Adamiak, Jaroslaw P., 2187
Adelberger, Eric G., 2579
Adis, Daria, 1737
Aguiar, Odylio D., 2448
Agullo, Ivan, 1437
Ahmad, Zahid, 2291
Ahmedov, Bobomurat J., 2098, 2122
Aksenov, Alexey, G., 1180
Alam, Ujjaini, 1797
Alekseev, George A., 543, 2252
Alexeyev, Stanislav O., 1251
Aliev, Alikram N., 1057, 1409, 2243,
2830
Alimi, Jean-Michel, 1785, 1831
Allen, Steve W., 1773
Aloy, Miguel Angel, 1589
Altamirano, Diego, 1198
Aman, Jan E., 1511
Amati, Lorenzo, 1965
Amato, Elena, 1561
Ambj0rn, Jan, 2779
Amelino-Camelia, Giovanni, 952
Amin, Mustafa A., 1773
Anderson, Matthew, 1579
Anderson, Paul R., 1497
Ando, Masaki, 2393
Anglada-Escude, Guillem, 2588
Angonin, Marie-Christine, 2407
Anninos, Peter, 1573
Ansoldi, Stefano, 2827
Ansorg, Marcus, 1600
Antoci, Salvatore, 1254
Antoniadis, Ignatios, 2054
Antonini, Piergiorgio, 2755
Anzalone, Evan, 1107
Arefiev, Vadim, 589
Arkhangelskaja, Irene V., 1968, 2015
Arkhangelsky, Andrey I., 1968
Aros, Rodrigo, 1317
Ashenberg, Joshua, 2530
Ashtekar, Abhay, 126
Astone, Pia, 2438
Atrio-Barandela, Fernando, 1677
Babichev, Eugeny, 1471
Bakala, Pavel, 1546
Bakry, Mohamed A., 2131
Balbi, Amedeo, 1674
Balcerzak, Adam, 2051
Barbero Gonzalez, Jesus Fernando,
2677
Barkov, Maxim V., 1615
Barrau, Aurelien, 1349
Barrett, John W., 2782
Barrow, John D., 1207
Barsuglia, Matteo, 2351
for the Virgo Collaboration, 177,
2351
Bashinsky, Sergei, 1659
Bassan, Massimo, 2359
for the ROG Collaboration, 2359
Basu, Prasad, 2343, 2500
Battat, James B., 2579
Battisti, Marco Valerio, 1890
Beciu, Mircea, 2140
Beesham, Aroonkumar, 1873
Beissen, Nurzada A., 2158
Belinski, Alexander, 543
Bengtsson, Ingemar, 1511
Benini, Riccardo, 1857, 1909, 2090
Bergamin, Luzi, 2686
Bergslioeff, Eric A., 2878
Beriiardini, Maria Grazia, 368, 1956,
1959, 1974, 1977, 1981, 1992,
1995
Berrocal Arellano, Aaron V., 2199
Berthier, Jerome, 2600
Bertolami, Orfeu, 2611
Bertoldi, Andrea, 2519
Bezerra, Valdir B., 2674, 2701
2911
2912
Bianco, Carlo Luciano, 368, 1956, 1959,
1974, 1977, 1981, 1989, 1992,
1995
Bicknell, Geoff V., 807
Bieli, Roger, 1767
Biermann, Peter L., 291, 985
Bilge, Ayse H., 2225
Bimonte, Giuseppe, 2749
Bini, Donato, 2104, 2113, 2137, 2152
Binkley, Mathew, 1497
Bishop, Nigel T., 1630, 1633
Bisnovatyi-Kogan, Gennadyl S., 2331
Bizouard, Marie-Anne, 177
for the Virgo collaboration, 177
Bjornsson, Gunnlaugur, 2003
Blaes, Omer M., 1573
Blanchet, Luc, 881
Blandford, Roger D., 1773
Bluhm, Robert, 1217
Boccaletti, Dino, 2261
Bogenstahl, Johanna, 2398
Bolejko, Krzysztof, 1847, 700
Bombaci, Ignazio, 605
Book, Laura G., 1078
Boonserm, Petarpa, 2285
Borchers, Marc, 972
Bostani, Neda, 1427
Bouhmadi-Lopez, Mariam, 1898
Boutloukos, Stratos, 1152, 1198
Bozza, Valerio, 1122, 1710, 2833
Bradley, Michael, 795
Braggio, Caterina, 2773
Brasileiro Formiga, Jansen, 1329
Breeveld, Alice, 1947
Bressi, Giacomo, 2755, 2773
Brill, Dieter, 2264
Brizuela, David, 1627
Broekaert, Jan, 1281
Bromm, Volker, 340
Brown, Duncan A., 1597
Brugmann, Bernd, 1612
Bruneton, Jean-Philippe, 1233
Brunnemann, Johannes, 2800
Bucciantini, Nicolo, 1561
Bucciarelli, Beatrice, 2543
Buchert, Thomas, 1831
Buhr, Henrik, 2515
Buonanno, Alessandra, 197
Burigana, Carlo, 1671
Burinskii, Alexander, 2101, 2246, 2631
Burrows, David, 1947
Butcher, J., 1549
Cacciapuoti, Luigi, 2519
Cagnoli, Gianpietro, 2379
Caito, Letizia, 368, 1959, 1974, 1977,
1981,1992, 1995
Calchi Novati, Sebastiano, 1694, 1700
Calderon, Hector, 1497
Camacho, Abel, 2639
Cannata, Roberto, 2261
Cantley, Caroline, 2379
Capone, Monica, 1755
Carloni, Sante, 1213
Carminati, John, 1549, 2128, 2268
Carugno, Giovanni, 2755, 2773
Carvalho, Carla, 2611
Case, Gary L., 1107
Castiheiras, Jorge, 2680
Catoni, Francesco, 2261
Cattoen, Celine, 2057
Cavero-Pelaez, Ines, 2727
Cembranos, Jose A.R., 2851
Cermak, Petr, 1546
Cerny, Slavomir, 1139
Chabrier, Gilles, 1189
Chakrabarti, Sandip K., 569, 1063,
1066, 1085, 1119, 1130, 1155,
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PARTC
PROCEEDINGS OF THE ELEVENTH
I I
Editors
Hagen Kleinert
Robert T Jantzen
Series Editor
Remo Ruffini
PROCEEDINGS OF THE ELEVENTH
MARCEL GROSSMANN MEETING
ON GENERAL RELATIVITY
The Marcel Grossmann Meetings are three-yearly forums that
meet to discuss recent advances in gravitation, general relativity
and relativistic field theories, emphasizing their mathematical
foundations, physical predictions and experimental tests. These
meetings aim to facilitate the exchange of ideas among scientists,
to deepen our understanding of space-time structures, and to
review the status of ongoing experiments and observations
testing Einstein's theory of gravitation either from ground or
space-based experiments. Since the first meeting in 1975 in
Trieste, Italy, which was established by Remo Ruffini and Abdus
Salam, the range of topics presented at these meetings has
gradually widened to accommodate issues of major scientific
interest, and attendance has grown to attract more than 900
participants from over 80 countries.
This proceedings volume of the eleventh meeting in the series,
held in Berlin in 2006, highlights and records the developments
and applications of Einstein's theory in diverse areas ranging
from fundamental field theories to particle physics, astrophysics
and cosmology, made possible by unprecedented technological
developments in experimental and observational techniques
from space, ground and underground observatories. It provides
a broad sampling of the current work in the field, especially
relativistic astrophysics, including many reviews by leading
figures in the research community.
World Scientific
.worldscientiTic.com
6997 he
ISBN-13 978-981-283-426-3 (set)
ISBN-10 981-283-426-5 (set)
9 "789812 834263"