ISBN: 0-19-851291-0

Текст
                    THE
INTERNATIONAL SERIES
OF
MONOGRAPHS ON PHYSICS
GENERAL EDITORS
R. J. ELLIOTT J. A. KRUMHANSL D. H. WILKINSON


THE MATHEMATICAL THEORY OF BLACK HOLES S. CHANDRASEKHAR University of Chicago Clarendon Press • Oxford Oxford University Press • New York 1983
Oxford University Press, Walton Street, Oxford 0X2 6DP London Glasgow New York Toronto Delhi Bombay Calcutta Madras Karachi Kuala Lumpur Singapore Hong Kong Tokyo Nairobi Dar Es Salaam Cape Town Melbourne Wellington and associate companies in Beirut Berlm Ibadan Mexico City © Oxford University Press, 1983 Published in the United States by Oxford University Press, New York All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging in Publication Data Chandrasekhar, S. (Subrahmanyan), 1910— The mathematical theory of black holes. (The International series of monographs on physics) 1. Black holes (Astronomy) I. Title. IL.'JSefles. QB843.B55C48 1982 523 82-7920 ISBN 0-19-851291-0 AACR2 British Library Cataloguing in Publication Data Chandrasekhar, S. (Subrahmanyan) The mathematical theory of black holes.—(The International series of monographs on physics) 1. Black holes (Astronomy) I. Title II. Series 523 QB843.B55 ISBN 0-19-851291-0 (y
TO THE READER First, my fear; then, my curtsy; last, my speech. My fear is, your displeasure; my curtsy, my duty; and my speech, to beg your pardons. If you look for a good speech now, you undo me; for what I have to say is of mine own making; and what indeed I should say will, I doubt, prove mine own marring. But to the purpose, and so to the venture. W. Shakespeare (Henry IV, Part II) Contents. The table of contents has been made sufficiently detailed that the topics covered and their organization can be gathered from it. Also, the opening section of each chapter provides the prospect and the closing section, often, a retrospect. Arrangement. The section numbers run serially through the entire book; and so do those of the tables and the figures. References to equation numbers are to those in the same chapter unless qualified by the prefix of the number of the chapter to which it belongs. Notation. The choice of a notation which will be strictly consistent throughout the book proved to be impossible: all the alphabets that are available are simply not adequate. Besides, one is constrained not to mar the appearance of an equation or a formula and obscure its meaning by a random assortment of symbols. A compromise had to be made. While certain symbols, easily recognizable from the context, maintain their meanings throughout, many are transient. It is hoped that going to the beginning of the chapter (if not the section) will resolve most of the ambiguities. Bibliographical Notes. Since the entire subject matter (including the mathematical developments) has been written (or, worked out) ab initio, independently of the origins, the author has not made any serious search of the literature. The bibliographical notes at the end of each chapter provide no more than the sources of his information. The book is an expression of the author's perspective with the limitations which that implies.
ACKNOWLEDGMENTS In my study of the mathematical theory of black holes, I have greatly benefited from my association with several colleagues. I am particularly indebted to John L. Friedman, for his critical judgement, perceptive comments, and constant encouragement, besides his help with Chapter 10 (§ 102) and 11 (§ 114); to Steven Detweiler, for his active collaboration in several investigations (incorporated in the book) and for his ever-readiness to be helpful both in matters requiring clarification and in matters requiring computations—Tables IV, VI, IX, X, and XI, as well as the tables included in the Appendix, are due to him; to Garret Toomey, for his assistance with the sections dealing with the geodesies in the Schwarzschild, the Reissner-Nordstrom, and the Kerr space-times and for providing the beautiful illustrations included in Chapters 3,5, and 7; and to Robert Wald, Robert Geroch, and James Hartle, for acting as referees on diverse matters. Of my indebtedness to Basilis Xanthopoulos, I can make no adequate acknowledgement. He undertook, gladly, the formidable task of reading the entire book in several stages of the manuscript and checking the mathematical developments. But his help went much beyond that: besides his original contributions included in Chapters 4 (§ 25), 5 (§ 42), 6 (§ 57), 7 (§ 60), and 11 (§ 109), his critical appraisal was always invaluable. I am also grateful to Ms. Mavis Takeuchi-Lozano for her assistance in the preparation of the manuscript and in its passage through the press. The writing of this book (during the period March 1980-January 1982) was supported in part by the National Science Foundation under grant PHY- 78-24275 with the University of Chicago. Also, during the three-month period (March-June 1980) I had the support of a Regents Fellowship of the Smithsonian Institution (Washington, D.C.) at the Center for Astrophysics, Harvard College Observatory. And finally, I am grateful to the Clarendon Press for bringing to this book (as to two earlier books of mine) that excellence of craftsmanship and typography which is characteristic of all their work. S.C.
CONTENTS Prologue 1 1. MATHEMATICAL PRELIMINARIES 3 1. Introduction 3 2. The elements of differential geometry 3 (a) Tangent vectors 4 (b) One-forms (or, cotangent or covariant vectors) 5 (c) Tensors and tensor products 8 3. The calculus of forms 10 (a) Exterior differentiation 12 (b) Lie bracket and Lie differentiation 14 4. Covariant differentiation 16 (a) Parallel displacements and geodesies 19 5. Curvature forms and Cartan's equations of structure 21 (a) The cyclic and the Bianchi identities in case the torsion is zero 25 6. The metric and the connection derived from it. Riemannian geometry and the Einstein field-equation 26 (a) The connection derived from a metric 28 (b) Some consequences of the Christoffel connection for the Riemann and the Ricci tensors 30 (c) The Einstein tensor 31 (d) The Weyl tensor 31 (e) Space-time as a four-dimensional manifold; matters of notation and Einstein's field equation 32 7. The tetrad formalism 34 (a) The tetrad representation 35 (b) Directional derivatives and the Ricci rotation-coefficients 36
k CONTENTS (c) The commutation relations and the structure constants 38 (d) The Ricci and the Bianchi identities 39 (e) A generalized version of the tetrad formalism 40 8. The Newman-Penrose formalism 40 (a) The null basis and the spin coefficients 41 (b) The representation of the Weyl, the Ricci, and the Riemann tensors 42 (c) The commutation relations and the structure constants 45 (d) The Ricci identities and the eliminant relations 45 (e) The Bianchi identities 48 (/) Maxwell's equations 51 (g) Tetrad transformations 53 9. The optical scalars, the Petrov classification, and the Goldberg-Sachs theorem 55 (a) The optical scalars 56 (b) The Petrov classification 58 (c) The Goldberg-Sachs theorem 62 Bibliographical notes 63 2. A SPACE-TIME OF SUFFICIENT GENERALITY 66 10. Introduction 66 11. Stationary axisymmetric space-times and the dragging of inertial frames 66 (a) The dragging of the inertial frame 69 12. A space-time of requisite generality 70 13. Equations of structure and the components of the Riemann tensor 73 14. The tetrad frame and the rotation coefficients 80 15. Maxwell's equations Bibliographical notes 82 84
CONTENTS xi 3. THE SCHWARZSCHILD SPACE-TIME 85 16. Introduction 85 17. The Schwarzschild metric 85 (a) The solution of the equations 88 (b) The Kruskal frame 90 (c) The transition to the Schwarzschild coordinates 92 18. An alternative derivation of the Schwarzschild metric 93 19. The geodesies in the Schwarzschild space-time: the time-like geodesies 96 (a) The radial geodesies 98 (b) The bound orbits (£2<1) 100 (i) Orbits of the first kind 103 (a) The case e = 0 106 (/3) The case 2/t (3 + e) = 1 107 (8) The post-Newtonian approximation 107 (ii) Orbits of the second kind 108 (a) The case e = 0 109 (/3) The case 2/t (3 + e) = 1 110 (iii) The orbits with imaginary eccentricities 111 (c) The unbound orbits (£2>1) 113 (i) Orbits of the first and second kind 113 (ii) The orbits with imaginary eccentricities 115 20. The geodesies in the Schwarzschild space-time: the null geodesies 123 (a) The radial geodesies 123 (b) The critical orbits 124 (i) The cone of avoidance 127 (c) The geodesies of the first kind 130 (i) The asymptotic behaviours of <px for P/3M—* 1 and P»3M 132 (d) The geodesies of the second kind 133 (e) The orbits with imaginary eccentricities and impact parameters less than (3J3)M 133 21. The description of the Schwarzschild space-time in a Newman-Penrose formalism 134 Bibliographical notes 136
4. THE PERTURBATIONS OF THE SCHWARZSCHILD BLACK- HOLE 139 22. Introduction 139 23. The Ricci and the Einstein tensors for non-stationary axisymmetric space-times 139 24. The metric perturbations 142 (a) Axial perturbations 143 (b) The polar perturbations 145 (i) The reduction of the equations to a one-dimensional wave equation 149 (ii) The completion of the solution 150 25. A theorem relating to the particular integrals associated with the reducibility of a system of linear differential equations 152 (a) The particular solution of the system of equations (52)- (54) 158 26. The relations between Vi+) and F(~> and Z<+) and Z(~> 160 27. The problem of reflexion and transmission 163 (a) The equality of the reflexion and the transmission coefficients for the axial and the polar perturbations 164 28. The elements of the theory of one-dimensional potential- scattering and a necessary condition that two potentials yield the same transmission amplitude 166 (a) The Jost functions and the integral equations they satisfy 169 (b) An expansion of lgT(o-) as a power series in o--1 and a condition for different potentials to yield the same transmission amplitude 171 (c) A direct verification of the hierarchy of integral equalities for the potentials V[±) = ±/3/' +/32/2 + k/ 173 29. Perturbations treated via the Newman-Penrose formalism 174 (a) The equations that are already linearized and their reduction 175 (b) The completion of the solution of equations (237)-(242) and the phantom gauge 180 30. The transformation theory 182 (a) The conditions for the existence of transformations with f = 1 and /3= constant; dual transformations 185
(b) The verification of the equation governing F and the values of k and /32 188 31. A direct evaluation of ^0 in terms of the metric perturbations 188 (a) The axial part of ^0 190 (b) The polar part of ^0 192 32. The physical content of the theory 193 (a) The implications of the unitarity of the scattering matrix 197 33. Some observations on the perturbation theory 198 34. The stability of the Schwarzschild black-hole 199 35. The quasi-normal modes of the Schwarzschild black-hole 201 Bibliographical notes 203 5. THE REISSNER-NORDSTROM SOLUTION 205 36. Introduction 205 37. The Reissner-Nordstrom solution 205 (a) The solution of Maxwell's equations 206 (b) The solution of Einstein's equations 206 38. The nature of the space-time 209 39. An altemative derivation of the Reissner-Nordstrom metric 214 40. The geodesies in the Reissner-Nordstrom space-time 215 (a) The null geodesies 216 (b) Time-like geodesies 219 (c) The motion of charged particles 224 41. The description of the Reissner-Nordstrom space-time in a Newman-Penrose formalism 224 42. The metric perturbations of the Reissner-Nordstrom solution 226 (a) The linearized Maxwell equations 226 (b) The perturbations in the Ricci tensor 227 (c) Axial perturbations 228 (d) Polar perturbations 230 (i) The completion of the solution 234
43. The relations between Viw and Vty and Zj<+) and Zf 235 44. Perturbations treated via the Newman-Penrose formalism 238 (a) Maxwell's equations which are already linearized 238 (b) The 'phantom' gauge 240 (c) The basic equations 241 (d) The separation of the variables and the decoupling and reduction of the equations 242 45. The transformation theory 245 (a) The admissibility of dual transformations 246 (b) The asymptotic behaviours of Y±i and X±j 248 46. A direct evaluation of the Weyl and the Maxwell scalars in terms of the metric perturbations 249 (a) The Maxwell scalars <\>o and <t>2 252 47. The problem of reflexion and transmission; the scattering matrix 254 (a) The energy-momentum tensor of the Maxwell field and the flux of electromagnetic energy 257 (b) The scattering matrix 258 48. The quasi-normal modes of the Reissner-Nordstrom black- hole 261 49. Considerations relative to the stability of the Reissner- Nordstrom space-time 262 50. Some general observations on the static black-hole solutions 270 Bibliographical notes 271 6. THE KERR METRIC 273 51. Introduction 273 52. Equations governing vacuum space-times which are stationary and axisymmetric 273 (a) Conjugate metrics 276 (b) The Papapetrou transformation 277 53. The choice of gauge and the reduction of the equations to standard forms 278
(a) Some properties of the equations governing X and Y 281 (b) Alternative forms of the equations 282 (c) The Ernst equation 284 54. The derivation of the Kerr metric 286 (a) The tetrad components of the Riemann tensor 290 55. The uniqueness of the Kerr metric; the theorems of Robinson and Carter 292 56. The description of the Kerr space-time in a Newman-Penrose formalism 299 57. The Kerr-Schild form of the metric 302 (a) Casting the Kerr metric in the Kerr-Schild form 306 58. The nature of the Kerr space-time 308 (a) The ergosphere 315 Bibliographical notes 317 7. THE GEODESICS IN THE KERR SPACE-TIME 319 59. Introduction 319 60. Theorems on the integrals of geodesic motion in type-D space-times 319 61. The geodesies in the equatorial plane 326 (a) The null geodesies 328 (b) The time-like geodesies 331 (i) The special case, L=aE 331 (ii) The circular and associated orbits 333 62. The general equations of geodesic motion and the separability of the Hamilton-Jacobi equation 342 (a) The separability of the Hamilton-Jacobi equation and an alternative derivation of the basic equations 344 63. The null geodesies 347 (a) The 0-motion 349 (i) -rj > 0 349 (ii) -q = 0 349 (iii)i7<0 349
(b) The principal null-congruences 349 (c) The r-motion 350 (d) The case a=M 357 (e) The propagation of the direction of polarization along a null geodesic 358 64. The time-like geodesies 361 (a) The 0-motion 362 (b) The r-motion 363 65. The Penrose process 366 (a) The original Penrose process 368 (b) The Wald inequality 370 (c) The Bardeen-Press-Teukolsky inequality 371 (d) The reversible extraction of energy 373 66. Geodesies for a2>M2 375 (a) The null geodesies 375 (b) The time-like geodesies 376 (c) Violation of causality 377 Bibliographical notes 379 8. ELECTROMAGNETIC WAVES IN KERR GEOMETRY 382 67. Introduction 382 68. Definitions and lemmas 383 69. Maxwell's equations: their reduction and their separability 384 (a) The reduction and the separability of the equations for ¢0 and ¢2 385 70. The Teukolsky-Starobinsky identities 386 71. The completion of the solution 392 (a) The solution for (/> 1 393 (b) The verification of the identity (80) 394 (c) The solution for the vector potential 395 72. The transformation of Teukolsky's equations to a standard form 397 (a) The r^r)-relation 399
73. A general transformation theory and the reduction to a one- dimensional wave-equation 400 74. Potential barriers for incident electromagnetic waves 404 (a) The distinction between Z, + 0>) and Z<-0>) 406 (b) The asymptotic behaviour of the solutions 408 75. The problem of reflexion and transmission 410 (a) The case <t>&c (= -aim) and a2>0 410 (b) The case &s<&<o*c 412 (c) The case 0 s£ & < o^ 414 76. Further amplifications and physical interpretation 417 (a) Implications of unitarity 419 (b) A direct evaluation of the flux of radiation at infinity and at the event horizon 421 (c) Further amplifications 425 77. Some general observations on the theory 427 Bibliographical notes 428 9. THE GRAVITATIONAL PERTURBATIONS OF THE KERR BLACK-HOLE 430 78. Introduction 430 79. The reduction and the decoupling of the equations governing the Weyl scalars ^0, *i, ¥3, and ¥4 431 80. The choice of gauge and the solutions for the spin coefficients k, a, A, and v 434 (a) The phantom gauge 435 81. The Teukolsky-Starobinsky identities 436 (a) A collection of useful formulae 441 (b) The bracket notation 442 82. Metric perturbations; a statement of the problem 443 (a) A matrix representation of the perturbations in the basis vectors 444 (b) The perturbation in the metric coefficients 445 (c) The enumeration of the quantities that have to be determined, the equations that are available, and the gauge freedom that we have 446
XV111 CONTENTS 83. The linearization of the remaining Bianchi identities 447 84. The linearization of the commutation relations. The three systems of equations 448 85. The reduction of system I 453 86. The reduction of system II: an integrability condition 454 87. The solution of the integrability condition 458 88. The separability of ¥ and the functions *3l and &~ 464 (a) The expression of £% and Sf in terms of the Teukolsky functions 465 89. The completion of the reduction of system II and the differential equations satisfied by £% and & 467 90. Four linearized Ricci-identities 470 91. The solution of equations (209) and (210) 472 (a) The reduction of equations (233)-(236) 475 (b) The integrability conditions 478 92. Explicit solutions for Zi and Z2 479 (a) The reduction of the solutions for Zi and Z2 482 (b) Further implications of equations (211) and (212) 484 93. The completion of the solution 485 94. Integral identities 487 (a) Further identities derived from the integrability condition (263) 489 95. A retrospect 497 96. The form of the solution in the Schwarzschild limit, a —»0 500 97. The transformation theory and potential barriers for incident gravitational waves 502 (a) An explicit solution 503 (b) The distinction between Z< + ot)and ZI-** 506 (c) The nature of the potentials 507
CONTENTS xix (d) The relation between the solutions belonging to the different potentials 512 (e) The asymptotic behaviours of the solutions 513 98. The problem of reflexion and transmission 514 (a) The expression of R and l in terms of solutions of Teukolsky's equations with appropriate boundary conditions 517 (b) A direct evaluation of the flux of radiation at infinity 520 (c) The flow of energy across the event horizon 523 (d) The Hawking-Hartle formula 526 99. The quasi-normal modes of the Kerr black-hole 528 100. A last observation 529 Bibliographical notes 529 10. SPIN-'/i PARTICLES IN KERR GEOMETRY 531 101. Introduction 531 102. Spinor analysis and the spinorial basis of the Newman- Penrose formalism 531 (a) The representation of vectors and tensors in terms of spinors 535 (b) Penrose's pictorial representation of a spinor £A as a 'flag' 537 (c) The dyad formalism 538 (d) Covariant differentiation of spinor fields and spin coefficients 539 103. Dirac's equation in the Newman-Penrose formalism 543 104. Dirac's equations in Kerr geometry and their separation 544 105. Neutrino waves in Kerr geometry 546 (a) The problem of reflexion and transmission for cr> <rs ( = -am/2Mr+) 548 (b) The absence of super-radiance (0< cr< crs) 550 106. The conserved current and the reduction of Dirac's equations to the form of one-dimensional wave-equations 552
CONTENTS (a) The reduction of Dirac's equations to the form of one- dimensional wave-equations 553 (b) The separated forms of Dirac's equations in oblate- spheroidal coordinates in flat space 555 107. The problem of reflexion and transmission 556 (a) The constancy of the Wronskian, [Z±,Z%~\, over the range of r, r+<r< °° 557 (b) The positivity of the energy flow across the event horizon 558 (c) The quantal origin of the lack of super-radiance 560 Bibliographical notes 561 11. OTHER SOLUTIONS; OTHER METHODS 563 108. Introduction 563 109. The Einstein-Maxwell equations governing stationary axisymmetric space-times 564 (a) The choice of gauge and the reduction of the equations to standard forms 566 (b) Further transformations of the equations 567 (c) The Ernst equations 569 (d) The transformation properties of the Ernst equations 570 (e) The operation of conjugation 572 110. The Kerr-Newman solution: its derivation and its description in a Newman-Penrose formalism 573 (a) The description of the Kerr-Newman space-time in a Newman-Penrose formalism 579 111. The equations governing the coupled electromagnetic- gravitational perturbations of the Kerr-Newman space-time 580 112. Solutions representing static black-holes 583 (a) The condition for the equilibrium of the black hole 586 113. A solution of the Einstein-Maxwell equations representing an assemblage of black holes 588 (a) The reduction of the field equations 590 (b) The Majumdar-Papapetrou solution 591 (c) The solution representing an assemblage of black holes 592
CONTENTS xxi 114. The variational method and the stability of the black-hole solutions 596 (a) The linearization of the field equations about a stationary solution; the initial-value equations 603 (b) The Bianchi identities 606 (c) The linearized versions of the remaining field equations 608 (d) Equations governing quasi-stationary deformations; Carter's theorem 609 (e) A variational formulation of the perturbation problem 614 (i) A variational principle 619 (ii) The stability of the Kerr solution to axisymmetric perturbations 620 Bibliographical notes 622 Appendix. Tables of Teukolsky and Associated Functions 625 Epilogue 637 Index 639
Roy Patrick Kerr (1934- ) A-4 Karl Schwarzschild (1873-1916)
PROLOGUE The black holes of nature are the most perfect macroscopic objects there are in the universe: the only elements in their construction are our concepts of space and time. And since the general theory of relativity provides only a single unique family of solutions for their descriptions, they are the simplest objects as well. The unique two-parameter family of solutions which describes the space- time around black holes is the Kerr family discovered by Roy Patrick Kerr in July, 1963. The two parameters are the mass of the black hole and the angular momentum of the black hole. The static solution, with zero angular momentum, was discovered .by Karl Schwarzschild in December, 1915. A study of the black holes of nature is then a study of these solutions. It is to this study that this book is devoted.
1 MATHEMATICAL PRELIMINARIES 1. Introduction In this chapter, we shall provide an account of the analytical methods that lie at the base of much of the developments that are to be described in this book.These consist of Cartan's calculus of differential forms and the tetrad and the Newman-Penrose formalisms. While none of these matters are novel in themselves, they cannot all be found in a coherent treatment in one place; and the account is included to make the book as self-contained as is possible. The presentation of the elements of differential geometry in §§2-6 is, however, not intended to replace the standard accounts of the subject available elsewhere: the presentation is confined to the barest essentials leading to Cartan's equations of structure. 2. The elements of differential geometry Differential geometry deals with manifolds. Manifolds are essentially spaces that are locally Euclidean in a sense which we shall first make precise. We recall that an Euclidean space of n-dimensions, R„, is the set of all n-tuples, (x1,. . . , x") (- oo < x' < + go), with open and closed sets (or, neighbourhoods) defined in the usual way. A manifold, M, is locally identical to Euclidean space in the sense that M is covered (i.e., a union of) neighbourhoods, 11^, and that associated with each <^a there is a one-one map, (/>„, which images each point p e 91^, to a point in an open neighbourhood of R„ (onto which <^a is imaged by fa) with the coordinates (x1,..., x"). Further, if two neighbourhoods, 11^, andf^, of M, intersect and have points in common (i.e., ^a <^^"a =/= 0), and if fa and fa are the associated maps onto neighbourhoods in U„, then the map fa0^/^1 images a point fa(p) (pef,nfa) with the coordinates (x1, ..., x"), say, to the point fa(p) with the coordinates (x1, ..., x"), then as a part of the definition of M, it is required that x' (i — 1,. . . , n) are smooth functions of (x1, ..., x"). (Smooth functions are those which have continuous partial derivatives of all orders.) The Cartesian product M x N of two manifolds M and N is, in the first instance, the ordered pair of points, (p, q), where peM and q e N; further, if ^fa and f~x are neighbourhoods in M and in N, t/>a and \j/e are the associated maps, and <px(p) = (x1,. . . , x") and i/^(<?) = (y1, ■ . • , /") (m not necessarily equal to n), then the map, (fax ipe) (p,q) = (x1, . . . ,x",y\ . . . , f),
4 MATHEMATICAL PRELIMINARIES suffices to complete the definition of M x N as a manifold of (m + «)- dimensions. We now consider a function f on M defined by a map /: M -» R1. We shall suppose that the combined map f° <j>~1 which images a point (x1, . . . , x") in W on to the reals, IR^isasmooth function of the coordinates (x1, . . . ,x").We define a smooth curve X on M by the map X: an interval I (a <t < b) in U1 -* X(t) = peM such that ((/)^)(0 = ^(0,-. .,x"(t)]; (1) and we require that x'(f) (i = 1, . . ., n) are smooth functions off. Finally, we may note that a function f, defined on the manifold, enables us to define the function, f° X on the curve X. With the aid of the map (/>a ° X we are led to consider the function f(X (t)) =f(x1(t), . . . , xn(t) (where (x*(0> • • • ,x"(t))are the coordinates of p = X(t) by the map t/>a. (a) Tangent vectors With the definition of/(x1 (f), ..., x"(f)) on a curve X on M, which we have just given, consider "A dt Jut) = limit -{f(X(t0 + e))-f(X(t0)) = " dxJ(0 ,= i dt t0\dxS)uto) \dt 8x> (2) where, in the last step, summation over repeated indices is assumed. (This summation convention will be adopted throughout the book.) It is now clear that by considering various curves X passing through a given point p, we can define a linear vector-space (at p) consisting of linear combinations of the coordinate derivatives djdx' of the forms x=x'L- (3) where the XJ,s are any set of n numbers. These tangent vectors arise by considering the curves X defined by x>(t)^x'(p) + X>t 0-=1,...,/1) (4) for t in some small interval — e < t < + e. The tangent vectors at p form a linear vector-space over U1 spanned by the coordinate derivatives, since the requirement for a linear vector-space, namely, («X+pY)f=«(Xf) + p(Yf), (5)
THE ELEMENTS OF DIFFERENTIAL GEOMETRY 5 is satisfied for all vectors X and Y, numbers a and ft, and functions f. Moreover, the vectors (d/dxJ')p are linearly independent; for, otherwise, there should exist numbers X'(j = 1, . . . , ri), not all zero, such that X = X'd/dx' applied to any smooth function is identically zero; but the application of X to the coordinate functions xk(k = 1, . . . , n) would lead to Xk = 0 for all k; and this is a contradiction. Finally, the definition of X by Xf=Xl]jJ = xJf.J> (6> for every smooth function f clearly satisfies the Leibnitz rule when operating on products of functions, thus X(fg)\m=(fXg + gXf)\Mt). (7) Tangent vectors may, in fact, be considered as directional derivatives. (In equation (6) we have introduced the notation of denoting derivatives with respect to x' by the index), following a comma, both as subscripts.) The space of tangent vectors (or contravariant vectors, as they are also called) to an n-dimensional manifold M at p, denoted by Tp(M) or simply Tp, is an n-dimensional vector-space. This space, which may be visualized as the set of all 'directions' at p, is called the tangent space at p. Instead of a basis determined by local coordinates, we may choose any other n linearly independent vectors ea(a = 1, . . . , n) (say). There must, then, exist linear relations of the form where the determinant of the matrix formed by <!>„* must be non-zero. The inverse relation is then given by ^=*V* (9) where [¢^] is the inverse of the matrix [¢/]: $ak$'j = 5kj and $.k$bk = dab. (10) Given any basis (ej), we can express any tangent vector at p in the form X=XJe;. (11) The XJ,s are the components of X relative to the basis (ej). (b) One-forms (or, cotangent or covariant vectors) A one-form, a>, at p is a linear mapping of the tangent space Tp on to the reals: a>. Tp^U\ (12)
6 MATHEMATICAL PRELIMINARIES In other words, given any tangent vector X at p, the one-form <o associates uniquely with it a number <o{X) which is also written as a>(X)=(a>,X}. (13) The required linearity of the map is expressed by the relation (a>,otX+pY> = a(a,,X> + p(a>,Y>, (14) where X and Y are any two tangent vectors and a and /? are any two real numbers. We, further, define multiplication of forms by real numbers and sums of forms by the rules that, for any XeTp and any real number a, (<xo})(X) = <x((o,X} and (a) +n)(X) = (<o,X) + (it, X), (15) where <o and n are two one-forms. By these rules, one-forms span a vector space which we denote by T*p; it is called the cotangent space at p and the dual of the tangent space. For this reason one-forms are also called cotangent vectors (or, covariant vectors). We shall now verify that a basis for T*p, associated with a basis (e}) for Tp, is provided by the one-forms (e')(i = 1, . . . , n) which map any tangent vector X = X'e} to its components; thus, e'(X)=(e',XJeJ) = Xi (i= 1,..., n). (16) From this last equation it follows that el(ej)=W,e}y=6lj. (17) The expression of any arbitrary one-form <o as a linear combination of the e"s is obtained by observing that (a>,X)=(a>,Xiei) = Xi(a>,eiy. (18) Now letting co;= (a>, ety =<o(ei), (19) be the numbers to which a> maps the basis vectors (e;) of the tangent space Tp at p, we may write (<o,X} = tOtX' = «(<>', X^j) = <a^',X>. (20) Since this last equation is valid for any XeTp, it follows that a> = a>ie'; (21) and this is the required expression of a> as a linear combination of the e"s. That the vectors ei are linearly independent is manifest from its definition. The bases (¢() and (e') are said to provide dual bases for the tangent and the cotangent spaces at p.
THE ELEMENTS OF DIFFERENTIAL GEOMETRY 7 If in place of the dual bases (e;) and (e') we should choose different bases, ev = d>; ;*; and ey = &' se\ (22) obtained by non-singular linear transformations represented by 3>;>J and Q>}'s, then the condition, that the new bases (er) and (e3) continue to be dual, requires = cDJ>rk^=cDJ>V; (23) in other words, the matrices [<&;-'] and [3>j;] are the inverses of one another. Finally, we may note that if we change from one set of local coordinates (x;) to another set (xl'), then the corresponding expressions for 3>r' and ¢-^ are Associated with any function f on the manifold, one defines a one-form d/ by requiring that df(X)=(df,X} = Xf, (25) for any vector XeTp. In a local coordinate basis, and by definition, in particular, X=X>1; (26) (dlXy^X'^^X'fj; (27) <dx^,£7> = ^i. (28) Hence, the one-forms (dxj) provide a local coordinate basis for the cotangent vectors which is dual to the local coordinate basis provided by the tangent vectors (dj = d/dx}) for the tangent space. The bases (dj) and (dxJ) are sometimes referred to as the canonical bases for the tangent and the cotangent spaces. We note that if d/= ajdxJ, (29) then it follows from Xi/i=<d/,^>=<aJdxJ,X'ai> = a,X'<dx',S, > = «,*', (30) that a, =/, and d/ = /;dx<; (31)
8 MATHEMATICAL PRELIMINARIES and this last equation is consistent with the conventional meaning one attaches tod/ (c) Tensors and tensor products LCt nsr = T*p x T*p x ... x T% x Tp x Tp x ... x Tp (32) v. ■> v / r factors s factors represent the Cartesian product or r cotangent spaces and s tangent spaces at some point p of a manifold, i.e., the space of ordered sets of r one-forms and s tangent vectors: (to1, . . . , a/, Xlt . . . ,XS). And consider a multilinear mapping, T, of the manifold nsr to the reals: T: ff^R1. (33) Precisely, what the mapping provides is an association (in some unique manner) of any given ordered set of r one-forms and s tangent vectors to a real number: T{(ol, . . . ,(or, Xu ..., Xs) = a real number. (34) The condition that the map is multilinear requires that nm\...,ar,aX+pY,X2,...,XM) = a7>\ . ..,&', X,X2, ..., Xs) + pT{a>\ . . . , wT, Y, X2, . . . , Xs), (35) for all a, /JelR1 and X, YeTp; and for similar replacements of all the other forms and vectors. A multilinear mapping so defined is said to be a tensor of type (r, s). Linear combinations of tensors of a given type (r, s) are defined by the rule (a7+ /?S)K, ...,<*„...,*,) = a7>\ ...,wr,Xu.. . ,^,) + 0,5(0.1. ...,aT,Xu.. .,XS), (36) for all a./JeR1, <o'eT*p, and XjeTp (i = I, . . . ,r; j = I, . . . ,s). By these rules, tensors of a given type (r, s) span a linear vector-space of dimension nr+s. And the space of such tensors is called the space of tensor products and denoted by Trs(P) = rp® ... ®rp® r*p®... ®r% (37) r factors s factors We shall presently verify that a basis for tensor products of type (r, s) is provided by the nr+s special mappings <?,,...,/'• ;'-K, ...,ar,Xlt...,X.) = 'u ...,/'' ■ H^kie\ ..., afKe\ 1,% ..., Xs'-e,) = <o1il...cfirX1i'...X.J: (38)
THE ELEMENTS OF DIFFERENTIAL GEOMETRY 9 These mappings are clearly linear in every argument and are tensors of type (r, s). An equivalent way of denning these mappings is «,,... ,/' ■■■ V', • • -,e\eh,. . .,et) = 8\d\ . . . 5\S\ . . . 5\. (39) That any tensor of type (r, s) can be expressed as a linear combination of the mappings (39) follows from noting T(a>\ ...,a>',Xu...,Xs) = Tip1 ^, . . . ,co,.V%X>A, . . . , A>Js) = co1,,. . . . a/, X,* . . . Xj-T(e\ ..., e\ eh, ..., eh\ (40) and letting T(e\..., e\ eJlt ...,^) = 7^---^ ...j,; (41) for, we can then write (cf equation (38)) r= ^--¾....;.««,. V,,"i"; (42) and this is the required expression for T as a linear combination of the mappings (39). It is manifest that the mappings (39) are linearly independent; they, therefore, provide a basis for tensors of type (r, s). The number of these basis elements, e^ /'■ ■ ■>', is nr+s (which is the dimension of the Trs). The coefficients, T'1- l'Si ^, in the expansion (42), are said to be the components of T relative to the chosen basis. One generally writes '(,... .'/V ■ ''' = «i, ® ■ ■ ■ ® eK®eh ® ■ ■ ■ ® e''> (43> as representing the tensor product of the dual bases (et) and (e>) of Tp and T*p. In this notation, the tensor product Y1®...®Yr®£l1® ...®QS (44) of r tangent vectors and s one-forms is that element of Trs which maps (to1 . . . , (or, Xt,. . . , Xs) to the number W.Y.y ... (o/.ijxtf,^)... <ns,*s>. (45) In particular, (<?,. <g>. . . ®eir®ej' <g>. . . ®ej') K, . . - , < Xu . . . , X.) = (^,^)... ««t><^, *,>... <e\X.> = a;1,- . . . a>W'. . .*,'■ = ^...,./"' -'-(a.1,. ..,a>',Xu..., Xs); (46) and this justifies the notation. If instead of the dual bases (e;) and (e'), we choose different dual bases (ev) and (eJ ), then it follows from equations (22) that the components of T, relative to the new basis, efl® ■ ■ ■ ®eVr®e>'i® . . . ®e>\
10 MATHEMATICAL PRELIMINARIES are given by 7"1"-^,.../. = ^, ■■■^V---^-^-^,...v (47) The contraction of a tensor of type (r, s) with the components P1 ■ ■ ■ trjt ...^, with respect to a chosen contra variant index ip and a chosen covariant index;,, is defined as the following tensor of type (r - 1, s - 1): 7-(,... «,.,«,♦,...,;._ Vi^<+i A<?ii® . . . ®elpi®eip+i® ■ ■ ■ ®<?,v ®eh® . . . ®<?'«->(x)<?Vi(g) . . . ®eA (48) where, as the notation indicates, summation over all values of k (of the i„- contravariant index and of the ;4-covariant index) is to be effected. It can be readily verified, with the aid of equations (23) and (47), that the process of contraction is independent of the chosen basis. A tensor of type (0,2) is said to be symmetric or antisymmetric if T(X, Y) = T(Y, X) or T(X, Y) = -T(F, X) for all X and Y in T„. (49) In terms of components, in an arbitrary basis, symmetry or antisymmetry implies _ _ _ _ .... Tls = Tn or T,j=-Tj,. (50) More generally, a tensor of type (r, s) is said to be symmetric or antisymmetric in its covariant indices i and j if 7>\. ..,a>',... ,ta>,. ..,wr,Xu...X,) = ±T(a>\ . . ,a>\ . . ,a>1, . . .,a>r,Xu . . . , Xs), (51) for all co's and ^'s. Symmetry or antisymmetry with respect to chosen contravariant indices are similarly defined. 3. The calculus of forms A particularly important class of tensors of type (0, s) is the class of totally antisymmetric tensors, i.e., covariant tensors which are antisymmetric in every pair of their arguments, i.e., T(Aj, ..., Xj, ..., Xj, ..., Xs) = — 1 \X\, ..., Xj, ..., A;, ..., As), (52) for all pairs of indices i and j and for all a"s. Tensors of this kind can be constructed out of a general tensor T of type (0, s) by applying to it the alternating operator A whose effect on it is to give the linear combination defined by ,47-(^,,...,^) = 4 I sgnU,, ...,;s)r(A-;.,, ..., A";.), (53)
THE CALCULUS OF FORMS 11 where the summation is extended over all s! permutations of the s integers (1, . . . ,s)andsgn(;!, . . . ,js) = + 1, according as (ju . . . Js) is an even or an odd permutation of (1, . . . , s); and equation (53) is to be valid for every (X j ,..., As). It is clear that if T is already totally antisymmetric, the effect of A on it is, simply, to reproduce T. Also, if s > n (the dimension of the vector space) the effect of/4 on T(Xi, . . ., Xs) is to reduce it to zero; in other words, there can be no totally antisymmetric tensor of type (0, s) for s > n. Totally antisymmetric tensors of type (0, s) are called s-forms. Since they must vanish when any two of their arguments coincide, it follows that the s-forms span a vector space of dimension n\/s\ (n — s)!. This space is denoted by at%. If Tj _j are the components of a tensor of type (0, s) relative to the basis, eh® ■ ■ ■ ®e>\ and if the tensor should be totally antisymmetric, then its n\/s\{n — s)! distinct components can be distinguished by arranging its indices in a strictly descending sequence in the manner: Th.-.j. where Ji >^> ■ ■ ■ >■/'»• (54) A basis for AST*P can be obtained by applying the alternating operator A to the basis elements of the tensor product: A(eh® . . . ®e>-). The resulting basis elements are written as the exterior or the wedge product of the eJ,s in the manner: «>' A «>'» A . . . A «>■ (ji >j2> ■■■ > j,). (55) A general s-form can then be written as SI = Q;i.. .,/>' A e1* A . . . A e>- (ji >j2> ..-> Js), (56) where the summation is now extended only over strictly descending sequences. Since interchanging a pair of indices is equivalent to interchanging the corresponding elements in the wedge product, it follows that interchanging the elements in a wedge product must be accompanied by a change of sign; thus e> Aek = -ekAej. (57) In a local coordinate basis, the expression for an s-form is " = £V .jdx'^A . .. Adx^. (58) Given any p-form il and a q-form il2, we can form their wedge (or, exterior) product by the rule il1 A il2 = A(Q* ®Q2) (59)
12 MATHEMATICAL PRELIMINARIES to obtain a (p + g)-form. (It must accordingly vanish identically if p + q > n.) Wedge products of forms clearly obey the associative and the distributive laws, but they are not in general commutative. For, by definition, ft1 A ft2 = (ft1^ jeh A . . . AeJ-)A (Q2k] ktek> A . . . A <?*.), (60) where (j\, ■ ■ ■ ,jp) and (/cl5 . . . , kq) are strictly descending sequences. Accordingly, ft1 A ft2 = (-l)M(n2fcl...fc<<?fc'A . . . A«k.)A(Q1.]Jf''A .. . Ae'<) = (-i)wn2An1, ' (61) since each of the q basis elements ek', . . ., ek* must suffer p interchanges before il1 A il2 can be brought to the form required of ft2 A ft1. So far, we have considered tensors and forms defined at a point on the manifold. We shall now enlarge the basic definitions in a way which enables us to envisage fields defined on M. Thus, a smooth tensor-fieldT' S(M) (or, simply Trs) of type (r, s) on M is an assignment of an element of T's(p) at each point p e M in such a way that the components of Trs relative to any local coordinate basis are smooth functions of the coordinates. This enlargement of the basic definitions is necessary if we are to formulate notions of differentiation. In the future we shall be concerned only with smooth tensor fields; and this is to be understood even if the qualifying words 'smooth' and 'field' are omitted. (a) Exterior differentiation Exterior differentiation is effected by an operator d applied to forms. It converts p-forms to (p+ l)-forms consistently with the following rules: (a) The operator d, applied to functions (or zero-forms) f, yields a one-form d/ defined by df(X)=(df,xy = xf for every Xe T10. In particular, in a local coordinate basis ^^ (b) If Ax and A2 are two p-forms, d(otA1+PA2) = otdA1 + pdA2 (a^eR1). (c) If A is a p-form and B is a q-form, d(A A B) = dA A B + (- 1)" A A dB. (d) Poincare's lemma, which requires that d(d/l) = 0, for every p-form A.
THE CALCULUS OF FORMS 13 To clarify that the operator d subject to the foregoing rules is well-defined, consider, first, the exterior derivative dA = d(A, i dx3' A . . . A dx>>) of a p-form A. By rules (a), (b), and (d) dA =dA: -, Adx^'A . . . AdxJ* dA: , = Jl: 'dx*AdxJ-A . .. AdxV (62) dx It is important to verify at this point that dA, as given by equation (62), is independent of the choice of the local coordinate system. For, if instead of the local coordinates (x3') we had chosen a different set of local coordinates (x1'), then by equations (24) and (47), dx3' dx >' Ai\...i=Ah.-U^'"^J,i (63» and we should conclude that d (/1,, j.dx3' A . . .Adx^) v Ji---jp ' ( dx'' dx3' = d[Aj i ——. . .—— dx^'A . . . AdxJ* \ 3'-3' dx3' dx3' dx3' dx3' = —- . • • ^r-rd^i j A dx;i A . . . A dx3' dx3' dx3' 3'--3' d2x3' dxh dx3' 4 v , ? , /• dxkdx3' dx3' dx3' 3'-3' dx3' dx3'-' d2x3' ., ■ ■■ + v~r- --^-^- -, t-, fAi idxk Adx3' . .. Adx3'. (64) dx3' dx3'-' dxkdx3' 3'--3' v ' All the terms on the right-hand side of equation (64) involving the second derivatives of x3<(i = 1, . . . ,n) vanish on account of their symmetry in/c'and./; and the antisymmetry of the basis elements, in these same indices, in the wedge product; and the sole surviving term is the first one which is clearly the same as dAi , Adx3' A . . . Adx3' = d(Ai , dx3' A . . . A dxM. Accordingly, d(An j.pdx3' A . . . A dx3'-) = d(Ah jdx3' A . . . A dx^); (65) and this is what we set out to verify. We next verify that the rule (c) is consistent with the expression (62) for dA.
14 MATHEMATICAL PRELIMINARIES For, by the rules (a), (b), and (d) d(A A B) = d(Aji j^dx'' A . . . A dxj- A Bki k^dxk> A . . . A dx*') dA, , = —-^dx' A dx'' A . . . A dx'- A Bk k dxk> A . . . A dx^ dxi kt...k, SBk k + A, f ^-?dx! A dxJl A . . . A dx'- A dxk' A . . . A dx*1* ;,...;„ dx, /dBk k \ = dA AB + (-l)M;-_ ..jdxJ'A . . . A dx'- Al '"," ' dx1' A dx k> A ... Adxfc. I = dA AB + (-l)"A AdB. (66) And finally, to establish the consistency of rule (d), we need only observe that dxk rdA: , d{dA) = d( a t dx* A d*;'' A . . . A dx^ d2A- = ^-^dx' A dxk A dx^'' A . . . A dx'- = 0. (67) This completes the demonstration that the operation d is, indeed, well-defined. (b) Lie bracket and Lie differentiation Given any two vector fields, Xand Y, their Lie bracket, \_X, Y~\, is defined by its action on any function /; and it is given by IX, F]/= (XY- YX)f= X(Yf)- Y(Xf). (68) The Lie bracket of any two tangent vectors is, again, a tangent vector, since IX, F] («/■+ fig) = «[*, Ylf+ fi\_X, Y]g (69) and \_X, F](fg) = g[X, Y1f+f\_X, Y^g, (70) where f and g are any two functions and a and fi are any two real numbers. The first of these relations is manifest while the second follows quite readily: \.X,Y2<Jg) = X(Yfg)-Y(Xfg) = X(gYf+fYg)- Y(gXf+fXg) = gXYf+ (Xg)(Yf)+(Xf)(Yg)+fXYg -{gYXf+(Xf)(Yg)+(Yf)(Xg)+fYXg} = g[X, Yy+flX, Y]g. (71) The relation (69) establishes the Lie bracket as a linear operator while the relation (70) establishes it as a differentiation.
THE CALCULUS OF FORMS 15 As may be readily verified, the Lie bracket satisfies the Jacobi identity, [[*, F], Z] + [[F, Z], XI + [[Z, XI, Y] = 0. (72) We have seen that the Lie bracket of X and Y is a tangent vector. Its components, relative to a local coordinate basis, can be obtained by its action on x'. Thus, IX, Yy = (AT- YX)x> = XY'-YX' = XkY\k-YkX\k, (73) where (as indicated earlier) a comma preceding an index denotes partial differentiation with respect to the local coordinate with that same index. In a local coordinate basis, the Lie bracket \_dk, df\ clearly vanishes. Considered as a differentiation, \_X, Y~\ is called the Lie derivative of Y in the direction X and is written as SexY = IX, F] = - \Y, XI = - <£YX. (74) More generally, we define the Lie derivative, if XT, of a tensor field T of a given type, as a tensor of the same type which satisfies the following rules: (a) Its action on a scalar field f is given by &xf= Xf= df(X). (75) (b) Its action on a tangent vector Y, as we have already defined, is given by <£XY=\X,Y\ (76) And (c) it operates linearly on tensor fields and satisfies the Liebnitz rule when acting on tensor products: ^X(S®T) = ^XS®T+S®^XT, (77) where S and T are arbitrary tensor fields. The last of the foregoing rules enables us to derive the effect of Z£x on a tensor of arbitrary type. Thus, its effect on a one-form <o can be determined by considering, for any arbitrary vector-field Y, the contracted version of the relation 2>x(<o®Y) = (tfx<D)®Y + a>®{2>xY\ (78) namely, ^<a.,F> = <^a.,y>+<a.,^y>. (79) Writing out this last equation explicitly, we have Xk(<o,Y'\k = {tfxa>)jY> + a>,{#xY)' (80) or, making use of equation (73), we obtain (^xa)tY' = Xk(a>J,kY> + a>iY>,k)-a>i(XkY1,k-YkX1,k) = (Xk(oJ,k + (okXktJ)Y'. (81)
16 MATHEMATICAL PRELIMINARIES Since this last equation is valid for an arbitrary Y, we conclude that (Sexm)l^(0l,kXk + <okX\J. (82) We may write equation (79) alternatively in the form &x\a>W\ = (&x«>){Y) + «>(&xV)- (83) By rule (c), equation (83) admits of generalization to a tensor of type (r, s). We have ^x\_T(<o\..., c/, y„..., ys>] = (sexT)(m\..., w, y„..., ys) + nsexa>\ a>2,...,a>',Yu...,Y,)+ ■■■ + 7-(^,...,^,1^,...,^}¾ (84) where all the terms in this equation, except the first one on the right-hand side, can be evaluated in terms of the known results (73), (75), and (82). The components of ifXT can, therefore, be deduced from equation (84). For later use, we shall derive here a simple identity relating the exterior derivative of a one-form to Lie derivatives. By making use of equation (82), we have <i?,a»,y>-y<a»,*> = ((Oj,kXk + cokXkj)Y>-Y'(cokJXk + cokXkj) = (a>j,k-a>kJ)XkY> = 2da>(X, Y). (85) Now, substituting for (yx<o,Yy from equation (79), we obtain the required result: da>(X, Y) = ${X(a>, Y} - Y(a>, X} - <o», \_X, F] >} (86) where we have written the Lie bracket \_X, Y~\ in place i£xY. 4. Covariant differentiation We shall now define a type of differentiation which, unlike exterior and Lie differentiation, requires that the manifold be endowed with an additional structure. This additional structure is an affine connection, V, which assigns to each vector field Xon M a differential operator, V*, which maps an arbitrary vector-field, Y, into a vector field VXY. Consistent with these requirements, we impose the conditions, (a) VXY is linear in the argument X, i.e., Vfx+gYZ=fVxZ+gVYZ (X,Y,ZeT\\ (87) where/and g are any two arbitrary functions defined on M; (b) VXY is linear in the argument Y, i.e., VX(Y+Z) = VXY+VXZ (X,Y,ZeT\), (88) (c) Vxf=Xf, (89)
COVARIANT DIFFERENTIATION 17 where/is any function on M; and, finally, (d) Vx(fY)=(Vxf)Y+fVxY. (90) It should be noted that, according to equation (89), in a local coordinate basis (dk), Vdk, when acting on functions, coincides with partial differentiation with respect to x\ With the action of V^ on vector fields YieT1^ specified by the rules (a)-(d), we now define the covariant derivative, V F, of F as a tensor field of type (1,1) which maps the contravariant vector-field X to V^F, i.e., VF(*)=<VF,*> = V*F, (91) for every XeT V In this notation, we can rewrite equation (90) in the form V(/F) = d/®F+/VF (92) To clarify what the assignment of a connection precisely means, it will be useful to rewrite VXY relative to some chosen dual bases (et) and (e'). Thus, making use of the rules (a)-(d), we have V*F = Vx(Y'ej) = (X Y>)e} + y'V^. (93) Since Vxes, for a particular e3, is a tensor field of type (1,1) we must have a representation, in the chosen basis, of the form Vxej = a>'j (X)e„ (94) where <olj (depending on I and;') are one-forms. Accordingly, we may write VxY^iXY^ej+ro'jiX)^. (95) Alternatively, we may also rewrite equation (93) in the form VxY=(XY')ej+YJVx^ej = (XP)ej+PXkVetej, (96) or, in conformity with the definition (94), VXY = (XYJ)eJ+YJXka>'J(ek)el. (97) Letting 0-,.(^) = 0^ (98) be the coefficient of ek in the expansion of <o'j in the basis (ek), we conclude that a connection V is specified by the n2 one-forms a>';. or, equivalently, by the n3 scalar fields to'Jk. Returning to equation (95) and rewriting it in the form VXY=\_XY> + e>;,(*)y']<?;, (99) we infer that (VXY)J = XYi + (D\(X)Yl. (100)
18 MATHEMATICAL PRELIMINARIES In a local coordinate basis (dk, dX'), equation (100) gives (V^YV^YJ + Y'o^Y^ + y'coV (101) In a local coordinate basis, it is customary to write F}lk in a place of (x)'lk; (102) and using semicolons to indicate covariant derivatives (in contrast to commas which indicate ordinary partial derivatives), we obtain the standard formula y^ = y^ + y'rv (103) The definition of covariant derivatives of vector fields can be extended to tensor fields, in general, by requiring that the operation of V satisfies the Leibnitz rule when acting on tensor products. Thus, we require that V(S®T) = VS®T + S®VT, (104) where S and T are two arbitrary tensor-fields. An immediate consequence of this requirement is (cf. equation (84)) Vx{T(a>\ . . . ,a»', Ylt . . ., Ys)} = (VxT)(a>\ . . . ,a»', Yu . . ., Y.) + T(Vxa>\a>2, ...,a.', ^,,...,^)+ ... + T(a>\ . . . , a.', Yu . . ., YS^,VXYS). (105) Thus, if il is a one-form, then, for every vector field Y, the foregoing equation gives Vx(il(Y)) = (Vxil)(Y) + il(VxY), (106) or, in terms of a local basis (et) and {e1), we have V*(QjYj) = (Vx^jY' + QjiVxYy. (107) Now making use of rule (c) and equation (100), we find (Vxil)jY' = (XQj)Y^ + Qj(XY')-QJ\_XY^ + YW,(*)] = (XQj)YJ^Qi(o'j(X)YK (108) We conclude that {vxn)} = xn,-a,a>l,(X), (i09) or, alternatively, Vxil = \_Xnj-n,o}'j(X)^e}. (110) Specializing this last equation to the case when SI = e', we obtain the formula v*<?;=-«>;,(*),?', (in) which is to be contrasted with the earlier formula (94). Equation (111) shows that a knowledge of the n2 one-forms <olj suffices to determine the covariant derivatives of one-forms, as well, once we accept the Leibnitz rule for tensor products.
COVARIANT DIFFERENTIATION 19 Also, we may note that in a local coordinate basis, equation (109) gives Q;,, = QM-Q,rV (112) An important result follows from equations (109) and (112) when applied to the one-form d/ Since the components of d/ in a local coordinate basis are/,-, we obtain from equation (112), in this case, f,j;k=fj,k-fjr'jk; (113) and by permuting the indices) and k in this equation, we obtain Since partial differentiations applied to functions permute, we find, on taking the difference of equations (113) and (114), fj;k-f*J= -fAr'fi-r'y). (115) The right-hand side of equation (115) is non-vanishing only for non-symmetric connections. On this account, it is customary to write 7"'*= -(r'/k-r'm). (116) From the occurrence of this quantity in equation (115), it is clear that T'jk are the components of a tensor of type (1, 2). It is called the torsion tensor. We define the torsion tensor more generally in §5; meantime, we may note that in terms of it, we can write equation (115) in the form fj:k-f.k-j = TlJkftl. (117) Returning to equation (105), we now observe that, with the aid of equations (99) and (109), we can readily write down the covariant derivative of an arbitrary tensor-field. Thus, s\, = s%+s^kr'ml + s^r^ - sW (i is) (a) Parallel displacements and geodesies Let Y represent a contravariant vector-field. Consider its variation along a curve Aon M. The change 5 Y in Y caused by a displacement along A resulting from an increment 5t in t (which parametrizes A) is, in a local coordinate system, given by (Syy = YJk^miSt (119) at In Euclidean geometry and in a Cartesian system of coordinates, one would say that Yis 'parallely propagated' along A if SY = 0. In a general differentiable manifold with a connection, one defines, analogously, that a vector Y is
20 MATHEMATICAL PRELIMINARIES parallely propagated along X, if {Dyy = ,^ ^mi„ = YK/^piit = 0, „20, 1 at at or, alternatively, if (1^+1^)^^ = 0. (121) In other words, for parallel propagation of Y along X, we require that (cf. equation (119)) (6Y)> = - Ylr'lkdxk(j(t)) 5t. (122) at In particular, for the tangent vector to the curve X, dxJ(X(t))/dt parallely propagated along X, J4*!wm\ = _ d^(t))d,»Wt))^ \ df / df df A curve A on M is said to be a geodesic if the tangent vector to A, parallely propagated, remains a multiple of itself. This condition, for X to be a geodesic, is, clearly, dx'(A(t)) , dx\X(t))dx\X{t)) j 1 Ik ~j j of where t/>(f) is some function of t. In the limit (5 f -> 0, the equation for a geodesic becomes d2xj . dx'dx* ,, dxj ip-+rJ*dTdT-*(t)dr- (125) It can be readily verified that if we reparametrize the curve X by the variable df"exp dt'4(t')\, (126) equation (125) becomes d2xj . dx'dx* n -r^+r^ —— = 0; (127 ds ds ds and when the equation for a geodesic is reduced to this form, we say that it is ajfinely parametrized. It should be noticed that the only freedom we have in the choice of s is its origin and its scale.
CURVATURE FORMS AND CARTAN'S EQUATIONS 21 5. Curvature forms and Cartan's equations of structure For a manifold endowed with a connection, we define the two mappings T(X, Y) = VXY-VYX-\_X, F] (128) and *(*,r) = V,Vy-VyV,-V[xy] (129) where Xand Fare two contravariant vector-fields. These mappings are called torsion and curvature, respectively. As defined, both are antisymmetric in their arguments. Considering torsion first, we readily verify that T is linear in the arguments X and Y; thus T(X+Y,Z) = T(X,Z) + T(Y,Z) (X, Y, ZeT\); (130) also T(fX,Y)=fT(X,Y\ (131) where /is any function. (In proving the second of these relations, we must make use of identity UX,Y~\=f\.X,Y-\-(Yf)X.) The relations (130) and (131) clearly imply that the mapping T: T'oxT'o^T'o (132) is multilinear. Accordingly, T is a tensor field of type (1, 2). Let (ej) and (el) provide dual bases for T„ and T*p. Then, as we have shown (equation (100)), (VxYy = XY}+Y'(o',(X) = Xes(Y) + e'(Y)(o',(X). (133) Therefore, VxY-VyX=\_XeJ(Y) + a>sl(X)e'(Y)-YeJ(X)-a>Jl(Y)e'(X)']eJ: (134) Hence, P(X, Y) = <e\VxY-VYX-\_X,Yiy = X(e\ Y> - Y(e\X}- <«', \_X, F] > + <0'l{X)e\Y)-<0il{Y)e\X), (135) or, making use of the general identity (86), we have \TS{X, Y) = (de' + a,1, A el)(X, Y). (136) Since this equation is valid for arbitrary X and X we conclude that ir^' = deJ + co^Ae' = SlJ (say). (137)
22 MATHEMATICAL PRELIMINARIES This is the first of Cartan's equations of structure. In the important special case, when the torsion is zero, equation (137) reduces to deJ! + a)', A el = 0. (137') In a local coordinate basis, de> — 0 (since e' = dx') and equation (137) reduces to T> = 2T\k dxk A dx' = (rjlk - r^dx* a dx', (138) in agreement with our earlier definition of this quantity in equation (116). Turning next to the curvature, by definition, we have R(X,Y)Z = VXVYZ- vyV,Z- Vw y]Z. (139) The expression on the right-hand side of equation (139) is manifestly linear in X, Y, and Z And, moreover, it can also be verified that R (fX, Y )Z = R (X, fY )Z = fR (X, Y )Z ) and > (140) R(X,Y)fZ=fR(X,Y)Z, J where/is an arbitrary function. Hence, the mapping R: T\xT\xT\^T\, (141) is a multilinear function of the arguments. Consequently, R is a tensor field of type (1,3): it is called the Riemann tensor. Now, making use of known relations, we obtain S/XVYZ= Vx{Yei{Z) + a>Sk(Y)ek(Z)}ej = \_Ye,(Z) + <olk(Y)ek(Zn\/xel + \XYe\Z) + *{o»',(lV(Z)}>, = [_Ye,(Z) + <olk(Y)ek(Z)^l(X)eJ + \_XYe' (Z) + e> (Z)X(o', (Y) + a>\ (Y)Xe' (Zftej = \_XYeJ(Z) + <o\ (Y)Xe1 (Z) + e1 (Z)Xoj', (Y) + ©>, (X)Yel (Z) + a,', (X)w'k {Y)t* (Z)]<?,.. (142) Consequently, VXVYZ- VYVXZ = {lXa>i,(Y)- Ya>\(X) + m\(X)a>k,(Y) -to\{Y)(ok,(X)y(Z)+\_X,Y^(Z)}eJ. (143) We also have VwnZ= {\_X,Yle'(Z) + a>\i\_X,Yl)e'(Z)}ej. (144) Now combining equations (143) and (144), we obtain R(X, Y)Z={X( m\ Y > - Y < m\ X > - < <o'u \_X, Y] > + a>lk(X)at,{Y)-a>\(Y)at,{X)}J{Z)ej, (145)
CURVATURE FORMS AND CARTAN'S EQUATIONS 23 or, making use of the identity (86), we have \R(X, Y)Z= (do.', + a>\ A a>\)(X, Y)el(Z)ej. (146) Accordingly, if a> is any one-form, R(a>,Z,X, Y) = Rilkmlej®el®{ek A em)-]{e>,Z,X, Y) = {R\kmek A em) (X, Y^(Z)ej(<D). (147) From a comparison of equations (146) and (147), we obtain the relation Wikm^ A «m = dcoJ, +e>\Ae>\. (148) Denning the two-form, il1, = &a>\ + to'k A a>\, (149) we have Carton's second equation of structure: W^e* A *" = HV (150) In a local coordinate basis ©', = T\mAxm; (151) therefore d«>j, = rv„dx« a dxm = i(rv„ - rj,„,m)dx" a dxm. (152) Also, o^Ao^r^r^dx^dx"- = i[r^„rk,m-rJkmrk,„)dx"Adx"'. (153) Accordingly, Cartan's second equation of structure is equivalent to the definition R^nn, = r v „ - r\, m + rv,r\m - rjkmr\„; (154) and this is the Riemann tensor as conventionally defined. It is of interest to relate the foregoing treatment of the Riemann tensor, following Cartan, to the more customary treatment of it, by evaluating the right-hand side of equation (139), ab initio, in a local coordinate basis. Thus, by making use of equation (103), we have S/XS/YZ = Vx(YkZ\kej) = Yk V*(Z V,.) + Z>,ke-^xYk = YkXlZ>,k.le)+Yk.lXlZ',kej. (155) Accordingly, VXVYZ-VYVXZ = YkX\Zlku- Zll.k)eJ+[Yk.lX' - XkuY'lZ>.ke} = YkX\ZKM-ZKA,k)eJ+lX,Yyz\ke1 + XlY»{Tknl-Ykln)Z-'.ke}- (156)
24 MATHEMATICAL PRELIMINARIES or, since v[xy]z=[*,y]kzv,, (157) R(X, Y)Z= (ZJ.k.l-Z}.l.k + T»lkZ'.n)X,YkeJ = RJilkZiXlYkej. (158) We thus obtain the relation ZAt:/-ZA/;t= -K^,Z! + r"wZ^„. (159) This is the Ricci identity; it is the customary starting point for the introduction of the Riemann tensor when the torsion is zero. It is of interest to contrast equation (159) with the equation, f,ku-f,i;k = T"uf,n, (160) which we derived earlier in §4 (equation (117)). A further result of some importance which follows from equation (129) is (cf. equation (114)) R(X, Y)/= v,(y'/,) - VYX%) - ix, y-\% = (xiY>.t- rx'.o/j+Y>xi (fju-iu)- ix, nj'fj = x'r(p„,-r;,„)// + x'YVr^ + r",,.)/„ = o. (iei) So far, we have considered the effect of R(X, Y) on contravariant vector- fields and scalar fields only. We shall now consider its effect on arbitrary tensor-fields. By virtue of the Leibnitz rule satisfied by covariant differentiation of tensor products, we readily verify that R(X,Y)(P®Q)=R(X,Y)P®Q + P®R(X,Y)Q, (162) where P and Q are arbitrary tensor-fields. With the aid of equation (162) we can, for example, find the effect of R(X, Y) on a one-form ft. Thus, if Zis any contravariant vector-field, it follows from equation (162) that R(X,Y)(RjZ>)= \_R(X,Y)£ll-Z* + £lj\_R(X,Y)zy. (163) Since R(X,Y) acting on a scalar field vanishes (by equation (161)), we conclude: IR{X,Y)S11jZ'= -njR'utZ'X'Y" = -SliR'^ZlX'Y1'. (164) Hence, \_R(X, Y)ill = - R'pOtX'Y*. (165) On the other hand, by evaluating the effect of R(X, Y) on il by the same procedure that was followed in deriving equation (158), we now obtain R(X, Y)il = XlYk[Qj.k.nQJ.l.k + r'lkQj.H]e>. (166)
CURVATURE FORMS AND CARTAN'S EQUATIONS 25 Now combining equations (165) and (166), we have n,-;t;/-ty;/;t = ^ j,,^ +^,,^. „. (167) Again, considering the effect of R(X, Y) on a tensor of type (2,0), we have (on making use of equations (161) and (162)) *(*, y)[S^®<g = *(*, Y)S"el®ej + el®R(X, Y)SlU} = (R^SV + U>Illll,Sa)X"Y-«i ®«j; (168) and we conclude that SlJ;kii-s'J;i;k= -Rlm»SmJ-RJMS-+^8^,,. (169) In a similar fashion, we find It is now manifest how we can write down corresponding formulae for tensors of arbitrary type. Finally, we note that by contracting the Riemann tensor (154) with respect to the second (or, the third) co variant index, we obtain the Ricci tensor (or, its negative); thus K^ = -*'*, = *■• (171) In a local coordinate basis, the expression for the components of the Ricci tensor is (cf. equation (154)) Rlm = rvpr^+r^rl - r\mrk,j. (172) (a) The cyclic and the Bianchi identities in case the torsion is zero In case the torsion is zero (cf. equation (128)), VxQ-VqX=\.X,Q] (X,QeT\), (173) the curvature tensor satisfies two important identities which we shall now establish. First, we verify that by virtue of the relation (173), vx\y, z] + vy[z, *] + wzix, r\ = (V,Vy-VyV,)Z+(VzV,-V,Vz)y+(VyVz-VzVy)*. (174) Therefore, R(X,Y)Z+ R(Z,X)Y+ R(Y,Z)X = v,[y,z] + Vy[z,jf] + vz[jf,y]-vWy]z-v[yZ]jf-v[zfly. (175) On the other hand, by writing Q = [Y,Z~\ in equation (173), we obtain V*[F, Z] - V[y>z]* = IX, \Y,Z\ ]. (176)
26 MATHEMATICAL PRELIMINARIES Accordingly, equation (175) reduces to R(X, Y)Z+R(Z,X)Y+ R(Y,Z)X = \X, [Y,ZH + [Y, [Z, XI ] + [Z, IX, Y] ] = 0, (177) by the Jacobi identity. In a local coordinate basis, equation (177) provides the cyclic identity K'ita + KJtai + KJ»» = 0. (178) Next, consider the exterior derivative of Cartan's two-form Q;, (cf. equation (149)). We have dil'i = dco \ A <okl — e> -^ A dco'', = (il^-a^Ac^Aa^-a^A^-a^Aa.",). (179) The triple wedge-products of the one-forms which occur in the second line of equation (179) are seen to cancel; and we are left with dilJ,- £ljkA<ok, + o>'\ Ailk, = 0; (180) and this equation expresses the Bianchi identities. We can obtain them in their standard forms by rewriting equation (180) in a local coordinate basis when ";i = i^JiMdxpAdx« and e>\ = r\rdxr. (181) With these substitutions, equation (180) gives (R}lpt,r- R}kptrklr + rVK'Wdx" Adx" Adx' = 0. (182) Since the connection F'kr is symmetric in k and r, when the torsion is zero, the additional terms we include in the following equation do not affect its validity: Wipn.r — R'kpqT lr + rjkrR lpq - R'mT^r- RJlpkrkv)dx> A dx" A dx' = 0. (183) But the quantity in parentheses in equation (183) is precisely the covariant derivative of R\pq with respect to xr. We conclude that RJlPq;r+Rilqr;p + Rilrp;q = ^ (184) and this is the Bianchi identity in its standard form. 6. The metric and the connection derived from it. Riemannian geometry and the Einstein field-equation A metric tensor g is a non-singular symmetric tensor-field of type (0,2). Thus, (a) tif.xf^R1; (b) ^FH^JJOforevery^Fer0,; \ (185) (c) g(X, Y) = 0 for every YeT0! implies that X = 0.
THE METRIC AND THE CONNECTION DERIVED FROM IT 27 The condition (b) ensures the symmetry of g while the condition (c) its non- singular nature. In a local basis, we may write g = gije'®e} and gt} = g}i; (186) and, similarly, in a local coordinate basis, g = giJdxi®dxi and gi} = g}i. (187) In terms of its components, the requirement that the metric tensor be non- singular is equivalent to the requirement that the determinant g of the matrix [#;,] is non-zero at every point of the manifold. The matrix [#;;] has then a unique inverse. We denote the elements of the inverse matrix by g1' so that g,Jg}k = slk. (188) This last equation guarantees that we may, in fact, regard gij as the components of a tensor field, g'1, of type (2,0) whose representation in the basis e;®£, (dual to the basis used in equation (186)) is given by g'1=gi}ei®eJ. (189) One uses the metric tensor to define a path length L, along a curve X on M, from X (a) to X (b) (for example) by the formula L = dx< (X (t))dx' (/1(f)) 9u 1/2 dt. (190) df df In conformity with this definition, it is customary to write ds2 =0i,-dx;dx'' (191) and consider ds2 as giving the square of the interval ds between neighbouring points of the manifold. Given a tensor field T of type (r, s), we may contract g ® T and g~1 ® T with respect to one of the indices of the metric tensor (or its inverse) to obtain tensors of ty pes (r— l,s+ l)and (r+ l,s — Irrespectively. The components of the contracted tensor are written as a-Tah■••'••" . = Tab••■.-••? . ^1 iJ ij cd . . . q ' j ca. . . q ] and I (192) nij-rab...p _-rab...p j J g ' cd . ..i.. .q — ' cd q- J The process can clearly be repeated. We regard tensors derived by such raising and lowering of indices as representing the same geometric quantity since by raising an index and subsequently lowering it, we recover the original tensor. An important notion concerning the metric tensor is its signature. It is defined as the difference in the number of coefficients that are positive and the
28 MATHEMATICAL PRELIMINARIES number of coefficients that are negative when gi} (at some point) is brought to its diagonal form. It can be shown that the signature so defined is the same at all points of a (connected) manifold. An Euclidean metric is one for which the signature is, numerically, equal to the dimension of the manifold. And the metric is said to be Lorentzian or Minkowskian if the signature is + (n —2)-, the plus or minus sign being a matter of convention. (a) The connection derived from a metric So far, we have considered the introduction of the metric as independent of whatever connection we may or may not have endowed the manifold. We shall now show that, associated with a metric, we can endow the manifold with a unique torsion-free connection by the requirement that V* = 0. (193) With such a connection v(*®r) = *®vr, (194) where 7" is any tensor field. The principal advantage of such a connection is that the operation of raising or lowering of indices commutes with the operation of covariant differentiation. To deduce the torsion-free connection which follows from the requirement (193), we evaluate ¥xg with g expressed in the form (186). Thus, we require VxQije'®^ = Xgt,l ® e> + gtjl(Wxe') ® e> + e1® (V*«'")] = 0, (195) or, making use of equation (111), we have \.Xgij-glj<o,i(X)-gil<o'J(X)-]ei®eJ = 0. (196) We conclude that (dgtj-g^af, - gu<o';)(X) = 0. (197) Letting X = dk in a local coordinate basis, we obtain from equation (197) the requirement gis,k=gijTllk + gllTlSk% (198) where by our assumption of zero torsion, the T -symbols are symmetric in their 'covariant' indices. From equation (198) we derive, in the usual fashion, that 9allk~2\dxk+-dJ dx'J' (199) or, equivalently, The connection is thus uniquely specified; and this connection underlies Riemannian geometry.
THE METRIC AND THE CONNECTION DERIVED FROM IT 29 The T-symbols (200), appropriate to a metric g, are called the Christoffel symbols; and the connection itself is called the Christoffel connection. Two elementary consequences of the Christoffel connection are the following. The first is that the scalar product, (X- Y), of two contra variant vector-fields, X and Y, defined by g(X,Y)={XY) = gijXlYK (201) remains unchanged as A'and Fare parallely propagated along a curve X on M. For (by equations (120) and (121)) 0^^)= (DgrfX'Y' + gulYiDX' + X'DY^^O, (202) since Dgtj vanishes by virtue of the condition Vg = 0 (from which the connection was derived) and DX1 and DX' vanish by the assumption of parallel propagation along X. A second consequence is that the geodesic equation (127), derived in §4 from the requirement that the tangent vector to the curve X, as it is parallely propagated along X, remains a multiple of itself, now emerges as the Euler- Lagrange equation of an extremal problem. Thus, consider the integral (cf. equation (190)) I ^l(X{s))dx'(X(s)) „M^ Lds, where L = gl}—^-^-—^^, (203) and the curve X is parametrized by the arc length, s, along X. The Euler- Lagrange equation, for the extremal problem associated with the integral /, is Af8L\ dL n , dxUX(s)) 3- tt- -v-7 = 0 where *'= r-^- (204) ds\dx>J dx' ds y Since dL . dL -^=2gi}xl and ^ = ff«M*'*. (205) the Euler-Lagrange equation reduces to M' + tey>jk-±0tti .,)*'** = 0, (206) or, alternatively, ffy3e' + itoy.* + 0ui-ff«M)*'*k = °- (207) Contracting this last equation with g'\ we obtain 3c' + r'ikxix'' = 0. (208) An alternative form of the foregoing equation, which we shall find useful, is to express it in terms of the 'velocity'' . dx> W = x> = —. (209) ds
30 MATHEMATICAL PRELIMINARIES Then xJ=u\kuk, (210) and the equation for the geodesic takes the form (<k + n'iku'V = "j;X = 0. (211) If the geodesic is not affinely parametrized, it will take the form (cf. equation (125)) u'.^ = <puJ, (212) where <j> is some scalar function. As may be directly verified(by contracting with u,), equation (211)allows the integral («•«) = constant, consistently with the fact that u is parallely propagated along the geodesic. (b) Some consequences of the Christoffel connection for the Riemann and the Ricci tensors When the connection is that compatible with a metric, the Riemann and the Ricci tensors have additional symmetries. Thus, equation (170) with gi} substituted for Stj gives g!n,Rmjk, + gn,jRm!k,=0; (213) or, with the index m lowered, we have RiJkl + RjM = 0. (214) Hence, the completely covariant Riemann-tensor is antisymmetric in the first pair of indices, as well. We deduce a further symmetry by the following sequence of transformations. Starting from the cyclic identity (cf. equation (178)), Rjkmn + Rjmnk + Rjnkm = 0, (215) making use of the known antisymmetry in the first and the second pair of indices and of the cyclic identity as well, we find successively, Rjkmn = ~~ ("jmnk + Rjnkm) = "mjnk + R-njkm = ~ (Rmnkj + ^mk/n) ~ (Rnkmj + Rnmjk ) = 2 Rmnjk + (Rkmjn + Rknmj) = 2 Rmnjk — R-kjnm = 2 Rmnjk ~ Rjkmn- (216) Hence Rjkmn = Rmnjk- (217) The Riemann tensor is therefore unchanged by the simultaneous interchange of the first pair with the second pair of its covariant indices. From these various symmetries, one concludes that the number of independent components of the Riemann tensor is n2(n2 — 1)/12 (i.e., 20 for the Riemann tensor of a four- dimensional manifold).
THE METRIC AND THE CONNECTION DERIVED FROM IT 31 From the symmetries of the Riemann tensor, the symmetry of the Ricci tensor follows; for Ru^fRw^fRm^Rji- (218) We may note in this connection the explicit expression for the Ricci tensor in a local coordinate basis. But first we observe that rJ;k = igJI (g,3,k + g,kJ- gjk,i) = hg*gXj, k = (WlfflU (219) where \g\ denotes the absolute value of the determinant of [#;;]■ With the foregoing expression for the contracted Christoffel symbol, equation (172) gives „ _r; S2\gy/\g\ d\gyf\g\ R,m-r"-;~ dSdx™ +^^r "-~r k™r »■ (220) (c) The Einstein tensor The Einstein tensor G;j is related to the Ricci tensor by Gn^Ru-hgtjR, (221) where R = Rl, = gilRu (222) is the contracted Ricci tensor (or, as it is usually called, the scalar curvature). The most important property of the Einstein tensor is that its covariant divergence vanishes: G'J:,= 0. (223) This identity follows from the Bianchi identity (184). Thus, contracting the relation, *ljkl:- - eiPRipln.;k + Rijmk;l = °' (224) with respect to the indices i and k, we obtain K;/:.-0*'V.:it-Kj-;/ = O. (225) Now raising the index; and contracting it with m, we obtain KJ/:j + K\:it-K/ = 0> (226> or (#,-^,11)., = 0, (227) which is the required identity. (d) The Weyl tensor We have seen that the Reimann tensor, R{Jkl, is antisymmetric in both pairs of indices (y) and (kl) and it is, moreover, unchanged by a simultaneous interchange of the two pairs of indices (y) and (kl). By virtue of these
32 MATHEMATICAL PRELIMINARIES symmetries, the only non-trivial contraction we can make, by raising one of its indices and contracting it with any one of the three remaining covariant indices, is that leading to the Ricci tensor. Accordingly, it will be convenient to separate the Riemann tensor into a 'trace-free' part and a 'Ricci' part. This separation is accomplished by the Weyl tensor, Cijki = R-ijki — tn_2\ ^ik ^A + Qj' ^ik ~ ®>k ^" ~~ &" ^ik' + fr-iHW-2)(gftgj,~gflg't)Jt- (228) This tensor has manifestly all the symmetries of the Riemann tensor; but 0y,CyH = O in contrast to gilRiJkl = Rik- (229) A further distinction is that while the Riemann tensor can be denned in a manifold endowed only with a connection, the Weyl tensor can be denned only when a metric is also denned. The importance of the Weyl tensor for the deeper problems of differential geometry (and of general relativity) is the conformal invariance of Cljkl, i.e., its invariance to thetransformationg -»Q2g where Q is some scalar function; but we shall not be concerned with those problems in this book. (e) Space-time as a four-dimensional manifold; matters of notation and Einstein's field-equation In our account of differential geometry, we have, up to the present, made no restrictions on the dimensionality of the manifold (or, of its signature). But space-time, in the general theory of relativity, is considered as a four- dimensional differentiable manifold with a Lorentzian signature; and our subsequent discussion in this chapter (as in the rest of the book) will be restricted to this case. In this section, we shall recapitulate the basic equations of the theory in the notation and with the conventions (regarding signs) we shall adopt. First, with regard to the signature, our convention will be that when the metric is brought to its diagonal form (at any point of space-time), the difference in the number of coefficients that are positive and the number of coefficients that are negative is — 2; in particular, in flat space, the Minkowskian metric will be taken to have the form ds2 = c2 dt2 - dx2 - dy2 - dz2, (230) where c denotes the velocity of light (which we shall generally set equal to 1 by a choice of units). With its Lorentzian signature, the metric of space-time is not positive-
THE METRIC AND THE CONNECTION DERIVED FROM IT 33 definite in the sense that g(X,X) can be positive, zero, or negative. And we call vectors as time-like, null, or space-like according as g(X,X) > 0, = 0, or < 0. Material particles (with finite rest-mass) can describe only time-like trajectories (i.e., curves along which the tangent vector is always time-like), while massless particles (such as photons, gravitons, and neutrinos) describe null trajectories (i.e., curves along which the tangent vector is always null). Freely falling particles describe geodesies—time-like if they have finite rest- mass and null if they are massless. Null geodesies cannot clearly be parametrized by the arc length even though they can be affinely parametrized by a change of variables (as shown in §4, equation (126)). The equations for the geodesies (time-like or null), when expressed in terms of the four-velocity, u, can always be reduced to the forms given in equations (211) and (212). Along time-like geodesies («• «) = 1, while along null geodesies («•«) = 0. We note that by our definition of the Riemann tensor, the Ricci identity has the form ^^ = ^1-¾^ (231> Further, the Ricci tensor is obtained by the contraction gJlRiJU = Rit- (232) And the Weyl tensor is now related to the Riemann and the Ricci tensors by Ciju = Run — i (Sik Rji + Gji Rtk — Qjk R-u — Gu Rjk) + i(gikgfl-gjk9u)R- (233) The Riemann tensor has twenty distinct components while the Weyl and the Ricci tensors have ten components each. And the Bianchi identity, R.;[W,m] = %(R ijkl;m + Rijlm:k + Rijmk;l) = °> (234)* includes 24 distinct equations corresponding to the six distinct index-pairs (i,j, i j= j), each of which can be associated with the four choices for k ^= I ± m. However, only 20 of these 24 equations are linearly independent since, as may be directly verified, the following four linear combinations identically vanish by virtue of the various symmetries of the Riemann tensor (including that of the cyclical relation): ^12[34,1]~ ^13[4l;2] + ^14[12;3] = 0, ^23[4l;2]~ ^24 [12; 3] + ^21 [23; 4] = 0, ^34[12;3]—^3l[23;4] + ^32[34;1] = 0, I ^ ' an<I R 41 [23; 4] ~ R 42 [34; 1] + R 43 [41; 2] = 0. * We are here adopting the notation that enclosing a group of indices in square brackets signifies that the quantity in question has been 'antisymmetrized' by applying the alternating operator A to them in the manner of equation (S3).
34 MATHEMATICAL PRELIMINARIES Einstein's equation governing the geometry of space-time is SnG Gu = RtJ -i»u«=-y 7-y, (236) where TVj denotes the energy-momentum tensor of the matter and of the fields (other than gravitational) which may be present and G is the constant of gravitation (which, like c, we shall generally set equal to 1 by a choice of units). An alternative form of equation (236) is Ris = - ^r Vu - ^SijT) (T = gUTu). (236') In a vacuum (i.e., in regions of space-time in which TtJ = 0), Einstein's equation reduces to Gu = 0, or, equivalently, K;, = 0; (237) and the Riemann and the Weyl tensors coincide. One generally considers Einstein's equation as an equation for determining the metric (more particularly, the ten metric coefficients gtj). In view of the identity (equation (223)), G'jj = 0, (238) it is clear that any solution of Einstein's equation must involve four arbitrary functions. The freedom we thus have in the choice of these functions is described as gauge freedom and traced to the 'general covariance' of the theory. And this gauge freedom allows us to impose four coordinate conditions on the metric. And finally, it should be noted that, in the vacuum, the number of Bianchi identities is reduced from 20 to 16 since four of the identities included in equation (238) are identically satisfied by virtue of the field equation (237). 7. The tetrad formalism The standard way of treating problems in the general theory of relativity used to be to consider the Einstein field-equation in a local coordinate basis adapted to the problem on hand. But in recent years, it has appeared advantageous, in some contexts, to proceed somewhat differently by choosing a suitable tetrad basis of four linearly independent vector-fields, projecting the relevant quantities on to the chosen basis, and considering the equations satisfied by them. This is the tetrad formalism. In the applications of the tetrad formalism, the choice of the tetrad basis depends on the underlying symmetries of the space-time we wish to grasp and is, to some extent, a part of the problem. Besides, it is not always clear what the 'relevant' equations are and what the relations among them may be. On these
THE TETRAD FORMALISM 35 accounts, we shall present the basic ideas of the theory without any prior commitments and derive the various equations which will later appear as the relevant ones of the formalism for the applications we have in view. (a) The tetrad representation We set up at each point of space-time a basis of four contravariant vectors, e(a)( (a =1,2,3,4), (239) where enclosure in parentheses distinguishes the tetrad indices from the tensor indices (which are not so enclosed). (Also, we shall reserve the earlier letters of the Latin alphabet (a, b, etc.,) for the tetrad indices and the later letters (i,j, etc.) for the tensor indices.) Associated with the contravariant vectors (239) we have the covariant vectors, e(a)i = 9 ike(a)k, (240) where g;k denotes the metric tensor. In addition, we also define the inverse, eib\, of the matrix [«(„>'] (with the tetrad index labelling the rows and the tensor index labelling the columns) so that eMlJb), = SmM and eMitPJ = 5,}, (241) where the summation convention with respect to the indices of the two sorts, independently, is assumed (here and elsewhere). Further, as a part of the definitions, we shall also assume that e(a)'em = riiam, (242) where r/(a)(fc) is a constant symmetric matrix. (243) One most often supposes that the basis vectors, e(a)', are orthonormal in which case rj(am represents a diagonal matrix with the diagonal elements, + 1, — 1, — 1, — 1. We shall not make this assumption though it should be stated that a formalism, more general than the one we shall describe in detail, can be developed in which the f/(a)(h)'s are allowed to be functions on the manifold. In some contexts this further generalization may commend itself. We consider it briefly in §(e) below; but, for the present, we shall proceed on the assumption that the r)(a)(b)'s are constants. Returning to equation (242), let ri("m be the inverse of the matrix [^(j,)]; then '/""'"W) = <5<V (244) As a consequence of the various definitions, »Z(.x« ela)i = elbV, r/<a)<fc) ew = tf\; (245)
36 MATHEMATICAL PRELIMINARIES and most importantly, *<„>,• ^,- = ^,, (246) Given any vector or tensor field, we project it onto the tetrad frame to obtain its tetrad components. Thus, AM=eMJA' =e(a)JAj, and A' = eM'AM = «<■>' A(o); and more generally, 1 ij— e ie j1 (b)(W — C i ' (B)/- (247) (248) It is clear from equations (241), (242), (244), and (245) that (a) we can pass freely from the tensor indices to the tetrad indices and vice versa; (b) raise and lower the tetrad indices with riia)ib) and r/(a)(fc) even as we can raise and lower the tensor indices with the metric tensor; (c) there is no ambiguity in having quantities in which the indices of both sorts occur; and (d) the result of contracting a tensor is the same whether it is carried out with respect to its tensor or tetrad indices. (b) Directional derivatives and the Ricci rotation-coefficients The contravariant vectors eM, considered as tangent vectors, define the directional derivatives (cf. equation (8)) . d I dxl and we shall write (249) ^,(.) = e<->'|? =<W </>■<' (250> where <j> is any scalar field. More generally, we define A(„),(b) - eib)' —{ A(a) - e(b)'^; e(a)J A} d . .. d \eto'Aj = e(bjVdilejAJ = emlleMU}.t + Akela)k;i-]. (251) We thus obtain A (a), <(,) = e(a)' A j .,«,„' + eMk;, e(b; e(c)k Alc), (252) making use of the various rules enunciated at the end of the preceding section and of the fact that the raising and the lowering of tensor indices permutes with the operation of covariant differentiation.
THE TETRAD FORMALISM 37 With the definition ?(c) (a) (t)=eW* «(<,)*;I«(*)'. (253) we can rewrite equation (252) in the form A {a), (ft) = <W Ai; i e(b) + 7 (c) („) (ft) Alc)- (254) The quantities, 7(c)(a),(,), which we have defined in equation (253) are called the Ricci rotation-coefficients. An equivalent definition of these coefficients is e(a)k;i = e{c\y[c)[a)(b/b\. (255) The Ricci rotation-coefficients are antisymmetric in the first pair of indices: y (c)(a) (b)= -y(a)(c)(b) (256) —a fact which follows from expanding the identity 0= 1(a)(b),i= [«(<,)*«(*)*];£. (257) (Notice that we could not have deduced this antisymmetry if the ri(a)(b)'s had not been constants.) By virtue of this antisymmetry, equation (255) can also be written in the form «M;i=-y(fl)\. (258) Returning to equation (254), we write it in the alternative form e«M;;e(ft/ = A(a),(b)~ »Z^7<„)(a)(ft)A(m)■ (259) The quantity on the right-hand side of this equation is called the intrinsic derivative of A(a) in the direction e(b) and written Aia)^b): A(a)m = e[a{ A,-Se(h> (or, A,j = «<■>, A^/"^). (260) We thus have the formula, A(a)\(b) = A(a);(b)-1(n)(m)y(nHaHb)A(m), (261) relating the directional and the intrinsic derivatives. It is clear from the definitions (260) that we can pass freely from intrinsic derivatives to covariant derivatives and vice versa. The notion of the intrinsic derivative of vector fields is readily extended to tensor fields in an obvious fashion. Thus, the intrinsic derivative of the Riemann tensor is given by R(a)(b)(c)(d)\(/) = Rijki;me(a)'e(b)Jeic) e(d)e(f)m. (262) Now expanding R(a)(b)(c)(d)\(f) = LR ijkle(a)1 e(b)3 C(c) e(d) ],me(/)m' (263) and replacing the covariant derivatives of the different basis-vectors by the respective rotation-coefficients (in accordance with the relation (258)) we find
38 MATHEMATICAL PRELIMINARIES (analogous to equation (261)) R[a)[b)[c)(d)\(f) = R [a){b){c){dUf) -l" m \_y{„)[a)[f)R {m)[b){c){d) + y{n)ib){f)R{a)(m){c){d) + 7in)ic)(f)Ria)ib)im)id) + V(n)(d)(/1R (a)(fc)(c)(m)]- (264) Finally, it is important to observe that the evaluation of the rotation coefficients does not require the evaluation of covariant derivatives (and, therefore, of the Christoffel symbols). For, denning ^(fl) (fc) (c) = e[byj [<W e(c)J - e(a)> e<c)'], (265) and rewriting in the form ^(<i) (fc) (c) = [e(fc)i,; - e{b]j,i] e[a) c(c/> (266) we observe that we can replace the ordinary derivatives of c(fc)i and e(fcy by the corresponding covariant derivatives (in accordance with equation (115) for symmetric connections) and write -*(a)(6)(c) = \-e(b)i;j- e(by-,i] «(«)'«(()' = y(fl)(fc)(c)-r(c)(fc)(fl)- (267) By virtue of this relation, we have V(B)(*)(C) = l[^(B)(*)(c) + ^(C)(B)(fc) — ^(*)(C)(B)] (268) and as is manifest from equation (266) the evaluation of A(a)(fc)(c) requires only the evaluation of ordinary derivatives. Notice also that the A-symbols are antisymmetric in the first and the third indices: ^•(B)(*)(C) = — ^(C)(*)(B)- (269) (c) The commutation relations and the structure constants The Lie bracket, \_e(a), eih)~\, plays an important role in the theory; and being a tangent vector itself, can be expanded in terms of the same basis, e(a). Thus, we must have an expansion of the form le(a),e(b)-] = Cic\ame(c). (270) The coefficients, C(c\a)(b), in this expansion are called the structure constants; they are antisymmetric in the indices (a) and (b) and there are 24 of them. The structure constants can be expressed in terms of the rotation coefficients as follows. Consider the effect of the Lie bracket on a scalar field/. We have I><b), «<*)]/= C(b/[C(*//;], i - <Wl><«///],i = [-W(«i + Wn)K; = [-WC,(B, + 7(B,<C,,fc)]^/'/;- (271)
THE TETRAD FORMALISM 39 Comparison with equation (270) now yields the relation C C\a)(b) = 7 C\b)(a) — 7 C\a)(b)- (272) Equation (270), written out explicitly with the structure constants expressed in terms of the rotation coefficients, provides the commutation relations; there are twenty-four of them. (d) The Ricci and the Bianchi identities Projecting the Ricci identity (equation (231)), e(a)i;kJ — e[a)i;l;k= ^mitt^a)"1* (273) on to the tetrad frame, we have R(a)(b)(c)(d) = R-mikle(a) e(b) e(c) e(d) = {-[y(.,(/,to,*(/V«VI;« + LyMlf^f)i^lk}elbilee^eld)'. (274) Now expanding the quantities in the square brackets, on the right-hand side, and replacing, once again, the co variant derivatives of the basis vectors by the respective rotation-coefficients, we obtain K<<oa»<c)(d) = — y<<oa»<o, (<f) + V(<D(*)(<i), <o + V(*)(<d(/)[V(o '(d)-7(d) V)] + V(/-)(b)(0V(*) (d)-7(/)(a)(d)7(fc) (c). (275) Because of the antisymmetry of the rotation coefficients in the first pair of their indices and the manifest antisymmetry of the operation by which the tetrad components of the Riemann tensor are constructed from them, it is clear that there are 36 equations of the kind (275). And finally, the Bianchi identity (234), expressed in terms of intrinsic derivatives and tetrad components, takes the form R{a)[b)Uc){d)\{f)1 = £ X {K(a)(fc)(c)(d), (/) ucwnm — 11nm [V(n)(B)(/-)^(m)(*)(c)(d) +V(n)(*)(/-)^(B)(m)(c)(d) + V(n)(c)(/-)^(B)(*)(m)(d) + V(n)(d)(/-)^(B)(*)(c)(m) ]}; (276) and, as we have noted earlier in §6(e), there are 20 linearly-independent equations of this kind. The basic equations of the tetrad formalism are the 24 commutation relations (270), the 36 Ricci identities (275), and the 20 linearly-independent Bianchi identities. It is not clear how many of these equations are independent, how they are to be ordered or used and, indeed, what they are for. These are all questions which we shall confront in due course.
40 MATHEMATICAL PRELIMINARIES (e) A generalized version of the tetrad formalism In the foregoing account of the tetrad formalism, we have explicitly assumed that f/(a)(!,), as denned in equation (242), represents a constant symmetric matrix. If this should not be the case and the rj(am's are functions in space-time, then (as we have already noted) we cannot infer the antisymmetry of the rotation coefficients in their first two indices; instead, we shall have (cf. equation (257)) V(b)(W(c) — V(W(b)(c) = 1(a)(b), <<:)• (277) And even more important, the process of raising and lowering of tetrad indices by riia)(b) and rimb) does not permute with the operation of directional or of intrinsic differentiation. A careful scrutiny of the analysis of the preceding sections shows that while the correct expression for the structure constant, in place of equation (272), is C^a><w = W%-)VV (278) the Ricci and the Bianchi identities as written in equations (275) and (276) continue to be valid. 8. The Newman-Penrose formalism The Newman-Penrose formalism is a tetrad formalism with a special choice of the basis vectors. The choice that is made is a tetrad of null vectors I, n, m, and m of which I and n are real and m and m are complex conjugates of one another. The novelty of the formalism, when it was first proposed by Newman and Penrose in 1962, was precisely in their choice of a null basis:* it was a departure from the choice of an orthonormal basis which was customary till then. The underlying motivation for the choice of a null basis was Penrose's strong belief that the essential element of a space-time is its light-cone structure which makes possible the introduction of a spinor basis. And it will appear that the light-cone structure of the space-times of the black-hole solutions of general relativity is exactly of the kind that makes the Newman-Penrose formalism most effective for grasping the inherent symmetries of these space-times and revealing their analytical richness. But it may be stated, here already, that the special adaptability of the Newman-Penrose formalism to the black-hole solutions of general relativity derives from their "type-D" character and the Goldberg-Sachs theorem—matters which will be considered later in this chapter (§9) and subsequently. * Penrose was originally led to consider the introduction of a null basis from his interest in incorporating in general relativity spinor analysis in an essential way. We briefly consider this alternative approach to the Newman-Penrose formalism in Chapter 10 (§102).
THE NEWMAN-PENROSE FORMALISM 41 (a) The null basis and the spin coefficients As we have already stated, underlying the Newman-Penrose formalism is the choice of a null basis consisting of a pair of real null-vectors, /and n, and a pair of complex-conjugate null-vectors m and in. They are required to satisfy the orthogonality conditions, /• m = /• in = nm = »■ m = 0, (279) besides the requirements, /• /= »• n = m- m = m- m = 0, (280) that the vectors be null. It is customary to impose on the basis vectors the further normalization conditions, / ii=1 and m- m = — 1. (281) It is strictly not necessary to impose these further conditions (cf. § 7(e)). Indeed, in some of Penrose's later work, he prefers not to impose the conditions (281) as more in harmony with his views on the null-cone structure of space-time. But for the purposes of this book, this additional freedom in the choice of the basis vectors does not have any advantage that overcomes the disadvantage of having rotation coefficients which are not antisymmetric in their first two indices, and the still more serious disadvantage of not being able to permute the operation of the raising and the lowering of the tetrad indices with the operation of directional and intrinsic differentiations. On these accounts, in this book, we shall retain the normalization conditions (281). Then, the fundamental matrix represented by r/(a)(fc) is a constant symmetric matrix of the form [»/<.><«] = lV"""] = 0 1 0 0 1 0 0 0 0 0 0 -1 0 0 -1 0 (282)* with the correspondence (cf. §7(a)), e\ = I, e2 = n, e3 = m, and e4 = m. (283) The corresponding covariant basis is given by el = e2 = n; e2 = e^ = /; e3 = —e*= —m, and e* = —e3= —m. (284) The basis vectors, considered as directional derivatives, are designated by * In the sequel, we shall dispense with the distinguishing parentheses for the tetrad indices so long as there is no ambiguity as to what is intended.
42 MATHEMATICAL PRELIMINARIES special symbols: ei = e2 = D; e2 = el = A; e3 = —e* = 5; and e4 = — e3 = 5*. (285) The various Ricci rotation-coefficients, now called the spin coefficients, are, similarly, designated by special symbols: K = y3iu p = y3i*, £ = 3(7211+7341); ^ = 7313; /* = 7243; 7 = 3(7212+7342); ^ = 7244; T = y312; a = 5(7214 + 7344); (286) v = 7242; 7t = y241; /^ = 3(7213+7343)- It is clear that the complex conjugate of any quantity can be obtained by replacing the index 3, wherever it occurs, by the index 4, and conversely. This is a general rule. (b) The representation of the Weyl, the Ricci, and the Riemann tensors The Weyl tensor is the trace-free part of the Riemann tensors; and its tetrad components are given by (cf. equation (233)) R-abcd = C abed-lilac R-bd — 7/bc^ad — Had^-bc + Ibd^-ac) + Ulac1bd-riadribc)R, (287) where KM denotes the tetrad components of the Ricci tensor and R the scalar curvature: Rac = rib''Rabcd and R = „">Rab = 2(Rl2 - RM). (288) The fact that Cabcd is trace-free requires that if Cabcd = Clbc2 + ^2bcl — C3bcA. — C^bci = 0. (289) Besides, we must also require that Cl234+Ci342+C1423=0. (290) The condition (289), written out explicitly for b = c, gives £-1314 = C2324 = £-1332 = C1442 = 0, (291) while the condition for b± c together with the cyclic requirement (290) give ^1231 = C1334; C124i = C1443; C1232 = C2343; C1242 = (-2434) C1212 — C3434; and C1342 — 3(Ci2i2 —C1234> = 3^3434 — Ci 234)-J Making use of the foregoing results, we find that the various components of
THE NEWMAN-PENROSE FORMALISM 43 the Riemann tensor are related to those of the Weyl and the Ricci tensors by ^1212 = ^1212+^12 — 5^; ^1324 = £-1324 + 12^1 ^1234 = £-1234i 3434 = ^-3434- ^34-S^i ^1313 = Ci3i3; R2 R ^1314 = ?R-1U ^2324 = 3^22; ^3132 = — 2^33i ^1213 = ^1213 +3^13; ^1334 = £-1334+2^13^ = c. ^1223 — C\223 _ 3^23» R 2334 = c 2334 + *K, (293) and the additional complex-conjugate relations obtained by replacing the index 3 by the index 4, and vice versa. In the Newman-Penrose formalism, the ten independent components of the Weyl tensor are represented by the five complex scalars, *o= -C1313= -CpqJ»mH'ms.} ^ = -c1213=-V»'W, ^2= -C1342= -Cppl'WiS'ir, ™3 = — £-1242 = VA= -c ■CpqJ<>n*rffns 2424 = — c «p rfnfin'rfC. (294) While it is clear on general grounds that the Weyl tensor is completely specified by the five complex scalars *P0, . . . , *F4, it will be convenient to have a general formula which expresses the different components of the Weyl tensor explicitly in terms of the five scalars. Such a formula can be obtained in the following manner. First, we shall introduce a symbol which indicates the construction out of products of four quantities a linear combination of them which will have all the principal symmetries of the Riemann and the Weyl tensors. Thus, we shall let {wpxtyrzs} = wpxqyrzs-wpxtzrys-xpwtyrzs + xpwqzrys + ypzqwrxs-ypzqxrws- zpyqwrxs + zpyqxrws. (295) The linear combination so constructed is manifestly antisymmetric in (p,q) and in (r, s)and unchanged for the simultaneous interchange of the index pairs (p, q) and (r, s). In this notation, it is clear that the tensor components, Cpqrs, of the Weyl tensor must have the representation Cpirs = G1212 {lPnqlrns} + Q3A3A{mpmllmrms} + Qi23*{lpntmrms} + Qi3i*{lpmtlrms} + Q232^{npmtnrms} + LQ1313 {lP™tlrms} +Q2323 {nP™tnrms} +Q1213 {lPntlrms} + Qi223{lPnqnrms} + Ql323{lpmllnrms} +Ql32^{lpmqtirms} + Qi33*{ipmtmrms} + Q233^{npmqmrms} + complex conjugates], (296)
44 MATHEMATICAL PRELIMINARIES where Q are coefficients, unspecified for the present, and the 'complex conjugates' of the eight quantities in the square brackets on the right-hand side are to be obtained by writing m wherever m occurs and index 4 wherever the index 3 occurs and vice versa. The coefficients Q in the representation (296) can be obtained by contracting both sides of the equation by appropriate products of the components of/, n, m, and m. Thus, by contracting with lpmqlrms, the left-hand side, by definition, yields — *P0» while the only surviving term on the right- hand side has the coefficient 62424; and, accordingly, 62424= — *P0. Similarly, by contracting the equation with l"mqlrms, the only surviving term on the right-hand side has the coefficient 62423; and this coefficient must vanish since the left-hand side yields C1314 which we have shown to be zero (cf. equation (291)). Proceeding in this fashion, systematically making use of the various definitions and requirements included in equations (291), (292), and (294), we find 61324 = ^2; 61212 = 63434= -(^2+^2*); 61234 = ^2-^2*); 6l313 = — ^4-1 62424 = ~^0l 6l213 = — 6l334 = ^3; 61224 = 62443=-^1: and 6l314 = 62324 = 6l323 =6l424=0. (297) The required formula is, therefore, C„„ =-(^2 + *2*)[{ Wr«.} + {mpmqmrms}] + (*2 -^2*){lPntmrms} + l-^o{npmqtirms} -vVA{lpmqlrms} +^211^^^} -*\Wpninrfns} + {npm,m,.ms}] + ^3l{lpnqlrms} -{lpmqmrms}'\ + complex conjugates]. (298) In particular, C1334 = *Pi; £2443 = 4^ Ct2i2 = C3434 = — (*P2 + ^2*); and ^234 = (^2-^). (299) Including the four components which vanish and allowing for complex conjugation, we have specified in equations (294) and (299) all the distinct components of the Weyl tensor. And finally, the ten components of the Ricci tensor are defined in terms of the following four real and three complex scalars: ¢00 = —3^11; ¢22 = —3^22; ¢02 = —3^33; ¢20 = —3^44;! ¢11 = -i(*i2 + «34); ¢01 = -3*13; ¢10 = -3*14; f (300) A =&R = MRi2 -K34); ¢12 = -i^23; ¢21 = -3^24- J
THE NEWMAN-PENROSE FORMALISM 45 (c) The commutation relations and the structure constants We shall now proceed to write down the explicit forms of the various equations of the theory. We consider first the commutation relation (cf. equations (270) and (272)) [«(«).«<*)] = (ycba-7cab)ec = Ccabec. (301) As an example, consider [A,£>] = [»,/] = l>2, *i] = (yci2-yC2iK = - 7l21 el +7212*2 +(7312- 7321 )*3 +(7412- 7421 )<?* = -7121 A +7212£)-(7312-732l)^*-(7412-742l)^ (302) or, giving the spin coefficients their designated symbols, we obtain AD-DA = (y + y*)D + (e + e*)A-(r* + n)5-(r + n*)5*. (303) In similar fashion, we obtain: SD~DS = (<x* + P~ii*)D + kA -(p* + e-e*)5-a5*, (304) <5a-A,5= -v*D + (r-<x*-p)A + (fi-y + y*)S + t*S*, (305) 5*5-55* = (jj.* - n)D + (p* - p)A +(ot-p*)5 + (P-ot*)5*. (306) By expressing the foregoing relations in the manner of equation (301), we find that the structure constants are related to the spin coefficients as in the accompanying tabulation: Cl2l=+(y + y*);Cl3l = +(«* + p-n*);Cl32 = -v*; C\3 = n*~n, C22l = +(£ + £*); C231 = +k; C232 = T-oi*-p;C\3 = p*-p, C\l=-(T* + ny,C33l = -(p* + £-£*);C332=fi-y + y*;C\3 = at-p*, C*21 =-(* + «*); C*31 = -<r, C*32=+X*; C\3=p-ot*. (307) (d) The Ricci identities and the eliminant relations We shall now write down the explicit forms which the Ricci identities take in the Newman-Penrose formalism. Considering the (1313)-component of equation (275), for example, we have (making use of the relations (293)) — M) = £-1313 = ^1313 = 7l33,l _7l31,3 + 7133(7121 +7431 -7413 +7431 +7134» -7l31 (7433+7123-7213+7231 +7132»; (308) or, substituting for the directional derivatives and the spin coefficients their designated symbols, we obtain Do-5k = o(3£-e* + p + p*) + K(n*-z-3p-tx*) + x¥0. (309)
46 MATHEMATICAL PRELIMINARIES [K1314] [K1313] [K1312] (a) (b) (c) As we have noted in the context of the standard tetrad formalism in §7(d), we can write down a total of 36 equations by considering the various different components of equation (275). But in the context of the Newman-Penrose formalism, it will suffice to write down only half the number of equations (by omitting to write down the complex conjugate of an equation). We list below a set of 18 equations, as originally derived by Newman and Penrose in 1962; and we have indicated in each case the component of the Riemann tensor which gives rise to the equation. Dp-S*K = (p2 + aa*) + p(e + e*) - k*z - k(3cc + /?* - n) + 0>00, Da -5k = o(p + p* + 3e-£*) -K(T-rc*+a* + 3/?) + ¥0, Dz- Ak = p(z + n*) + a(r* + n) + r(e-e*) -^ + ^) + ^+¢01, Da -6*£ = (x(p + E*- 2e) + /?<7* - /?*£ - kX -K*y + n(£ + p) + <t>l0, [i(K3*i*-Ki2i*)] (d) Dp-5e = a(a + n) + P(p*-e*)-K(n + y) -£(«*-**) + *!, Cl(«i2i3-«34i3)] (e) Dy- Ae = <x(t + 7t*) + /?(t* + n)-y(e+£*) -ely+y^ + rn-VK + Vt+Qu-A, [^1212-^3412)] (0 DX-S*n = {pX + o*ix) + n(n + a-fl)-vK* -A(3£-£*) + (D20, Dn-5n= (p*n + aX) + n(n* - a* + /?) -H(£ + £*)-VK + V2+2A, Dv - An = n(n + t*) + X(n* + t) + n(y-y*) -v(3£ + £*) + ^3+(D21, AX-5*v= -X(n + n* + 3y-y*) + v(3a + /?*+rc-T*)-¥4, ^p-^*ff = p(a* + /?)-ff(3a-/?*) + T(p-p*) + ^-^)-^+(1)0,, 5a - 5*0 = (up - Xa) + aa* + /?/?* - 2a/? + y(p-p*) + £(H-H*)-x¥2+(I>ll+A, [1(^1234-^3434)] 0) [^244l] [^243l] [^242l] [^2442] [^3143] (g) (h) (i) (J) (k)
THE NEWMAN-PENROSE FORMALISM 47 6A- 6*p = v(p - p*) + n(p- p*) + p(<x + P*) + X[a*-3fi)-V3 + <b21, [R2443] (m) 6v- Ap = (p2 + M*) + p(y + y*)-v*n + v(T-3/?-a*) + 0>22, [R2423] (n) Sy — A/? = y(z — a* — /}) + pz — av — ev* -p{y-y*-H)+*X* + 4>l2, [i(«l232-*3*32)] (0) Sz- Aa = (po+X*p) + z(z + p-(x*) -a(3y-y*)-Kv* + %2, [«1332] (P) Ap-S*z = -(pp* + oA) + z(P*-a-z*) + p(y + y*) + vK-V2-2\, [R132*] (q) Aa- 5*y = v(p + e)- Hz + 0) + a(y* - p*) + y(P*~Z*)-V3. [1(^,242-^3442)] W (310) While each of the foregoing equations contains one or more of the Weyl and the Ricci scalars, it is clear that one should be able to obtain from them a total of sixteen real equations which are independent of them and involve only the spin coefficients. The reason is that, while the right-hand side of equation (275) (from which all of the foregoing equations were derived) is antisymmetric in the first pair and in the second pair of indices, it has not the in variance of the Riemann tensor (on the left-hand side) for the simultaneous interchange of the first and the second index of pairs. It is precisely this latter in variance (together with the cyclic identity) which reduces the number of independent components of the Riemann tensor from thirty-six to twenty. For this same reason, we must be able to eliminate the Riemann tensor, altogether, from sixteen of the thirty- six equations included in equation (275) (and (310)). This elimination can be carried out directly from the equations listed. Thus, the imaginary part of equation (310,a) will not contain d>00 since it is real. Similarly, from equations (310,c), (310,e), and the complex conjugate of equation (310,d) we can readily eliminate 4^ and d>01 (=d>01*), and obtain a single complex equation involving only the spin coefficients. By such systematic eliminations, we obtain the following set of four real and six complex eliminant relations: D(p-p*) + 5k*-S*k = (p-p*)(p + p* + £ + e*)+K(z* + n-3ot-p*) -K*(z + n*-3a*-P), (a) D(p-p*) + 5(<x + p*-n)-5*((x*+p-n*)=(y + y*)(p-p*) + a(n* -2P)-ot*(n-2p*) + K*v* -kv + fin-P*n*+ (p+ p*)(p-p*)(b) D(p-p*-y + y*)+ A(£-£*)-Sn + S*n* = (£ + £*)(p*-p)
48 MATHEMATICAL PRELIMINARIES + t*(a* + n* - /J)-r(a + 7r- /?*) + Xa - X*o* + p*p- pp.* + 2(ey-£*y*), (c) A(p*-p) + <5v-c5*v* = (p-p*)(p + p*+y + y*) + v(T-3/?-a* + 7r*) -v*(t*+rt-3/?*-a), (d) D(T-a*-/J)- A/C + ^(£+£*) = p(T*+rt) + K*/l*+ff(T*-a-/?*) + £(T-rt*)-p*(/? + a* + rt*) + £*(2a*+2/?-t-rt*) + »c(/i-27), (e) ^(p-£ + £*)-^*ff +£»(/?-a*) = p(a* + /? + T)-p*(t-/? + a* + rt*) + (e*-e)(2<x*-n*) + a(n-2<x) + K(y*-y-n*) + K*A*, (f) + /l(p-p*-3£ + £*)-/?7T-T*/?*, (g) Dv + A (a + /J* - rc) - c5*(y + y*) = v(p - 2e) + /1(tc* - a* - /J) + p(rc + t*) -/i*(a + /?* + T*) + y(»-T*) + y*(2a + 2/?*-w-T*) + ff*v*, (h) A(/?*-a) + 5A + 5*(y-y*-/i) = v(e*-e-p*)+A(T-2/?) + a(/i + /i*) -p*(rt + T*+/?*) + p(rt + /?*)+(7-7*)(T*-2/?*) + ff*v*, (i) £>p+ Ap-^-^*T = p*p-pp* + rt(rt*-a* + /?) + t(/?*-a-T*) + p(y + y*)-p(£ + £*). (J) (311) (e) The Bianchi identities As we have stated earlier in § 7(d), there are, altogether, twenty linearly- independent Bianchi-identities. A complete set is provided by the eight complex identities ^13[13|4] = °; Kl3[21|4] = CI; ^13[13|2] = 0; ^13[43|2] = °! j -^ ^42[13|4] = 0; ^42[21|4] = 0; ^42[13|2] = 0; #42[43|2] = 0; i and the four real identities which follow from »/*"(«*-WKJlc-O. (313) Explicitly written out, equation (313) provides the two real equations, ^ll|2+^34|l—^13|4—^14|3 = 0, ] and I (314) ^22|l + ^34|2 —-^2314--^2413 = 0> J and the one complex equation ^33|4+^12|3_^31|2_^32|l = 0- (315) We shall now write down the explicit forms of the various identities in terms of the spin coefficients and the Weyl and the Ricci scalars. As an example,
THE NEWMAN-PENROSE FORMALISM 49 consider the first of the identities listed in equation (312). We have ^1313|4+^133411+^1341|3 = °- (316) By making use of the relations given in equations (293), we can write instead, ^131314+(^1334 + 3^13)11-3^1113 = 0- (317) Considering the terms in the Weyl tensor, we have ^- 131314 = ^1313,4- I"™ (7nl4Cm3i3 + 7„34^-imi3 + 7nl4^-13m3 + 7n34Cl31m) = ^1313,4- 2(7214 + 7344)^-1313 + 273i4(Ci2i3 + C4313) = -6*V0 + 4aV0-4pVl, (318) and C 1334|1 = ^1334,1- '/""I[7nllCm334 + 7n3l(Clm34 + Ci3m4) + 7n4lCi33m] = ^-1334,1—C 0*211 + 7341 ) Cl334 + 7l 3 l(Cl 234-C3434) + 7231C1314 + 7141C1332 + 7131^-1324 + ^241^--1331] = Dx¥l-2ex¥l + 3K'i>2-n'V0. (319) Similarly, we find i(K13|1-KU|3)= -/^0i+ <5<D00 + 2(£ + p*)<I>0i+2<T<I>10 -2/cd)ii-»c*(I)02 + (rt*-2a*-2/?)(I)00. (320) Combining equations (318), (319), and (320), we obtain the required explicit form of the identity (316). The remaining identities can be derived in similar fashion. We give below the explicit forms of the eight complex identities (312): -5*X¥0 + Dx¥l + (4a - rt)»P0 - 2 (2p + e)x¥l + 3kV2 + [Ricci] = 0; + d*xVl-DV2-mi0 + 2(n-a.)xVl + 2>pxV2-2KV3 + [Ricci] = 0; -5*X¥2 + DX¥3 + 2Wl - 2>nV2 + 2(e- p)¥3 + k¥4 + [Ricci] = 0; + 5*V3 - DXVA- 3/1*2 + 2(2n + a)*3 - (4e- p)¥4 + [Ricci] = 0; -A^o + ,5^ + (4y- fi)V0-2(2r + p)V t +3aV2 + [Ricci] = 0; ^13[13|4] ^13[21|4] ^42[13|4] ^42[21|4] ^13[13|2] = 0, = 0, = 0, = 0, = 0, (a) (b) (c) (d) (e)
50 MATHEMATICAL PRELIMINARIES - A*! + 5XV2 + vV0 + 2(y-/*)¥,- 3t¥2 + 2oV3 + [Ricci] = 0; Ki3[43|2] = 0, (f) - A¥2 + 6>¥3 + 2v¥, - 3/*¥2 + 2(0-t)^3 + <t¥4 + [Ricci] = 0; «42[13|2] = 0, (g) - A¥3 + <5¥4 + 3v¥2 - 2(7 + 2/^)^3 - (t - 4/?)*4 + [Ricci] =0; «42[43|2] = 0, (h) (321) where the terms in the Ricci tensors (enclosed in square brackets), in the respective equations, are: - £>0>01 + (5<I>00 + 2(e + p*)O01 + 2a®l0 - 2KOn - k*%2 + (n*-2a*-2p)d>00, (a) + ^*(D01- A(D00-2(a + T*)d)01+2p(I)1 ,+ff*O02 -(/i*-2y-2y*)«»00-2T<I»10-2DA, (b) -£)<D2! + ^<D20 + 2(p* -£)0>21 - 2/*0>10 + 2710),, - k*0>22 -(2a*-2/?-7c*)0)20-2^*A, (c) - A0>20 + <5*0>21 + 2(a -t*)<D21 + 2vd>10 + ff*0)22 -21^, -(fi* + 2y-2y*)(t>20, (d) - £)(D02 + 5% ,+2(7^-0)4)0,-2^- A *0)00 + 2a<t>, t + (p* + 2£-2£*)<D02, (e) A0)0i - <5*<D02 + 2(/i* - y)<D0i - 2pO)12 - v*a)00 + 2t<D, , + (T*-2/?* + 2a)<D02 + 2<5A, (f) - £>0>22 + M>2, + 2(tc* + /?)0>2, -2/iO),, -/l*0>20 + 2tc<I)12 + (p*-2£-2£*)0)22-2AA, (g) Ad>2, -c5*a>22 + 2(/i* + y)0>2! -2V0)!, - v*0>20 + 2X<t>l2 + (T*-2a-2/?*)(D22. (h) (321') The explicit forms of the contracted identities (314) and (315) are: <5*<D01 +^0-0(4^ +3A)-A<I>oo = k*0>12 + k<D21 + (2a + 2t* - 7c)<D0i + (2a* + 2t - 7t*)0>10 -2(p + p*)<D11-ff*«»02-ff<I»20 + |> + /i*-2(y + y*)]<I»00, (i) (5*<I>12+c5<I>21- A^n+SAJ-D^z = -v(DOi-v*a)1o + (T*-20*-27c)(I)12 + (T-20-27c*)(I)21 + 2(/i + /i*)(D11-(p + p*-2£-2£*)(I)22 + A(I)o2+A*a)2o, (j)
THE NEWMAN-PENROSE FORMALISM 51 5(CD1S -3A)-DCD12- &<b01+6*<b02 = K<t>22 -v*<t>00 + (z*-n + 2(x-2/?*)4>02 - ff021 + A*4>10 + 2(t-«*)«), 1-(2p + p*-2e*)<D12 + (2/i*+/i-2y)<D01. (k) (322) In the vacuum, the Ricci scalars vanish and the relevant Bianchi identities are given by the eight complex equations (321, a-h) with the Ricci terms set equal to zero; also, in this case, we need not concern ourselves with the contracted identities (322). In general, however, the Ricci terms in equations (321, a-h) must be included; and in this formalism they are replaced by the components of the energy-momentum tensor in accordance with Einstein's equation, 8nG Ru= —^-(Tij-tgijT). (323) The basic equations of the Newman-Penrose formalism are comprised in the commutation relations (303)-(306), the Ricci identities (310, a-r), the eliminant relations (311, a-j), and the Bianchi identities (equations (321) and (322)). As we have remarked earlier, it is not clear what these equations are for and in what sense they replace Einstein's equation or are equivalent to it. (f) Maxwell's equations In the Newman-Penrose formalism, the antisymmetric Maxwell-tensor, Fy, is replaced by the three complex scalars </>o = -Fi3 = Fijlimi, </>i = i(*"12 + F«) = iFy(/V + iii'm>), and </>2 = f42 = FlJmln>; and Maxwell's equations, Fum = 0 and gikFij;k = 0, (325) expressed in terms of tetrad components and intrinsic derivatives, namely, F[ofc|c] = 0 and >TFo„|m = 0, (326) are replaced by the equations </>l|l-</>0|4 = 0> </>2|l-</>l|4 = 0> 0113-0012 = °. </>2|3-</>l|2 = °-. The explicit forms of these equations are readily found. Thus, </>i|i = KFi2, i - nnm (y„.. Fm2 +yn2l Flm) + F43,! - nnm (7„41 Fm3 + yn3l F4m)] = 01,1-(7131^42+7241^13) = D0i+K02-rc0o; (328) (324) (327)
52 MATHEMATICAL PRELIMINARIES and, similarly, </>0|4= <5*(/>0-2<x(/>o + 2p<j)l. (329) The first of Maxwell's equations (327), therefore, takes the form D^-S*<p0 = (n-2a)<p0 + 2p$l-K$2. (330) And the remaining equations are D(j)2-5*(j)l= -/1(/)0 + 2710! + (p-2e)(j)2, (331) <5</>i- A </)0 = (/*- 2y)(/)o + 2t(/)1-ff(/)2, (332) ^</>2-A (/)1= -v(/)0 + 2^(/)^(1-2/^)(/)2. (333) Turning next to the energy-momentum tensor of the Maxwell field, we have ^ = if JV FM - ir/ofc F„ F"; (334) and in terms of the Maxwell scalars (/>0, (/>i, and (/>2, we readily find that -iTu = </>o</>o*; -i^i3 = </>o</>i*; -i(7"i 2+^34) = (/)1 </>!*; -2^23 = </>i</>2*; (335) -?T22 = <p2<p2*; -iT33 = <p0<p2*; and the trace of rafc is, of course, zero. In accordance with equations (300) and (323), we can, therefore set A = 0 and replace <Sm„ by (/>„,(/>„*, (336) apart from the constant of proportionality, — SnG/c*. In the Bianchi identities listed in equations (321, a-h), the Ricci terms (.321', a-h) simplify considerably when use is made of Maxwell's equations (330)-(334). Thus, considering the terms in the derivatives of the Ricci scalars in equation (321', a), we have -D<t>0l+d<t>00^ -0(^^) +d(<fi0$0*) = -(^0(/)^-^)-(/)1^0 + (/^^0 = -<fi0{(n* -2«*)(t>0* + 2p*<j>l* -K*(t>2*} -(/),*£>(/)o+(/)o*<5(/>o; (337) and inserting this last expression in (321', a), we find that we are left with - (/>i*0(/>o + (/>o*<5(/>o + 2(£(/>o(/>i* + ff(/>i (/>o* - *(/>! (/>i* -/?(/>o(/>o*)- (338) The other terms listed in (321') allow of similar simplifications; and the Ricci terms to be included in equations (321, a-h) for an Einstein-Maxwell field take the following forms (apart from a constant of proportionality):
THE NEWMAN-PENROSE FORMALISM 53 - </>i*0</>o + </>o*<5</>o + 2(£(/>0(/>i* +o<pi <p0* - K(/>, (/>,*-/? (/)0 (/>o*), (a) + <pl*S*<p0-<p0*A<po + 2(-a<p0<pl*+p<pl<pl*+y^^*-^^0*), (b) -(/),-0(/)2 + (/)o*^2+ 2(-£(/)2(/)^ -/*(/), (/)0*+/?(/)2</>0*+™/), </>!*), (C) -(/)0* A (/)2 + (/),*^*(/)2+ 2(a</)2(/),* + v(/)l(/)o*-7(/)2(/)0*-^</>i </>i*), (d) -</>2*£></>0 + </), *<5</)o + 2(-K(/),(/)2*-/J(/)o</>,* + ^(/), </), * + £</)0 (/)2 *), (e) + (/),* A (/)o-</>2*^*</>o + 2(-p(/),(/)2*-7(/)o(/>i* + T(/), (/),*+a</)0(/)2*), (f) -(/)2*£»(/)2+(/),*^(/)2+2(-£(/)2(/)2* -/"/>, </>,* +/?</>2</>, * + ™/), </>2*), (g) + (/),* A(/)2-</>2*<5*</>2 + 2(-a(/)2(/)2*^v(/), (/),*+7</)2</),* + ^</>i</>2*)- (h) (339) And since the vanishing of the covariant divergence of the energy-momentum tensor is automatically guaranteed by Maxwell's equations, we need not concern ourselves with equations (322). (g) Tetrad transformations Having chosen a tetrad frame—either an orthonormal frame, as in the conventional tetrad formalism, or a null frame, as in the Newman-Penrose formalism—we can subject the frame to a Lorentz transformation at some point and extend it continuously through all of space-time. Corresponding to the six parameters of the group of Lorentz transformations, we have six degrees of freedom to rotate a chosen tetrad frame. In considering the effect of such Lorentz transformations on the various Newman-Penrose quantities, we shall find it convenient to regard a general Lorentz transformation of the basis vectors l,n,m, and m as made up of the following three classes of rotations: (a) rotations of class I which leave the vector / unchanged; (b) rotations of class II which leave the vector n unchanged; and (c) rotations of class III which leave the directions of/and n unchanged and rotate m (and m) by an angle 6 in the (m, /w)-plane. Associated with these three classes of rotations, we have the following explicit transformatiqns (which, as may be readily verified, preserve the underlying orthogonality and normalization conditions): I: /-» /, »i-» m + al, wi-nw+ a*7, and, n -* n + a*m + am + aa*l; II: n-^n,m-^m + bn,m-^m + b*n, and, / -> l + b*m + bm + bb*n; HI: /-> A~ll,n^> An,m-^eiem, and m-^e-®m\ where a and b are two complex functions and A and 6 are two real functions on the manifold. The effect of a rotation of class I on the various Newman-Penrose
54 MATHEMATICAL PRELIMINARIES quantities is readily found. Thus, considering the Weyl scalars *P0 and *?[, we have — *P0 — C1313 ■CpqrslI'(rrf + al'i)lr(rrf + als --Cpvsl"mnrms= -*0, and 4», = C1213 ->Cp.„/"(«* + a*m'> + am* + aa*l*)lr(ms + als) = CMrsl<>{n* + a*m«)l'ms = C1213 + a*C1313 = -(^!+a*^0), (340) (341) where in the reduction for x¥l, apart from the obvious symmetries of the Weyl tensor, the fact that C1413 = 0 (cf, equation (291)) has been used. The effects of the transformation on the remaining Weyl scalars are similarly found; and the result is *o-»*o,*i-»*i + a**o,*2-»*2 + 2fl**1 + (fl*)2*0. ¥3 -> 4>3 + 3a*¥2 + 3(a*)2^! + (a*)3¥0, vP4^-vP4 + 4a*vP3 + 6(a*)2vP2 + 4(a*)3vP1 + (a*)4vP4. (342) In a similar fashion, we find that the spin coefficients transform as follows: k -* k; a -* o + cik; p -* p + a*K; e -* s + a*K; x -* z + ap + a*a + aa*K; n-* n+2a*e+ (a*)2K + Da*; cc-+ a + a*(p + e) + (a*)2k; fi ->/? + ae + a*a + aa*K, y-^y + aa + a*(P+T)+aa*(p + E)+(a*)20 + a(a*)2K; X -> X + a* (2a + n) + (a*)2 (p + 2s) + (a*)3K + 6* a* + a*Da*; H->H + an+ 2a* /J + 2aa*s + (a*)2 a + a(a*)2K + da* + aDa*; v->v + aX + a*(n + 2y) + (a*)2 (t + 2/?) + (a*)3a + aa* (n + 2a) + a(a*)2 (p + 2e) + a(a*)3K + (A + a*5 + ad* + aa*D)a*. (343) And the scalars representing the Maxwell field become </>o -»</>o, </>i -»</>i + a*</>o, </>2 -»</>2 + 2a*(/)! + (a*)2(/>0. (344) The corresponding effects of a rotation of class II on the various Newman-Penrose quantities can be readily written down from the foregoing formulae, since the effect of interchanging /and n results in the transformation 4>0 * 4>4*, *1 **3*, *2 * ^2*, </>0 * ~ </>2*, </>. * -</>!*, I k «± — v*, p *± — /x*, a *± — X*, a «± — /?*, £ +± — y*, and rc+±—t*.J (345)
OPTICAL SCALARS, CLASSIFICATION, AND THE THEOREM 55 In particular, the effect of a rotation of class II on the Weyl scalars is ^0-^0 + 4^1+6^2+4^3+^4, 1 V i -> Vl + 3bV2 + 3b2 ^3+^4, [ (346) ^2 - ^2 + 2bV3 + b2¥4, ¥3 -> ^3 + b¥4, ¥4 - ¥4. J And finally, we write down the effects of a rotation of class HI on the Newman-Penrose quantities. ^3-..^-^3, and ^4-^^-^4-, (/>o->;4~V0(/>o, </>i -*(j)i,<p2^Ae-w(f)2; k -> A'2ewK, a -^ A'1 e2i6a, p -> A'1 p,z-^ ei6z, % ->e'wn, X -> 4e~2'"/l, /i -> ^, v -» ^2e"''ev, E-^A'1e-^A'2DA + $iA~lD6, tx->e-w<x + $ie-ie6*6-$A-le-ie6*A, P->eiep + $ieie66-±A-leie6A. (347) This completes our account of the Newman-Penrose formalism. 9. The optical scalars, the Petrov classification, and the Goldberg-Sachs theorem The physical significance of the spin coefficients that were formally introduced in §8 in the context of the Newman-Penrose formalism becomes apparent when we consider the propagation of the basis vectors along / or n. Thus, the first-order change in a basis vector e(a), when it suffers an infinitesimal displacement £, is, by definition, Sew = e{a)l;jZ} = e<w»y(fc,(.,(C,e<c)y^ = -y<„><»<c>*V. (348) Therefore, the change, Se(a) (c) in e(a), per unit displacement along the direction c, is given by SeM(c)=-ymbmelb\ (349) In particular, for the change in /, per unit displacement along /, we have (cf. equation (284)) ^/(1)=-71,,,,1^=-7121^-7131^-7141^ = -7i2i/+7i3i« + 7i4i»», (350)
56 MATHEMATICAL PRELIMINARIES or, giving to the spin coefficients their designated symbols, we have 81(1)= (e + e*)l-tern-K*m. (351) Similarly, we find <5n(l)= -y2meib)= -(e + e*)n + nm + n*m (352) and c5»i(l)= -y3meih)= +(e-e*)m + n*l-Kn. (353) Several important conclusions follow from equations (351)-(353). Thus, rewriting equation (351) in the alternative form (cf. equation (348)), I..JJ =(e + £*)/, - Knii - K*mh (354) we conclude from equations (211) and (212) that the l-vectors form a congruence of null geodesies if, and only if, k — 0; and, further, that they are affinely parametrized if, and only if, in addition e = 0. If k = 0 the latter requirement for affine parametrization (namely, £ = 0) can always be met by a rotation of class III described in §7(g) (cf, equations (347)) which will not affect the direction of I nor of an initially vanishing k. If the vectors I should initially be defined as a congruence of null geodesies affinely parameterized, so that k = e = 0, then by a rotation of class I (which will not affect I nor the initial vanishing of k and e), we can arrange that n = 0. After such a rotation, the newly oriented vectors n, m, and m will, according to equations (352) and (353), suffer no change for displacements along I. In other words, under these circumstances, all the basis vectors I, n, m, and m will remain unchanged as they are parallely propagated along I. (a) The optical scalars Some further properties of a congruence of null geodesies can be derived by writing out explicitly the summations on the right-hand side of the equation, '.;; = e<°\y(an(b)e<»j. (355) We find /,;j =(e + e*)linj+(y + y*)lilj- (a* +fi)limj-(a + fi*)hmj — Kifiinj — /c*mjM;- + amiifij + ff*m,m;- — rrfiilj — T*m,/j + p mimj + p* mirhj. (356) First, we observe that if we contract this equation with V we recover equation (354). Since the /'s form a congruence of null geodesies affinely parametrized, k = e = 0, and equation (356) becomes ',;; = (7 + 7*) hh- («* + P)Um,- (a + /?*) /,my- rmjj + orhirhj + o*mimj + pmimj + p*mimj — z*milj. (357)
OPTICAL SCALARS, CLASSIFICATION, AND THE THEOREM 57 From this last equation, we find kan= -(a* + ^-T)/[jm;]-(a + ^*-T*)/[im;] + (p-p*)m[im;] (358) and kuM =(P~ P*)™[im;4]- (359) From the standard theorems of the subject (see the references in the Bibliographical Notes at the end of the chapter) it follows that the congruence of the null geodesies will be hyper-surface orthogonal (i.e. I will be proportional to the gradient of a scalar field) if, and only if, the spin coefficient p is real; and I will be equal to the gradient of a scalar field if, and only if, in addition a* + /? = x. From equations (357), (358), and (359) we find */';.-=-i(P + P*) = 0 (say), (360) i/ti:j]/'y'= -i(P-P*)2 = o)2 (say), (361) and iW':''=02 + M2. (362) The quantities 0, to, and a, as defined by the foregoing equations, were first introduced by R. Sachs and are called the optical scalars. Alternative definitions of 6 and co are 6 = — Rep and co = Imp. (363) The geometrical meanings of p and a are as follows. But first, we recall that I is the tangent vector to a null ray, say, N, and m is a complex vector orthogonal to I. At a point P on N, the real part of m spans with /a 2-plane. Now consider a small circle with its centre at a point P on N and lying in the 2-plane orthogonal to I. If we follow the rays of the congruence I, which intersect the circle, into the future null-direction, the circle may become contracted or expanded, rotated or sheared (into an ellipse). The expansion (or contraction), the rotation, and the shear are measured, respectively, by — Rep, Imp, and a (see Fig. 1); and \a\ measures the magnitude of the shear and j arg a the angle the minor axis of the ellipse makes with the assigned 2-plane. The fact that Imp is a measure of the rotation is consistent with the requirement that this quantity vanish for the congruence to be hypersurface orthogonal. The essential roles which the spin coefficients p and a play in describing the behaviour of bundles of light-rays as they traverse a gravitational field, are, further, manifested by the change experienced by I, in the orthogonal direction m, as it propagates. This change, in accordance with equation (349), is given by <5/(3) = - y i (fc,3 em = («• + /?)/- p*m - am. (364) The equations governing the variation of p and a, along the geodesic, are given by equations (310, a) and (310, b) in which we can now set k = £ = 0.
58 MATHEMATICAL PRELIMINARIES Fig. 1. The geometrical interpretation of the optical scalars in terms of the effect of propagation of a small circle perpendicular to the beam. Thus £>p = (p2 + M2) + <I>oo- (365) and Da = a(p+p*) + V0, (366) where it may be recalled that ¢00= -3K11 = -iV'/y and ^0= -Confirm?. (367) Some useful alternative forms of the foregoing equations governing the optical scalars may be noted here. Thus, equation (366) may be rewritten in the form £)(7=-20(7 + ^0- (368) An analogous equation for co follows from taking the imaginary part of equation (365) and remembering that <S0o ls real- Thus, Deo = i (p + p*) (p - p*) = - 200). (369) And the real part of equation (365) gives DO = o)2 -62 -I (712- (Dqo- (370) Equations (368), (369), and (370) are the standard forms in which they are commonly used. (b) The Petrov classification We have seen how, with respect to a chosen null tetrad-frame, the Weyl tensor is completely specified by the five complex scalars, *P0, *?!, ..., *P4. But the values which these scalars take are dependent on the choice of frame and on the six-parameter group of Lorentz transformations to which it is subject. The question now arises as to which of these scalars and how many of them can be made to vanish by a suitable orientation of the frame. The answer to this question gives rise to an algebraic classification of the Weyl tensor into types called the Petrov classification and the Petrov types.
OPTICAL SCALARS, CLASSIFICATION, AND THE THEOREM 59 Let *P4 ± 0. (If *P4 should happen to be zero in the chosen frame, then by a rotation of class I, we can make it non-zero so long as space is not flat conformally and not all of the Weyl scalars vanish.) Consider now a rotation of class II with a parameter b. The Weyl scalars 4*o, ^, etc., become (cf. equation (346)) ^d) = 4\> + 4^4*! + 6b2 4»2 + 4b3 4»3 + bA 4<4, 4/^)=4^+3^2 + 3^3 + ^4, (371) 4>2(1>= 4*2 + 2bx¥3 + b2 4»4, 4V1' = *P3 + bVA, 4V1' = ^4, where the superscript "(1)" distinguishes the new values of the scalars from the old. It is clear that by a rotation of class II, 4,0<1) can be made to vanish if b is chosen as a root of the equation 4\ b* + 44V3 + 6¾^2 + 4*! b + V0 = 0. (372) This equation has always four roots and the corresponding new directions of/, namely, l+b*m + bm + bb*n, are called the principal null-directions of the Weyl tensor. If one or more of the roots coincide, the tensor is said to be algebraically special; otherwise, it is said to be algebraically general. And the various ways in which the roots can coincide, or be distinct, leads to the Petrov classification. (a) Petrov type I—In this case, all four roots of equation (372) are distinct. Let them be bl,b2,b3, and b4. Then by a rotation of class II with parameter b = bl (say) we can make *P0 = 0. Then by a rotation of class I (which does not affect 4*0) we can make ¥4 vanish.* With 4*0 and 4*4 made to vanish in this manner, 4%, 4*2,and 4*3 will be left non-vanishing and of these scalars 4% 4*3 and 4*2 are invariant to the remaining rotations of class HI (which cannot, however, affect the vanishing of 4*0 and 4*4). (/?) Petrov type II—Let equation (372) allow two coincident roots bx = b2 (=^ b3± b4 and b3± bA). In this case, besides equation (372), its derivative (with respect to b), namely 4^ + 34^ + 34^ + ^1=0, (373) will also be satisfied when b = b2. Therefore, by a rotation of class II with parameter b = bl (= b2), *Po and *i will vanish. Then by a rotation of class I (which will not affect the initial simultaneous vanishing of 4*0 and 4%) (see equations (342)) we can make *P4 vanish. Only x¥2 and 4*3 will be left non- vanishing; and of them only *P2 will be invariant to rotations of class III. (7) Petrov type D—Let equation (372) allow two distinct double roots bl * Note that the roots of the equation (cf. equation (342)) T0(a*)4 + 4T1(a*)3 + 6T2(a*)2+4T3a* + T4 = 0 (372) are the reciprocals of the roots of equation (372).
60 MATHEMATICAL PRELIMINARIES and b2. In this case, we shall show how, by a rotation of class II followed by a rotation of class I, we can make *P0» ^1.^3. and ^4 vanish simultaneously and leave *F2 as the only remaining non-vanishing scalar. By assumption, *P0» after a rotation of class II with parameter b, will have the value *o(1) = **(*-*i)2(*-*2)2- (374) The values of the remaining scalars can be obtained by successively differentiating the expression for ^V1' and renormalizing, at each stage, to have the same coefficient *F4 for the highest power of b. We, thus, find V/l) = ^t(b-bl)(b-b2)(2b-bl-b2) ] V2W = $V4l(b-bl)(b-b2) + H2b-bl-b2)2l (375) ^>3^ = ^A(2b-bl-b2), and ¥4a) = ¥4.J With the choice, b = bu ^ = ^ = 0, *2a) = ^(^-¼)2. 1 \ (376) V3ll)=$V4(bl-b2), and *P4(1) = »P4J Now subject the frame to a rotation of class I with a parameter a*. Then, the new values of the Weyl scalars are (cf. equation (342)) *2<2) = ^2'U = i**(*i-*2)2, *P3<2> = »P3<1> + 3a*»P2<1)=|vP4(fc1-fc2)+^4a*(fc1-fc2)2 = \VA{bl-b2)[_\+a*{bl-b2)-], (378) and »P4<2> = vp4 + 4a*.|»P4(fc1-fc2) + 6(a*)2.^4(fc1-fc2)2 = »P4[l + a*(fc1-fc2)]2. (379) Accordingly, with the choice a* = (b2-bl)~\ (380) we can make *P3<2) and »F4<2) also vanish. Thus, by a rotation of class II with parameter bx followed by a rotation of class I with parameter a* = {b2—bl)'l,xV0,yVl,xV3, and *P4 have been reduced to zero and *P2 is the only non-vanishing scalar. And *P2 is invariant to rotations of class III. (5) Petrov type HI—If three roots of equation (372) coincide and bx = b2 = b3 ^= bA, then by a rotation of class II with parameter b = bl(= b2 = b3)v/e can make *P0, ^Vand *P2 vanish simultaneously; and by a subsequent rotation of class I we can make *P4 vanish (without affecting the vanishing of *P0' *i > and *P2). And *P3 will be the only non-vanishing scalar whose magnitude can be altered by rotations of class III. (377)
OPTICAL SCALARS, CLASSIFICATION, AND THE THEOREM 61 (e) Petrov type N—If all four roots coincide and there is only one distinct root b (say), then, by a rotation of class II with parameter b, we can make *P0, *?!, *P2, and *P3 vanish simultaneously and *P4 will be the only non-vanishing scalar. We shall now derive some simple conditions for / to be a principal null- direction of the Weyl tensor and for the occurrence of the various Petrov types. Using the general expression (298) for Cpqrs, we find that CpqJqlr = (¥2 + V2*)lpls + V0mpms + V0*mpms -4M/pm, + /,»»p)-¥i* (/,«, + ',«,)■ (381) Accordingly, C^t,/,]/'/^ (*o*«p-*i*/p)«[s/(]+(*o'»p-*i'p)'S[,',] (382) and kuCpwW*1' = *<>/[,,«,]«[,/,] + Vo*ku*Pl*isitr (383) Consequently, the condition, that / is a non-degenerate principal null-direction belonging to a Weyl tensor of type I, is ^,^/,^ = 0, if and only if *o = 0; (384) and the condition, that / is a doubly-degenerate principal null-direction belonging to a Weyl tensor of type II, is C^lqP V = 0, if and only if V0 = ¥, = 0. (385) And when 4*0 and x¥l vanish, CM„/'/r=(*2 + *2*)/p/,. (386) If the Weyl tensor is of type D and n is the second doubly-degenerate null- direction, then, besides the condition (385), we must also have ^[^,:^ = 0-, (387) and when ¥4 = ¥3 = 0, C„„n<nr=iV2+V2*)npni. (388) If the Weyl tensor is of type III and »P0 = x¥l = *P2 =0, then it follows from equation (298) that <WS= [*3 (lpmq-lqmp) + V3*(lpmq -/,mp)]/P; (389) and we conclude that CMr[s/t]/' = 0, if and only if 4>0 = ¥, = ¥2 = 0. (390) And, finally, if the tensor is of type N and ¥0 = 4», = *¥2 = *P3 = 0, then CM„= -**{/pm,/rm,}-***{/pm,tm,}, (391)
62 MATHEMATICAL PRELIMINARIES and C„„l' = 0, if and only if 4>0 = 4>, = 4>2 = ¥3 = 0. (392) (c) The Goldberg-Sachs theorem We restrict ourselves to the vacuum when the Ricci tensor vanishes and the Riemann and the Weyl tensors coincide. With this restriction the theorem states: If the Riemann tensor is of type II and a null basis is so chosen that I is the repeated null direction and *P0 = 4*, = 0, then k = a = 0; and, conversely, if k = a = 0, then 4^ = 4*, = 0 and the Riemann tensor is of type II. The importance of the theorem consists in its relating, reciprocally, the type-II character of a field with the existence of a congruence of shear-free null geodesies. To prove that, when *P0 = 4*, = 0, the congruence formed by the vector /is geodesic (k = 0) and shear-free (a = 0), is straightforward. Thus, when *P0 = 4*, = 0, the Bianchi identities (equations (321, a, b, c, e, f, and g)) are 3k4>2=0, (a);£>¥2= -2kV3 + 3PV2, (b); ^ D>¥3-5*>¥2 = -^-2(8-^3 + 3^2, (c); I (393) 3aV2 = 0, (e); -8V2= -3zV2 +2aV3, (f); M>2 -SV3= -3nV2-2(r-fi)V3+aV4,{g). If space is not flat (i.e., not all of *P0, 4^, 4^. *3» an(^ ^4 are zero), then it follows from equations (393,a,b,c) that k = 0 and from equations (393,e,f,g) that (7 = 0. To prove the converse, namely, that if k = a = 0 then 4*0 = x¥l = 0, is somewhat less direct. First, we shall suppose that by a rotation of class III (which will not affect the vanishing of k and a), s has been made to vanish. Then the Ricci identities (310,a,b,c,e, and k) for k = a = 0 give Dp = p2, (a); 4>0 = 0, (b); Dx = (t* +7^ + 4-, (c); } > (394) Dp = p*p +^^ (e); dp = (a* +p)p + (p-p*)r-Vl, (k);\ and the Bianchi identities (equations (321,a and e), with 4*0 set equal to zero in accordance with equation (394,b)) are D*,=4p*1,(a); and 5^,=2(21 + 0)^,,(6). (395) And finally, the commutation relation (304) gives DS-SD = (it*- a* -/?)£» +p*S. (396) From equations (343) it is clear that by a rotation of class I (which will not affect 4*0 nor the vanishing of k, a, or e), we can arrange that t = 0 provided
BIBLIOGRAPHICAL NOTES 63 p ± 0. (If on the other hand p = 0, it would follow from equation (394, k) that *Pj = 0, which is the result that we wish to prove.) Assuming then, that z has been made to vanish (and p ± 0), we have by equations (395) Dlg^l=4p and 5lgx¥l=2^. (397) Accordingly, (DS-SD)lg'¥l =2£>/?-4c5p. (398) Now, using the relations (394,e and k) and remembering that z has been made to vanish, we have (DS-SD)lg'¥l =2p*/?-4(a*+/?)p+6*,. (399) On the other hand, by applying the commutation relation (396) to lg 4%, and making use of equations (395,a and e), we obtain (D6-5D)\g'Vl = (7r*-a*-/J)Dlg4'1 +p*dlgx¥l = 4(rc*-a*-/?)p + 2/?p*. (400) Equating now the right-hand sides of equations (399) and (400), we obtain ¥, =!**p, (401) whereas, equation (394,c) with z = 0 requires 4», = -n*p. (402) Since we have assumed that p ^= 0 it follows from equations (401) and (402) that x¥l = 0; and the proof of the theorem is completed. A corollary to the Goldberg-Sachs theorem is that if the field is algebraically special and of Petrov type D, then the congruences formed by the two principal null-directions, I and n, must both be geodesic and shear-free, i.e., k = a = v = A=0 when 4*0 = x¥l = *P3 = *P4 = 0; and conversely. It is a remarkable fact that the black-hole solutions of general relativity are all of Petrov type D and, therefore, enable their analysis in a null tetrad-frame in which the spin coefficients k, a, X, and v and all the Weyl scalars, except *P2» vanish. It is this circumstance which accounts for the special effectiveness of the Newman-Penrose formalism for their consideration. BIBLIOGRAPHICAL NOTES §§ 1-6. There are many good books on differential geometry; but most of them are either more advanced or more extensive than is strictly necessary for our purposes. The account in these sections is adapted to our particular needs. In style and outlook, it is similar to the accounts in: 1. D. Lovelock and H. Rund, Tensors, Differential Forms, and Variational Principles, Appendix, 331-52, John Wiley & Sons, Inc., New York, 1975.
64 MATHEMATICAL PRELIMINARIES 2. S. W. Hawking and G. F. R. Ellis, The Large-Scale Structure of Space-Time, chapter II, 19-55, Cambridge, England, 1973. § 7. Brief accounts of the tetrad formalism will be found in: 3. L. D. Landau and E. M. Lifschitz, Classical Fields, §98, 291-94, Pergamon Press, Oxford, 1975. 4. S. Chandrasekhar in General Relativity—An Einstein Centenary Survey, chapter 7,371-91, edited by S. W. Hawking and W. Israel, Cambridge, England, 1979. § 8. The Newman-Penrose formalism is central to the book; and as we shall see in Chapters 8, 9, and 10, it is peculiarly well adapted to the study of the black-hole solutions of general relativity. The basic paper on the subject is: 5. E. T. Newman and R. Penrose, J. Math. Phys., 3, 566-79 (1962). An account which more fully motivates the underlying concepts is: 6. R. Penrose, An Analysis of the Structure of Space-Time, Cambridge University Adams Prize Essay, Cambridge, England, 1966. An alternative version of the formalism which exploits the underlying symmetry with respect to the basis vectors I and n and with respect to m and m is given in: 7. R. Geroch, A. Held, and R. Penrose, J. Math. Phys., 14, 874-81 (1973). For certain practical considerations (explained in the Bibliographical Notes for Chapter VIII) we have preferred to adhere to the original less symmetric version. For other accounts of the Newman-Penrose formalism see: 8. F. A. E. P irani in Lectures on General Relativity, Brandeis Summer Institute in Theoretical Physics, I, 249-373, Prentice Hall, Inc., Englewood, New Jersey, 1964. 9. R. A. Breuer, Gravitational Perturbation Theory and Synchotron Radiation, Lecture Notes in Physics, 44, Springer Verlag, Berlin, 1975. See also reference 4. § 9(a). The optical scalars were introduced by Sachs: 10. R. K. Sachs, Proc. Roy. Soc. {London) A, 264, 309-37 (1961). 11. , ibid. 270, 103-26 (1962). See also: 12. R. K. Sachs, Relativity, Groups, and Topology, 523-62, Les Houches Summer School in Theoretical Physics, Gordon and Breach Science Publishers, New York, 1964. For a condensed and an illuminating account see: 13. R. Penrose in Perspectives in Geometry and Relativity, 259-74, edited by B. Hoffmann, Indiana University Press, Bloomington, Indiana, 1966. The significance of the optical scalars, as spin coefficients, emerged, of course, only after the Newman-Penrose formalism had been developed. (See also reference 6.) § 9{b). The original papers in which the Petrov classification (sometimes also called the Petrov-Pirani classification) is described are: 14. A. Z. Petrov, Sci. Nat. Kazan State University, 114, 55-69 (1954). 15. F. A. E. Pirani, Phys. Rev., 105, 1089-99 (1957). For more extensive accounts see: 16. A. Z. Petrov, Einstein Spaces, translated by R. E. Kelleher and J. Woodrow, Pergamon Press, Oxford, 1969. 17. F. A. E. Pirani in Gravitation: An Introduction to Current Research, 119-226, edited by L. Witten, John Wiley & Sons, Inc., New York, 1962. See also reference 8.
BIBLIOGRAPHICAL NOTES 65 The account in the text is an elaboration of the one gven in reference 5. (See also reference 4.) § 9(c). The Goldberg-Sachs theorem is proved in: 18. J. N. Goldberg and R. K. Sachs, ActaPhys. Polonica, Supp. 13, 22,13-23 (1962). But the proof given follows reference 5.
2 A SPACE-TIME OF SUFFICIENT GENERALITY 10. Introduction The black-hole solutions of general relativity are static and spherically symmetric (as the Schwarzschild and the Reissner-Nordstrom solutions are) or stationary and axisymmetric (as the Kerr and the Kerr-Newman solutions are). The perturbed state of these solutions, which are of equal concern to us, will be neither stationary nor axisymmetric. As a basis for the treatment of the various problems that are suggested in these latter contexts, it will be convenient to have on hand the Einstein and the Maxwell equations in a space-time of sufficient generality which will encompass the different situations we shall encounter in the course of our study. In this chapter, we shall consider a space-time which appears to have the requisite generality; and we shall provide expressions for the basic tensors which will enable us to write down, without any ado, the relevant equations appropriate to the different contexts. 11. Stationary axisymmetric space-times and the dragging of inertial frames We consider first the metric appropriate to stationary axisymmetric space- times as a preliminary to our consideration of more general space-times which are neither stationary nor axisymmetric. For the description of a stationary axisymmetric space-time, it is convenient to take as two of the coordinates the time t( = x°) and the azimuthal angle <p( = x1) about the axis of symmetry. The stationary and the axisymmetric character of the space-time require that the metric coefficients be independent of t and <p and that gu = gu(x2>x3)> (1) where x2 and x3 are the two remaining spatial coordinates. Besides stationarity and axisymmetry, we shall also require that the space- time is invariant to the simultaneous inversion of the time t and the angle <p, i.e., to the transformation, t -> —t and <p -* — (p. The physical meaning of this additional requirement is that the source of the gravitational field, whatever that may be, has motions that are purely rotational about the axis of symmetry; for, in that case, the energy-momentum tensor of the source will have the stated invariance. In other words, the space-time we are considering is that associated
STATIONARY AXISYMMETRIC SPACE-TIMES 67 with a 'rotating body'. In any event, the assumed invariance requires that 002 = 003 = 012 = 013 = 0, (2) since the terms in the metric with these coefficients change sign under the inversion, t-* —t and q> -> — (p. Therefore, under the assumptions made, the metric must have the form ds2 = g00(dx0)2 + 2g0ldx°dxl +gil(dx1)2 + [022 (d*2)2 +2023 dx2dx3+033 (dx3)2]- (3) where all the metric coefficients are functions of x2 and x3 only. A further reduction in the form of the metric can be achieved by making use of the following theorem. THEOREM. The metric ds2= 011(dx1)2 + 2gl2dxldx2 + g22(dx2)2 (4) of a two-dimensional space, (x1, x2), with a positive- or a negative-definite signature ( + ,+ or —, —) can always be brought to the diagonal form, ds2= +e2"[(dx1)2 + (dx2)2] (5) by a coordinate transformation, where e2)i is some function ofx1 and x2. Proof To prove the theorem, it will clearly suffice to show that there exist transformations, xl' = $(x\x2) and x2'= ^(x1, x2), (6) which will reduce the contravariant form of the metric, ds2 = »11(dx1)2 + 2&12dx1dx2 + &22(dx2)2, (7) to a diagonal form with equal coefficients for (dxt)2 and (dx2)2. For a transformation to achieve the stated ends, it is necessary and sufficient that 01'2' =011 </>,ifi+012(</>,i<A,2 + </\2f i) + 022</>,2«A,2 = O (8) and 011'-02'2' = 0^(^-^) + 20^((/),1(/^-^,2) + 022(</>,22-«A,22) = O (9) We readily verify that equation (8) is identically satisfied by the substitutions (/^=^(0^,1+022^) and </>>2= -K(0u<A,i+012f 2), (10) where k is any arbitrary function. But the same substitutions in equation (9)
68 A SPACE-TIME OF SUFFICIENT GENERALITY yield the condition [k2{&11»22-(&12)2}-i][»1V,i2 + 231V,i'A,2 + 32V,22] = o (li) For a positive- or a negative-definite metric, the term in square brackets in equation (11) cannot vanish; and the equation can be satisfied only by the choice *2 = gllg22_(gl2)2=gllg22-(gl2)2 =g, (12) where g denotes the determinant of the covariant form of the metric. We conclude that equations (8) and (9) are satisfied by the substitutions ¢.1 = +0%2>,i + 022f2)= +g*g2k>P,k and (13) ¢.2 = -aHgll^.i +012f 2) = -0W.k. The integrability condition for these equations requires (0W,k)., = O. (14) In other words, \p can be any solution of Laplace's equation in the 2-space considered. The existence of transformations which will reduce the metric to a diagonal form with equal coefficients for (dxt)2 and (dx2)2 (and, therefore, also for (dx1)2 and (dx2)2 in the covariant form of the metric) is established. It is worth noticing that by rewriting equations (13) in the covariant form, ¢.1 = 9^9^^^, (15) (where £i;is the alternating symbol in two indices), the gradients of (/) and of ip appear as 'duals' of one another. It follows from the theorem we have proved that the part of the metric in square brackets in equation (3) can be reduced to the form + e2"[(dx2)2 + (dx3)2], (16) where fi is some function of x2 and x3. With the various simplifications available, we shall write the metric appropriate to a stationary axisymmetric space-time in the form ds2 = e2v(dt)2 -e2*(d<p -codt)2 -e2^(dx2)2 -e2^(dx3)2, (17) where v, \j/, co, fi2, and /x3 are functions of x2 and x3. It will be noticed that we have not specialized the metric as fully as we might have, within the requirements of our basic assumptions of stationarity, axisymmetry, and invariance under simultaneous inversion of t and <p, sinee we have left the functions /i2 and fi3 to be different. We have done so in order that we may, at a later stage, avail ourselves of this gauge-freedom to restrict \i2 and /i3 by a coordinate condition which may appear advantageous.
STATIONARY AXISYMMETRIC SPACE-TIMES 69 (a) The dragging of the inertial frame Associated with the metric (17), we can define a tetrad frame by the basis vectors: = (*" 0 e(0)i eav = (<oe*, -e* 0 0 e(2)i c(3)i (0 = (0 0 0 0), 0), -«"», 0), 0 , -e^). (18) The corresponding contravariant basis-vectors are: e{0j = (e-v , coe-* ,0 , 0), 0 0), -<2) ' = (0 «(3)' = (0 e""2, 0), 0 , e~ (19) For a tetrad frame so defined e{a) £(fc)i — '/(BXfc) — 0 -1 0 0 0 0 -1 0 (20) Thus, in the chosen frame, the metric is Minkowskian: it, therefore, represents locally an inertial frame. Now consider a point in space-time which is assigned the four-velocity, u° = where and df e v , d</> „ dxa „ d(p dx" ~dT (a = 2,3), (21) (22) (/2 =c2^-2v(fi_ajj2 + c2^,-2v(1;2j2+c2^3-2v(1;3j2_ (23.) The same point will be assigned, in the local inertial frame, the four-velocity, M<"> = e{a\ul = n^memu\ (24) or, as one readily finds, 1 ,(0) _ ~Jv-v2y ^-v(fi-to) 7(1 - ^2) «(1) = ,.. ..,/, and u(a) = //t T/2 To"-172) (a = 2,3). (25)
70 A SPACE-TIME OF SUFFICIENT GENERALITY Therefore, a point, considered as describing a circular orbit (with the proper circumference ne^) with an angular velocity Q in the chosen coordinate frame, will be assigned an angular velocity, e*-v(fi-co), (26) in the local inertial frame. Accordingly, a point which is considered as at rest in the local inertial frame (i.e., ua) = ui2) = u|3) = 0), will be assigned an angular velocity co in the coordinate frame. On this account, the non-vanishing of co is said to describe a 'dragging' of the inertial frame. In Chapter 6, we shall show that in an asymptotically flat space-time with the metric (17), a)->2Jr"3 (27) where J is a constant which will be interpreted as the angular momentum of the source. 12. A space-time of requisite generality We now wish to generalize the line element (17) to encompass situations which are non-stationary and non-axisymmetric. Our principal interest in these generalizations is to be able to treat, in full generality, perturbations of space-times which are either static and spherically symmetric or stationary and axisymmetric. In the first instance, we shall restrict ourselves to space-times which retain their axisymmetry at all times, i.e., we shall allow the metric to be dependent on t but restrict it to be independent of (p. Let the contravariant form of the metric be g = gtJdtd,. (28) Then by assumption, all the components, g'J, of the metric are functions only of x°, x2, and x3. We shall now show how, by a local coordinate- transformation involving only x°, x2, and x3, we can bring the 3 x 3-matrix, [0U']> (i,j — 0,2,3), to a diagonal form. THEOREM (COTTON-DARBOUX). The metric g = giididj (i,j = 0,1,2), (29) in a three-dimensional space, (x^x^x2), can always be brought to a diagonal form by a local coordinate-tranformation. Proof. First, we observe that by the choice of a geodesic system of coordinates, the metric can be brought to the form g = e°d0 + q<*t>Jdf (e° = + 1; a, p = 1, 2). (30) [A geodesic system of coordinates is constructed by considering a surface
A SPACE-TIME OF REQUISITE GENERALITY 71 f(x°,xl,x2) such that g'jfti fj}J= 0, letting the geodesies, normal to f= 0, be the coordinate lines x°, and choosing the coordinates x1 and x2 on the surfaces geodesically parallel to /] Consider the coordinate transformation, x1' = tf'{x°,x\x2) (f = 0,1,2), (31) where (/>' are functions of x°, x1, and x2 which we wish to determine by the conditions that the metric (30), in the new coordinate system, is diagonal. The conditions clearly are <?' = <°%%^'%% = « ^/. = (0,1,.(..2,,^, (32, or, written out explicitly, yw2 fl„aW _Ko ,_> ' ^^--° -W^ ~K '(Say)' (33) Hence, 3(/)° /KlK2V12 d<j)1 /K2K°V12 d<j)2 /K°KlV12 and dx° \ K° J ' dx° \ Kl J ' dx° V K' (34) Now, suppose that t/>°, t/)1, and t/>2, as functions of the two variables x1 and x2, are specified on a surface x° = 0 (say); and, further, that K°, Kl, and K2 are nowhere zero on the surface and satisfy the usual conditions of smoothness. Then, by the Cauchy-Kowalewski theorem, there exist unique functions, t/>°, (/)1, and (/)2 which satisfy the system of equations (33) and which reduce to the values specified on x° = 0. The existence of local coordinate-transformations, which will bring the metric in a 3-space to the diagonal form, is thus established. Returning to the consideration of a metric appropriate for a non-stationary axisymmetric space-time, we can, in accordance with the theorem we have established, set g02 = g03 = g23 = 0. (35) And we shall write the diagonal coefficients in the forms g00 = e-2\ g22=-e~2"\ and g33 = -e~2^, (36)
72 A SPACE-TIME OF SUFFICIENT GENERALITY where v, /i2, and fi3 are functions of x°,x2, and x3. Let the remaining coefficients of g1' be and a0l=coe-2\ g12 = -q2e~2^, g13 = -q3e~2^) (3?) g11 = co2e~2v - e~2* - q22e~2^ - q23e~2^, J where u>, q2, q3, and ij/ are further functions of x°, x2, and x3. With the contravariant form of the metric, chosen in the manner we have described, the covariant form of the metric is ds2 = e2v(dt)2 -e2*(d(p - q2dx2 - q3dx3 -codt)2 - e2^(dx2)2 - e2^(dx3)2. (38) Comparison with the metric (17) shows that the metric (38) is, in some sense, a natural generalization of it to allow for non-stationarity. Also, it should be noted that we do not now have the freedom of gauge to impose on fi2 and fi3 any additional coordinate condition—a freedom we did have in the context of the stationary metric (17). The chosen form of the metric (38) involves seven functions, namely, v,{j/,fi2, /x3,u),q2, and q3. On the other hand, since Einstein's equation is covariant and provides only six independent equations for the metric coefficients, it should be the case that these seven functions occur in the field equations only in six independent combinations. As to how this might arise can be inferred from considering the coordinate transformation x1 =x1'+/(x°',x2',x3') and x' = xv (/=0,2,3). (39) Under this transformation, only the metric coefficients gn(i =0,2,3) will change; thus, gvv = gn+9iif,i or i0vv-9ti)l9ii=f.i (» = 0,2,3). (40) From the integrability conditions of these relations, it follows that «.2-42,0» «3-93,0, and ^2,3-^3,2, (41) are invariant to the transformation (39). And it is a fact, which we shall verify, that the functions to, q2, and q3 occur in the field equations only in the combinations (41). But among the combinations (41), we have the identity, (^,2-92,0),3-(^,3-93,0),2 + (92,3-^3,2),0 = 0- (42) In this manner, the seven functions, in terms of which we have expressed the metric coefficients, occur in the field equations (as we have indicated) only as six independent quantities. The form of the metric (38), which we have derived for non-stationary axisymmetric space-times, can be applied to a class of non-axisymmetric space-
EQUATIONS AND COMPONENTS OF RIEMANN TENSOR 73 times whose metric coefficients are separable in the variable x1 (= <p), i.e., when 0u(x°, x1, x2, x3) = gli{x°, x2, x3)h(xl), (43) where ^(x1) is some function of x1. For, in this case, it is clear that by a coordinate transformation, involving only x°, x2, and x3 and leaving xl unchanged, we can make g02, g03, and g23 vanish: the common factor, ^(x1), which will occur in equation (33) will not, of course, make any difference. To allow for this possible generalization, we shall formally let the seven functions, v, \p, fi2, fi3, q2, q3, and w be functions of x1 as well. It will appear in later chapters that the form of the metric has the requisite flexibility and generality to encompass all the situations to which we shall apply the equations and the expressions appropriate for the metric (38) with the metric coefficients allowed to be functions of all four variables x°, x1, x2, and x3. Also, it should be remarked that, even with this generalization, the seven functions in terms of which the metric coefficients are expressed occur in the field equations only in six independent combinations (see §§13 and 14). The remaining sections of this chapter are devoted to writing down the basic expressions and equations which should be used in conjunction with the chosen form of the metric. 13. Equations of structure and the components of the Riemann tensor For the purposes of writing down the explicit form of Einstein's equation for the most general form of the metric chosen in §12, we shall obtain the components of the Riemann tensor via Cartan's equations of structure. In deriving the relevant equations, we can preserve maximum symmetry by writing dt=—idx4, <3, = i<34, v =/i4, and to = iq4. (44) The metric (38) then takes the form ds2= -Jje2^(dxA)2-e2,''(dxl-JjqAdxA)2, (45) A A where the capital Latin indices (/1, B,. . .) and summations over them are restricted to the values 2, 3, and 4. In the form (45), the metric is entirely symmetric in the indices 2,3, and 4. But it should be noted that in this complex version, its signature is ( — ,—,—, —). Also, it should be recalled that ip, nA, and qA are functions of all four variables x1 and xA(A = 2, 3,4). To avoid ambiguity, summation over a repeated capital Latin index (restricted to its range) will not be assumed: summation will be explicitly indicated (as in equation (45)) whenever it is needed, In applying Cartan's method, we shall take coA = e^dxA and ml = e^(dxl-^qAdxA) (46) A
74 A SPACE-TIME OF SUFFICIENT GENERALITY as our basis of one-forms (denoted by el in Ch. 1, §5). These forms correspond to the choice of an orthonormal tetrad-frame with signature ( —, —, —, —). The inverse relations expressing the one-forms dxA and dx1 in terms of co^and to1 are dxA = e-^coA and dx1 = e'^o)1 +Jje-^qAcoA. (47) A The first step in deriving Cartan's equations of structure (Ch. 1, equations (137) and (149)) is to express the exterior derivatives of the ofs in terms of the basis of the two-forms, namely, a/ A co1 (i j=j, i,j = 1, 2, 3, 4). We find dcoA = £ e**iiA BdxB A dxA + e»«nA1dxl A dxA B = Xe-"«/iAiBcoB A aA + nA ![e-^co1 + I>~"B<JBa)B] A a/ B B = le-^{^AB + qBflAl)(0BA(0A+e-<pflAitoiAcoA (48) B In writing compactly equations such as the foregoing, we shall find it convenient to define a colon derivative of a function f(xl,x2,x3,x4) with respect to xA (A = 2,3,4) by f:A=f,A + qAf,i- (49) It is a differentiation, since it satisfies the Leibnitz rule: (fg):A=f9:A + gf:A- (50) In this notation, equation (48) takes the form dtoA= -Y^e-^BHA-B^ /\mB-e-'1' nAlmA r\ml. (51) B Similarly, we find dm1 =Jje-»«(il/.A + qAA)coAAcol- £ e* ~"A-"BqB:A^A A «B- (52) A ' A,B Now, when the torsion is zero, Cartan's first equation of structure (Ch. 1, equation (137)) gives dtol= -Jj(olAA(oA (53) A and da>A= -JjcoABA(oB-(oAlA(ou, (54) B
EQUATIONS AND COMPONENTS OF RIEMANN TENSOR 75 and these equations determine the connection one-forms co1 A and co^gfrom our knowledge of do;1 and dcoA. Also, since the one-forms co1 and coA provide the basis for an orthonormal tetrad-frame, it follows from Ch. 1, equation (197) that co'j=-coJt (i,; = l,2,3,4). (55) From a comparison of equations (51) and (52) with equations (53) and (54), we conclude that <o1a= -a>Ai =e-»A(\l>;A + qA,\W -e'*Va,iO>A + lle*-K*-K°QAB(oB (56) B and coAB = ~(oBA= -^e*-"*-i"QAB(o1+e-il'>liA:B(oA-e-il*iiB.A(oB, (57) where Q.AB = lA:B-lB;A- (58) Now, letting ^ = ^ + 4,4,1, (59) we have the following explicit expressions for the different connection forms. a,l2 =e-^xV1ojl -e-^ 11^^ + ^-^-^Q^w3 +^-^-^Q^o/, (ol3 =e-x>x¥3col -e-*n3Aco3 + ^-10-^()340)4 + ^-^-^632^, 0^ = ^-^^4^-^-^/14^^+^^-^-^642^ + ^-^-^6430)3. a,23 = -^^-^-^6230)1 +e-^ii2W -e-^^.2co3, o)34= -^-^-"'Qs^oo1 +e-"*nyA(o3 -e-"'n4.3(o\ 0)42= -^^-^-^642^^^-^^4:2^-^-^^2:40)2. (60) To obtain the components of the Riemann tensor from Cartan's second equation of structure, namely, ^R'juco1' A co' = n'j = do/,- + o/k A tokh (61) we have to evaluate the exterior derivatives of the connection forms listed in equations (60). (Actually, it will appear that, by taking into account the underlying symmetry of the metric in the indices 2,3, and 4, we can write down all the components of the Riemann tensor from a knowledge only of, for example, Cll2 and fi23.) The required exterior derivatives of o/,- can be written down with the aid of the following readily verifiable lemmas.
76 A SPACE-TIME OF SUFFICIENT GENERALITY LEMMAS. If F is any function of xl,x2,x3, and x4, then A + $YjFe*-K*-x>>QABcoAAcoB, (62) A,B and d(FcoA) = Jje-^-"B(e^FyB(j0BA wa B + c-^-^(e^F)>1aj1AtoA, (63) where Q A is an operator, which acting on any function f (x1, ..., x4), yields ®Af=f:A + qA,J=f,A + (qAf),l- (64) (Note that SiA is not a differentiation; and also that (cf. equation (59)) e-*9A{e*)=*xA + qA,l = 'VA. (65) Now making use of the foregoing lemmas in conjunction with the connection forms (60), we find that dcol2 = ^-^-^^4(^-^^2)0/^ 0^-6-^-^(^-^2.1).1^ m2 A -ie-*-"'(e*-feG23).i«3A 0/-2^-^-^(^-^24)^0/^ co1 + 031 A ^{^-^-^^2^23+^-^-^(^-^623):2 + e-K>-K>(e-* + K>»2,lh} + co2A ^{^-^-^^2624 + ^-^-^(^-^624):2 + e-^-"*(e-*+^2,l):4} + t03A ^{^-^-^-^^634+^-^-^(^-^2^).3 _|e- «.-ft (e* -^623):4), (66) 0,^ A C023 = [e-2"3VP3^2:3-|e2'/'"2''2"2''3e23]«,A ^ -[e-^-"^3W:2+^-^-"^3.lG23]«lA«3 + ^-^-^-^634623^ A ^-^^-^-^634^2:3^2 A «0* + 1^-^-^-^634/^:20/^0/, (67) co\A co\ = [¢-^4^2:4-1^-^-^624]^ «2
EQUATIONS AND COMPONENTS OF RIEMANN TENSOR 77 -^-2"*-"'043/i2:4G>2A CO3 + ^ ~ "^ "'~ "'Q43 m:2(0* A ft)3 and dco23 = -^e-*-**S>A(e2,l/-^-^Q23)toA A to1 A -e-*-K(e^-*>H2:3),i(j°2a a>1 +6-*-">(<*>-^h3:2),i<o3/\ ft)1 _e-K2-K3(eK3-^3:2):2} + ft)2A ^{-^-^-^-^623624-6-^-^(^-^/12:3):4} + ft)3A ^{-^-^-^-^623634 + ^-^-^^3-^^^)^}. (69) (Ol2A C0l3 = 1^-^-^-^^^3^-^-^-^-^^^2^(0^ 0)4 + ^-^-^^2632 + 6-^-^^^.l]^1 A ^2 + 1:-2-^-^-^3 023-^-^2^3 ,l]«lA«3 + [i6^-2^-2"3Q23+e-2^2>1/l3,l]0)2AO)3 + [i6^-2^-"3-^Q24Q23-2-6-"3-"4e34^2,l]«2AO)4 + [ie2*-2"3-to-/«4e23e34+ie-fc-/«.e24/,3il]a)3A o)4, (70) 0)34A 0)42 = [-1^-^-^-^634^:2-^^^^642^4:3]^ 0)4 + 1^-^ -^^34/12:4^1 A ft)2+^-^^^642/^3:4^1 A ft)3 + 6-^/^3:4/12:4¾)2 A ft)3 - e ~^3 ~*'H4.3 H2-A<>>2 A ft)4 + e-^-^/i3.4/i4.2ft)3A ft)4. (71) Now (cf. equations (55) and (61)) \Ri2U(Ok/\(o' = ill2 =dft)12-ft)13 Aft)23-ft)14Aft)24. (72) Therefore, combining equations (66), (67), and (68) in accordance with the foregoing expression for Qi2t we can read of the components of Rl2kl of the Riemann tensor by simply collecting the coefficients of mk A o)'. Thus, Rl2l2, which is the coefficient of ft)1 A o)2 in ili2, is given by Rl2l2 = -r*-^2(^-^2)-c-2^3to-r*-^-V2,i),. + ^-2^-^623+^^-^^^624-^-^^4^2:4: (73) and, lowering the index, we obtain equation (a) in the list of equations (75)
78 A SPACE-TIME OF SUFFICIENT GENERALITY given below. Equations (g), (j), and (s) in the list are, similarly, the coefficients of co1 A co3, co2 A co3, and co3 A co4 in £ll2. In the same way, equations (d)and (p) are obtained by combining equations (69), (70), and (71) in accordance with the definition ^R23klcok A co' = Q23 = dco23 - co12 ^ co13 - co34 A a>42. (74) The remaining components of the Riemann tensor are obtained by suitable permutations of the indices in equations (a), (d), (j), (p), and (s) by making use of the underlying symmetry in the indices (2), (3), and (4). -e-*-">(e">-V2.i).i+4«2,''"2''2[«"2''3G23 + «"2''4G24]; (a) -U1313 = -^-^-^^3(^-^^3)-^-^^2^3:2-^-^^4^3:4 -c-^^^-^^.o.i+i^-^tr^Q^ + r^e^]; (b) -*1414 = -^-^04(^-^4)-^-^2^4:2-^-^3^4:3 -e^-^ie^-tnt^+ieW-^le-^Qlt + e-^QW]; (c) -^2323 = -^-"2-"3 [(^-^^2:3):3+ (^3-"^3:2):2]-^-2>3:4^2:4 _le2*-2,2-2,}Q23_e-2^2 ^3i. (d) -^2424= -^-"2-^C(^-^2:4):4 + (^-"^4:2):2]-^-2"^4:3^2:3 -V^-^-^ei*-*-2^,!^,!; (e) -^3434 = -^-^-^:(^-^^3:4):4+(^-^^4:3):31-^-^^4:2^3:2 _l^-2,3-2,4G24_e-2^3^4i. (f) -*i2i3 = -e-t-^g^-^J+e-^-^i^a +^-^-^:623.1+623(^3-^2+^,1]; (g) -K1214 = -^-^-^^4(^-^^2)+^-^-^^4^:2 +^^-^-^-^634632 + ie-"'-^[624.1+624(^4-^2 + ^,1]; (h) -«1314 = -^-^-^^(^-^^3) + ^-^-^^4^4:3 + le2*-2fc-ft-«4g24g23 + ^""'-^[634.1+ 634(^4-^3 + ^.1]; (i) -«1223 = ^-^-^623(^2-^3:2) + ^-^-^(^-^623):2 + ^-^-^643/^2:4
EQUATIONS AND COMPONENTS OF RIEMANN TENSOR 79 + ^-^-^^2,1):3-^-^3,1/^ (J) ^1224 = ^-^-^224(^2-^4:2) + ^-^-^(^-^224):2 + ^-^-^634^2:3 + e-ft-^(«-* + "^2,1):4-«"*~'<4^.1^2:4; (k) «1334 = ^-^-^634(^3 -^4:3) + ^-"3-"4(^-"3e34);3 + ^-^-^624/^3:2 + e-^-^(e-* + ^3tl).A-e-*-i^^1^.A; (1) «1332 = ^-^-^632(^3 -^2:3) + ^-^-^(^-^632):3 + ^-^-^642/^3:4 + ^-^-^^3,1):2-^-^2,1/^3:2; (m) ^1442 = ^-^-^642(^4-^2:4) + ^-^-^(^-^642):4 + ^-^-^632/^3 + e-^-"2(e-^ + "^4,l):2-^-^-"^2,1^4:2; (") ^1443 = ^-^-^643(^4-^3:4) + ^-^-^(^-^643)..4 + ^-^-^623^2 + «-"*-'"(e-* + ^4.i):3-e"*-''^3.1/^4:3: (°) «2334 = e-"2""4 [/^3:2:4 + /^3:2(/^3 -/^2):4-/^3:4/^4:2] -1^-^-^-^623634-^-^-^624/^3,^ (P) «3224 = e -"3 " "' [/^2:3:4 + /^2:3 (/½ ~ /^):4 ~ /^2:4/^4:3] -^-^-^-^632624-^-^-^634/^2,1: (q) «3442 = e-"3_"2 [/^4:3:2+ /^4:3 (/^4-/^3 ):2-/^4:2/^2:3] -^-^-^-^634642-2^^-^632/^4,1: W «1234 =^-"3-"4 [(^-^624):3-(^-^623):4] + ^-^-^-^634(2^2-/^3:2-/^4:2): (S) «1423=^-fe-"3C(^-^643):2-(^-'Ue42):3] + ^-"'-«»-«.623(2*4-^2:4-^3:4); (*) «1342 =^-^-^^(^-^632):4-(^-^634):2] + i^-«2-«3-^Q (2vp3 -^3-^:3)- (U)
80 A SPACE-TIME OF SUFFICIENT GENERALITY It should be noted that we can obtain formally different expressions for the same component of the Riemann tensor (modulo its symmetries) by differently ordering the permutations of the indices 2, 3, and 4. For example, by combining equations (68), (70), and (71), as required by the definition of Q23, we obtain -R23i3=U'*~"'S3{.e2,l'-^-i1'Q23)-e-*-i"{.eil'-^ii3.2ll + ±e*-K>-2^3Q23 + e-*-^2Li3A-y<l>-2>"-KiQ42Li3.A,(76) as the coefficient of co1 A a)3.Itisnotmanifestthatthisexpressionfor#23l3 is, apart from sign, the same as the expression for Rl332 given in equation (75, m). Nevertheless, by expanding the two expressions, we can verify that, in fact, they agree as required if proper use is made of our definitions of the colon derivative and of the operator Q)A. The establishment of other similar equalities is not always easy: they often depend on identities which are not obvious, as the following: 2[23:4]-G[23<74],l = 0. (77) Turning next to the components of the Ricci and the Einstein tensors, we can obtain them by combining appropriately the components of the Riemann tensor. Thus, -#n = #1212 + Ki3i3 + #i4i4, etc., + #12 = + Gi2 = #1332 + #1442, etc., -# — #1212 + #1313 + #1414 + #2323 + #2424 + #3434, 2 and ^11 = #2323+ #2424+#3434, etc. (78) We shall not write out these expressions explicitly: we shall have occasions to consider various special cases in later chapters. Finally, we may note that we can revert from the (complex) coordinate x4 to the (real) space-time coordinate x°(= t) by the correspondence -i4->0,; ,4-» -',o, /U->v; and qA-+-iw. (79) 14. The tetrad frame and the rotation coefficients In some contexts, as in obtaining the explicit forms of Maxwell's equations in §15, it is convenient to apply the tetrad formalism to the space-time with the metric chosen in §12. For such applications, a knowledge of the Ricci rotation- coefficients in a suitable tetrad-frame is, of course, essential. The tetrad frame we shall select is the real version, with a signature ( + , —, —, —), of the complex frame with the signature ( —, —, —, —), used in §13. Explicitly, the
TETRAD FRAME AND THE ROTATION COEFFICIENTS 81 covariant vectors, which provide the basis for the orthonormal frame selected, are (cf. equations (18) and (19)) (80) e(o„ = («v , 0 , 0 , «(i,i =ifoe* , -e* , q2e* , «(2,« = (0 , 0 , -e»> , e(3„ = (0 , 0 , 0 , The corresponding contravariant vectors are «(o,' = («-v ,o»«-v ,0 «(!,' = (0 ,e-* , 0 «(2)! = (0 , q2e-"2 , e'K> «(3)i = (0 - «3«~"3 , 0 0), q*e* , 0), , -«"')• , 0), ,0), ,0), ,e-"3). (81) As explained in Chapter 1 §8(b), the most convenient method of evaluating the rotation coefficients is via the symbols (Ch. 1, equation (266)) ^•(b,(*,(c, = e(a) le{b)i,j~ e{b) j,i:Je(c) ■ (82) The evaluation of these symbols is direct and straightforward. The results can be expressed in compact forms in the notation of'colon derivatives' introduced in §13. However, since we do not wish to introduce the imaginary coordinate, x4 = ix°, we shall now restrict the capital Latin letters (A, B, etc.) to the indices 0, 2, and 3. Also we shall write q0 = to. (83) DEFINITIONS. (1) The colon derivative ofa function f(x°,x\ x2, x3) with respect to xA(A = 0,2,3) is given by f:A=f.A + qAf.l. (84) (2) The operator Q)A acting onf(x°, xl, x2, x3) gives ®Af = f.A + qA,if = f,A + (qAf),u (85) (3) QAB = qA:B~qB:A, (86) and (4) %VA~^.A + qA,i~e-*3A(e*). (87) It is of interest to notice in this connection that the colon derivatives are simply related to the directional derivatives along the tangent vectors e(0). Thus, emf=e~vf:0, ewf=e~^f2, e{3)f=e~^f,3 and (88) W=«-*/!i.
82 A SPACE-TIME OF SUFFICIENT GENERALITY With the foregoing definitions we find that the A-symbols have the values listed below; ■fi ^110 ^0e V> ^223 — ~^2:3e" ^112 = ~Vie-»\ A233 = +/*3:2e~"2, Aii3=-*3e-"', A21o = G2o^-fe-v, (89) ^220= ~fi2:0e-\ A31o = G30^-"3-V, ^330 = ~M:0e-\ ^213 = Q23e*-^-l". The rotation coefficients ymb)(C)are now determined in terms of A{oKW{c) by the formula (cf. Ch. 1, equation (268)) 7(B)(fc)(c> = 2 [^(b)(W(c) + A(c)(b)(*) — A(k)(c)(B)]• (90) We find: ^100 ^200 ^300 ^122 ^133 = = = = = - - - /<2 V3 V,le v.2e v:3e Ae~ Ae~ -^ -fi -¾ *, 7ioo 7200 7300 7l22 7133 7233 7102 7103 = = = = = = = = -v Ae — v.2e — v;3e V2,le~ H3,\e~ V3:2e~ 7201 = 7301 = -0 -V; -e fl - - , 7ioi . 7121 , 7131 7202 73 03 7232 7120 = 7130 = = = = = = = ie ie -V0e~ -V2e- -V3e~ -/*2:0e -/*3:0«~ -/^2:3^ 20«*-"J 30«*-"' V P-2 9 ^3 — V 9 - V 9 ~Hs — V — V (91) 7132 = 7231= -7123=^23^-^3. 15. Maxwell's equations We conclude this chapter by deriving Maxwell's equations in the space-time with the metric we have chosen. The derivation is most conveniently carried out in the tetrad frame of §14. Maxwell's equations, 1WWf<*,|W = 0 ^d F[(fl)(D)|(c)] = 0, (92) in the orthonormal frame of §14, become c ' a0:0 c '«1,1 c ' «2:2 c * a3:3 = - -foi (7obi - 7ibo) ~-F02 (7ob2 - 72bO> - Foi (70b3 - 73bO> + ^12(71b2-72bi) + ^23(72b3-73b2)+^3i(73b1-7ib3) + ^Bl(-7lOO+7l22+7l33) + ^B2(-7200+7211 +7233» + ^b3 ( - 7300 + 73 11 + 7322» + ^bo(7i01 + 7202 + 7303» (93)
MAXWELL'S EQUATIONS 83 and F[a,b,c-]+ Z (_ 70acF0b + 7 lac F\b + 7lacF2b + 7lacF3b [a Ac] ~7obcFaO + 7lbcFal + 72bc Fa2 + 7ibc Fa3 ) = 0. (94) The particular form of the left-hand side of equation (93) derives from the fact that the directional derivatives, along the contravariant basis-vectors (81), are simply related to the colon derivatives (as we have noted in equations (88)). Now, substituting in equations (93) and (94) the values for the rotation coefficients given in equations (91), we find, after some lengthy but straightforward reductions, the following set of eight equations: ^3(^ + ^12)-^2(^+^13) + (^+^23),1=0, (a) ^2(^ + ^01) + ^0(^ + ^12)-(^+^02),1 =0, (b) ®3^ + ,foi) + @o^+^i3)-(«' + "Jfo3),i=0, (c) = e*^F02Q02 + e*^^F03Q03-e1'^F23Q23, (d) 1^^+^02) + ^3(^+^03) + (^+^01),1 =0, (e) -1^ + ^3) + 1^+^03)-(^ + ^13),1 = (), (f) + ^^ + ^23) + ^0(^+^02)-(^ + ^12),1 =0, (g) (^ + ^2):3-(^+^03):2+(^+^23):0 = ^ + V^0lG23+^+^12G03-^+^l3G02- (") (95) The foregoing eight equations are not all linearly independent; and that they are not depends on the following commutation rules (which are readily verified): ®A<J.l)=f:A,l ] and [ (96) (SA@B-@B@A)f=-(QABf\l.\ Thus, by applying the operators QJ0, — Qs3, and &2 to equations (a), (b), and (c), respectively, adding and simplifying with the aid of the commutation rules, we obtain \.-Q<*e* + »>Fl2 + Q02e*+i»Fl3-Q23e* + "FQl + (^+^23):0+(^ + ^02):3-(^+^03):211 =0. (97) But this equation is identically satisfied by virtue of equation (h). Similarly, by applying the operators Qj0, — &3, and — &2 to equations (e), (f), and (g), respectively, adding and simplifying with the aid of the commutation rules (96), we find that the resulting equation is identically satisfied by virtue of equation (d). We have now the means to write down the explicit forms of the Einstein and
84 A SPACE-TIME OF SUFFICIENT GENERALITY the Einstein-Maxwell equations for the non-stationary non-axisymmetric space-time with the metric (38). BIBLIOGRAPHICAL NOTES §11. Bardeen appears to have been the first to recognize the special appropriateness of the form of the metric chosen in this section to describe stationary axisymmetric space- times: 1. J. M. Bardeen, Astrophys. J., 161, 103-9 (1970). See also: 2. J. M. Bardeen, W. H. Press, and S. A. Teukolsky, ibid., 178, 347-69 (1972). §12. In spite of its elementary character, the theorem (which we have designated as the Cotton-Darboux theorem) that in a 3-space one can always set up locally, i.e., in finite open neighbourhoods, a triply orthogonal system of curvilinear coordinates, seems to have escaped mention in any of the standard sources on differential geometry. Indeed, the reference to 3. E. Cotton, Ann. Fac. Sci. Toulouse, Ser. 2, 1, 385^138 (1899) (see particularly p. 410) was traced by Professor A. Trautman to an 'example' in: 4. A. Z. Petrov, Einstein Spaces, translated by R. E. Kelleher and J. Woodrow, Pergamon Press, Oxford, 1969. See Example 5 on p. 44. We have adjoined Darboux's name with that of Cotton, in designating the theorem, since Cotton's proof is modelled on one given by Darboux for Euclidean spaces: 5. G. Darboux, Lecons sur les Systemes Orthogonaux et les Coordinriees Curvilignes, Gauthier-Villars, Paris, 1898. See pp. 1-2. I am grateful to Professor Trautman for his interest and advice in these and related questions. §13. The tetrad components of the Riemann tensor for a metric of the same form but under conditions of axisymmetry are given in: 6. S. Chandrasekhar and J. L. Friedman, Astrophys. J., 175, 379-405 (1972). §§14-15. The analysis in these sections generalizes the treatment in: 7. S. Chandrasekhar and B. C. Xanthopoulos, Proc. Roy. Soc. (London) A, 367,1-14 (1979).
3 THE SCHWARZSCHILD SPACE-TIME 16. Introduction A full understanding of the Schwarzschild space-time, as consisting of an event horizon and an essential singularity at the centre, was achieved only comparatively recently. In this chapter, we shall bypass the common historical route and provide, instead, a derivation of the Schwarzschild metric (in essence, due to Synge) which addresses itself directly to the essential features of the space-time. Also, since the geodesies in the Schwarzschild space-time illuminate some basic aspects of space-times with event horizons, we shall include an account of them —perhaps more complete than is strictly necessary. We conclude the chapter with a description of the Schwarzschild space-time in a Newman—Penrose formalism and a proof of its type-D character. 17. The Schwarzschild metric The Schwarzschild metric is a spherically symmetric solution of Einstein's equation for the vacuum. Following Synge, we shall define a spherically symmetric space-time as a manifold which is the Cartesian product, S2 x U2, of a unit two-dimensional sphere S2 and a two-dimensional manifold U2 with an indefinite metric. On S2, we take the usual polar coordinates, (0, <p), with the metric dQ2 = (d0)2 + (d<p)2 sin2 0. (1) On U2, since it is characterized by an indefinite metric, null lines u = constant and v = constant (2) must exist. We shall take them as a basis for a coordinate system on U2. With this choice of coordinates on S2 and U2, we can write the most general metric for a spherically symmetric space-time in the form ds2 =4/dudt;-e2"3[(d0)2 + (d(/>)2sin20], (3) where/and fi3 are functions of u and v. The metric (3) can be brought to a form in which it becomes a special case of the general metric considered in Chapter 2 by the substitutions u=±(x° + x2) and v = ^(x°-x2); (4)
86 THE SCHWARZSCHILD SPACE-TIME for, the metric, then, takes the form ds2 = /[(dx0)2 - (dx2)2] - e2^ [(d0)2 + (d<p)2 sin2 0]. (5) In order that we may transcribe the formulae of Chapter 2 directly, it is convenient to write f= e2"\ e2* = e2^ sin2 0, dx1 = d<p, dx3 = d0, and dx4 = idx°, (6) when the metric becomes -ds2 = e^^dx*)2 + e2<l'(dx1)2 + e2^(dx2)2 +e2^(dx3)2. (7) The metric is now of the form considered in Chapter 2, §13 with 1i = 13 = «4 = 0, /U = /<2, and e* = e^ sin 0. (8) Accordingly, for the present choice of the metric, ^4=/0,4, ^2=/0,2, and y3=cot0, (9) and /i2 and /x3 are functions of x2 and x4 only. We can now run down the list of the components of the Riemann tensor given in Chapter 2 (equations (67)) for the special case we are presently considering. We find that the only non-vanishing components of the Riemann tensor are: -R1212= -^2323= -^-^(^-^3,2),2-^^3,4^2,4 = -e_2"2|> 3,2,2 +^3,2(^3-^2),2 + ^3,4^2,4], (a) -^1414= -^3434= -^-^(^-^3,4),4-^-^3,2^2,2 = -e"2"2[^3,4,4+/^3,4(/^3-^2),4 + ^3,2/^2,2]. (b) -K1214 = +R23M= -e-"'-"1(«'""''^3.2).4 + «"2''^3.4^2.2 = -e_2"2[/i3,2,4+^3,2(^3-/^2),4-/^3,4/^2,2], (c) -Rl3l3=e-2K>-e-2»>l(n3,2)2 + (n3,4)2l (d) -^2424= -e"2"2(/i2,4,4+ /^2,2,2)- (e) (10) The field equations now require that the components of the Ricci tensor vanish. These conditions give (cf. Ch. 2, equations (70)) — ^22 = ^1212 + ^2323 + ^2424 = 2^1212 + ^2424 = 0. (H) — K44 = K1414 + K2424 + K3434 = 2K1414 + ^2424 = 0. (12) -^11=^1212+^1313 + ^1414 = 0, (13) -«33=^1313 + ^2323 + ^3434 = 0; (14) and the only non-diagonal component of the Ricci tensor, which does not
THE SCHWARZSCHILD METRIC 87 vanish identically, is #24 = #2114+ #2334 = 0. ^5) Equations (10, b) and (11)-(15) are readily seen to require #1212 = #1414 = — 2#2424 = — 2#1313 and \ (16) #2334 = #1214 = 0- Reverting to the space-time variable x°, we have #1212 = — #1010 = 2#2020 = _I#1313 and \ (17) #1210 = 0, where K1212 = e"2"2[/i3,2,2+/^3,2(/^3-/^2),2-/^3,0/^2,0], (a) #1010 = ^^[^3,0,0 + /^3,0(/^3-/^2),0-/^3,2/^2,2]. (b) #1210 = e~2"2|>3,2,0+^3,2(/i3-^2).0-^3,0^2.2]. (C) #1313= -e-2"3 + e-2"2C(/<3,2)2-(/<3,o)2], (d) #2020 = e_2"2(/^2,0,0-/^2,2,2)- (e) (18) We consider equations (17) in the combinations #1212 + #1010 = 0, #1210 = 0, #1212 + #1313 —#1010 = 0, an(I #0202 + #1313 = 0- (19) Inserting from equations (18), for the components of the Riemann tensor, in the foregoing equations, we find that they can be reduced to the forms [(e"3),2 e~fc]i2 + [(«">) 0e-">).„ + («">) 0 (e -*).<> + (e>"\2 (e-*).2 = 0, (20) («"').o.2 - («"').2^2.o - (^),0/^2,2 = 0, (21) ^2+{^3(^3),o,o-^3(^3),2,2 + CK3),o]2-C(^3),2]2} = 0, (22) and ^2"2(/^2,0,0-/^2,2,2)-^2"3+^2''2~2''3{C(^3),2]2-C(^3),o]2}=0. (23) Reverting to the variable f= e2^ and letting Z = e">, (24) we can rewrite equations (20)-(24) in the forms /(Z 2.2 + Z 0,0)-(Z2/2+Z o/o) = 0, (25)
88 THE SCHWARZSCHILD SPACE-TIME 2/Z o.2-(Z 2/0 + 2.0/2) = 0, (26) /+ Z(Z 0,0 - 2,2.2) + (Z,o)2 ~ (Z,2)2 = 0, (27) ^C(lg/).o,o-(lg/),2.2]-^2- + ^yC(2,2)2-(2,o)2]=0. (28) Equations (25) and (26) can be combined to give /(Zo.o±2Zo,2 + Z2,2)-(Z2±Z0)(/.2±/o) = 0; (29) while equation (28) can be simplified, with the aid of equation (27), to give K(lg/),o,o-(lg/),2,2] +^(Z 0,0-2,2,2) = 0. (30) Finally, reverting to the original variables u and v, we have the equations /Z,„,„-Z„/„ = 0, (31) /Z,,,-Z,/, = 0, (32) /+ZZ,„„ + Z,„Z, = 0, (33) and i(lgAu.„ + |2„„ = 0. (34) (a) The solution of the equations Rewriting equation (31) in the form 2^, =4 (35) •^.u J we conclude that f=B(v)Z,u, (36) where B(v) is an arbitrary function of 1;. Similarly, from equation (32), we conclude that f=A(u)Z,v, (37) where A (u) is an arbitrary function of u. Now writing equation (33) in the two alternative forms, /+(ZZ,,),„ = 0 and /+(ZZ„),, = 0, (38) and substituting for/from equations (36) or (37), we obtain [B(t;)Z + ZZ,,],„ = 0 and [>1(U)Z + ZZ,„],, = 0. (39) Therefore, Flu) G(v) Z_u=-A(u)+-^- and Z, = -B(v)+-^-, (40)
THE SCHWARZSCHILD METRIC 89 where F(u) and G(v) are further arbitrary functions of the arguments specified. By making use of equations (36), (37), and (40), we obtain Z,UZV= -A(u)Z.„ + F(u)^= -f+F(u)^ = -B(i;)Z, + G(»)|=-/+G(i))|!. (41) Accordingly, F(u)^ = G(v)^. (42) Together with equations (36) and (37), equation (42) requires Z„ F(u) A(u) Z.v G(v) B(vY Therefore, F(u) G(v) -~ = —^ = a constant = 2M A(u) B(v) Thus, we obtain the solution Z..= -A(u)U-^f) and Zv=- and ( 2M\ f=-A(u)B(v)ll- — \. (say). -B(v)(l- \ 2M~ (43) (44) (45) (46) It can now be verified that the solution of equations (31)-(33) expressed by equations (45) and (46) satisfies the remaining equation (34). The meaning of the solution for Z given by equation (45) becomes clearer in the form ( 2M\ dZ=- 1—— \[A(u)du + B(v)dv]. (47) According to this equation, Z is not a function of u and of v independently: it depends jointly on them in the sense that we can draw curves of constant Z in the (u, y)-plane. The latter statement is essentially the content of Birkhoff's theorem to which we shall return in §(c) below and also in §18. Finally, we may note that the solution for the metric itself is given by ds2 = -4[ 1--^- jA(u)B(v)dudv-Z2dil2. (48)
90 THE SCHWARZSCHILD SPACE-TIME (b) The Kruskal frame The occurrence of the two arbitrary functions, A (u) and B(v), in the solution (48) for the metric, is entirely consistent with the freedom we have in defining, as coordinates in U2, u and v or some function of u and some function of v. The availability of this gauge freedom is in accord with the fact that the two arbitrary functions, A(u) and B(v), occur in the combinations A(u)du and B(v)dv both in the metric (48) and in the solution (47) for Z. However, some care must be exercised in the choice of the functions of u and v which we wish to designate as coordinates in U2 if we are not to introduce spurious coordinate- singularities in the metric (particularly at Z = 2M, as is manifest from the form of the metric). The occurrence of singularities might only signify that the coordinate functions chosen are valid in certain well-defined neighbourhoods of the manifold and not valid globally. A choice of the coordinate functions that avoids spurious singularities is obtained b} letting - A(u)du-^2Md\gu and - B(v)dv -> 2Mdlgi; Then dZ = 2Mil-—\d[\g(uv)l and 16M2 f 2M\ , , ds2 = 1 dudt;-Z2dQ2. uv \ Z J On integration, equation (50) gives \uv\ = C\Z-2M\ez>2M, where C is a constant of integration. With the choice C = (2M)~\ (53) and replacing Z by the customary radial-coordinate r, the metric (51) takes the Kruskal form 32M3 ds2 = - -^- e ~r'2M du dv - r2 dQ2, (54) r where r is related to uv by uv = (l-r/2M)er/2M. (55) We shall now examine the domain in the (u, t;)-plane in which the metric is well-defined. First, we observe that the (r, ut;)-relation (55) is double-valued if we allow negative values for r. The two branches of the relation, for r > 0 and for r < 0, must, therefore, refer to entirely distinct physical situations. Since reversing the sign of r has the same effect as reversing the sign of M, it is clear that the branch (49) (50) (51) (52)
THE SCHWARZSCHILD METRIC 91 Singularity Fig. 2. The Schwarzschild geometry in the u- and v- coordinates (equivalent to the Kruskal coordinates). The light cones at any point are defined by the null geodesies, u = constant and v — constant, passing through that point. The curves of constant t and constant r are the radial lines from the origin and the hyperbolae, respectively. The dashed line is a possible time-like trajectory crossing the horizon. of the relation (55) for r < 0 is that appropriate to a negative value of M; it is, therefore, not of much interest for, as we shall presently see, M has the physical meaning of mass. We shall, accordingly, restrict ourselves to r > 0 and M > 0. In the (u, t;)-plane, the curves of constant r are rectangular hyperbolae with the coordinate axes as asymptotes. Also, the 'iso-radial' curves for r > 2M are confined to the quadrants I and III (in Fig. 2), while the curves for r < 2M are confined to the quadrants II and IV; and the axes themselves define the loci for r = 2M. The light cones at any point are defined by the null geodesies, u = constant and v = constant, passing through that point. It is clear from the disposition of the light cones in Fig. 2 that no time-like (or, null) trajectory, in the direction of increasing r in the quadrant II, can ever emerge into the quadrant I. These geometrically obvious facts imply (as we shall see more explicitly in §§19 and 20 below) that the surface r = 2M is an event horizon. A further essential feature of the space-time which Fig. 2 exhibits, is that a
92 THE SCHWARZSCHILD SPACE-TIME future-directed time-like trajectory (in the direction of decreasing r), once it crosses the horizon (at r = 2M) into the quadrant II, cannot avoid intersecting the locus r = 0 where the element of proper volume vanishes. In other words, anything which crosses the horizon cannot escape being crushed to 'nothing' at r = 0. For this reason, the centre emerges as a singular line of the Schwarzschild space-time—a fact which is further exemplified by all the non- vanishing components of the Riemann tensor (in a local inertial-frame) diverging at r = 0 and nowhere else (see equations (75) below). (c) The transition to the Schwarzschild coordinates Let u/v = e"2M. (56) This substitution is strictly valid in the quadrants I and III. In these quadrants, the curves of constant t are straight lines through the origin. In particular, t = 0 along u = +v, and t-* + oo along the y-axis (approached in the counter-clockwise direction) and t-* — oo along the u-axis (approached in the clockwise direction). Restricting our considerations to the quadrants I and III, we have /du diA df=2M(v-vJ- (57) We also have (replacing Z by r in equation (50)) r&r -,., /du diA = 2M —+— . (58) r — 2M \u vr From these equations, it follows that r2 (dr)2 , du di; and the expression (51) for the metric becomes ds2 = (l -™^J (At)2 - i {^/r -r2[(dfl)2 + (d<p)2sin2fl]. (60) This is the Schwarzschild metric in its most familiar form; it is also the form in which Schwarzschild originally wrote it. The present derivation, however, emphasizes that it is strictly applicable only in the quadrants I and III. But the form (60) has several advantages. It makes it manifest, for example, that the radial coordinate r is a 'luminosity distance' in the sense that the surface area of a sphere of 'radius' r is Anr2 and that t is a measure of the proper time for an observer at rest at infinity. Further, the asymptotic form of the metric for r -> oo shows that it represents the space-time external to a spherical distri-
ALTERNATIVE DERIVATION OF SCHWARZSCHILD METRIC 93 bution of inertial mass M. And finally, since the metric coefficients are explicitly independent of time and there is, in addition, no dragging of the inertial frame (cf. Ch. 2, §11(a)) the space-time is static as experienced by an observer external to the horizon. And this is Birkhoff's theorem as commonly stated. Returning to the (u, t;)-plane, we now observe that any time-like or null trajectory, crossing into quadrant II from quadrant I, will, according to any observer stationed in quadrant I, take an infinite time to do so. (We shall provide illustrations of this phenomenon in §§19 and 20.) Also, since no timelike or null trajectory can ever emerge from quadrant II into quadrant I, events in the space interior to r = 2M are inaccessible to all observers in quadrant I: the events taking place interior to r = 2M are incommunicable to the space outside. It is in this sense that the surface r = 2M is an event horizon; and it is also the sense in which the Schwarzschild metric represents the space-time of a black hole. 18. An alternative derivation of the Schwarzschild metric In view of the important role which the Schwarzschild metric in its standard form (60) plays in subsequent developments, it will be useful to have an ab initio derivation of it starting with the coordinates r and t as they eventually emerged in §17(c). Besides, it will provide the occasion to obtain directly the various components of the Riemann tensor, in these coordinates, which we shall need. The arguments, which led us to postulate a metric of the form (1) for a spherically symmetric space-time, will permit us to write ds2 = f00(dt)2 + 2f02drdt+f22(dr)2 + gdil2 (61) where/oo>/o2>/22» and S are functions of the variables r and t only. By a coordinate transformation involving these variables, we can arrange that/02 vanishes and g is replaced by the square of the luminosity distance r. Such a transformation is clearly possible in some open neighbourhood; but it is not evident that it may be possible globally. Indeed, as our discussion in §17 has shown, such a transformation is not, in fact, possible for the entire space-time. With this understanding, the metric can be reduced to the form ds2 = e2v(df)2-e2^(dr)2-r2[(d0)2 + (d(p)2sin20], (62) where v and \i2 are functions off (= x° = — ix4)andr( = x2). The metric (62) is again of the general form considered in Chapter 2. In the present instance v = /^4, /½ = lgr, iA = lgr+ lgsin0, n32 = r~l, | I (63) ¥4 = 0, VP1 = 0, V2 = r~l, and ¥3 = cot0. J
94 THE SCHWARZSCHILD SPACE-TIME Running down the list of the components of the Riemann tensor given in Chapter 2 (equations (67)), we find that the only non-vanishing components of the Riemann tensor are e ~ 2^ ~ *M212 = — #2323 = "I ^2,r' — #1414 ~ —#3 #2334 — + #1214 — -2|j2 "V.r. . -V-V2 -/^2,4. (64) ■K,3,3 = >-2(l-e-H and -#2424 = -e-^-V[(e"^V2,4),4+(eV-fcV,r),r], where it should be remembered that these are the tetrad components in a frame with the basis vectors c(a)i e" 0 0 0 0 -rsinfl 0 0 0 0 -e^ 0 0 0 0 -r (65) The field equations now follow from setting the various components of the Ricci tensor equal to zero. Thus, — #42 _ #1412 + #3432 ~ 2- -/^2,4 = 0. Hence, (66) (67) V-2 = Hl(r)- Considering next the equation, -#44+ #22 = #1414+ #3434-(#1212 + #2323> = 0' (68) we find Therefore, e-2»>(v + fi2\r = 0. v= -ii2(r)+f(t\ (69) (70) where/(f) is an arbitrary function of f. There is no loss of generality in setting /(f) = 0, since it can be 'absorbed' in the definition off, i.e., by replacing e/("df by df. With this redefinition of the time coordinate V = ~H2, (71)
ALTERNATIVE DERIVATION OF SCHWARZSCHILD METRIC 95 and both v and /x2 are functions of r only. The fact that the metric coefficients, originally allowed to be functions of t as well, are now seen to be independent of t is a further demonstration of Birkhoff's theorem: the space-time external to a spherical mass is necessarily static. The solution can now be completed by considering the equation — «33 = «1313 + «2323 + «4343 = 0. (72) Substituting for the components of the Riemann tensor from equations (64), we find e ~ 2^ 1 ^2 -2^r- = (i_e-2^). (73) The solution of this equation is e~2^ = i_2M/r(=e2v), (74) where M is a constant. It can now be verified that the vanishing of the remaining components of the Riemann tensor is ensured by the solution (74). We have thus recovered the solution (60) for the metric. Finally, we note that, with the solution for /*2 and v given by equation (74), the non-vanishing components (64) of the Riemann tensor reduce to _-^(1)(3)(1)(3) = -^(2)(0)(2)(0) = 2Mr J and V (75) «(1)(2)(1)(2) = «(2)(3)(2)(3) = _«(1)(0)(1)(0) = _ «(3)(0)(3)(0) = Mr > J where we have enclosed the indices in parentheses to emphasize that these are tetrad components in a local inertial-frame specified by the basis vectors (65). We observe that all these components of the Riemann tensor diverge at r = 0—a manifestation of the singular nature of the Schwarzschild space-time at r = 0. To obtain the covariant tensor-components, in the coordinate frame in which the Schwarzschild metric is expressed, we must subject the tetrad components (with respect to each of the indices) to the transformation (65); and applying this transformation to the components listed in equation (75), we find «oioi = -Mr~le2v sin20, R0303 = -Mr~le2\ Rl2l2= + Mr~ le~2v sin2 6, R2323 = + Mr~ le~2\\ (76) Rl3l3= -2Mrsin20, K0202 = 2Mr"3, where it may be recalled that e2v = 1 - 2M/r, (77) and, further, that the indices 0,1,2, and 3 refer to the coordinates t, q>, r, and 6, respectively.
96 THE SCHWARZSCHILD SPACE-TIME 19. The geodesies in the Schwarzschild space-time: the time-like geodesies We have shown in Chapter 1 (§6(a), equation (203)) that the equations governing the geodesies in a space-time with the line element, ds2 = gijdxidx\ (78) can be derived from the Lagrangian dx;dxj 2* = '«**> (79) where t is some affine parameter along the geodesic. For time-like geodesies, t may be identified with the proper time, s, of the particle describing the geodesic. For the Schwarzschild space-time, the Lagrangian is 2 I1-™)*-!^-^-^2'*2 (80) where the dot denotes differentiation with respect to t. The corresponding canonical momenta are d£> ( 2M\. d£> 2 . pr= -- = ^1--) r, and Pe=--^^r6. (81) The resulting Hamiltonian is je=pti-{prf + ped + pil,M-& = &. (82) The equality of the Hamiltonian and the Lagrangian signifies that there is no 'potential energy' in the problem: the energy is derived solely from the 'kinetic energy' as is, indeed, manifest from the expression (79) for the Lagrangian. The constancy of the Hamiltonian and of the Lagrangian follows from this fact: jt = £> = constant. (83) By rescaling the affine parameter t, we can arrange that 2I£ has the value + 1 for time-like geodesies. For null geodesies, if has the value zero. (We shall not be concerned with space-like geodesies.) Further integrals of the motion follow from the equations ^ = ^ = 0 and ^=-^ = 0. (84) dt dt dx dep
THE TIME-LIKE GEODESICS 97 Thus, ( 2M\dt p. = 1 — = constant = E (say) (85) V r /dT and p„ = r"1 sin"10— = constant. (86) v dt Moreover, from the equation of motion, it follows that if we choose to assign the value n/2 to 6 when 0 is zero, then 0 will also be zero; and 6 will remain constant at the assigned value. We conclude that the geodesic is described in an invariant plane which we may distinguish by 0 = n/2. Equation (86) then gives 7d<» p9 = ir -7- = constant = L (say), (88) where L denotes the angular momentum about an axis normal to the invariant plane. With i and ip given by equations (85) and (88), the constancy of the Lagrangian gives F.2 r2 L2 --^ = 2^= +1 or 0, (89) l-2M/r \-2Mjr r depending on whether we are considering time-like or null geodesies. In this section we shall restrict ourselves to time-like geodesies. (Null geodesies are considered in the following section.) For time-like geodesies, equations (88) and (89) can be rewritten in the forms £)' + (.-^)(.+£)-E. and d<p L dt r (91) By considering r as a function of q> (instead of t), we obtain the equation drV , r4 2M , , dW =(£-^Vr-r+m (92) Letting u = r'\ (93) as in the analysis of the Keplerian orbit in the Newtonian theory, we obtain the
98 THE SCHWARZSCHILD SPACE-TIME basic equation of the problem: du\2 , , 2M l-£2 -J =2MU3-^+_u-_. (94) This equation determines the geometry of the geodesies in the invariant plane. Once equation (94) has been solved for u(<p), the solution can be completed by direct quadratures of the equations, dt _ 1 df d<p~Lui an Aq> Lu2(l-2Mu) 2 and 77=,,.2,, ,„..,■ (95> (a) The radial geodesies The radial geodesies of zero angular momentum, while simple, illustrate some essential features of the space-time to which references were made in §17. The equations governing these geodesies are (cf. equations (85) and (90)) drV 2M , „, dt E 1-)= l-£ ) and — = -———. 96) Ax J r ' dt \-2Mjr ' We shall consider the trajectories of particles which start from rest at some finite distance r, and fall towards the centre. The starting distance is related to the constant E by 2M \^E' r, = ^-^2 (r = r> when ^ = 0)- (97) The equations of motion are most conveniently integrated in terms of a variable r] where M 2M r = 1_£2(1 + cosr/) = 1_£2cos2i>? = r;cos2ir/. (98) Clearly, rj = 0 when r = r;; and the values of r\ when r crosses the horizon at r = 2M and arrives at the singularity at r = 0 are rj = r/H = 2 sin-1 E when r = 2M and (99) r] = n when r = 0. In terms of r\, the equations to be integrated are '-) =(l-£2)tan2^ and ^ = „„27.7^2... » (100) together with dr\2 2 21 df Ecos2^r/ dt/ 2 dt cos2 ^-cos2 ^H' dr . — =— r sin^r/cos^r/. (101) dr/
THE TIME-LIKE GEODESICS 99 Since we are considering infalling particles, dr /2M\1/2 — = -(1-E2)itan|r/= - tan^. (102) dr V r, / From equations (101) and (102), we now obtain dt / r? \1/2 / r3 \1/2 sr(js) -^-(si) (1+CMrt ,103' Therefore, / r3 \ 1/2 T = V8m) {r,+Sinr])' (104) where we have assumed that t = 0 at the starting point r] = 0. From equation (104) it follows that the particle crosses the horizon and arrives at the singularity at the finite proper times, <°-w- 8M ' ('/H + sin'/H) and To = ( ^77 I n- (105> The situation, as we shall see presently, is very different when we consider the equation of the trajectory in coordinate time, t. The equation to be integrated to obtain t is (cf. equations (100) and (103)) jj = £fzLV/2 «****, t , (106) drj \2M/ cos \r\ -cos \r]H On integration, this equation gives v2m; 1 = £(;fbY L^+sinr/J + U-E2),/] + 2Mlg tanir/H+tanjr/ (107) tan jrjH- tan ^r\ According to this equation, f->oo as rj -frjn — 0, (108) in sharp contrast with the behaviour of the proper time t. This is an example of what we concluded on general grounds in §17(c), namely, that with respect to an observer stationed at 'infinity', a particle describing a time-like trajectory will take an infinite time to reach the horizon even though by its own proper time it will cross the horizon in a finite time. And after crossing the horizon, the particle will arrive at the singularity, again, at finite proper time. These facts are illustrated in Fig. 3 in terms of the solutions (104) and (108).
100 THE SCHWARZSCHILD SPACE-TIME 6 5 4 r/M 3 2 1 0 5 10 15 20 25 30 time/A/ Fig. 3. The variation of the coordinate time (r) and the proper time (t) along a time-like radial- geodesic described by a test particle, starting at rest at r - 6M and falling towards the singularity. (b) The bound orbits (E2 < 1) The consideration of equation (94), governing the geometry of the orbits described in the invariant plane, is conveniently separated into two parts, pertaining, respectively, to E2 < 1 and E2 > 1. These two classes of orbits are characterized by energies (exclusive of the rest energy) which are negative or positive (and zero). As we should expect, this distinction is one which will determine whether the orbits are bound or unbound (i.e., whether along the orbits r remains bounded or not). In this section (b), we shall restrict ourselves to bound orbits (E2 < 1). These orbits are governed by the equation (£)' =f(u), (109) where 2M l-£2 f(u) = 2Mu3-u2 + -^u jj- (£2<1). (110) It is clear that the geometry of the geodesies will be determined by the disposition of the roots of the equation/(u) = 0. Since/(u) is a cubic in u, there are two possibilities: either, all the roots are real or one of them is real and the two remaining are a complex-conjugate pair. Letting uu u2, and u3 denote the roots of the cubic equation/(u) = 0, we have ulu2u3 = (l-E2)/2ML2 (111) and ul + u2 + u3 = l/2M. (112) Since we have assumed that (1 — E2) > 0, the equation/(u) = 0 must always allow a positive real root. From the further facts,/< 0 for u = 0 and/(u) -> t, Schwarzschild time
THE TIME-LIKE GEODESICS 101 Fig. 4. The disposition of the roots of a cubic/(u) = 0 for E1 > 1. The various cases (a), (/!), etc., are distinguished in the text. + ooforu-> + oo,itfollows that wemust consider the fivecases distinguished in Fig. 4. These different cases lead to the following possibilities.* Case (a): For every pair of values E and L, which allows for u three real roots 0 < ul < u2 < u3, there exist two distinct orbits confined, respectively, to the intervals ul ;% u < u2 and u ^ u3, i.e., an orbit which oscillates between two extreme values of r( = u^1 and u2 l) and an orbit, which, starting at a certain aphelion distance (=U3X), plunges into the singularity at r = 0 (i.e., as u -> oo). We shall call these two classes of orbits as of the^zrsf and second kinds, respectively. The orbits of the first kind are the relativistic analogues of the Keplerian orbits and to which they tend in the Newtonian limit. The orbits of the second kind have no Newtonian analogues. However, we shall find that the orbits of both kinds (and the unbound orbits, as well, as we shall see in §(c) below) are most conveniently parametrized by an eccentricity, e ^ 0 and a latus rectum, I, even as the Newtonian orbits are. Case (/?): In this case the orbit of the first kind is a stable circular orbit (of zero eccentricity) while the orbit of the second kind, even though labelled as of zero eccentricity, still plunges into the singularity. Case (y): In this case, the orbit of the first kind starts at a certain aphelion distance, uj-1, and approaches the circle of radius u3l, asymptotically, by spiralling around it an infinite number of times. The orbit of the second kind is, in some sense, a continuation of the orbit of the first kind in that it spirals away from the same circle (towards the centre) to plunge eventually into the singularity. Case (5): In this case all three roots coincide; and the principal difference with case (e) below is that the equations also allow an unstable circular orbit of radius u\ l( = u2 l = u3 l). * By combining the relations (111) and (112) with the further relation uiu1 + u1u1 + u1ui = l/L2, it can be shown that the equation / (u) — 0 for £2 < 1 does not allow two negative real roots beside a positive root. All the allowed possibilities are distinguished in Fig. 4.
102 THE SCHWARZSCHILD SPACE-TIME 1.04 1.02 1.00 Y 0.98 0.96 0.94 0.92 -r/M Fig. 5. The effective potentials appropriate for time-like trajectories (cf. equation (113)). The minima in the potentials correspond to the stable circular orbits while the maxima correspond to unstable circular orbits. At the point of inflexion the last stable circular orbit occurs. Case (e): In this case, we have only one class of orbits: they all plunge into the singularity after starting from certain finite aphelion distances. We shall find that these orbits are most conveniently parametrized by an imaginary eccentricity; otherwise, they are similar to the radial geodesies. The different cases we have distinguished can also be inferred by interpreting r'.(x-2-^U*K\ 013, which occurs on the left-hand side of equation (90), as a 'potential energy' in the sense that, together with f2, interpreted as 'kinetic energy', it is a constant of
THE TIME-LIKE GEODESICS 103 the motions. (See Fig. 5 in which we have displayed the potential-energy curves for some typical values of the parameters E and L.) (i) Orbits of the first kind Orbits of the first kind occur in the cases (a), (/?), (y), and (5); and these cases require that/(u) = 0 allows three real roots, all of which are positive; and we shall write them as 11 12 M,=T(l-e), u2=-(l+e), and "3=777-7. (H4) where the latus rectum, I, is some positive constant and the eccentricity e is less than 1 for ux > 0, as required by the condition E2 < 1: 0<e<l for E2 < 1. (115) (We have assigned to u3 the value consistent with the requirement (112)). It is important to note that to be in conformity with the ordering, ux ^ u2 ^ u3, we must require ^7-7^7(1+8) or l>2M(3 + e). (116) 2M I I Defining li = M/l, (117) we have the important inequality ^^TTT or l-6/*-2/ie^0. (118) 2 (3 + e) The condition, that 'w-'K-^X-^X-^t) <»" agrees with its definition (110), yields the relations, Ml , 1 - E2 1 73 = ^[/-M(3 + e2)] and -^E^(|-4M)(l-e2), (120) or, in terms of fi, 7^ = 7^1-^(3 + ^2)] and ^#^ =^(1-4^)(1-^2). (121) It follows from these relations that /*<(3 + e2)-1 and n<l (122) It can be verified that these inequalities are guaranteed by (118).
104 THE SCHWARZSCHILD SPACE-TIME An alternative expression for E2 may be noted here: fJ = y^[(2^-l)2-Ve2]. (123) Returning to equation (109), we now make the substitution u = -(1+ecosx), (124) where ^ is a new variable which we may call the 'relativistic anomaly', following Darwin. According to equation (124) at aphelion, where u = (1 — e)/l, x = n ) and > (125) at perihelion, where u = (1 + e)/l, # = 0. J We readily verify that the substitution (124) reduces equation (109) to the simple form = [1-2/43 + ecosx)] d(p J = [(1 - 6 /i + 2 fie) -4 fie cos2 &], (126) or, alternatively, + -^=(1-6/^ + 2fie)i/2 (1 - /c2 cos2 ;h)1/2, (127) d<p where *2= X ■ (128) 1 —Ofi + z/xe The inequality (118) guarantees that /c2 < 1 and 1 - 6/i + 2/*e > 0. (129) It is now apparent that the solution for q> can be expressed in terms of the Jacobian elliptic integral * Ay F(ip,k) = where (130) 0 ^/(1-fc2 sin2?)' * = ±(«-X). (131) Thus, we may write 2 (l-6/* + 2/*e)' where the origin of <p has been chosen at aphelion passage where % = n. The » = /i «... 1..,.11/2 F(**-**,fc). (132)
THE TIME-LIKE GEODESICS 105 perihelion passage occurs at x = 0 where \j/ = n/2. (Orbits derived on the basis of equation (132) are illustrated in Fig. la, (a), (b), (c), pp. 116 and 117.) The solution can now be completed by direct integration of the equations (95) for t and t. Thus, and t = 1 L % E L *d<p 1 Cdcp dx u2 L dv u2 *d<p dx dXu2(l- -2Mu) (133) (134) By making use of equations (121), (123), (124), and (126), we obtain dx(1 + ecosx)~2 [1-2/i(3 +ecos*)]~1/2 ^[1-/,(3 + ^)]1^ M (135) and /3/2 ' = T7T72[(2/<-l)2-Ve2] 2^11/2 d^(l +ecos^)" x [1 -2/i(3 + ecosx)T112 [1-2/^(1 +ecos^)]_1. Expressing t and t in units of the Newtonian period, 4n2l3 Newton (l-e2)3GM 1/2 (136) (137) of a Kepler orbit with the same eccentricity and latus rectum, we find that the factors in front of the integrals on the right-hand sides of equations (135) and (136) are, respectively, 1 2% ^,0,,(1-^1:1-/43+^)]^ and 2% ^Newton (l-^2)3/2 [(2/1-l)2-4/.2 e2] (138) (With rNewton given by equations (137), the integrals on the right-hand side of equations (135) and (136), with the foregoing factors, give t and t in seconds: the units in which c = 1 and G = 1, which we have adopted hitherto, is abandoned in this instance for comparison with the results of the Newtonian theory.) We now consider two special cases: the case e = 0 when the two roots u t and u2 coincide (the case (/?) of Fig. 4) and the case 2/^(3 + e) = 1 when the two roots u2 and u3 coincide (the case (y) of Fig. 4).
106 THE SCHWARZSCHILD SPACE-TIME (a) The case e = 0: In this case the orbit is a circle with the radius rc = I and n = M/rc. (139) Equations (121) and (123), relating the angular momentum L and the energy E of the orbit to the parameters e and I of the orbit, now give 1 l-3M/r, E2 (2M/rc-l)2 -^ = — and —, = - — -. (140) L2 rcM L2 rcM V ' Rewriting the first of these equations in the form r2-^rc + 3L2=0, (141) we conclude (as is evident from Fig. 5) that an orbit of zero eccentricity is compatible with the one or the other of the two roots, rc = ^ll±(l-12M2/L2)l'2l, (142) and, further, that no circular orbit is possible for LjM < 2^3. (143) For the minimum allowed value of LjM, rc = 6M and E2 = f for LjM = 2^3. (144) It is clear that the larger of the two roots of equation (142) (for LjM > 2 ,/3) locates the minimum of the potential-energy curve y(r) defined by equation (113), while the smaller root locates the maximum of the potential-energy curve. Therefore, the circular orbit of the larger radius will be stable in contrast to the circular orbit of the smaller radius which will be unstable. The allowed ranges for the radii of these two classes of orbits are 6M < rc(stable) < oo and 3M ^ rc (unstable) ^ 6M. (144') Of the two orbits of zero eccentricity allowed for LjM > 2,/3, it is the stable o ne with the larger radius (smaller u) that is included among the orbits of the first kind we are presently considering. The periods for one complete revolution of these circular orbits, measured in proper time and in coordinate time, are (cf. equations (136)-(138)) /1-3/A1'2 _ _1/2 Tperiod — ^Newton! , _ , ) and fperiod — ^Newton(1 — fyi) (145) (Notice that r^od -> oo when \i ->^ and r,-»6M.)
THE TIME-LIKE GEODESICS 107 (/?) The case 2/^(3 + e) = 1: In this case, the perihelion, rp, and the aphelion rap, distances are given by I 3 + e 3 + e r„ = -— = 2M—- and r ap = 2M-^~. (146) 1 + e 1 +e 1 — e It follows that the perihelion distances for these orbits are restricted to the range AM «5 rp < 6M. (147) Also, for these orbits L2 =1 (3 + e)2 M2 (3-e)(l+e) 2 =4^ v wi^ and l-E2=- r (148) The modulus, /c, of the elliptic integral, in terms of which the solution (132) is expressed, becomes l in this case and it is convenient to go back to equation (126); it gives ^J =4^sin2|Z or ^ = -2(^e)l/2smh, (149) where the negative sign has been chosen so that <p may increase (from zero) when x decreases from its aphelion value n (to its perihelion value 0). The required solution of equation (149) is <P= ~ i, lg(tanix) («>=0 when x = n). (150) Equation (150) shows that <p -> oo when #->0 and the perihelion is approached. In other words, the orbit approaches the circle at rp, asymptotically, spiralling around it (in the counter-clockwise direction) an infinite number of times. In §(ii, /?) below, we shall show that this orbit 'continues' into the interior of the circle as an orbit of the second kind to plunge eventually into the singularity. (See Fig. la, (d), p. 117.) (y) The post-Newtonian approximation: The first-order correction, of relativistic origin, to the Keplerian orbits of the Newtonian theory can be readily deduced from equation (126) by noting that under conditions of normal occurrence, \i = M/l is a very small quantity, it is essentially the ratio of the Schwarzschild radius (~ 2 km) to the major axis of a planetary or a binary- star orbit ( ~ 106-108 km). Expanding, then, equation (126), to the first order in ix, we obtain -dcp = dx(l +3/^ + ^e cos x)', (151) or, in integrated form, — <p = (1 + 3/i)x + l^esinx +constant. (152)
108 THE SCHWARZSCHILD SPACE-TIME From equation (152) we infer that the change in q>, after one complete revolution during which x changes by In, is 2(1 + 3/*)tl Therefore, the advance in the perihelion, A<p, per revolution is Acp 6n— = 6n— I a{\ M (153) where a denotes the semi-major axis of the Keplerian ellipse; and this is the standard result (first derived by Einstein). (ii) Orbits of the second kind As we have already explained, orbits of the second kind have their aphelions at u Jl and eventually plunge into the singularity at r = 0. It is important to observe that, since ul + u2+u3 = 1/2M and ul + u2> 0, u3 < 1/2M; and that, therefore, all these orbits start outside the horizon. To obtain the solution for these orbits, we now make the substitution in place of (124). By this substitution, 1 2 « = «•» = 2M~ 1 WhCn £ = 0 and (155) (156) u -> oo as £ -»• n. We further find that equation (109) now reduces to d£ V -^ =(l-6/* + 2^)(l-/c2sin2!0, d<p/ with the same definition of k2 as for the orbits of the first kind. The solution for <p can, accordingly, be expressed in terms of the same elliptic integral (130). Thus, we may now write (cf. equation (132)) 2 <P (l-6n + 2ne)1'2 FQZ.k). (157) At aphelion, £ = 0 and q> = 0; and at the singularity £ -> n and q> takes the finite value ^ = (1-6^2^/^^ (158> where K(k) denotes the complete elliptic integral, K(k) = o y(l-/c2sin27)' (159)
THE TIME-LIKE GEODESICS 109 (Examples of orbits derived on the basis of equation (157) are illustrated in Fig. la, (a), (b), (c), pp. 116 and 117.) The solution for the proper time t and the coordinate time t can be obtained by integrating the equations dr _ 1 dcp dt _ E dcp di~Lu2di and d£~ Lu2(l-2Mu)df (16U) The occurrence of the factor (1 — 2Mu)~l on the right-hand side of the equation for df/d<^ shows that the integral for t diverges as«-» 1/2M. The parts of these orbits for r < 2M are, therefore, inaccessible to an observer stationed outside the horizon; and this is a manifestation of the same phenomenon which occurred in the case of the radial geodesies: it is a further example of what was deduced on general grounds in §§17(£>) and (c). It is also important to observe that all these orbits which cross the horizon necessarily end at the centre—a manifestation of the singular nature of the space-time at this point. We now consider the two special cases, e = 0 and 2/^(3 + e) = 1. (a) The case e = 0: In this case (since k2 is also zero) equation (156) integrates to give ¢ = (1-6^(^.-^0), (161) where cp0 is a constant of integration. The corresponding solution for u is « = j + (^r-|)sec2[|(l-6/i)1'2(<p-<p0)]. (162) This orbit of'zero eccentricity' is not a circle! Starting at an aphelion distance u Jl (in the range 3 M ^ u^l < 6M), when <p = <p0, it arrives at the singularity at r = 0, when <p-<Po = w/(l- 6/*)1'2, (163) after circling one or more times depending on how close \i is to 1/6. The circle at u^l is the envelope of these solutions, so that the circular orbit, predicted for this radius, is a singular solution of the equations of motion. (See Fig. la, (e),p. 118.) The case e = 0 and /x = 1/6 must be treated separately. In this case, all three roots of/(u) = 0 coincide (case (S) of Fig. 4) and ut = u2 = 1/6M. As we have shown in §(i, a), a circular orbit of radius 6M is allowed by the equations of motion as a special solution; it is, in fact, the lower bound for the radii of stable circular orbits (see equations (144') and (146)). The general solution can be obtained from the equation appropriate to this case, namely, (^-(-^
110 THE SCHWARZSCHILD SPACE-TIME and the solution of this equation is 1 2 6M M((p — (p0 U = -ZT7 + Tm——3- (165) This orbit approaches the circle at 6M, asymptotically, by spiralling around it an infinite number of times (see Fig. la, (f), p. 118). (/?) The case 2/i(5 + e)= 1: We cannot obtain the solution for this case by simply letting \i = 1/2(3 + e) in the analysis leading to equation (156): the coefficient of tan2 ^, in the initial substitution (152), vanishes. We must, therefore, consider this case ab initio. When 2/^(3 + e)= 1, the roots of/(u) = 0 are l~e a \ \-e \+e «i = — and "2 = "3 = 4M—2T=-r; (166) and the substitution that is suggested is u = -(l + e + 2etan2K). (167) By this substitution, u = u2 = u3 = (1 + e)/l when £ = 0, and u-»oo when £ = n. (168) We further find that equation (109) reduces to ^-J =4/*esin2if. (169) We observe that this is exactly the same equation we obtained when considering the orbits of the first kind in the same context, u2 = u3 (see equation (149)). And, as before, we shall write <P= T^--lg(tanK). (170) Along this orbit, <p = 0 when ^ = n and r -> 0; and <p -> oo as ^ -»• 0 and we approach the aphelion at r = //(1 + e). In other words, the orbit approaches the circle at //(1 + e), asymptotically, by spiralling around it (in the counterclockwise direction) an infinite number of times. This behaviour is the same as of the orbits of the first kind. But there is one important difference: the circle at //(1 + e) is the perihelion for the orbits of the first kind, while it is the aphelion for the orbits of the second kind. If we wish to consider the orbits of the two kinds as continuations of one another (when u2 = u3), we must suppose that the circle at //(1 + e) is approached by the orbit of the first kind by spiralling around it an infinite number of times and then spiralling inwards before its ultimate fall into the singularity. (See Fig. la, (d), p. 117.)
THE TIME-LIKE GEODESICS 111 (iii) The orbits with imaginary eccentricities Finally, we have to consider the case when/(u) = 0 allows only one real root (which is necessarily positive for the bound orbits we are presently considering) and a pair of complex-conjugate roots (the case (e) of Fig. 4). It is evident that the corresponding orbits, while starting at some finite aphelion distances, will fall into the singularity though they may circle the origin one or more times before doing so. We shall characterize these orbits by an imaginary eccentricity ie (e > 0) and write the roots of/(u) = 0 as "1 = 2^-? "2=7(1 + /e)' and "3 =y(l-ie). (171) It is manifest that with the present definitions, we can obtain the equations which replace equations (121) and (123) by simply writing — e2 wherever + e2 occurs and conversely. We thus obtain and 1 1 Tm [1-/.(3-e2)], \-E 1 L2 =^(1-4/^)(1+e2), — = J-l(2n-l)2+4n2e2l (172) The last of these equations shows, incidentally, that I must be taken to be positive. Moreover, the fact that we are presently considering bound orbits with (1 — E2) > 0, requires that \i < 5; and this inequality guarantees l-3fi + /ie2 > 0. (173) Also, we observe that we can set no upper limit to e2. With the assumption (171) regarding the roots of/(u) = 0, the equation we have to consider is d"V „ ( 1 2 — =2M u-:— + - . d<p) \ 2M I And the substitution we now make is 1\2 e2' u =-(l+etan^). Since the range of u is 1 < u < 00, 2M I the corresponding range of £ is (174) (175) (176) (177)
112 THE SCHWARZSCHILD SPACE-TIME where or, equivalently, ,1^=-.^-1 tan|£0 = ^ , (178) 2\xe . , 6u — 1 . 2ue smHo=—^ and cos|^0=^, (179) where A = \_(6fi-l)2+4fi2e2y<2. (180) We find that with the substitution (175), equation (174) reduces to -~= + Jl[ (6^ - 1) + 2/^sin £ + (6/^-1)cos£]1/2- (181) By a standard formula in the theory of elliptic integrals, the solution for <p can be expressed in terms of the Jacobian elliptic integral. Thus, where > dy (182) and ,/(1 -k2 sin2 y)' ^2=^(A + 6/i-l), (183) sin2 \p = —— { A —2/iesin^ -(6/^- l)cos<^)} = A + 6 _1 { A + 6p - 1 - 2[2/*e sinK + (6/* - l)cos±{]cos^}. (184) From equations (179) and (184), it follows that sin2 ij/ = 1 both when £ = £o (at aphelion) and £ = n (at the singularity). (185) Moreover, sin2i^=0 when £ = tan_1——. (186) Therefore, ip assumes the value zero within the range (177) of £. We conclude that the range of \p associated with the range of £ is - w/2 < \j/ < + w/2 (io^i^ w). (187) Accordingly, we may write the solution for <p as <p=-lK{K(k)-FW,k)}, (188)
THE TIME-LIKE GEODESICS 113 where K(k) denotes the complete elliptic integral and F(^i, k) (as usual) the incomplete Jacobian integral. In writing the solution for q> in the form (188), we have assumed that q> = 0 at the singularity where £ = n and \j/ = n/2. The value of <p at aphelion, where £ = £0 and \p = — n/2, is <pap = 2K(k)/s/A. (189) (See Fig. la, (g), (h), p. 119, where examples of orbits derived on the basis of equation (188) are illustrated.) This completes our discussion of the bound orbits. We now turn to the unbound orbits with E2 ^ 1. (c) The unbound orbits (E2 > 1) When E2 > 1, the constant term in/(u) is positive. The equation/(u) = 0 must, therefore, allow a negative root; and the cases we must distinguish (fewer than for bound orbits) are those shown in Fig. 6. [Again, it can be shown that the case of all three real roots being negative cannot arise (see footnote on p. 101).] So long as there are three real roots and two of them are positive (distinct or coincident), we must continue to distinguish between orbits of two kinds: orbits of the first kind restricted to the interval, 0 < " < u2 (which are the analogues of the hyperbolic orbits of the Newtonian theory) and the orbits of the second kind with u ^u3 (which are, in essence, no different from the bound orbits of the second kind). When u2 = u3, the two kinds of orbits coalesce as they approach, asymptotically, a common circle from opposite sides by spiralling round it an infinite number of times. And finally, when the equation f(u) = 0 allows a pair of complex-conjugate roots (besides a negative real root), the resulting orbits can be considered as belonging to imaginary eccentricities; and they differ from the bound orbits considered in §(b, iii) only in that they are not bound! (i) Orbits of the first and second kinds When all three roots are real, we shall continue to express them in terms of an eccentricity e as in the case of the bound orbits, with the only difference that now e#:l. Thus, we shall write (cf. equation (114)) u,= -l-{e-\), u2 = l-{e+\), and "3=^7-7 (*>!)• (190) The inequality (118), namely, 1-6/^-2/w >0, (191) continues to hold, since it is a consequence only of the assumed ordering of the roots: ut < u2 ^ u3. The relations (121) also continue to hold, again with the
114 THE SCHWARZSCHILD SPACE-TIME /(") Fig. 6. The disposition of the roots of the cubic equation/(u) = 0 for E1 > 1. difference that now e^l. We, therefore, write and ^ = ^D-M3 + ,2)] ^1 = 1(1-4/,)(^-1). (192) Since L2 > 0 and E2 — 1 > 0 (by assumption), l-H(3 + e2)> 0 and ^ (193) (194) The requirement, n ^ £, is guaranteed by both the inequalities (191) and (193). With regard to these latter two inequalities, one readily verifies that (193) ensures (191) for e < 3 while (191) ensures (193) for e > 3. In this connection, we may note that when 2/^(3 + e) = 1 (i.e., when the two roots u2 and u3 coincide) the relations (192) become (cf. equation (148)) L2 A (3 + e)2 e2-l —7 = 4—^ —— and E2 - 1 = j. M2 (3-e)(e+l) 9-e2 (195) Accordingly, for these special orbits, the allowed range of e is 1 ^ e < 3: (196) and the corresponding perihelion-distances must lie in the range (cf. the range (147) for the bound orbits) 3M < rp ^ AM. (197) Quite generally, we may also note that since the orbits of the first kind (and the orbits belonging to imaginary eccentricities, as well) 'arrive from infinity', we can associate with them an impact parameter, D, and a velocity at infinity, V; these are related to L and E by D2 = L2£2 E2 = 1 (198) V2 E2-\ V" 1-V2 Turning to the analytical representation of these orbits, we shall make the
THE TIME-LIKE GEODESICS 115 same substitution as before, namely, u=-(l+ecosx)- (199) However, since e > 1, u = 0 when x = cos"1 (-e"1) = xx (say)- (200) But the perihelion passage still occurs when x = 0. The allowed range of x is, therefore, 0<X<X„ =cos-1(-e"1). (201) Apart from this restriction on the range of x, the analysis in § (b, i) is applicable as it stands; and the solution for x can be expressed in terms of the elliptic integral (130) with the same modulus k. Keeping in mind the restriction in the range of x, we may now write the solution for <p in the form (cf. equation (132)) ^ = (1-6, + 2^^^^-^^-^^^ (2°2) In writing the solution for q> in the form (202), we have taken the origin of q> at perihelion passage where x = 0. The trajectory goes off to infinity, asymptotically, along the direction V = ^°° = (i -6, + 2/^2 [K(fe)~f (l^' fe)] («Aoo=icos-1e-1). (203) The limiting case of these orbits, when the two roots u2 and u3 coincide, is described by the same equation (150) (except, again, for the restriction in the range of x )• And as was stated in the context of the bound orbits, the unbound orbits approach the circle at rp, asymptotically, by spiralling around it an infinite number of times. The discussion of the bound orbits of the second kind in §(b, ii) applies to the unbound orbits without any change; only we are now concerned with values of e>l. (ii) The orbits with imaginary eccentricities As we have shown in §(b, iii), the bound orbits which fall into the singularity are best characterized by imaginary eccentricities. The unbound orbits, which similarly fall into the singularity, can likewise be characterized by imaginary eccentricities; and the analysis of §(b, iii) can be adapted to the present context with only minor changes to allow for the orbits arriving from infinity rather than from finite aphelion-distances. The relations (172) for unbound orbits are J__ ! -- ~ -- . £ -i 1 1?~Tm 2- [1-,(3-^)] and -3- = -(4^-1)(1+^). (204)
-24 -18 -12 -6 0 (a) e=Vi,(=\\,M= 6 12 18 24 3_ 14 (b) e='/2, (=7.5, M
6 4 -2 -4 -6 -8 -8 -6 -4 (c) e=Vi,t=3,M=-^ 3 2 1 0 1 2 3 1 - - - - I 1 1 1 /^ "" fs~\ vC J V^/ 1 1 1 1 ■\ - - - 1 -3-2-1 0 l : (d) e=Vi, 1=-^, M=^ 117
2.0 1.5 1.0 0.5 0 -0.5 -1.0 -1.5 -2.0 i i i i i i i _L _L -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 (e) e = 0, (=j-,M=^ (f) e=0)M=— M=^
(g) e = 0.01/,f=l,M=-^ (h) e = OM, t=l,M= 14 Fig. la. Various classes oftime-like geodesies described by a test particle with E2 < 1: (a), (b), (c): geodesies of the first and the second kind with eccentricity e = 1/2 and latera recta, I = 11,7.5 and 3 respectively; (d): an example of a trajectory in which the orbits of the first and the second kind coalesce (e — 1/2, I — 3/2) for which 2fi (3 + e) = 1); (e): an example of a circular orbit (e = 0, I = 3/2) and the associated orbit of the second kind; (f): the last unstable circular orbit when the orbit of the second kind spirals out of the orbit of the first kind (e = 0, ft — 1/6); (g), (h): bound orbits with I = 1 and imaginary eccentricities e = 0.01 i and 0.1 i. (In all these figures M = 3/14 in the scale along the coordinate axes.) 119
3 2 1 0 -1 -2 -3 -4. -4 -3 -2 -1 (a) e = Y> (=4.5, M =^ (b), = f^=f,M=-i 120
3 2 1 0 -1 -2 -3 -4 (c) e = Y> l= !-94> M=u (d) e = 0.0011, i'=l,M=0.3 (e) e = 0.1i, i'=l,M=0.3 Fig. lb. Various classes of time-like geodesies described by a test particle with E2 > 1: (a), (b), (c): orbits of the first and the second kind with eccentricity e = 3/2 and laterarecta, 4.5,2.5, and 1.94 respectively (Af = 3/14 in the scale along the coordinate axes); (d), (e): unbound orbits with I = 1 and with imaginary eccentricities e = 0.001 i and 0.1 i (Af = 0.3 in the scale along the coordinate axes). 121
122 THE SCHWARZSCHILD SPACE-TIME Accordingly, we must now require \i^\ and 1 - 3/i + y.e2 > 0. (205) For 3 > n > i, the second inequality imposes no restrictions on e2; but for /x > 3, it is necessary to impose the restriction e2> 3-1//1 (,*>$). (206) For obtaining the solution for these orbits, we make the same substitution (175) as before. However, in view of present requirement, \x ^ \, the range of t, must be terminated at £x (before £<, as denned in equation (179)) where tan i ^ = -e~\ or, (207) sin" ^ = -7(17^) and cosH-=7(iT?)- For, when £ = £^, u becomes zero and when £ < £x, u becomes negative. Therefore, as ¢-+^+ 0, u -> 0 and r -> oo. But the upper limit on £, namely n (when u -* oo and r -> 0), is unaffected. The allowed range of £ is, therefore, £00 < ^ «. (208) Apart from these changes, the solution for <p can again be written as (cf. equations (183), (184), and (188)) q>=-j-K{K(k)-FW,k)}. (209) The origin of q> (as in the earlier solution) is at the singularity (where £ = n and \j/ = + n/2). But the lower limit on ip, say ^w, as the orbit goes off to infinity, can be obtained from equation (184) by inserting for sin \^ and cos \£ the values appropriate for ^ given in equations (207). We find sin2^ = ^h^{A+6"-l-2e24^r) (210) (Note that for fi > i, the right-hand side of equation (210) is less than 1; and that it is equal to 1 for \i = \, and ^/^=- n/2 as in the case for the bound orbits.) In Fig. lb, a number of examples of the different classes of unbound orbits are illustrated. This completes our discussion of the time-like geodesies in the Schwarzschild space-time.
THE NULL GEODESICS 123 20. The geodesies in the Schwarzschild space-time: the null geodesies As we have already explained in §19, the Lagrangian must be equated to zero for the null geodesies. Therefore, equation (89) must now be replaced by E2 f2 L2 = 0, (211) + -r 1 ) = £■ (212) (213) l-2M/r l-2M/r r or dr\2 L2 ( 2AA *) +A ~) This equation must be considered together with ' 2M\dt dw L l--j-=E and -1 = -,. Again, by considering r as a function of <p and replacing r by u = 1/r as the independent variable, we obtain the equation (cf. equation (94)) -J = 2Mu>-u2+-2 where D = L/E (215) denotes the impact parameter. We observe that in contrast to the earlier definition, f(u), as now defined, has no term linear in u; and, in addition, the constant term is always positive. (a) The radial geodesies We begin our consideration of the null geodesies with the radial geodesies. The relevant equations are (cf. equations (212) and (213)) dr _ - (\ 2M\dt_ :) =2Af«3-«2+-j=/(«) (say), (214) , =±E and 1 — = E. (216) dt \ r Jdz Accordingly, d dt 7-±('-™} or, in the integrated form, t = +r+ + constant ±, (218) where r+ = r + 2Mlg(^-l). (219) The variable r+ to which we have now been led plays an extremely important
124 THE SCHWARZSCHILD SPACE-TIME role in subsequent developments. It is customarily defined by the equation (cf. equation (74)) d Ad h- = -*;P (220) dr+ r ar where A = r2-2Mr = rVv (221) is the horizon function. With r+ as defined in equation (219) /■„ -> - oo as r -» 2M + 0 and /■„ -* r as r -> + oo. (222) dt r2 — = —E, dt A ' Ar dr~ ±E, de — = o, dt The importance of the variable r+ arises from this fact: its range from — oo to + oo exhausts the entire part of the space-time that is accessible to observers outside the horizon. Equation (218) must be contrasted with the equation r= +£t +constant ± (223) which relates r to the proper time t. Equations (218) and (223) show that while the radial geodesic crosses the horizon in its own proper time without ever noticing it, it takes an infinite coordinate-time even to arrive at the horizon: its 'crossing the bar' is not within the realm of experience of a mere observer outside the horizon. All of this is made manifest in Fig. 8 where the null cones included between the geodesies (217) are illustrated. The tangent vectors associated with the radial geodesies (217) are and -r- = 0. (224) dt We shall find in §21, that these null vectors provide the basis for constructing a null tetrad-frame towards a description of the Schwarzschild space-time in a Newman-Penrose formalism. (b) The critical orbits Returning to the general equation (214), we first consider the different cases that must be distinguished. They clearly relate to the disposition of the rootsof the cubic equation f(u) = 2Mu3-u2+^ = 0. (225) The sum and the product of the roots u1, u2, and u3 of this equation are given by U! + u2 + u3=— and Ul"2"3= "^MD1' ^226'
THE NULL GEODESICS 125 /■ = 0 r=2M r Fig. 8. Illustrating the ingoing and outgoing radial null-geodesies in the Schwarzschild coordinates. Clearly the equation/(u) = 0 must allow a negative real root; and the two remaining roots may either be real (distinct or coincident) or be a complex- conjugate pair. The different cases that must be distinguished are, therefore, essentially the same as those for the unbound time-like geodesies. In the present instance, the case when two of the positive real roots* coincide plays a specially decisive role in the discrimination of the null geodesies. On this account, we shall consider this critical case first. The conditions for the occurrence of two coincident roots can be obtained as follows. The derivative of equation (225), namely, /'(u) = 6Mu2-2u = 0, (227) allows u = (3M)~' as a root; and u = (3M)"' will be a root of equation (225) (indeed, a double root) if D2 = 27M2 or D = (3 J3)M. (228) * The impossibility of three real negative roots follows from the equation ui+u2 + ui = 0.
126 THE SCHWARZSCHILD SPACE-TIME From the condition on the product of the roots, we infer that the roots of /(h) = 0 are u, = - 1/6M and u2 = u3 = 1/3 M and D = (3y/3)M. (229) Moreover when D has the value (288), du/d<p vanishes when u = (3M)_1. Therefore, a circular orbit of radius 3M is an allowed null-geodesic. This circular orbit cannot, however, be a stable one. Its place in the family of the null geodesies can be understood by considering the full equation, appropriate to this case, namely, We verify that this equation is satisfied by the substitution U= -^ + ^4^^-^ (231) where <p0 is a constant of integration. If <p0 is so chosen that tanh2i<p0=i, (232) then -> oo when <p = 0. (233) But we u=0 also notice that u and 1 ~3M when q> -* oo. (234) Therefore, a null geodesic arriving from infinity with an impact parameter D = (3 y/3)M approaches the circle of radius 3M, asymptotically, by spiralling around it. Associated with the orbit (231), we must have an 'orbit of the second kind' which, originating at the singularity, approaches, from the opposite side, the same circle at r = 3M, asymptotically, by spiralling around it. Such an orbit can be obtained by the substitution in the same equation (230). We find that the equation then reduces to j|)2=sin^; (236) and we may take p = 21g(tanif) or tani{ = e*/2 (237)
THE NULL GEODESICS 127 as the appropriate solution. Inserting this solution in equation (235), we obtain 1 lef 3~M + M(ef-l)2 Along this orbit U = WT~. + .,,., M2- (238> u -* oo and r ->0 when <p -* 0 "] and > (239) u-»(3M)_1 as (p -* oo; J it has accordingly the attributes that were stated. As we have explained in a similar context in §19(4», ii), the solution (239) with the sign of q> reversed may be considered as a 'continuation' of the solution (231). In Fig. 9 the various classes of null geodesies that can occur are illustrated. (i) The cone of avoidance At any point we can define a 'cone of avoidance' whose generators are the null rays, described by the solution (231) of equation (230), passing through that point, since, as is clear on general grounds and as we shall establish analytically in §(e) below, light rays, included in the cone, must necessarily cross the horizon and get trapped. If \p denotes the half-angle of the cone (directed inward at large distances), then ldr cot ^= +-3-, (240) ra<p where dr = (l-2M/r)"1/2dr (241) is an element of proper length along the generators of the cone. Therefore, 1 dr 1 du C°t{l/~ +r(l-2M/r)ll2d(p~ «(1 -2Mu)1/2 d<p' ( ' where u = l/r. In equation (242) we may substitute for du/d<p from equation (230). In this manner we obtain (r/2M-l)ll2\l 3m)\1+6m) co^ = -,■„., ,,./2( 1-TT7 )( ! +ZT7 J (243> or tan^ MM-I)1'2 (244) tan[p-(r/3M-l)(r/6M+1)1^ (244) From this last equation it follows that 3x/3 d/ ' ^-M as r -* oo; \l/ = \% for r = 3M, r and \p = 0 for r = 2M. (245)
3 2 1 0 1 2 3 4 U \ \ \ r>) J / / i / -4 -3 -2 -1 (a) P=l,M=y^ (c) e = 0.001 /, f=l,M=y Fig. 9. Various classes of null geodesies in the Schwarzschild metric (a): a null geodesic with P — 1 (cf. equation (251)) illustrating orbits of the first and the second kind; (ft): the critical null- geodesic, with D = Dc — 3 ,/3. M, for which the orbits of the two kinds spiral towards the unstable 128
2.0 1.5 1.0 0.5 0 -0.5 -1.0 -1.5 -2.0 i r j_ j_ _L (b) D = Dc = 3y/3.M, M= J_ -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 3 14 (d) e = 0.01 /, £=1, M=y circular orbit at 3M (M = 3/14 in the scale along the coordinate axes for (a)and (ft)); (c), (d): orbits with imaginary eccentricities (e — 0.001 i and 0.01 i with I = 1) (M = 1/3 in the scale along the coordinate axes).
130 THE SCHWARZSCHILD SPACE-TIME 2.25M 3M 5M 6M Fig. 10. The cone of avoidance at different distances from the Centre. Therefore, the cone of avoidance is narrow and, at large distances, its angle is that subtended by a disc of radius (3 ^/3) M. The cone opens out fully at r = 3M; and for r < 3M, it is directed outward and becomes narrower asr-> 2M; and at r = 2M, everything is blotted out (see Fig. 10). (c) The geodesies of the first kind We now consider the case when all the roots of the cubic equation/(u) = 0 are real and the two positive roots are distinct. Let the roots be P-2M-Q, ^ 1 J P-2M + Q , „ where P denotes the perihelion distance and Q is a constant to be specified presently. The sum of the roots has been arranged to be equal to 1/2 M as required (cf. equation (226)). Also, it should be noted that the ordering of the roots, u, < u2 < u3, requires that Next, evaluating Q + P - 6M > 0. f(u) =2M(u-u1)(u- u2)(u - u3) (247) (248) with «i, u2, and u3 as specified in equations (246), and comparing the result with the expression (225), we obtain the relations Q2 = (P-2M)(P + 6M) and 1 D1 8MP: [g2-(P-2M)2]. (249) (250)
THE NULL GEODESICS 131 With the aid of the first of these relations, the second simplifies to give P3 D' = (251) (252) (253) P-2M Combined with the relation (249), the inequality (247) gives (P-2M)(P + 6M) > (P-6M)2, or, after simplification, P>3M. From equation (251), we now obtain the further inequality £>>(3v"3)M = £>c (say). (254) Therefore, the orbits we are now considering have impact parameters in excess of the critical value leading to the special solutions considered in §(b) above. Moreover, they are entirely outside the circle r = 3M. We now make the substitution 1 Q-P+6Mn , U~P= SMP—{1+C°SX)' or, equivalently, 0-P+2M Q-P+6M u + * ,,.„ = * „,jr> (1-cosx). AMP 8MP By this substitution, u = — when x = ^. and u = 0 and r -» oo when sin2 \x = Q-P+2M . = sm 1X«> (say). (255) (256) (257) 2A Q-P + 6M We further find that the substitutions (255) and (256) reduce equation (214) to (258) where (¾ -?a-*'-'h>. k2=— (Q-P + 6M). 2Q (259) The solution for q> can, therefore, be expressed in terms of the Jacobian elliptic integral in the form /p\l/2 ¢ = 2(-) LK(k)-F(h,k)l (260)
132 THE SCHWARZSCHILD SPACE-TIME where the origin of q> has been chosen at perihelion passage when i = n (see equations (257)). The asymptotic value of <p, as r -> oo, is given by <Poo=2[^\ LK(k)-F(±Xx,k)-\, (261) where x<x> is specified in (257). (See Fig. 9(a), p. 128.) (i) The asymptotic behaviours of (pwfor P -> 3M and for P/3M > 1 It is of interest to obtain the asymptotic behaviours of <px for P -* 3M and for P/3M -> oo. First, we observe that when P = 3M, Q = 3M, D = (3y/3)M, k2 = 1, and F(}Zk,1) sin2koo = i, (262) Next, we find from equations (249), (251), and (259) that if P = M(3 + S), Q = M(3 + |c5), k'2 = 1 -k2 = f<5, and D=DC + SD where 3D = {\J3)M52. Also, we have the known asymptotic relation, K(fc)-»lg(4/fc') = lg6-±lg5 (/c'^0). Inserting these various relations in equation (261), we find that or c5£> 6^3(^3-1) 2 (73 + 1} llg Letting we obtain 6*j3{j3-\)2 (73TTF M 2 ,(73-1) 7J-^^^---3^823^ (263) (264) (265) (266) (267) (268) and this is the required relation. It should be noted that geodesies, that are deflected by an angle © given by equation (268), include not only the one that has suffered this deflection, but also the ones which have been deflected by © + Inn (n = 1,2, . . .): they derive from geodesies with the impact parameters Dn = Dc + 3.4823 Afr(0 + 2""). (269) By similar (and simpler) arguments, it can be shown that for P |> 3M, the
THE NULL GEODESICS 133 deflection © (as defined in equation (267)) is given by 0^_ and £,^P(i+__j. (270) The first of these relations is the celebrated one which provided the basis for one of the early (and spectacular) confirmations of the general theory of relativity. (d) The geodesies of the second kind To obtain the null geodesies, whose ranges are u3 ^ u < oo, we make the substitution 1 Q + P-6M ,, U = P+ 4MP Sech> (271) in place of (255). By this substitution, u is at aphelion when u = u3 = (Q + P — 2M) and y = 0 4MP \ (272) and u -* oo and r ->0 when x = n- The substitution (271) reduces equation (214) to the same form (258) and with the same value of k2; and we may now write (cf. equation (260)) <P = 2[q) Fi-2X,k), (273) where the origin of <p is now at aphelion passage. (e) The orbits with imaginary eccentricities and impact parameters less than (3,/3) M Finally, we consider the orbits when the equation/(u) = 0 has a pair of complex-conjugate roots, besides a negative real root. As in §19(4», iii), we shall now write the roots in terms of an imaginary eccentricity ie as in equations (171). On evaluating f(u) = 2M[ u —+- Jy ' l 2M I IV e2 (274) and comparing it with/(u) given in equation (214), we obtain the relations /-M(3-e2) = 0 and —L— = (- -^^^- (275) v ' 2MB2 \\ 2M) I2 v ' or, alternatively, in terms of n = M/l, we have 7 1 D2 1 e2=-(3u-l) and —, = ^. (276)
134 THE SCHWARZSCHILD SPACE-TIME These relations require fi>^ and £»<(3v/3)M. (277) Therefore, these orbits have impact parameters less than the value which distinguishes the orbits considered in §(c) above. The analysis of §19(4», iii) applies equally to the present with the only proviso that e2 and \i are no longer to be considered as independent parameters but related: e2 = 3 — \x~'. Since these orbits are unbounded, the solution for <p is given by the same equation (209) as in §19(c, ii). In particular, <p„=-j-K{K(k)-FW„,k)}, (278) where \px (on substituting for e2 its present value, 3 —/i-1) is now given by A + l sln^=AT6^T' (279) where it should be remembered that fi> h and further that A = (48/i2-16/i + l)1/2. (280) Examples of orbits derived on the basis of equations (230), (238), (260), (273), and (278) are illustrated in Fig. 9, (c) and (d) p. 129. This completes our considerations relating to the geodesies in the Schwarzschild space-time. 21. The description of the Schwarzschild space-time in a Newman-Penrose formalism We shall conclude our consideration of the Schwarzschild space-time by a description of it in a Newman-Penrose formalism. For the construction of a null-tetrad frame needed for such a description, we start with the null vectors (224) representing radial null-geodesies. Precisely, for the real null-vectors / and n of the Newman-Penrose formalism we shall take and /' = (/', /',/fl,/*) = ^(r2, +A, 0, 0) n' = («', nr, ne, n<0) = —^ (r2, - A, 0, 0), (281) where the vector, associated with the negative sign for dr/dt in (224), has been multiplied by A/2r2 to satisfy the required orthogonality condition I n = 1. (282) On this account, the vector n, unlike the vector /, is not affinely parametrized.
THE NULL GEODESICS 135 We now complete the tetrad basis by adjoining to / and n the complex null- vector, rri1 = (rri, rrf, me, m*) = —7-(0, 0, 1, i cosec 0), (283) which is orthogonal to /and n and satisfies the further normalization condition m m= -1. (284) The contravariant vectors /', n', m', and m' provide the required basis. The corresponding covariant vectors are 7 "N /,. = (1,--,0,0), 1 ni = ^(\ + r2, 0,0), 1 (285) m, = —^(0, 0, -r2, - jr2sin 6). The spin coefficients (as defined in Chapter 1, equations (286)) with respect to the chosen basis are most conveniently evaluated via the A-symbols (Chapter 1, equation (266)) even as the rotation coefficients (in a different basis) were evaluated in Chapter 2, §14. We now find that the non-vanishing X symbols are M r-2M 1 _cot0 ^122--^2- ^243- 2P~' 341__r' and 334_7V2' (286) and the spin coefficients, derived with their aid, are K = a = X = v = e = n = r=0, (287) and 1 „ 1 cot0 r-2M M - 2,/2 r 2r2 2r2 (288) r The fact that the spin coefficients k, a, X, and v vanish shows that the congruences of the null geodesies, / and n, are shear-free as is, indeed, obvious from the purely radial character of these null geodesies in a spherically symmetric background. From the shear-free character of these congruences, we can conclude on the basis of the Goldberg-Sachs theorem that the Schwarzschild space-time is of Petrov type-D. The Goldberg-Sachs theorem also allows us to conclude that in the chosen basis, the Weyl scalars, *P0, x¥1, *P3, and *P4 vanish; and that the only non- vanishing scalar is *P2. These conclusions can be directly verified by
136 THE SCHWARZSCHILD SPACE-TIME contracting the non-vanishing components of the Riemann tensor listed in equations (76) with the vectors I, n, m, and m in accordance with the definitions of the Weyl scalars in Chapter 1, equations (294) (remembering that the tensor indices 0, 1, 2, and 3 correspond to the coordinate indices t, <p, r, and 9, respectively). The vanishing of the scalars *P0, 4*,, *P3, and 4*4 is readily verified. Considering the scalar *P2, we have xV2 = Rij\illiminkrhl r2 = ^(Rojo,-e2vRo}2i-e2vR2jo,-e*vR2j2l)mJm' e~2v = ^r(RoioiCosec2e + R0303-e4vR2323-e*vR1212cosec26), (289) or, inserting for the components of the Riemann tensor their values given in equations (76), we find 4>2 = -Mr'3. (290) With the specification of the null basis, the spin coefficients, and the Weyl scalars, we have completed the description of the Schwarzschild space-time in a Newman-Penrose formalism which exemplifies its special algebraic character. BIBLIOGRAPHICAL NOTES K. Schwarzschild's (1873-1916) original derivation of the solution, known by his name, was published in: 1. K. Schwarzschild, Berliner Sitzungsbesichte (Phys. Math. Klasse), 189-96, 3 Feb. 1916 (Mitt. Jan. 13). See also: 2. K. Schwarzschild, Berliner Sitzungsbesichte (Phys. Math. Klasse), 424-34, 23 Mar. 1916 (Mitt. Feb. 24). There can hardly be a book on general relativity which does not include some discussion of Schwarzschild's solution. Nevertheless, the heroic circumstances under which the solution was discovered seem hardly to be known. The first published account (so far as the present author is aware) is in: 3. S. Chandrasekhar, Notes and Records of the Royal Society of London, 30,249-60 (1976). And the story is retold in: 4. W. Sullivan, Black Holes, 61-2, Anchor Press, Doubleday, Garden City, New York, 1979. Since these accounts may easily have been overlooked by most students of relativity, the account given in reference 3 is briefly abstracted below: Schwarzschild's paper, in which he derived his solution, was communicated to the Berlin Academy by Einstein on 13 January 1916just about two months after Einstein himself had published the basic equations of his theory in a short communication. In acknowledging the manuscript of Schwarzschild's paper on 9 January 1916, Einstein wrote: 'Ihre Arbeit habe ich mit grossten Interesse durchgesehen. Ich hatte nicht
BIBLIGRAPHICAL NOTES 137 erwartet, dass man so einfach die strenge Lossung der Aufgabe formulieren konnte. Die rechenrische Behandlung des Gegenstandes gefiilt mir ausgezeichnet' [I have read your paper with the greatest interest. I had not expected that one could formulate the exact solution of the problem so simply. The analytical treatment of the problem appears to me splendid.] The circumstances under which Schwarzschild derived his now famous solution are the following: During the spring and summer of 1915, Schwarzschild was serving in the German army at the eastern front. While at the eastern front with a small technical staff, Schwarzschild contracted pemphigus—a fatal disease; and he died on 11 May 1916. It was during this period of illness that Schwarzschild wrote his two papers on relativity [besides a fundamental one on the Bohr-Sommerfeld theory]. Among the many books which discuss Schwarzschild's solutions, the reader can find excellent accounts in: 5. L. D. Landau and E. M. Lifschitz, Classical Fields, §§100-1,299-321, Pergamon Press, Oxford, 1975. 6. C. W. Misner, K. S. Thorne, and J. W. Wheeler, Gravitation, chapter 25,655-78, and chapters 31-32, 819-41, W. H. Freeman and Co., San Francisco, 1970. Figs. 3 and 5 in the text are taken from reference 6. §§17-18. It is customary (as, for example, in references 5 and 6) to derive Schwarzschild's solution in the coordinate system (t, r,d,<p) (of §18) and then transform it to the 'Eddington-Finklestein' or the 'Kruskal' frame to show that there is no real singularity at the Schwarzschild-radius, Rs = 2GM/c2, and to clarify the nature of the surface at R s as an event horizon. We have inverted this common procedure and obtained the solution ab initio in the Kruskal frame and then transformed it to the Schwarzschild frame as a matter of convenience. The manner of presentation is due to Synge: 7 J. L. Synge, Annali di Matematica pura ed Applicata, 98, 239-55 (1974). As Synge remarks: "If the problem of spherical symmetry had been attacked originally in this way, it would never have occurred to anyone to think of a singularity here [i.e., at r = Rs]-" However, the basic idea that the nature of the Schwarzschild space-time is best clarified in a system of null coordinates is due to: 8. A. S. Eddington, Nature, 113, 192 (1924). 9. D. Finklestein, Phys. Rev., 110, 965-7 (1958). 10. M. D. Kruskal, ibid., 110, 1743-5 (1960). For fuller references to the literature see reference 6. §§19-20. The geodesies in the Schwarzschild space-time have been discussed widely in the literature ever since Einstein's first evaluation, in a post-Newtonian approximation, of the deflection of light and the precession of the Kepler orbit in the field of a central spherically symmetric source of gravitation. The interest in the geodesies as a means of understanding the nature of the space-time itself is, however, relatively recent; and the accounts in references 5 and 6 are adequate. On the purely analytical side, an exceptionally complete treatment of the geodesies is contained in an early investigation by: 11. Y. Hagihara, Jap. J. Astron. Geophys, 8, 67-175 (1931). But Hagihara's treatment makes the subject much more complicated than is necessary. The account in the text is essentially a completion of the program set out by: 12. C. G. Darwin, Proc. Roy. Soc. (London) A, 249, 180-94 (1958). 13. , ibid., 263, 39-50 (1961).
138 THE SCHWARZSCHILD SPACE-TIME In particular, by classifying the orbits into those of 'the first kind' and those of 'the second kind,' and allowing the eccentricity to be imaginary, we are able to treat all cases in a coherent and a unified manner. The geodesies illustrated in Figs. 7, 8, and 9 have all been evaluated and drawn by Mr. Garret Toomey to whom the author is most grateful. For a complete listing of references bearing on geodesies in the Schwarzschild space- time see: 14. N. A. Sharp, General Relativity and Gravitation, 10, 659-70 (1979). For a beautiful pictorial visualization of the Schwarzschild black-hole with a luminous accretion disc see: 15. J. P. Luminet, Astron. Astrophys., 75, 228-35 (1979). §21. The first applications of the Newman-Penrose formalism for a study of the Schwarzschild space-time are due to: 16. R. H. Price, Phys. Rev. D, 5, 2419-38 and 2439-54 (1972). 17. J. M. BARDEENand W. H. Press, J. Math. Phys., 14, 7-19 (1972). The actual applications considered in these papers, however, are in the context of the next chapter.
4 THE PERTURBATIONS OF THE SCHWARZSCHILD BLACK-HOLE 22. Introduction This chapter is devoted to the study of the perturbations of the Schwarzschild black-hole. On the physical side, the principal question to which the study is addressed is the manner in which gravitational waves, incident on the black- hole, are scattered and absorbed. The answer to this question has clearly some astrophysical interest. On the theoretical side, the answer has a more transcendent interest: it provides insight, in its simplest and purest context, into the deeper aspects of space and time as conceived in general relativity; and it reveals the analytical richness of the theory. There are, at present, two avenues of approach to the study of the perturbations of space-times. One can either study, directly, the perturbations in the metric coefficients via the Einstein or the Einstein-Maxwell equations linearized about the unperturbed space-times; or, one can study the perturbations in the Weyl and in the Maxwell scalars via the equations of the Newman-Penrose formalism. While the latter avenue appears specially suitable to the study of the perturbations of the space-times around black holes (on account of their special algebraic character), it will emerge that the theories developed along both avenues complement each other very effectively in disclosing inner relationships which will have remained shrouded otherwise. 23. The Ricci and the Einstein tensors for non-stationary axisymmetric space-times In studying the perturbations of any spherically symmetric system, one can, without any loss of generality, restrict oneself to axisymmetric modes of perturbations. For, non-axisymmetric modes of perturbations with an elmv- dependence on the azimuthal angle <p (where m is an integer positive or negative) can be deduced from modes of axisymmetric perturbations with m = 0 by suitable rotations since there are no preferred axes in a spherically symmetric background. Thus, an axisymmetric mode, evaluated at a point (6, <p) on the sphere with respect to a chosen polar axis, when expressed in another frame with its polar axis pointing in a direction (6', <p') will be assigned a polar angle © given by cos© = cos 6 cos 6' + sin OsinO' cos (</>' — <p). (1)
140 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE When the chosen axisymmetric mode is subjected to this transformation, it will be decomposed into non-axisymmetric components, each of which may be considered as representing different non-axisymmetric modes; and we may do this by virtue of the linearity of the underlying perturbation theory. In practice, the above mentioned decomposition will occur in the following way. The equations governing the perturbations of a spherically symmetric system will be separable in all four of the variables t, r, 0, and <p. Ignoring the dependence on t and r, we may expect the dependence of an axisymmetric mode on the angle 0 to be expressible in terms of the Legendre functions P, (cos0). When a mode belonging to a particular P, (cos0) is subjected to the transformation (1), it will become m= +1 P,(cos0) = ^ ^(0080)^^^080^-^, (2) m= -l where P™ (cos0) are the associated Legendre functions. Thus an axisymmetric mode belonging to a particular / is decomposed into a complete set of (2/ + 1) modes with angular dependences given by Pi,(cosO)e"mp. The functions describing the radial dependence will, of course, be unaffected by this decomposition. It is this same reason which ensures the independence of the radial wave-function of an electron in a central field on the magnetic quantum number m and its dependence only on the orbital angular momentum specified by/. In considering the perturbations of the Schwarzschild black-hole, we shall, in accordance with the foregoing remarks, restrict ourselves to time-dependent axisymmetric modes. It will suffice then to consider the Schwarzschild solution as a special, spherically symmetric time-independent solution of the field equations appropriate to the line element (Ch. 2, equation (38)), ds2 =e2v(dt)2-e2'l'(d<p-codt-q2dx2-q3dx3)2 -e2">(dx2)2-e2^(dx3)2, (3) and obtain the relevant perturbation equations by linearizing the field equations about the Schwarzschild solution. For this purpose (and other purposes, as well) it will be convenient to have the explicit expressions for the various components of the Ricci and the Einstein tensors for the line element (3) when v, \j/, /*2, /*3, u>, %2, and <?3 are functions only of t, x2, and x3. These components can be readily obtained by suitable contractions of the Riemann tensor whose components (when the metric functions are dependent on q>, as well) are listed in Chapter 2 (equations (75)). The results of such contractions are given below.
THE RICCI AND THE EINSTEIN TENSORS 141 -R00 =e-2vl(ij/ + n2+ /i3),0,0 + "A,o(<A-v),o+ ^2.0(/½ -v),o + /^3,0(¾-v),0] -e-2"2[v,2,2 + v2(^ + v-/i2 + /i3)i2] -e-2"3[v, 3,3 +V,3 ("A+ V+ ^-/^3),3] + ^-2v[e-2,2Q2o+e-2,3Q2o]) (a) -Kn =e-2^[f2,2 + f2(^ + v + ^3-^2)>2] + e-2"3[f3, 3 +f 3(^ + V + /<2-/<3 ),3] -^-2vCf0,0 + «A,0(«A-V + /^2+/^3),0] -^2^-2"2-2"3G223 + ^-2v[e-2,3G2o+e-2,2Q2o]) (b) - «22 = e-^W + V + ¾ ),2,2 +<A,2(«A -/^2),2 + /^3,2(/^3 -/^2),2 + v,2(v-/*2),2] + ^-2"3 [/^2,3,3 +/^2,3 (>A + V+ /^2 -/^3),3] -e-2VC/^2,0,0 + /^2,o(>A-V +/^2 +/^3),0] + le2^-2fe[e-2ftG23_e-2vQ2o]) (c) -R01 =^-^-^-^^(^-^^ + ^220),2 + (^^-^^^30).3]. (d) -^12=^-^-^^^(^^-^-^632),3-(^-^^-^602),0]. (e) -K02 = e~"2~v[(iA + /^3),2,0 + ^,2(^-/^2),0 + /^3,2(/^3 -/^2),0 -W +/^3).0 V,2)]-|^-V-2"3-"2G23G30, (f) - K23 = «-"!-"> [(^ + V)i2>3 -(^ + V),2^2,3 -(^ + V),3 H3<2 + f 2 f 3 + V,2V3]-l^-2v-^-^G2oG30, (g) Goo = -^-2"2[("A+/^3),2,2 + >A,2(lA-^2+^3),2+^3,2(/^3-/^2),2] -e^ttf + ftb.a + filf-ft+ftb + ft.afe-ftb] + e-2vC«A,o(/^2+ /^3),0+ /^3,0/^2,0]-i^-2vC^-2"2G22o + ^-2"3G32o] (h) .le2,A-2^-2ftQ23) Gn = e~2"2[(v +113),2.2+v,2(v -/^2 + /^3),2 +/^3,2 (/^3- /^2),2] + e-2"3[(v + /i2),3,3+V,3(v-^3+/i2),3+/i2i3(/i2 -^3).3] - e"2v[ (/½ + /*3 ),0.0 + /*2,0 (/*2 ~v),0 + /*3. o(/*3~ V>, 0 +/*2.0/*3 ,0] + ie2*le-2^-2^Q223-e-2^-2vQ220-e-2^-2vQ230l (i)
142 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE G22=e-2"3[(1A+V),3,3 + (>A + V),3(v -ll3),3 +f3f3] + e-2"»[v2W+/*3),2 + f 2/*3i2] -e~2v[(i^ +H3\0,0 + (>A +/^3).0(^3 -v),o + >A,o>A,o] -i^[e-2^-2^Q223-e-^-^Q|0+e-2"3-2vG2o]) (j) (4) where Qab = (Ia,b-(Ib,a and £40 = <Uo-«,4 U, B = 2,3). (5) The components R33,Rl3,R03, and G33 are not listed; they can be obtained by interchanging the indices 2 and 3 in the components R22, K12, R02, and G22 which are listed. 24. The metric perturbations The Schwarzschild line-element considered as a special case of the line element (3) has the metric coefficients e2v = e-2ni = i_2M/r = A/r2, e^ = r, e* = rsin0, (6) and co = q2 = q3=0 (A = r2 - 2Mr; x2 = r; x3 = 0). (7) Consequently, a general perturbation of the Schwarzschild black-hole will result in co,q2, and q3 (8) becoming small quantities of the first order and the functions v, fi2, fi3, and ^ experiencing small increments, 5v,Sfi2,5fi3, and 5$. (9) It is clear that the perturbations leading to non-vanishing values of co, q2, and q3 and perturbations leading to increments in v, ^2, /i3, and ip are of very different kinds: the former induce a dragging of the inertial frame and impart a rotation to the black hole while the latter impart no such rotation. For this reason we shall call them axial and polar perturbations, respectively. The terminology is justified when we consider the effect of a reversal in the sign of </> on the metric. It has no effect when the perturbations are polar while, when the perturbations are axial, the signs of co, q2, and q3 must also be reversed if the metric is to remain unchanged. As one may expect from these different behaviours of the perturbations of the two kinds, they must decouple in the sense that they can be considered independently of each other. We shall find that this is indeed the case.
THE METRIC PERTURBATIONS 143 (a) Axial perturbations As we have stated, axial perturbations are characterized by the non- vanishing of co, q2, and q3. The equations governing these quantities are given by Rl2 = R13=0. (10) From the expression (4, e) for Rl2, for example, it is apparent that in equations (10), we may insert for v, /i2, ix3, and ip their unperturbed values (6). The resulting equations are (<3*+'-ft-'"eM)13 = -<3'-,+',s-,"eo21o (5*12 = 0), (ii) {e^ + v-^-^Q2i).2= +e3*~v + "2-"3Go3,o (5*13=0). (12) Letting Q(t, r, 9) = AQ23 sin3 6 = A(q2,3 -q3,2)sin3 9, (13) and substituting for v, ix2,ix3, and ty their unperturbed values (6), we obtain the pair of equations 1 8Q = -(«, 2-¢2.0).0, (14) r4 sin3 6 d6 A 8Q r4 sin3 6 dr =+(«,3-93,o).o- (I5) In our further considerations, we shall assume that the perturbations have the time-dependence eiat, (16) where a is a constant (generally real). This assumption corresponds to a Fourier analysis of the perturbations and considering the Fourier component with the frequency — a. Retaining the same symbols for the amplitudes of the perturbations with the foregoing time-dependent factor, we can rewrite equations (14) and (15) in the forms 1 5Q . r4sin30 30 and = +1(760,3 + ff q3. r4sin30 dr Eliminating co from these equations, we obtain (17) '^£)+-'4U£K«- <18>
144 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE The variables r and 0 in equation (18) can be separated by the substitution Q(r,e) = Q(r)C,-+3{2(8), (19) where C"n denotes the Gegenbauer function governed by the equation -^sin2v0— + n(n + 2v)sin2vo da da c;m = o. (20) 4(4^W<We=0, ,23, It may be noted here that the Gegenbauer function C,+32/2 (o) is related to the Legendre function Pi(0) by the formula r-^™-,i„3fld 1 dP,(6) on* Cl + 2 (0)-sin <>__-_ (21) or Cf+32/2 (0) = (Pus- P,,e cot o) sin2 6. (22) With the substitution (19) in equation (18), we obtain the radial equation d /AdQ\_ 2A ^d7^~d7j_^ r~^ where /i2 = 2« = (/-1)(/ + 2) (24) specifies the associated angular dependence.Changing to the variable (cf. Ch. 3, equation (219)) >. = r + 2MW2M-l) (£-££) (25) and further letting Q(r) = rZ<-\ (26) we find that Z(_) satisfies the one-dimensional Schrodinger wave-equation A2 where the potential V{~) is given by V{"l=^l(n2 + 2)r-6Ml (28) Equation (27) governing the axial perturbation was first derived (though by an entirely different procedure) by Regge and Wheeler; and it is often referred to as the Regge-Wheeler equation. * With the usual normalization of the Gegenbauer and the Legendre functions, there is an additional factor 3[((-1)((( + 1)(( + 2)]1 on the right-hand side of this equation. 2 +ff2 )Z(->= K(_)Z(_), (27) *
THE METRIC PERTURBATIONS 145 Table I The potential barrier, V{~\for axial perturbations/or different values of I and n rjM 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.2 3.4 3.6 / = 2; n = 2 0. 0.03394 0.06147 0.08362 0.10127 0.11520 0.12605 0.13435 0.14057 0.14506 0.14815 0.15106 0.15086 0.14861 /=3; n = 5 0. 0.09872 0.17417 0.23156 0.27488 0.30720 0.33087 0.34773 0.35923 0.36647 0.37037 0.37079 0.36458 0.35437 / = 4,- n = 9 0. 0.18511 0.32443 0.42881 0.50637 0.56320 0.60397 0.63224 0.65077 0.66169 0.66667 0.66376 0.64954 0.62871 r/M 3.8 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 12.0 14.0 16.0 18.0 20.0 / = 2,- n = 2 0.14503 0.14063 0.12803 0.11520 0.09259 0.07497 0.06152 0.05121 0.04320 0.03183 0.02436 0.01923 0.01555 0.01283 / = 3; n = 5 0.34185 0.32813 0.29264 0.25920 0.20370 0.16243 0.13184 0.10882 0.09120 0.06655 0.05060 0.03973 0.03201 0.02633 / = 4,- n = 9 0.60428 0.57813 0.51212 0.45120 0.35185 0.27905 0.22559 0.18564 0.15520 0.11285 0.08559 0.06708 0.05396 0.04433 In our subsequent work, we shall find it convenient to introduce the operators d d2 A±=—±io and A2 = A+A_ = A_ A+=-^ + (72. (29) .T , -+ia and A2 = A+A_ = A_ A+=—T + a2. - drt dr2 In this notation, the equation satisfied by Z( 'is A2Z(-)= yi-)Zl-) (30) Table I provides a brief tabulation of the potential V'' for I = 2, 3, and 4; and they are exhibited in Fig. 11. (b) The polar perturbations Polar perturbations are characterized by non-vanishing increments in the metric functions v, /*2, /*3, and i//. Examining the expressions (4/, g, b, and j) for #02, #23» #i i, and G22, we find that the Q^b's occur quadratically in them; they can, accordingly, be ignored in a linear perturbation-theory. Thus, as expected, the equations governing the axial and the polar perturbations do decouple. Linearizing the expressions for R02, R03, #23» #11» and G22 about the Schwarzschild values, we obtain the equations (^+^3).,+ 1 vr l(^ + fy3)--fy2 = 0 (<5#02 = 0), (31) (a* + ty2).fl+W-fy3)cot0 = O (<5KO3 = 0), (32)
146 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE V(- Fig. 11. The potential barriers surrounding the Schwarzschild black-holes for axial perturbations. The curves are labelled by the values of I to which they belong. (Sip + 5v\rtg+ (dip-5fl3\rCOt0 + ( v,-- )Svt8 -2|j2 V,+ - )^2.0 = 0 05*23=0), ^V,r + (;+V,rW + ^),r-2W-V,r+^ + -3 [ (Sij/ + dv\6tg +(2Sij/ + Sv-Sn3 ),0COt 0 + 2<5/i3 ] -e~2*(5ip + ^3).0.0=0 (SG22=0), and (33) (34) e+2vj^.r.r + 2[-+v,r)^,r + -(# + ^v-^2 + ^3).r -2^^ + 2v.r^|+l{5ffl>fl+(25^ + 5v + ^2-^3)|flcot0 + 25/i3} -e-2Mf0,o=0 (5*11=0). (35) We shall assume once again that the perturbations have a time-dependence e"" so that "0" is replaced by the factor ia; and we shall suppress the factor e"" in our subsequent considerations.
THE METRIC PERTURBATIONS 147 The variables r and 6 in equations (31)-(35) can be separated by the substitutions (due to J. Friedman) 5v = N(r)Pl (cos 6), (36) 8p2 = L{r)Pl (cos 0), (37) Sfi3 = [T(r)P,+ V(r)PIAel (38) and 5^ = [T(r)Pl+V(r)Pltecotei (39) We may note that, according to these substitutions, Sil/ + S^ = l2T-l(l+l)V^P„ (40) 6tfi + W-6n3)cotO = (T-V)Pl,e, (41) and fyAe + (2^ - 5fi3 \e cot 9 + 26^ = [2-/(/ + l)]rP, = -2rt77>,. (42) Returning to a consideration of equations (31)-(35), we first observe that equation (32), by the substitutions (37) and (41), leads to the relation T-V+L = 0 (5R03=0). (43) Accordingly, only three of the four radial functions we have defined are linearly independent. We shall choose N, L, and V as the independent functions. Making use of equations (40) and (41), the (0,2)- and the (2,3)-components of the field equations—equations (31) and (33)—give ^- + 1-vr)[2r-/(/ + l)K]--L = 0 (44) ar r ' J r and (T-V + N\,-(^-v\n-(^ + v\l = Q, (45) or, after the elimination of T, we have JVr-Lr = (J-v,rW(^ + viP)L (46) and Lr + (--vr)L = X,+ (i-v,U (47) where we have replaced V by X = nV = \(l-\)(l + 2)V. (48)
148 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE Similarly, we find that by making use of equations (40) and (42), equation (34) gives -N r+ - + v r • \r 2T-l(l+\)V 2/1 r\r + 2v,r)L e~2v.. . e~2v -1(1+ 1)^-JV - 2n—r-T + o2e~4v[2T-1(1 + 1) V\ = 0, r r or, after the elimination of T, we have ^-/(/+1)1^ JV-;(J + 2vrJL-2^ + vrj(L + «C).r e~2v -272^-(^-/.)-2^-^ + ^) = 0. (49) (50) Finally, considering equation (35), we find that it consists of terms in P, and P/_e cot 6. These terms vanish separately by virtue of the preceding equations; and the terms in P( e cot 0 give 1 \ e~2 Vrir + 2[- + vtr)Vr + —r-(N+L) + <T2e-*,V = 0. (51) This equation, as we have stated, is not independent of equations (46), (47), and (50); it is, nevertheless, useful in this form in our further reductions. It will be observed that equations (46), (47), and (50) provide three linear first-order equations for the three radial functions L, N, and V (or X). By suitably combining them, we can express the derivatives of each of them as linear combinations of themselves. Thus, we find N<r=aN + bL + cX, Lr = (a — + v,)JV + (b v, IL + cX, x,r= -(a-^ + v./W-N» + --2vr JL- (c+--vir )X, where, for the sake of brevity, we have written n + 1 M a = V r = r-2M' 'r r(r-2M)' b = 1 ■ + M - + M- + a' r r-2M r(r-2M) r(r-2Mf (r-2M) 1 1 C = + ^TT + M + oA r r-2M r(r-2Mf (r-2M) (52) (53) (54) (55)
THE METRIC PERTURBATIONS 149 We shall find the following alternative forms of equations (47) and (54) useful: (L + X) =-(^-v,rV-f--vr )X i -Vr7rC(2r-5M)L + (r-3M)Z] (56) r(r — 2M) and nr + 3M X = -- —-N- _ __\. M M2 + <r2r4 r(r-2M) \_r-2M ~ r(r-2M) + r(r-2M)2 (L + X) n+\ + ^2ML <57> (i) The reduction of the equations to a one-dimensional wave equation It is a remarkable fact that Z<+>= T\ (—X-l\ (58) nr + 3M \nr J by virtue of equations (52)-(54), satisfies a one-dimensional wave equation similar to Z(_). This reduction of the third-order system of equations (52)-(54) to a single second-order equation emerged empirically. Its origin must, however, lie in some deeper fact at the base of equations (52)-(54). As to what this may be is clarified in §25 below. But first, we shall verify that Z(+), as defined in equation (58), does in fact satisfy a one-dimensional wave equation. Rewriting Z(+) in the form r2 Zl+) = rV —-(L + X), (59) nr + 3M ' v ' differentiating it with respect to r+ and substituting for (L + X) r from equation (56), we find r(nr + 5M) (nr + 5M) Differentiating this expression with respect to r and making use, once again, of equation (56), we obtain Z^ = (r-2M)V,r,r+V,r+™{r-2^V,r • r(nr + iM) -nr2+4Mnr + 6M2 3Mn r2(nr+3M)2 + (nr+ 3M nr2-3Mnr-3M2 — in -T hm ni-\-vm Jivin _ ,_ -,,.., + 3M .2/..,,,^2 ^ + ,...,, „«W + n)r-MUL + X) r(r-2M)(nr + 3M) T[(2r-5M)L + (r-3M)Z]. (61)
150 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE On substituting for Kr r from equation (51), we shall be left with an expression, which, besides L, X, and N, involves only the derivative of V for which we can substitute from equation (57). After making these substitutions and simplifying, we find that we are left with (after some remarkable cancellations) (d^+ff2)z<+)= J/<+)z<+)' (62) where K(+> = ^- r[n2(n+l)r3+3Af«V + 9M2nr + 9M3]. (63) rs(nr + 3M)2 Equation (62) was first derived (by an entirely different procedure) by Zerilli; and it is often called the Zerilli equation. We shall rewrite equation (62) in the alternative form (cf. equation (30)) A2Z(+>= J/(+>Z( + >. (64) Table II provides a brief tabulation of the potential V{+) for I = 2, 3, and 4; and they are exhibited in Fig. 12. A comparison of the tabulations of V{ + ) and V{') shows that they differ remarkably little over the entire ra.nge of r+ even though their analytical forms (28) and (63) are so different. (ii) The completion of the solution With the reduction of the third-order system of equations (52)-(54) to the single second-order equation (61) for Z(+), it is clear that the solution for L, X, Table II The potential barrier, V{ + \for polar perturbations for different values of land n r/M 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.2 3.4 3.6 1 = 2; n = 2 0. 0.04183 0.07311 0.09647 0.11383 0.12660 0.13584 0.14235 0.14673 0.14946 0.15089 0.15094 0.14848 0.14452 /= 3; n = 5 0. 0.10289 0.18018 0.23805 0.28106 0.31263 0.33536 0.35122 0.36171 0.36802 0.37106 0.37003 0.36272 0.35172 1 = 4; n = 9 0. 0.18766 0.32809 0.43272 0.51004 0.56639 0.60656 0.63420 0.65211 0.66246 0.66691 0.66314 0.64827 0.62355 rjM 3.8 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 12.0 14.0 16.0 18.0 20.0 1 = 2; n = 2 0.13970 0.13443 0.12079 0.10787 0.08617 0.06974 0.05734 0.04786 0.04050 0.03002 0.02310 0.01832 0.01487 0.01231 / = 3 n = 3 0.33866 0.32458 0.28876 0.25542 0.20054 0.15993 0.12988 0.10728 0.08997 0.06575 0.05005 0.03934 0.03172 0.02611 / = 4; n = 9 0.60225 0.57591 0.50974 0.44891 0.34997 0.27758 0.22444 0.18475 0.15449 0.11239 0.08527 0.06685 0.05379 0.04420
THE METRIC PERTURBATIONS 151 j/( + > 3.0 2.5 2.0 1.5 1.0 0.5 Fig. 12. The potential barriers surrounding the Schwarzschild black-holes for polar perturbations. The curves are labelled by the values of I to which they belong. and N will require a further quadrature. Thus, rewriting equation (47) in the form d , d — (r2e_vL)= -nr~(re'vV) dr dr and replacing V by 2>Mr 3M we obtain the equation (l+^)^:(^-^) =-^7A[e-v(nr + 3M)Z<+)]. 'drv " ' 3Mdr This equation yields the integral relation 2 -v f nr d ~ ~ J(nr + 3M)dr or, after an integration by parts [e"v(nr + 3M)Z(+)]dr, r2e'vL= -nre"vZ(+) + 3Mn nr+3M Z( + >dr. (65) (66) (67) (68) (69)
152 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE Defining we have the solution d> = nev ■vZ( + ) nr + 3M 3M dr, r{ + ) + —¢. r With this solution for L, equation (66) gives Z = -(Z<+» + (D). r As a consequence of these solutions for L and X, L + Z=-T(nr + 3M)0). (70) (71) (72) (73) To obtain the solution for N, we return to equation (60) and substitute for V,r{— X,r/n) on the right-hand side from equation (57). In this manner, we obtain Zrt = -(r-2M) n nr + 3M ' r(r-2M) N- 1 M + M2 + <tV r-2M r(r-2M) r(r-2M)2 ,, v, " + 1 J 3M{r-2M)v nr2-3Mnr-3M2 tl vx X (L + X) + r-rrLV +--^ ^#+ ——= (L + X). r-2M nr(nr + 3M) (nr + 3Mf (74) Simplifying this last equation and substituting for L and L + X their solutions (71) and (72), we find N= nr Z<+> n nr + 3M ,r* (nr + 3M)2 "6M2 + 3Mn + n(n+ \)r r{ + ) + M- M2+ff2r4\0) r-2M f (75) This completes the formal solution of the basic equations. We shall return to a consideration of the wave equations for Z(+) and Z(_) in §§26 and 27. 25. A theorem relating to the particular integrals associated with the reducibility of a system of linear differential equations In our treatment of the polar perturbations in §24, we saw that the system of three linear equations (equations (52)-(54)) governing N, L, and X could be reduced to a single second-order equation for a particular combination of them which we denoted by Z(+). This reducibility of the system was a verifiable
REDUCIBILITY OF A SYSTEM OF LINEAR EQUATIONS 153 but an empirically discovered fact; and in view of the complexity of the potential in the wave equation for Z(+), this is a remarkable coincidence shrouded in mysterious cancellations. In this section, we shall try to locate the origin of the mystery. In principle, the reducibility of a system of linear equations to one of lower order is, of itself, nothing extraordinary: it can always be achieved if one has a prior knowledge of some particular solutions of the equations. The reduction in §24 was, however, achieved without any such prior knowledge. A general question which occurs is whether, from a knowledge of the reducibility of a system of linear equations to one of lower order, we can, conversely, devise an algorism for deriving the particular solutions which made the reduction, in the first place, possible. Such an algorism was recently devised by Xanthopoulos. We shall give an account of it, since the procedure has applications in other contexts which we shall encounter. We consider a system of n linear equations, d^=AjkXk (j= 1,..., «), (76) where the Ajks are some known functions of x and summation over repeated indices is assumed. Letting Xdenote the column vector (with components Xj) and A the (n, n)-matrix (whose elements are AJk) we can write equation (76) in the form AX — = AX. (77) We define the associated adjoint system of equations by AX .-( AX: - \ -=-^ ^ot_L^_AkiXky (78) where At denotes the transposed of the matrix A. If Xj and Xj are solutions of equations (77) and (78), then it readily follows that — (XjXj) = 0 or XjXj = constant. (79) dx We define the adjoint of the operators Ai and A&- (80> in the customary manner by Jf A AA* J Jf A2 AA* A A2A^ ~A a -a— and A a-^ + 2-a—a- + ^-^> 81 dx dx dx^ dx dx dx2
154 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE respectively. Also, we shall find it convenient to write equation (76) in the form d \„ /„ d EX=\ITx~A)x= \3jkdx~Ajk)Xk = 0' (82) where I denotes the unit (diagonal) matrix. Suppose that there exists a linear combination, Z = LjXj =LX, (83) of the Xfs which, by virtue of equation (77), satisfies the second-order equation OZ=(d^ + ^ + C)L^ = 0' (84) where B and C are some known functions of x. The problem now is to devise an algorism which will enable us to discover the particular solutions of equations (77) which made this reduction possible. More generally, one may suppose that there are m combinations Za = LxJXj (a = 1, . . . m; j = 1, . . . n; 2m < n) (85) which satisfy the set of m equations d2 d 'dx^^'dx t«l>T^ + B°l>^- + C°l>)Zl> = 0 («=1, ■■■,«), (86) where Bxfi and CaP are certain known functions of x. We shall, however, restrict ourselves to the simpler case when we have only one combination to consider. No essentially new ideas are required to treat the general case; but a suitable notation must be devised. The fact that equation (84) is satisfied, by virtue of equation (82), implies a certain operator identity which can be derived as follows: ^ = ~tLiXj=(^t+LkAkJ)Xj=r,Xj (Say)' (8?) where T, = ^ + LkAkj, or r = ^+ LA. (88) Differentiating equation (87), we find d2Z (AT ^ Vd,2 + r^'J*'- (89) Inserting the foregoing expressions for the derivatives of Z in equation (84), we obtain oz = (^+ rkAkj+bt j+clAxj = o; (90)
REDUCIBILITY OF A SYSTEM OF LINEAR EQUATIONS 155 and we infer the identity dr — + rA + Br + CL=0. (91) dx We now seek an operator S which will satisfy the identity SE = OL, (92) or, explicitly, d 'dx "*>)"' Vdx2 " "dx S«\*<<J7Z-A«j)XJ = \7Z2+B7Z + C)LJXJ- (93> Then and (cf. equation (87)) d2Z / dr, dx dZ dx7 ~ + rkA Wj U,V + LjAj, f+rtAt To determine S, we evaluate the right-hand side of equation (93) when X} satisfies the inhomogeneous equation -^-=AJkXk + \j. (94) (95) ^(LjAj). (96) Therefore, by making use of the identity (91), we now find OZ-OL^-^+L^ + BLj+rty, ={^+L<£+BL<+r>){s»£-A»)x>-(97) Comparison with equation (92) now shows that d AL S=L — + — +BL + r. (98) dx dx The adjoint of this operator, S, is 5+= -L+ — + BLt + Tt. (99) dx Now consider Xj = Sj</, = - Lj^-+ (BLj + r})4 (j = 1, . . . , ri), (100) where <j> is some function of x. We shall now suppose that the foregoing n
156 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE equations determine <f> and <j>tX uniquely. This is possible if, and only if, the (3. nl-matrix (3, n)-matrix *i + Ll + L2 ■(BLi + ro (BL2 + r2) X„ +L„ -(BL„+r„) or, equivalently, the matrix, x, lx r\ x2 l2 r2 *„ l„ r„ is of rank 2, i.e., all the third-order determinants (101) (102) Dijk = x, u r, {i+) + k + i\ (103) formed out of any three rows selected from the matrix (102) vanish. We shall now show that the vanishing of these determinants is preserved for the X/s varying in accordance with the adjoint system of equations dX, dx — AkJXt (104) To show that this is the case, we differentiate the determinant Dijk and replace the derivatives of X, L, and T in accordance with equations (104), (88), and (91), respectively. We thus obtain d d~xD^ = + An Xi A-ikXi Xi L, X; L, Lt Lj Lk + Xi (Ti-LiAh) Ti Xj (Tj-LiAu) Tj {TiAn + BTi + CLi) (riA,j+Brj+CLj) Lk -(r,Alk + Brk + CLk) = -An x, l, r, Xj Lj Tj Xk Lk rk -A>J X; xt xk Lt L, Lk r< r, rk — Aik x> u r< X; Lj Tj xt ^ r, -BDijk. (105)
REDUCIBILITY OF A SYSTEM OF LINEAR EQUATIONS 157 Therefore, — Dijk= -AliDIJk-AIJDilk-AlkDiJl-BDijk. (106) We conclude that if all the third-order determinants, £)i;k's, vanish, their derivatives also vanish. Without loss of generality, we may assume that <j> and (/> x are determined by the first pair of equations, j = 1 and 2, in (100). Consider the (n — 2) determinants £>12a (a = 3,..., n). (107) Since any determinant DiJk can be expressed as a linear combination of the ^i2a's (with coefficients independent of the X/s), we can rewrite equation (106) in the form ^£>i2a= /a/,^12/, (a = 3,..., n), (108) where /ap are coefficients which are independent of the X/s and are explicitly known. Equation (108) guarantees that the vanishing of all the Dl2/s is preserved by the adjoint system (104). Equation (108) provides a linear system of (n — 2) first-order equations for the Dl2/s. We shall suppose that the integrating factors for these equations can be found and the solutions expressed in the forms QJX = kxpDl2p = Ca = constant (a = 3, . . . , n), (109) where the /cap's are some known functions. Since the determinants, Dt 2p, are themselves linear combinations of the X/s, the solutions (109), written out explicitly in terms of the X/s, will be of the forms @. = XmiJt, = C. (a = 3,..., n), (110) where the Xm/s are determinate linear combinations of the k^^s. Equation (79) now enables us to conclude that X(0l)J(j = 1, . . . , n), for each a, represents a particular solution of the original system of equations (77). It is the existence of these particular solutions that underlies the original reducibility of the system. We have thus solved the problem, in principle, which we set ourselves. It is of interest to note that in the particular case of an initial third-order system, reducible to a second-order equation, equation (106) takes the particularly simple form ^1=-(B + All+A22 + A33)Dl23 (111) —an equation which allows of immediate integration.
158 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE (a) The particular solution of the system of equations (52)-(54) We have seen in §24 that the system of equations (52)-(54) for N, L, and X is reducible to an equation of the second order for Z< + ) = 3M , L + X |. nr nr + 3M< To apply the algorism we have developed, we let x -> r, and Xu X2, X3 -> N, L, X. With this correspondence L1=0, L2 = and U = 3M (112) (113) (114) nr + 3M' " J nr + 3M nr Also, the wave equation satisfied by Z( + ), in terms of the variable r, is d27( + ) 1M A7( + ) r2 ^^+ 2M dZ + ,(tr2-J/< + »)Z< + » = 0. (115) dr2 r(r-2M) dr (r-2M)2V ' V ' We next consider equation (111) satisfied by ,2 £»123 = nr + 3M According to equations (52)-(54) and (114) n o r\ l -l r2 X 3M/nr T3 2 5M-2r ^11 + ^22 + ^33 =a + fc-c— = - ——, r r(r — 2M) and B = 2M r(r-2M) The equation for Dl2i, therefore, becomes d£)123 _ 2r-lM dr ~r(r-2M) The integrated form of this equation is £>12 -£)123 = constant (e2v = 1 -2M/r). (116) (117) (118) (119) (120) It remains to express Dl23 in terms of N, L, and X. By equation (116), -2 r x 3M_ \.~. _ /3M £»123 = nr + 3M r3+—r2 N + rw^^L+x nr \ nr (121)
REDUCTIBILITY OF A SYSTEM OF LINEAR EQUATIONS where (cf. equation (88)) I\ = L2A2l + L3A3l, T2 = L2<r+ L2A22 + L3A32, and r, = L3r + L2A23 +L3A J3,r 2^23 + ^3 A33-> 159 (122) The coefficients A2l, etc., in the foregoing equations can be read off from equations (52)-(54). On evaluating the various expressions, we find that we are finally left with 0123= — PN + 3ML + nrX), where M2 + a2r* r — 2M The integral (120) now takes the form :(pN + 3ML + nrX) = constant. Therefore N=/=p3, L = g = 3M^, r r and X =h = nr- (123) (124) (125) (126) represent particular solutions of equations (52)-(54) which were discovered by Xanthopoulos by this method. With the aid of the particular solution (126), equations (52)-(54) can be reduced to one of order two by the substitutions, N=fv, L = gX, and X = hX, (127) when they become v.r = Al2-().-v) + Al3-(x-v), ^,r = A2lJ-(v~k)+A23-(X-k), 9 9 f 9 X,r = A3l-(v-x)+A32~(k-x). (128) By letting P = v-A, Q = k-x, and R = x~v (P + Q + R=0), (129)
160 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE the equations take the forms dP ( 9 /\„ h h -7-= -(Al2-f+A2lJ- )P + A23-Q + Al3-R, dr \ f gj g f dQ f h g\ f fn ~T= -[A23-+A32l)Q + A3iJ-rR + A2lJ-P, dr \ g hj h g dK ( f h\ a g (130) and these equations are consistent with the requirement (cf. equation (129)) P + Q + R = 0. (131) The reducibility of the system to one of second order is now manifest. Since nr + 3M\nr J nr+5M nr+3M it is clear that the solution can be completed and expressed in terms of Z<+) and its derivative in the manner of §24(ii). 26. The relations between K<+) and K<~> and Z<+) and Z<_) In §24, we reduced the equations governing the polar and the axial perturbations to one-dimensional Schrodinger equations for two functions Z<_) and Z<+) with potentials Vi~) and V(+). It is a remarkable fact that, in spite of appearance, the two potentials Vi~) and V{ + ), given by equations (28) and (63), are very simply related. They are in fact given by (as one may directly verify) Vi±)=±p£L + p2f2 + Kft (133) where /J = constant = 6M, k = constant = An(n + 1) = \x2 (n2 + 2), (134) and A _ A r3(fi2r + 6M) ~2r3(nr + 3M)' /=.3,,,2,.^=^3, ,.„• (135> It should be particularly noted that in equation (133) P and k: are constants and/is a function which vanishes both at the horizon (r = 2M) and at infinity (with an inverse-square r~2-behaviour). There is no obvious reason, at this stage, to have expected (or, indeed, to expect) that the potentials V{+) and Vi~) are simply related in this fashion. The origin of the relation will emerge when we come to treat the perturbation problem via the Newman-Penrose formalism in §§28 and 29. Meantime, we
RELATIONS BETWEEN ^(t|AND K1"1 AND Z1*1 ANDZ1"1 161 shall accept the relation (133) as a directly verifiable fact and show that it implies a very simple relation between the solutions Z< + ) and Z<_). Quite generally, we shall consider the two wave equations d'Zl + a2Zl = V,Z, = (+p?£+P2f2 + Kf)zi (136) dx2 l l l \' rdx and d^ + a2Z2 = y2Z2 = (-li^+li2f2 + Kf]z2, (137) where /? and k are some real constants and/is an arbitrary smooth function which, together with its derivatives of all orders, vanish for both x -* + oo and x -> — oo, and whose integral over the entire range of x is finite. (For convenience of notation, we are temporarily replacing r„, by x and Z< + ) and Z'-'byZi andZ2.) There is clearly no restriction to supposing that, given a solution Z2 of equation (137), Zx = pZ2 + qZ'2 (138) is a solution of equation (136) where Z'2 denotes the derivative of Z2 (with respect to x) and p and q are certain suitably chosen functions. The equations that must govern p and q in order that Zu given by equation (138), is a solution of equation (136), can be derived as follows. Differentiating equation (138) and making use of equation (137) satisfied by Z2, we obtain Z\ = \_p' + q(V2-a2ftZ2 + (p + q')Z'2. (139) Differentiating this equation, once again, we similarly obtain Z'l = \_p" + (p + 2q')(V2-<T2) + qV'2lZ2 + [p' + q(J/2-^2) + p' + q"]Z'2; (140) and this expression must identically be the same that follows from equations (136) and (138), namely, Z'l={pZ2 + qZ'2)(Vl-a2). (141) Equating the coefficients of Z2 and Z2 on the right-hand sides of equations (140) and (141), we find q{Vl-a2) = 2p' + q" + q{V2-a2) (142) and P{V,-o2) = p" + (p + 2q')(V2-G2) + qV'2; (143) or, alternatively, q(J/i-J/2) = 2p' + q" (144) and p{Vl-V2) = p" + 2q'{V2-a2) + qV2. (145)
162 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE Eliminating (Vl — V2) between these two equations, we obtain 2pp'+pq"-p"q-2qq'(V2-a2)-q2Vi=0. (146) This equation provides the integral p2 + (pq' -p'q)-q2{V2-a2) = constant = C2 (say). (147) Equations (144) and (147) are the equations which p and q must satisfy if the combination pZ2 +qZ'2 is to be a solution of equation (136). In general (i.e., for arbitrarily specified Vx and V2) one cannot expect to solve these equations explicitly. But for the case on hand, we can! Indeed, for Vx and V2 of the forms specified, one can readily verify that q = 2/? ( = constant) and p = k + 2/?2/, (148) do satisfy equations (144) and (147) with C2 = k2+4/?2<t2. (149) Accordingly, the solutions Zt and Z2 of equations (136) and (137) are related in the manner (k + 2iap)Zl = (k + 2p2f)Z2 + 2pZ'2, (150) where we have chosen a relative normalization of Zy and Z2 such that the inverse relation in the same normalization is given by (K-2iop)Z2 =(»c + 2/?2/)Z1-2/?Z'1. (151) For the particular case of Z( + ) and Z(_) when /?, k, and/are given by equations (134) and (135), the relations (150) and (151) take the forms |>2(/<2 + 2) + 12ktM]Z< + ) = and \_fi2(fi2 + 2)-l2iaM'\Z{-) = H2(fi2+2)+J2M2 fi2(fi2+2) + 12M2 rJ(/i2r + 6M) Z<~> + 12MZ!;#» (152) rJ(^r + 6M) Z( + ) 12MZ<+) (153) We shall return in §27 to the implications of the foregoing relations and to the further consequences of the potentials being of the forms (133). An interesting consequence of the relations (152) and (153) is the following. By inserting for L, X, and N their solutions (71), (72), and (75) in the expression &=-(nr + 3M) r (n + l)r + (/1+1)- (L + X), (154)
THE PROBLEM OF REFLEXION AND TRANSMISSION 163 we find that the terms in <$> cancel (which is the reason for considering the expression 2£ in the first place) and we are left with Therefore, V(n + 1) ,,, r-2M y +3Mn- 3M r2(nr + 3M) Z(+). (155) —&= -12MZ< + > + 4n(n+l) + 36M2 r>r + 3M) r{+) (156) From a comparison of this last equation with equation (153) (and remembering that ix2 = In), we conclude that the expression for 2£, formed out of the radial functions describing the polar perturbations, will satisfy the wave equation appropriate to the axial perturbations] 27. The problem of reflexion and transmission We return to the wave equations satisfied by Z( + ) and Z(_) to consider the nature of the solutions of these equations and the quantities of physical interest which they determine. We first observe that the potentials V{ + ) and V{~) are smooth functions, integrable over the range of r#, ( + oo, — oo), and positive everywhere. Besides, they have an inverse-square behaviour for r„, -> + oo and vanish exponentially as we approach the horizon and /■„,-> — oo. Thus, J/'±)^-2(n+l)r-2 as r-^r^-^ + oo "] and \ (157) K<±)-» (constant) ±er*l2M as r#-»—oo.J Since K(±' fall off more rapidly than r~l for r% -» + oo, the asymptotic behaviour of the solutions, Z(±), for /■„,-> + oo, is given by e±ior. (^-,.+00). (158) For real a, the solutions, therefore, represent ingoing and outgoing waves at + oo. The underlying physical problem is, therefore, one of reflexion and transmission of incident waves (from + or — oo) by the one-dimensional potential-barriers, V(±\ This is the problem of the penetration of one- dimensional potential-barriers with which we are familiar in elementary quantum theory. Precisely, in the context of the problem on hand, we must seek solutions of the wave equations which satisfy the boundary conditions, Z(±) -> e + iar' + R{±\o)e-iar' (/■„,-> + oo), -> T^\a)e + iar' (r„, -> - oo). * ' " (159)
164 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE These boundary conditions correspond to an incident wave of unit amplitude from + oo giving rise to a reflected wave of amplitude Ri:t\a) at + oo and a transmitted wave of amplitude r(±'((7) at — oo. (Note that, in conformity with physical requirements, the boundary conditions we have imposed do not allow for waves emerging from the horizon.) Since the potentials are real, the complex conjugate of the solution, satisfying the boundary conditions (159), will satisfy the complex-conjugate boundary conditions, Z(±)*^e-'<"-. + K<±)*(ff)e + ''ffr. (^-+00), r<±>»e-ffr. (r^-oo). (lW) Since the Wronskian, [Z<±>,Z<±>*] =Z<±>Z<±>*-Z<±>Z<±>; (161) of the two (independent) solutions, Z(±' and Z(±'*, must be a constant, we obtain by evaluating at + oo and — oo, -2«t(|K(±V)|2-1)= +2ia\T^\a)\2, (162) or, 0^)((7) + 1^)((7) = 1, (163) where 03^)((7) = 1^)((7)12 and J^\a) = \T^(a)\2 (164) are the reflexion and the transmission coefficients. (a) The equality of the reflexion and the transmission coefficients for the axial and the polar perturbations Consider, quite generally, the problem of reflexion and transmission associated with the pair of wave equations (136) and (137). From equations (150) and (151) relating the solutions of these two equations, it follows, from our assumption that/vanishes for x -* + oo, that solutions for Zt, derived from solutions for Z2 having the asymptotic behaviours Z2^e + iax and Z2-><r,ffX (x-»+oo), (165) have, respectively, the asymptotic behaviours Zl^e + iax and Zt -> *" „ e~iffx (x -> + oo). (166) Accordingly, for the associated problems of reflexion and transmission of incident waves by the potential barriers Vx and V2, 7^((7) = ^((7) and ^i(t7) = K~^K2((7). (167) K + 2iap
poopooppooooop <^a>a><z>c>c>c>c>c>c>c>c>c> © o — ■& K> N V) ^ .¾ *P * «8 o o o o — ^-1 s* K> oo 4a. K> 4a- O O p O p O k) Ift OO « VO SO si oc 4^ u oo « * * ^4 a oo ^ Ui O " w oo w O O O p p O x o> S P P P * w w O oo a\ O O O p p U> U> l»J k> N* 4a. K> O O O P O p p O O O 8"© ©©■-■- "u> 4a. " K) 7i O 00 " OO U. XI Ui w ^ O ■- 4a. U> U> sj n- K> s| » 1^1 w o 4a. <_/i 4a. O 2 VO O K> v© P OO oo v© <_/l O 4a. <_/i p r *s VO Q — © ^ a_ (¾ V Ifl ^ \C U w 0i s| O Is) l/i U| Q w "si Q A * b 4a bo h 4a.'"- ©4a.l*>sj^4K><^isJOSu>— — OOK>*sJ— 4a. OO K> K> wwWMMwwww^^^^wiUiyip O -J a ^ w w ' u> \© © o^ oo <_/» O k) 0i bo ■• ■- O U> K> 4a. ' p p n* *- K> U> 8 4a. "oo -- 4a. "o K> — © sj \© ©■> sj "st> c § o <5' Oo *>j T *^ 1 (¾ ^ 2-¾ -! J* a a a o y o ^¾ -! Q_ <-. (¾ a- a (¾ <-► » S X <Z- a' a — a a *■ a a. »■ r 3.¾ "O rS the reflected amplitudes, waves for I = 2 and fo\ H > 03 r tn 5 H X tn *a 50 O 03 r tn 2 o 50 tn r tn x O Z > z o H 50 > z 2 o z
166 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE Thus, while the amplitudes of the transmitted waves are identically the same for the two potentials, Vl and V2, the reflected amplitudes differ only in their phases: ^(0) = ^^(0) where eu = K~l\ap (168) K + 2iap The equality of the reflexion and the transmission coefficients follows from these facts. For the particular case of Z'+' and Z' -' on hand, the phase difference in the reflected amplitudes is given by (cf. equation (134)). H (/r+ 2) + 12ioM In Table III we provide a brief listing of the reflexion and transmission coefficients (common for both the axial and the polar perturbations) and the difference in phase of the reflected waves given by equation (169). It should be further noted that, from the relations (150) and (151) between the solutions Zt and Z2, it also follows that should Vx allow a discrete set of states (with solutions falling off exponentially as x -* + 00), then V2 will also allow the same set of discrete states. However, since the potentials, K,+' and Vl~\ are positive everywhere, they do not allow any discrete states. 28. The elements of the theory of one-dimensional potential-scattering and a necessary condition that two potentials yield the same transmission amplitude In §27 we have seen how the pair of potentials (133) yield the same transmission amplitude for all a. The question arises whether there are some necessary conditions that will characterize potentials with this property. The question is related to the theory of'inverse scattering' dealing with the problem of determining the potential from a knowledge of the S-matrix (see equation (178) below) and its ramifications in the theory of solitons and the Korteweg-de Vries equation. Besides, it will emerge from our study of the associated problem in the context of the Kerr black-hole in Chapter 9 that we shall have to deal with cases of complex potentials and potentials with singularities. For these cases there is, as yet, no comprehensive theory. For this reason, we shall give a brief account of the elements of the extant theory and obtain necessary conditions for two potentials to yield the same transmission amplitude. The theory of one-dimensional potential-scattering is concerned with the solution of Schrodinger's wave-equation —- + a2)f=Vf (-oo<x< +00), (170) >dx I
ONE-DIMENSIONAL POTENTIAL SCATTERING 167 where V(x) is a smooth function of x. We shall also suppose that all polynomials, constructed out of V and of its derivatives of all orders, are integrable over the entire range, (-00,+ 00), of x. With these restrictions on V, the asymptotic behaviours of the solutions of equation (170), for x -* + 00, are given by e±iax. (171) We now consider two particular solutions, fx (x, a) and f2 (x, a), with the asymptotic behaviours /i(x,ff)-KJ-ta (x^+00) and (172) f2(x,a)^e + iax (x->_oo). Now/,(x,i7) and/i(x, — a) are two solutions of equation (170) which are independent, since their Wronskian [/1 (x, o\ A (x, - ff)] = f/i' (x, a)fx (x, - a) -ft (x, a)^ ' (x, - a)} = constant = lim {eb'x(-ia)e-iax-e-iax(ia)e + btx} = -2io±0. (173) Similarly, Ui {x, a\f2 (x, - «7)] = + 2i«7 ( ^ 0); (174) and/2(x, ff) and/2(x, —°) are also independent solutions of equation (170). It follows that there exist unique functions ^i(ff), R2{o), 7\(<r), and T2(a) such that /2 (x, ff) = ^1 A (x, «7) + -J- /1 (x, - a) (175) and /1 (x, ff) = ^ /2 (x, ^) + ^-/2 (x, - a\ (176) for a ± 0. From the assumed asymptotic behaviours/t(x, <r) and/2(x, a), it follows from the foregoing representations that K,(ff) 1 lim /2(x, .7) = -¾"»* + ——« + "»< (177) x-+oo 7\(<r) 7\(<r) and lim /i (x, a) = *^fl e + i°* + -±-e- •". (178) Accordingly, 7\(<t)/2 (x, a) corresponds to an incident wave of unit amplitude from + 00 (cf. equation (159)) giving rise to a reflected wave of amplitude Ri{o) and a transmitted wave of amplitude 7\(<r). Similarly, T2(a)fi (x, a)
168 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE corresponds to an incident wave of unit amplitude from — oo giving rise to reflected and transmitted waves of amplitudes R2(o) and T2(a), respectively. We define T,(a) R2{a) Rd°) T2(a) S(a) = (179) for a j= 0, as the scattering or the S-matrix. From the representations (175) and (176) and the relations (173) and (174), it follows that ^r— = -XT-[/i {x, o\f2 (x, ff)] = ——, and Therefore, ^r—- = -T^[/2(x,ff),/1(x, -(7 ], / t ((7) 2i(7 T2 ((7) 2iff (180) ^((7) = ^(.7) = ^(7) (say), Jtt((7)= Jt2(-(7) Jtl(-g) r(ff) r(-ff) ' ^ 7-(-(7) RAa) 7"((7) (181) (182) Besides, T* ((7) = 7(-(7), 7^((7) = 7^(-^), and Rf (-a) = 7?2((7). (183) Next, inserting fovf2(x,a) its representation (175) in equation (176) for fi (x, a) and equating the coefficients of/\ (x, a) and/! (x, — a) in the resulting expression (which we may since fx (x, a) and /t (x, — ff) are independent solutions) we find l=R2(a)R1(a) + 1 7-((7)7-((7) 7-((7)7-(-(7)5 (184) in addition to the relation (182). Making use of the relation s (182) and (183), we can rewrite equation (184) in the alternative forms \Rl(a)\2 + \T(a)\2 = \R2(a)\2 + \T(a)\2 = \. From equation (185) it follows that l*i (ff) I, 1*2 HI, and 17» I are <i; (185) (186) and further that 7?t((7) and R2(o)can differ only in phase. The relations (181), (182), and (183) establish the symmetry and the unitarity of the S-matrix.
ONE-DIMENSIONAL POTENTIAL SCATTERING 169 (a) The Jost functions and the integral equations they satisfy In the theory of potential scattering one defines the Jost functions, m1(x,(7) = e + iffYi(x,ff) and m2 (x, a) = e - ">xf2 (x, a), (187) which satisfy the boundary conditions (cf. equation (172)) mt(x, <t)-»1 as x -> + oo and m2(x, <r)-> 1 as x -> — oo. (188) Also, we shall now let a be a complex variable with Imff < 0 so that the functions,/t (x, <r) and f2(x,a), as defined in equation (172), vanish for x -* + oo and x -* — oo, respectively. In terms of the Jost functions, we can rewrite equations (175) and (176) in the forms T(o)m2(x,o) = Rl(o)e-2i<!Xml(x,o) + mi(x, -a) (189) and r^Jm^x.ff) = R2(a)e+2iaxm2(x,a) + m2(x, -a). (190) From the behaviours (188) it follows in particular that m2{x,a) = R^-e-2i"x + ~ + o{\) as x -» + oo (191) T{a) T(a) and «i(x,ff) = ^^ e + 2iffX + J- + o(l) as x^-oo. (192) / (a) / (a) We also note that the Jost functions satisfy the differential equations d2mt dml dx2 '" dx and — 2ht—— = Km^ d2m2 dm2 (193) We shall now obtain an integral equation for m2(x, <r). Letting f2(x,o) = eiax + ip(x,o) (ip-^O as x -» - oo), (194) we find that t// satisfies the differential equation d2 \ -^ +a2 U = (eiax + ¢)V. (195) dx J Solving this equation with the aid of the Green function, G(x-x') = -sinff(x-x')0(x-x'), (196) a
170 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE where 0(x — x') is the step function, 0(x — x') = 1 for x > x' and =0 for x < x', (197) we obtain (remembering that i^ -> 0 as x -> —oo), 1 ^/(x, a) = 2ia ry<r(x-x')_e-;ff(x-x')-j j/(x')[eiffX' + ^(x',ff)]dx'. (198) Rewriting this equation in terms of the Jost function, m2(x, a) = e-iaxf2(x, a) = 1 + e-iax\j/(x, a), we obtain m2(x, a) = 1 1 2io (e2iff(x' -x) _ j j v(x')m2 (x'5 ff) dx' (199) (200) Equation (200), for a ± 0, is a Volterra integral-equation for m2 (x, <r); and, for Im a < 0, its solution obtained by repeated iterations always converges for any smooth V which is integrable. It is also apparent that the solution obtained by such iterations provides an expansion for m2(x, a) in a power series in inverse a. Based on these facts, it can be shown that m2(x, a) is an analytic function in the lower-half complex <r-plane (i.e., for Im a < 0); and, further, that it is continuous for Im a < 0 (a ± 0). From equation (200) it follows that m2(x, a) = — e — liax 1 2io + 1 + 2ia = —e -liax 2ia e + 2iax'V(x')m2(x',a)dx - OO V(x')m2(x\ ff)dx' °° e+2iax'V(x')m2(x\a)dx' + 1 + 2ia V(x')m2(x', ff)dx' + 0(1) for x->oo. (201) Comparison of this last result with equation (191) shows that *i(g) = __L "' 2io and 7» 1 e2,ffXK(x)m2(x,ff)dx 1 = 1 + T(a) 2ia V(x)m2(x, a) Ax. (202) (203) It is clear that similar equations for R2 (a)/T(a) and 1 /T(a) can be written down in terms of the Jost function mt(x, a). From equation (203), we may draw one important conclusion. For Im<7 < 0,
ONE-DIMENSIONAL POTENTIAL SCATTERING 171 we can obtain a convergent expansion for 1/T(a), by inserting for the Jost function, m2(x, a), its solution obtained from the Volterra integral- equation (200) by repeated iterations. An expansion for 1 /T(o), obtained in this fashion, will be a power series in inverse a with coefficients which will be bounded so long as V(x) satisfies the requirements we have imposed on it, namely, the boundedness of integrals of all polynomials constructed out of V and its derivatives. In particular, it follows from equation (203) that r^=1"i V(x)dx + 0((7-2). (204) (b) An expansion of\gT{a) as a power series in a 1 and a condition for different potentials to yield the same transmission amplitude We have seen how, by combining equations (200) and (203), we can obtain an expansion for 1/T(a) as a power series in a~l. But it appears that to obtain the coefficients in this expansion explicitly, it is more convenient to follow a different route due to Faddeev. With the substitution, m2 (x, a) = ew<x' "\ (205) equation (193) satisfied by m2(x, a) becomes w" + Haw' + (w'f - V = 0, (206) where primes are used to denote differentiations with respect to x. From the fact that m2 -* 1 as x -> — oo, we conclude that w->0 as x-» -oo. (207) Also, from equation (191) we similarly conclude that w-> — \gT(a) as x -* + oo for lm<r<0. (208) Now letting w(x, a) = we obtain the equation v(x',a)dx (w' = v), (209) v' + 2iov + v2- V = 0. (210) The requirement (208) now implies that lg?» = v(x, a) Ax. (211) We now seek a solution of equation (210) by expanding v in a power series i a~l; thus v = ) , (212)
172 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE where, according to equations (204) and (211), Vl = v. Inserting the expansion (212) in equation (211), we obtain \gT(a) = - ^ c„<x-\ where (2i)aca = v„(x)dx. (213) (214)* (215) Now substituting the expansion (212) in equation (210) and equating the coefficients of the different powers of l/a, we obtain the recurrence relation, V„= ~v'n-l- X ^1^-1-1- (216) Using this recurrence relation, we can solve for the t;„'s successively starting with »!= V. We find v2= -v\ = - V, v3= -v'2-vl= V"-V2, vA= -v'3-2Vlv2= -V'" + 2(V2)', v5 = -v'A-2Vlv3-v2 = V""-3(V2)" + (V')2+2V\ etc. J (217) (218) The coefficients of odd orders, c2n+l, in the expansion for \gT(a) are, therefore, 2icl = Vdx; -(2i)3c3 V2dx; (205c5 = (2K3+K'2)dx;etc, (219) while all the coefficients of even order vanish. Now if two different potentials should yield the same T{a\ then the derived series expansions for \gT{o) (convergent for Im<7 < 0) for the two potentials must coincide term by term. In this manner, we find that the integrals of the * 1 am grateful to Dr. Roza Trautman for pointing out to me that this result follows directly for the following alternative expansion for lg7"(<r) due to V. E. Zakharov and L. D. Faddeev, Funct. Anal. Appl., 5, 280 (1971): ^^) = ^ f ^rj Vr'lg|7>')l2d<r'-
ONE-DIMENSIONAL POTENTIAL SCATTERING 173 following quantities for the two potentials should be the same: (i) V; (ii) V2\ (iii) 2V3 + V'2; (iv) 5VA + WVV'2+ V"2; (v) UV5 + 10V2V'2+UVV"2 + V'"2; etc. The integrals we encounter here are formally the same as the conserved quantities of the Korteweg-de Vries equation, ",r-6"",x+",x,x,x = 0- (221) This coincidence is not an accident; but it will take us too far afield to state the reasons. The interested reader may wish to consult the relevant literature quoted in the Bibliographical Notes at the end of the chapter. (c) A direct verification of the hierarchy of integral equalities for the potentials K<±) = + /?/' + /?2/2 + «f The various integral equalities required of the potentials V{ + ) and V{~), as given in equations (136) and (137), can be verified, individually, by evaluating the expressions (220) for V = pf' + P2f2 + Kf (222) and showing that the terms odd in /? are expressible as derivatives of combinations of/and its derivatives and, therefore, vanish on integration (by virtue of our assumption that/and all its derivatives vanish for x-» + oo). The equality of the integrals of V{+) and K'"' follows, for example, from the fact that the integral of /?/' vanishes. Considering next V2, for the terms odd in /?, we have 2/?(/?2/2 + Kf)f = lp3(f3y+Kp(f2y. (223) The integral of this expression clearly vanishes; and the remaining terms, even in /?, are the same for K< + ) and V{~). In the same way, the terms odd in /?, in the expression for 2V3 + V'2, can be reduced to the form 2(}3(ff'2)' + Kp(f'2)' + 6/?(/?2/2 + «f)2f; (224) and the integral of this expression also vanishes. The equality of the integrals of the two remaining quantities listed in (220) can be similarly verified, although the complexity of the reduction rapidly increases. A direct evaluation of the integrals of the expressions (220) fbr the potentials Vi+) and V{~) appropriate for the Schwarzschild black-hole yields the (220)
174 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE following results: ® i(2p"3); (H) ^(V-18P + 18); (ni)^l^(16p3_83p2+l50p-87); . 1 / 1 4 53_ 3 j263_ 2_}4T_ 444 \ (1V) 128M71V168P ~1386P + 2772P 1430P + 10010/ 1 ( 14 , 41 . 56 , 2557 , (v) 77^779 77^P-^^rP+' " - 512M9\6435K 2574 K 1155F 34320* 1203 723 \ /#v_ + 19448p-38896j' (225) where p = 2(n+ 1). 29. Perturbations treated via the Newman-Penrose formalism The treatment of the metric perturbations in the previous sections leaves one with a sense of bafflement: relations have emerged whose origins are not clear. Thus, why should the polar and the axial perturbations define the same reflexion and transmission coefficient? Why should the solutions describing them be related as simply as they are? And why, indeed, should the potentials V{ +' and V{ ~' be derivable from the same formula with only a change of sign in one of the terms? While the deeper physical origins of these relations are still obscure, they emerged, first, from the alternative treatment of the perturbations via the Newman-Penrose formalism. And to this study we now turn. We shall assume that the perturbations in the various quantities have a time (t) and an azimuthal angle (<p) dependence given by eHot + m<p)^ (226) where a is a constant and m is an integer positive, negative, or zero. The directional derivatives D, A, S, and S* along the basis null-vectors set up in Chapter 3 (equation (281)), when acting on functions with a t- and a <p- dependence given by (226), become 1 1 m^(5=—/r-i?,1,, and m = 5* -—xr^o. ryj2 r^/2 (227)
PERTURBATIONS VIA NEWMANN-PENROSE FORMALISM 175 where ir2a r — M £#„ = or+ —— + 2« —-—, i£n = 8e + n cot 6 + m cosec 6, t ir2a r — M t Si I = d,—— + 2n—-—, and i£ \ = de + ncot6-mcosec0. (228) It will be noticed that while Qin and 3s\ are purely radial operators, y„ and Z£\ are purely angular operators. The differential operators (228), which we have defined, satisfy a number of identities which we shall find useful; they are Si = (&.)*; <el(6)=-Sen{n-6\ AS>„+1 = S>„A, sin 0 ifn+1 = if „ sin 0. (229) Consistent with the type-D character of the Schwarzschild space-time (and as we have verified directly in Chapter 3, §21), the Weyl scalars 1*0,1^, T3, and 1*4 and the spin coefficients, k, a, A, and v vanish in the background. The only non-vanishing Weyl scalar is 1*2; and it has the value (Ch. 3, equation (290)) ¥2= -Mr'3; (230) and the non-vanishing spin-coefficients have the values (Ch. 3, equation (288)) A r-M _ M 2?' 1 „ cot e r r2s/2 2,.3- 7 = ^+--,.2 2r2 (231) (a) The equations that are already linearized and their reduction Among the various equations of the Newman-Penrose formalism listed in Chapter 1 (§8(c), (d), and (e)) there are six equations—the four Bianchi identities (Ch. 1, equation (321) (a), (d), (e), and (h)) and the Ricci identities (Ch. 1 equation (310) (b) and (j))—which are linear and homogeneous in the quantities which vanish identically in the background. The equations are and (5* - 4a + w)4»0 - (D - 2e - 4p)4*! = 3k¥2, (A - Ay + /*)4»0 - (5 - 4t - 2/?)^ = 3<r¥2, (D - p - p* - 3e + e* )a - (S - x + n* - a* - 3p> = T0; (D + 4e - p)T4 - (<5* + An + 2a)¥3 = - 3/14»2, (^ + 4/?-t)T4-(A + 27 + 4^),I'3 = -3vT2, (A + ^ + ^*+3y-7*)/l-(^*+3a+/?*+rt-T*)v = -T4. (232) (233) The foregoing equations are already linearized in the sense that since Tq, IV 1*3,1*4, k, ff, A, and v, as perturbations, are to be considered as quantities of the first order of smallness (with a t- and a (^-dependence given by (226)). We may
176 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE accordingly, replace all the other quantities (including the basis vectors and, therefore, also the directional derivatives) which occur in them by their unperturbed values given in equations (227), (230), and (231). We thus obtain ^(^o + 2cot0)¥o-(^o + *V = "7^' 2r A rJ-° V^l 'o+r)ff-^^(^o-cot0)»c = To; (234) and ^0 + -)^4 r I r 1 (i?0-cot 0)¥3= +^ A, r 7~2 ^(*J+2ca.*)*4+£(*l-^ + J)p,- ^ 3M 2(r-M) 4 (235) These equations take simple and symmetrical forms if we write them in terms of the variables (236) In this manner, we obtain the following basic set of equations: 3N and i?2O0-1 ^0 + -)01 = -6Mk, d®\—jOo + ifliO! = +6Ms, 0--|(D4-i?-i$3 = 6M/, if^4 + A(®t_1+-j*3 = 6Mn, a(V-i+^)/ + ^^=^. (237) (238) (239) (240) (241) (242)
PERTURBATIONS VIA NEWMANN-PENROSE FORMALISM 177 We can eliminate $t between equations (237) and (238) by applying the operator if t1 to equation (237) and the operator (3i0 + 3/r) to equation (238) and adding. The right-hand side of the resulting equations, apart from a factor 6M, is precisely the quantity which occurs on the left-hand side of equation (239). We thus obtain the decoupled equation seU&i + (®o+\\4®\-\ (D0=_ (D0. (243) Similarly, we obtain from equations (240)-(242) (by the elimination of ¢3) the decoupled equation if _ t if I + a(V_ t + ^) (^0-;) 0>4= ¢4. (243') r The identity, 6A 6M .2 r a(®!+-)(^5--)-- = h®^l+'l-ir20 + (r -M)-] = A@l@l-6ior, (244) reduces equation (243) to the form 1^-^2 + (A^i@\ - 6KTr)]d>0 = 0. (245) Similarly, equation (243') reduces to the form [if _t if ^ +(A® 1^0 + 6^)] ¢4 = 0. (246) Equations (245) and (246) allow a separation of the variables. Thus, by the substitutions, ®0 = R + 2(r)S+2(6) and ¢4 = K_2(r)S_2(0), (247) where R±2 and S±2 are functions, respectively, of r and 0 only, we obtain the two pairs of equations ifttif2S + 2= -,<2S+2, (248) (AS!lSil-6iar)R + 2= +fi2R + 2; (249) and ^.^1$^ = -^$.^ (250) (A$ll@0 + 6ior)R-2= +n2R.2, (251) where fi2 is a separation constant. It will be noticed that we have not distinguished the separation constants that derive from equations (245) and (246). The reason is the following: considering equation (248), we first observe that fi2 is a characteristic-value parameter that is to be determined by the requirement that S + 2(0) is regular at
178 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE 0 = 0 and 0 = n. On the other hand, since the operator acting on S_2 on the left-hand side of equation (250) is the same as the operator acting on S+2 in equation (248), if we replace 0 by n — 0, it follows that a proper solution, S + 2 (0; \x2), of equation (248), belonging to a characteristic value ix2, provides a proper solution of equation (250), belonging to the same value /i2, if we replace 0 by % — 0 in S + 2(0; n2). In other words, equations (248) and (250) determine the same set of characteristic values for \i2. The characteristic values of \x2 can be ascertained, without loss of generality, by considering the case m = 0. (The reasons are the same that allow us to restrict ourselves to axisymmetric perturbations of an initially spherically symmetric system.) On expanding the equation governing S + 2, for m = 0, we find d S+2 , „„,Qd5+2 -,,„^2 a , 2mc _ _ „2 do2 de By the substitution, + cot0~7^-2(cot20 + cosec20)S + 2 = -fi2S + 2. (252) S+2(0) = C(0)cosec20, (253) the equation becomes Sin3^sin^^^2C = 0- (254) Comparison with equations (20) and (21) shows that C(0) = C^212 (0) and fi2 = In = (I - 1) (I + 2). (255) Therefore, S+2(0) = Cr+T (9) cosec2 0 = sin 0 A J*£ d0 sin 0 d0 = pi,e,e-pi,ecote (m = °>' (256> where Ph as usual, denotes the Legendre function. When m ± 0, S+2(0) becomes a 'spin-weighted' spherical harmonic; but the value of fi2 is unaffected. Returning to the radial equation (249), we observe that A2R + 2 satisfies the equation (A2-M-6i<rr)A2R + 2 = fi2(A2R + 2); (257) and this is the complex conjugate of equation (251) satisfied by R-2. We shall transform equation (257) to a standard recurrent form in the theory. First, we observe that 2 r2 £>0 = -r A+ and ®\ = ~r A- (258) A A
PERTURBATIONS VIA NEWMANN-PENROSE FORMALISM 179 where (cf. equation (29)) d d A d — ±ia and =-=- — dr+ dr+ r dr A+ = —+ jff and - = ^-. (259) (260) Accordingly, A2-M = &2®o\®l = r2AA+ (^-)- Also, replacing A2R + 2 by Y+2 = r~3A2R + 2, (261) we can rewrite equation (257) in the form A + T2A-(r3r+2) ■6ia~Y+2=fi2^Y+2. (262) On further simplification, equation (262) can be brought to the form A2Y+2 + PA_Y+2-Gy+2=0, (263) where d r8 4 ^ = 3^72= 121'-3**) (264> and drt °A2 r2 S=4(A + 6M). (265) Equation (263) is the form in which we shall mostly use the equation for l^; and it is a form which we shall encounter frequently. A similar reduction of equation (251), governing R-2, will lead to the complex conjugate of equation (263). Thus, with the substitution Y_2 = r-3K_2, (266) we find A2Y_2+PA + Y_2-eY-2 =0- (267) With Y+2 and Y*2 satisfying the same equation, we readily deduce that r8 -2(7*2A- Y+2 - Y+2 A_ Yf2) = constant. (268) Since y_*2A_y+2-y+2A_y_*2 = y+2.,.y*2-y+2y*2.,,, (269) it follows that the Wronskian, [Y+2, Y-2], of the-solutions Y+2 and Y*2, is given by IT+2, y-2] = constant r-8A2; (270)
180 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE or, in view of the substitutions (261) and (266), \_A2R + 2,R*-2~\ = constant r 2 A2. (271) This last relation must clearly imply a 'conservation law', even as the constancy of the Wronskian, [Z(±), Z(±)*], led to the conservation law incorporated in equation (163). Indeed, we shall find in §32 that the implications of equation (271) are the same as those of equation (163). (b) The completion of the solution of equations (237)-(242) and the phantom gauge While we shall not pursue in this chapter the complete solution of all the relevant equations of the Newman-Penrose formalism—we shall do so in Chapter 9 in the more general context of the Kerr black-hole—we shall, nevertheless, complete the solutions of equations (237)-(242) since they will be needed in some of the considerations of Chapter 5. Two preliminary observations:^'^, we are provided with only six equations for the eight unknowns which they involve—a fact which implies that their solution must involve two arbitrary functions; and second, equations (237)-(239) governing 3>0, $t, k and s and equations (240)-(242) governing 3>4, ¢3,I, and n are decoupled. This latter decoupling of the two sets of equations has far-reaching implications—implications whose consideration we shall again postpone to Chapter 9. Returning to equations (237)-(242), we have shown how these equations lead to independent equations for <S>0 and ¢4. Clearly, the solutions for the remaining quantities must involve two arbitrary functions. The origin of this arbitrariness is clear. From our discussion of the effect of tetrad rotations on the various Newman-Penrose quantities in Chapter 1 (§8(3)), it follows that To and T4 are affected only in the second order by first-order infinitesimal rotations of the tetrad basis in a background in which *¥0, IV ¥3, and ¥4 vanish. But this is not the case with 1^ and ¥3: they are affected in the first order (see particularly Ch. 1, equations (342) and (346)). In other words, in a linear perturbation-theory, Tq and ¥4 are gauge-invariant quantities while 1^ and T3 are not. Consequently, we may, for instance, choose a gauge (i.e., subject the tetrad basis to an infinitesimal rotation) in which 1^ and ¥3 vanish (without affecting ¥0 and ¥4). If a choice of gauge, in which 1^ and ¥3 are zero, is made, then the corresponding solutions for k, s, I, and n can be directly read off from equations (237), (238), (240), and (241); -6Mk= R + 2y2S+2, 6Ms = S + 2A(@l-3/r)R + 2C\ and \ (272) + 6Mn=R_2y\S-2, 6M/ = S_2(S>0-3/r)K_2. J
PERTURBATIONS VIA NEWMANN-PENROSE FORMALISM 181 One important piece of information is, however, missing in the foregoing solutions: we do not, as yet, know the relative normalization of the radial functions A2R + 2 and R-2. We shall obtain this information in §32. Other choices of gauge, besides the one in which 1^ and ¥3 are zero, can be made; and one in particular which brings to equations (237)-(242) a symmetry which they lack. Thus, considering equations (237)-(239), the symmetry of these equations in 3>0, k, and s is only partially present in 4>u k, and s. Equation (239) is, for example, the 'right' equation which allows us to obtain a decoupled equation for 3>0 after the elimination of $t between equations (237) and (238). But a similar elimination of <t>0 does not lead to a decoupled equation for ¢^ since we do not have a 'right' fourth equation. However, exercising the freedom we have to subject the tetrad frame to an infinitesimal rotation, we can rectify the situation by supplying (ad hoc?) the needed fourth equation. Thus, with the additional equation, 3\ 2 i\—)k + S'2s = -^l, (273) r J r we can eliminate <t>0 between equations (237) and (238) to obtain the decoupled equation >J-*Vs0+^) + ^2^-1 $! = 12 — d>t. (274) r On expanding this equation, we find \_(&9\20-6ier) +#2^-^1 =0. (275) Similarly, supplementing equations (240)-(242) by the additional equation ^0-^/1-^1/ = ^3, (276) and eliminating ¢4 between equations (240) and (241), we obtain the decoupled equation [(A®i®t_1+6iffr) + i?Ji?'_1]O3=0. (277) Equations (275) and (277) are clearly separable: by the substitutions <i>,=R + l{r)S+l{Q) and ¢3 = /^(1-)5^(6) (278) —we justify the designation of these functions with the subscripts + 1 and — 1 presently—we obtain the pair of equations and (A®t@0-6ior)R + l = +fi2R + u &2&tiS+1 = ~n2S+l; (279) (A®!®^+6^)/^ = +/^2K_1,if^if_1S_1 = -/i2S_i. (280)
182 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE It will be noticed that in equations (279) and (280), we have used the same separation constant ix2 as in equations (248) and (250). The reason is the functions S+1 and S_ t are simply related to the functions S + 2 and S_2. The following relations are manifestly true if the functions are suitably normalized: &2S + 2=+nS+1, ^t1S+1=-liS + 2; (281) and SetS.2=-iiS.lt ^-,S.,= +/iS_2; (282) and the regularity of S±2 at 6 = 0 and n guarantees the regularity of S±l as well. We shall find (cf. Chapter 5, §46) that the functions ¢^ and ¢3, as defined by equations (278)-(280), describe Maxwell's field in Schwarzschild's geometry*. Thus, we have, in effect, derived Maxwell's equations (appropriate for photons with spin+ 1) by finding a gauge which rectifies the truncated symmetry of equations (237)-(242) in the quantities which occur in them. The designation of the relevant radial and angular functions by the subscripts + 1 is, therefore, justified. Because of the apparent veiled 'awareness' of Schwarzschild's geometry to the existence of Maxwell's field, we shall call the gauge, in which equations (273) and (276) are true, the phantom gauge. Finally, we may note that the solutions for the spin coefficients in the phantom gauge are -6Mk = S+ll + i*R + 2-{S0 + 3/r)R + l], + 6Ms = S + 2l-pR + l+A(Dt-3/r)R + 2], I + 6M/ = S_2[-^_1 + (^0-3/r)K_2], J + 6Mn = S_ t[ -fiR-2 + A(S>t t + 3/r)K_ J., We notice that by virtue of the relations (281) and (282), the foregoing solutions for the spin coefficients continue to be separable in r and 6. We shall find that this separability plays an important role in the treatment of the perturbation of the Reissner-Nordstrom black-hole in Chapter 5. 30. The transformation theory In the preceding section, we have shown how the Newman-Penrose formalism leads to a pair of complex-conjugate equations for the radial functions, A2R + 2 and K_ 2. As we shall see in detail in §32, there is, associated with equations (263) and (267) and the Wronskian (271), a problem of reflexion and transmission of incident gravitational waves which has the same physical content as that associated with the equations governing Z(±) and the Wronskian (161) leading to the conservation law (163). Indeed, the require- * We treat Maxwell's equation, in the more general context of the Kerr geometry, in Chapter 8.
THE TRANSFORMATION THEORY 183 ment, that the theories developed along the two avenues—the Newman-Penrose formalism and the linearized Einstein equation—be consistent with one another, is ultimately the reason why the axial and the polar perturbations determine the same reflexion and transmission coefficients. A corollary to this inference is that it should be possible to express the functions Y±2 and Z(±), explicitly, in terms of one another. This possibility—rather, the certainty—that a solution, Y+2, of equation (263) can be transformed, simultaneously, to provide solutions of either of the equations governing Z(+) or Z(""\ implies that there is a special feature of equation (263) which will make this possible. We shall now set out to find what this special feature is. The problem is to express the solution of an equation of the form A2Y+PA_Y-QY=0 (284) in terms of the solution of a one-dimensional wave-equation A2Z = VZ, (285) where n d f, r drA BA lg72 I- (286) and Q and V are certain functions which we shall, for the present, leave unspecified. Since Y and Z both satisfy equations of the second order, there is no restriction to assuming that Yis a linear combination of Z and its derivative. But instead of making this assumption simply as we did in §26 when considering a similar problem, we shall now assume that Y= /A + A + Z+WA+Z, (287) or, equivalently, Y= fVZ + (W + 2iaf)A + Z, (288) where f and Ware certain functions of r+ to be determined. It may appear odd that the-expression of Y, as a linear combination of Z and its derivative, is made in this oblique fashion. That it is the most appropriate one to make in the context of equations (284) and (285) emerged in a different context which will be clarified in §31 below. Applying the operator A _ to both sides of equation (287) and making use of the fact that Z has been assumed to satisfy equation (285), we find A_Y= or, with the definitions, A-{fV)+WV z + dr* A + Z; (289) .p^. = A-(fV)+WV (290)
184 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE and , R= fV + — {W+2iaf), (291) dr* we can write A2 A_y= -p-jrZ + RA + Z. (292) r° Next, applying the operator A _ to equation (292), we obtain (again making use of the equation satisfied by Z) A2 dB A2 d /A2 A_A_y= -p-^-wz-JL^z-pz-f^ + RVZ + -R-\ + Z. (293) On the other hand, by equation (284), A_A_y= - (P + 2kt) A _y+Qy, (294) or, substituting for Y and A_yfrom equations (287) and (292), we have (cf. equation (286)) ( d r8\/ A2 A_A_y= -(^ + - lg^-^Z + KA + Z + QlfVZ + (W+2iaf)\ + Z-\. (295) Since the right-hand sides of equations (293) and (295) must identically be the same, we can equate the coefficients of Z and A + Z. In this manner, we obtain RV-^^- = QfV, (296) r drt and dR A2 ( d r8\ — -p- = Q{W+2iaf)-(2ia + — lg-^JR. (297) We shall rewrite equations (296) and (297) in the forms -%■¥- = (Qf-RW, (298) r° dr+ and ^r(^R) = ^lQ(W+2iaf)-2i<TRl + p. (299) It can now be verified that equations (290), (291), (298), and (299) allow the integral r8 j-jRf V + P(W+ 2io f ) = K = constant. (300)
THE TRANSFORMATION THEORY 185 This integral is the present analogue of the integral (147) which was found in §26 in a similar context. The integral (300) enables us to write the inverse of the relations (288) and (292) in the forms KZ=~RY-^(W+2iof)A-Y, r8 KA + Z=pY+jIfV\_Y. (301) (a) The conditions for the existence of transformations with f = 1 and p = constant; dual transformations We may formally consider any four of the five equations (290), (291), (298)-(300) as equations governing the five functions p, f, R, V, and W. We now ask for the conditions when the transformation equations will be compatible with P = constant and / = 1. (302) We shall find that transformations compatible with the requirements (302) are possible only if Q satisfies a certain non-linear second-order differential equation even in p. By virtue of this last fact it will emerge that the requirements (302) are precisely those which will permit transformations of the equations governing Y, simultaneously, to either of the equations governing Z( + ) and Z(_) with V{ + ) and V{') of the forms stated in §26 (equation (133)). When P is a constant and / = 1, equation (298) requires that R = Q. (303) Equation (291) then gives AW V = Q-—. (304) ul * The remaining equations—which we may take to be (299) and (300 d r8 \ (r 2R = [ Q)W+P (305) and Letting dr+\A2 / IA ^jQ)V + pW=K- 2iap = co nstant = k (say). (306) F = ^G, (307)
186 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE we can rewrite equations (305) and (306) in the forms and W=l-i^--P) (308) FV + PW=F[Q-—) + PW=k. (309) Eliminating W between these last two equations, we obtain d/ldF A JJiF_ \ or, after some simplifications, we have 1/dFV d2F A2 . p2 ,„((1 F{dr-J-*i+SF =T + K- (3U) Since F must be assumed as given, equation (311) provides a necessary and sufficient condition for the transformation equations to be compatible with the requirements (302): there must exist constants P and k such that equation (311) is satisfied by the given Q = A2F/rB. Since P occurs as p2 in equation (311), two transformations, associated with + P and — /?, are possible when the equation is satisfied. We shall call them dual transformations; and, as we shall presently verify, they lead directly to the equations governing the axial and the polar perturbations. Distinguishing the transformations associated with +P and — P by superscripts ( + ), we can write and J/(±) = Q-^—. (313) dr+ Rewriting this last equation in the form and making use of equation (310), we obtain r{±) = i;*-*i\Z-*ri (315) 1 T-P-(— F F2\dr, Letting f=F~\ (316)
THE TRANSFORMATION THEORY 187 we obtain the formula dr„ The explicit forms of the associated transformations are Y = J/<±»Z<±»+(H^<±» + 2Jff)A+Z<±), and where A_y= + fi-s-Z^ + QA + Z^\ K'i'Z**' = -I[Qy-(^<±» + 2iff)A_ Y], 8 K<±)A + Z(±»= +)3Y+-1K<±»A_Y, K(±>=K + 2iffjS. (317) (318) (319) (320) And finally, we may note that by making use of the relations (318) and (319), we can relate the solutions for Z( + ) and Z(_). Thus, letting Z(-) denote a solution of the wave equation with the potential V{'\ we may use the first of the equations (319) to express it in terms of Y and A _ Y and then express these in terms ofZ( + ) and A+Z(+) with the aid of equations (318). By this procedure, we obtain A2 K(-)Z(-) = Gy_(^(-) + 2jff)A_y = Q[l/( + ,Z( + , + (^( + , + 2Jff)A+Z( + )] -(W^ + lio) -jS-8-Z( + , + eA+Z<+> (321) or, after some regrouping of the terms, we have K(-)Z(-) = QV{ + ) + p{W<> + ) + 2ia) + P{W^)- W<> + )) Z( + ) + f(H/( + >_H/(->)A+z< + >. (322) The expression on the right-hand side of equation (322) can be simplified by making use of equations (306), (312), (316), and (320); and we find (k- 2/(7)3) Z<-> = (K + 2p2f)Z{ + )-2p dZ< + > dr* " The inverse of this relation is (K + 2iaP)Z{ + ) = {K + 2p2f)Z{')+ip dz<-> dr* (323) (324)
188 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE We have thus recovered the relations (150) and (151) which we derived earlier by the transformation directly relating the solutions of equations (136) and (137). (b) The verification of the equation governing F and the values of k and p2 For the case on hand, Q is denned in equation (265). Therefore, F = ^Q=j(n2r + 6M) = ~. (325) We shall now verify that this expression for F satisfies equation (311) for certain determinate values of fi2 and k. Thus, considering the left-hand side of equation (311), we find -F r2 dr \r2 dr J r8 = fi2 (fi2 +2) + 36M2 r3(fi2r + 6M) = ^(^+2) + ^--, (326) and this must equal k + p2/F. Therefore, F as defined in equation (325) does satisfy equation (311) with P2 = 36M2 and k = ^2 (n2 + 2). (327) Therefore, equation (263) governing Y allows the dual transformations considered in §(a) above: and the expression (317) for K(±) with P = 6M and/ as defined inequation (325) are what was quoted in §26, equations (133)-(135). Besides, with k, P, and F as presently defined, the relations (323) and (324) are the same as those derived earlier on the assumption that V{ + ) and V{~) have the forms that we have now derived for them. And finally, we may note that for F given by equation (325), 2 ,_ ,.., „_, „„ + , ^2r2-3»2Mr-6M2 r2(n2r + 6M) W^) = ^(r-3M) and W^ + ) = 2M \ ^t^ . (328) 31. A direct evaluation of T0 in terms of the metric perturbations In the preceding section, we have related the Weyl scalar, %> = ^+2(r)S + 2(0), (329) to the metric perturbations by showing how the function Y+ 2 can be expressed in terms of Z(+) and Z(-) by the dual transformations incorporated in
EVALUATION OF "F0 IN TERMS OF METRIC PERTURBATIONS 189 equations (318) and (319). We shall now complete the full circle by evaluating To directly in terms of the metric perturbations. By definition ,^'o = K<P><,><r>w'(P,»I(,,'(r,»I(s,•- (330) We have already shown in Chapter 2 (§21) that ¥0 (as well as 1^, ¥3, and ¥4), for the Schwarzschild space-time vanishes as required by its type-D character. Therefore, in evaluating the perturbed value of Tq, it will suffice to contract the perturbed Riemann-tensor with the unperturbed basis-vectors since the contributions arising from the contractions of the unperturbed Riemann- tensor with the perturbed basis-vectors I and m (expressed, in turn, as linear combinations of the unperturbed basis-vectors) vanish by virtue of the vanishing of unperturbed Weyl-scalars Tq, IV ¥3 and IV Thus, we may write T0 =-SR (p)(«)(r)(s) /(P)m(«)/(r)mW (331) where we have enclosed the indices in parentheses to emphasize that they signify tetrad components in the frame in which we have evaluated the Riemann tensor in Chapter 2 (equations (75)). The tetrad components of the basis vectors can be obtained by applying to the tensor components (given in Chapter 3, equation (281)) the transformation represented by „(«> _ ev 0 0 0 0 r sin0 0 0 0 0 e~* 0 0 0 0 r (332) where e2v = A/r2 = (1 -2M/r). We find IIP) = (/<o /W /w /(«)) = (e-»f0,e'\0), n<p> = („«>,„{<p\„M „(«)) = \(e + \0,-e + \0), m(P) = (mW m(9\ mw m(0)) = _L (o, if 0, 1). (333) Contraction of the Riemann tensor with these vectors gives T0 = -ie-2*(5R030l +SR232l +5R230l+5R032l) -$e-2*(5R0303+25R0323+5R2323 -SR0l0l-2SR0l2l-SR2l2l). (334) It will be observed that Tq is decomposed into an axial part—the first group of terms, odd in the index 1, which reverses in sign with a reversal in the sign of <p, and a polar part—the second group of terms, even in the index 1, which is invariant to a reversal in the sign of <p.
190 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE (a) The axial part of Tq The relevant perturbed components of the Riemann tensor that are included in the axial part of Tq can be read off directly from the list given in Chapter 2. We find ^0301=^-^-^603.0-^-^-^623^, ^2301 =^-V-"2-"3 (;+5vr)g23 + ±g23.. 6o3.2-6o2,3+ 6o3(--V.r )-602 COt0 5*o32i = -ie*-"-"'-'" 620.3 + 620 cot 6 - Q23.0 + Q30l — v. , (335) where it may be recalled that we are now restricting ourselves to axisymmetric perturbations (which, as we have explained in §23, implies no loss of generality). Inserting the expressions (335) in the terms which occur with the factor — j in equation (334), we find .ImTn = ^ + ^G23r+le-2vQ23o 1 + 2e~ 6o3,r + 2(^-V.r ]Q03 + 2e-4v6o3.ohin0- (336) On the other hand, according to equations (13), (14), (15), and (19), 623 = Q(r)cr+y2(Q) ia Asm3 6 e and 203.0=^^^0.^,-^(^. (337) (338) Inserting these expressions in equation (336), we find -----^(1) + ½^ + ^T 2ioA d/AdQ\ A/1 sin20 (339) where we have suppressed the common factor e"". Simplifying this last
EVALUATION OF "F0 IN TERMS OF METRIC PERTURBATIONS 191 equation with the aid of equation (23), we obtain "1 d - Im ¥n = 1 ia = <ia 2dr\AJ rA 2A dr n n r-3MdQlCf+32/2(0) ^A" drj ia sin2 6 IdQ Q r-M A dr rA A2 W -ffVG + 3TG + 'Ve | /i r-3Mdg|Cr+32/2(e) r2A r2A dr J iff sin2 0 " Now letting e = »-z«->, (340) (341) as in equation (25), we find, after some further simplifications, that equation (340) becomes , r/ r-3M\dZ(-» + A' (H2 + 2)r-6M . r(r-3M) 2r2A - + ia - '(-) cr+y2(0) ia sin2 8 ■ (342) Recalling the definitions (27) and (328), namely, 0 A W{~)=^(r-3M) and V{') = -^[(n2 +2)r-6M], (343) r r we can now rewrite equation (342) in the form - 2ia Im ¥n = (2i<7+ W(->) dZ< > dr* + (K(_)+i<7 H^(-,-2ff2)Z(-) or, alternatively, c(-+32/2(0) sin20 ,cr+32/2(e) -2iffIm¥0=^[F<->Z(-> + (^(-> + 2iff)A + Z<->]-^ (344) (345) In view of the substitutions (256) and (261), the foregoing equation is in complete accord with equation (318) which we arrived at via the Newman-Penrose formalism and the transformation theory.
192 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE (b) The polar part of Tq The relevant perturbed components of the Riemann tensor, which are included in the polar part of Tq in equation (334), can be obtained by linearizing the corresponding components listed in Chapter 2 about the unperturbed Schwarzschild-metric. We find 2v /z \ i -<5K0303 = -2^2^rvir + e2vf-^ + vir^3>rJ + -I^vee + e"2vff2^3, -<5Koioi = -2&n2 — vr + e2v^+e2vvr5il/ r + C^-5v() + a2e-2v5il/, 1 v <W,r + ( " + V,r )dn3,r ^ -SRl2i2 =2(5¾ — v,r-e2v r W,rM-r + \)w,rJjf COt0 -^2-^2,( -^^0323 = ~'° — <5^0121 1"' $H3.r + ~ (<5/*3 - dUl) - V,r<5/*3 -^,r+ -(^2 -Sil/) + vrStl/ (346) Inserting these expressions in the terms which occur with the factor — \e 2v in equation (334), we find Re¥0=:h? (5il/-5fi3) \^-rTr) + 2l\Tr+-r-V') -ff2e-2.W_^3) + _/__cot0_j(5v_^2) (347) In accordance with equations (36)-(39), we may substitute in these equations ^-^3 = - V(PlAe-PL9cot6) = - V—^f1 and dv-dn2 = (N-L)Pl We thus obtain 1 (348) -Re ^0 = d2 2d „. .(A 1 -~ + -— + 2iae 2v — arA rar \dr + — vr |-e"4V V ^(N-L)|C-(0) ,-2v ~r~2 sin2 6 (349)
THE PHYSICAL CONTENT OF THE THEORY 193 or, making use of equation (51) and rearranging the terms, we have -2v -Re¥0=' v.rVtr + o2e-*vV + -—N -Viae Kr + |~v, W cr+V2(0) sin2 6 (350) Simplifying the first group of terms (without the factor ia) with the aid of equation (57), we find Re¥0 = lar r-2M r-3M V r + - ^r-V + •r ' r(r-2M) M (r-2M)2 n(r-2M) (L + X)-V + nr2-3Mnr-3M2 M[nr2 -M(2n- l)r-3M2] nr2(r-2M)2 nr2(r-2M)3 M (r2 - 3Mr + 3M 2) , 1 Cf+32/2 (0) r2(r-2M)3 K sin20 (351) After some further reductions in which we make use of equations (57) and (60), the foregoing equation can be brought to the form 1 f2n2(n+l)r3 + 6Mn2r2 + 18M2nr+18M3 -Re^o-^| r2(nr + 3M)2 + 2ia : + 2 nr2-3Mnr-3M2 r-2M (r-2M){nr + 3M) (Z^ + i°Z) cr+y2(Q) sin2 6 ' (352) or, recalling the definitions (63) and (328) of V{ + ) and W{ + \ we have -ReT0=-T[l/(+,Z( + , + (^( + , + 2Jff)A+Z( + )] sin20 ' (353) again, in complete accord with what we derived earlier via the Newman-Penrose formalism and the transformation theory. We have now come the full circle. 32. The physical content of the theory We stated at the outset (in §22) that, on the physical side, our principal concern in this study of the perturbations is to elucidate the manner in which incident gravitational-waves are reflected and absorbed by the black hole.
194 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE While it would appear on general grounds that the reflexion (R) and the transmission (T) coefficients, derived from the equations governing Z(±), are precisely the quantities which are sought, the matter requires clarification, since at no point in the analysis or in the arguments was there an occasion to mention the incidence of gravitational waves. And the matter is not a very simple one either: it will require us, as we shall find, to relate the theories, developed on the basis of the linearized Einstein-equations and the Newman Penrose formalism, at a deeper level than we have done hitherto. First, we observe that the 'constant' in the Wronskian relation (270) is simply related to the Wronskian [Z, Z*], whose constancy led to the conservation law, U + J = 1 in §27. Thus, considering the Wronskian, K2[Z1,Z2]=K2(Z2A + Z1-Z1A + Z2), (354) of any two independent solutions, Zl and Z2, of equation (285), substituting for KZ and KA + Z from equations (301), and making use of the integral (300), we find K2[Zt,Z2]= A2 A2 \2 ~RfV + P(W+2iaf^(Y2A^Yl-YlA^Y2) KIYUY2]. (355) Accordingly, tiITi-^] =KlZuZ2-\ = constant. (356) This last relation clearly implies that we should be able to determine the reflexion and the transmission coefficients from the equations governing Y+2 and Y_2 (or, equivalently, from the equations governing A2R + 2 and R-2). However, with our present knowledge, we cannot obtain any useful information from the Wronskian [Y+2, Y*2~\ since V+2and V_2 satisfy independent equations—a fact to which attention was drawn, already, in §29(b). In Chapter 9, we shall find, in the more general context of the Kerr black-hole, that the equations of the Newman-Penrose formalism, which we have not so far considered, do in fact determine the relative normalizations of the functions Y+2 and V_2 so that the Wronskian of Y+2 and Y*2 can, indeed, be used to determine U andT. But in the present context, we shall circumvent the need for that analysis, by obtaining the requisite relations directly from the equations provided by the transformation theory of §30. We know that the solutions for Z( + ) and Z(_) have the asymptotic behaviours e±,ar* both for /■„ -► + oo and for /■„ -> — oo. We may use this knowledge to obtain the associated asymptotic behaviours of Y+2 and V_2 with the aid of equations (318) and (319) and their complex conjugates.
THE PHYSICAL CONTENT OF THE THEORY 195 Remembering that V(±\ W(±\ and Q all vanish for r+ -► + oo, we find, from the first of each of the pairs of equations (318) and (319), that for Z<±>- and z'i'-ve-1'"-. (r+ -► + oo), (357) the asymptotic behaviours of Y+ 2 and Y_ 2 are Y+2-> -4ff2e + Iffr., K<±)* 4ff2 r4 4ff2 r4 -4a2e -'""*. (358) Similarly, remembering that I7'1' and Q vanish on the horizon and that (cf. equation (328)) W<±)-» -(2M)-1 for r = 2M, (359) we find that, for Z(±)_>c + «f. and z(±)->e-'<"■« (r+ -► - the behaviours of Y+ 2 and Y_ 2 are K(±)*A2e + i<"'« ■oo , (360) Y+2--4/(7(/(7-l/4M)e+""-.; Y_2 -► n2- K'i'A2^-""'' (2M)"4(i(7- 1/2M) (/(7- 1/4M) (2M)B4(/(7+ 1/2M) (/(7+ 1/4M) ; y_2 -► 4/(7(/(7 + l/4M)e-""■•. (361) Making use of the asymptotic behaviours of Y+2 and Y_2, we can write down the boundary conditions for the Weyl scalars, Q0 = ei°t + i»"eR+2S+2=ei°t + im^Y+2S+2 (= 4»0) (362) and <l>4 = eu" + >"vi>R_2s_2=ei'n + imi>r3Y_2S-.2 (=r4T4), (363) corresponding to the boundary conditions (cf. equation (159)) Z(±)->e + i^. + K<±»((7)e-''<"'' (/■„-► +00) -► r(±V)c+"r* (^---oo), (364) appropriate for determining the reflexion and the transmission amplitudes K<±'((7) and r<±>((7); thus, <l>o->e"" + imi>S+2(0)[ y| ■eiat + im^S+2(e)Y^: ?ml r e+ior. , y(ref)4 '+ '+2 —J- (r„ -+ + oo) (r+ -► - oo) (365)
196 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE and <D4->c«rt+ »«<(> S_2(0) yUni ^(■n) + Y(_re/ V3 e - ^. (^-+00) where Y<i"]=-4<T2; y(«f)=_^-ii(±)(ff); 4tr2 y(in]= -^-j y(Ie2f)= -4o2RM{p); 4ff Y^\ = (2M)Hia{ia - l/4M)T(a); K(±)Y(ff) y(?) = . (2M)54(i<7 + 1/2M) (iff + 1/4M) (366) (367) Now, it is clear that the coefficients of the leading terms, with an r~ ^behaviour in the asymptotic expansions of the Weyl scalars Tq and ¥4 at infinity, will determine the flux of energies in the incoming and the outgoing gravitational waves. The relationships are established in Chapter 9 (98(b)). The result of the analysis is that the flux of energies, per unit time and unit solid-angle (Q), in the incoming (incident) and the outgoing (reflected) waves are given by d2£<in> (S+2)2 1 dfdQ 2n 2ff y('n)i2 2 I ' +2 I and d2£(ref) (5_2)2 j dfdQ 2% 2ff2 ir'ifl2, (368) (369) where the angular functions, S+2 and S_2, are assumed to be normalized to unity. From the expressions given in equations (367), we find that, in agreement with equations (368) and (369), = l*(±V)|2 = y_ (ref) or 256ff8 y(in) 1 +2 y(ref) '(ref) y(in) y(ref) y(ref) 1 -7 ' +7 y(in)y(in) IK**'!2 y,in) |K'±)|2 256ff8 y(ref) 1 -2 y(ta) (370) (371) The expressions for IR given in equations (371) will enable us to determine the reflexion coefficient directly from the equations governing Y+2 or Y_2 (or, equivalently, from the equations governing R + 2 or K_ 2); and it will, of course, be in agreement with that determined by the equations governing Z(±).
THE PHYSICAL CONTENT OF THE THEORY 197 The corresponding expressions for the transmission coefficient,!, are (2M)6(ff2+l/16M2 y(tr) 1 +2 y(in) 1 +2 (2M)1 a2 + 1 16M2 a2 + 1 4M2 y[tr> y(in) = (2M)2 y(tr) y(tr) (<72 + l/4M2)1/2 (372) y(in) v(in) 1 +2 ' -2 While the physical meaning of T, as the fraction of the incident flux of energy that is absorbed by the black hole, is manifest from the conservation law IR + T = 1, a direct proof requires some sophisticated arguments which we postpone to Chapter 9. (a) The implications of the unitarity of the scattering matrix The unitarity of the scattering matrix (179) established in §28 requires, among other things, that the solution for Z satisfying the boundary conditions, r* + R{-a)e+iar* (r*->-oo) T(-a)e'iar* (^-^+00) (373) determines the same reflexion and transmission coefficients, IR and T, as the solution satisfying the boundary conditions (364). Making use of the same relations (358) and (361), we find that the corresponding boundary conditions for the Weyl scalars 3>0 and 3>4 are (J)0 _> eiot + im<PS + 2 yOn)e-.ffr, + y<ref) eiet + imVS+l y(tr)« and (r+ -► - oo), (r+ -► + oo), (J)4 _> eM + im<P S __ 2 (yOn)e - iar, + y(ref)^ + iar, ) (^ _, _ ^ (374) where . e<<x + <»«f>S-2(Y{±Ilr3e-i'"'*) K (r+ -> + oo), (375) y(in) + 2 (2M)54(fff - 1/2M) (ia - 1/4M)' 7¾ = (2M)34ia(ia- 1/4M); Y%f = (2M)34ia(ia- l/4M)R(-a); v(tr) _ 1 +2 — 2n-°\ K ~4o yLtr2» = -4a2T(-a), } (376)
198 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE From these relations, we obtain \R(-o)\2 = \T(-a)\2 y(ref) y(in) 1 +2 y(ref) 2 n (ref) y(in) 1 +2 IKI (2M)lb256(72 (ff2 + 1/16M2)2 (a2 + 1/4M2) y(in) m 256 IKI2 (2M)16ff2(ff2 + l/16M2)2(ff2 + 1/4M2) (377) (2M)10 (a2 + 1/16M2) (ff2 + 1/4M2) fM 2(2M)6(ff2 + l/16M2) y(in) (378) 33. Some observations on the perturbation theory The theory of the perturbations of the Schwarzschild black-hole, described in the preceding sections, has so many intertwining strands that it is difficult to unravel them and reveal a coherent pattern. But certain elements of the pattern do emerge. The centre from which the pattern seems to radiate is the equation provided by the Newman-Penrose formalism for the radial function Y+2 (or Y_2) belonging to the Weyl scalar Tq (or V4). As we have seen in §32, we can determine, in terms of the function Y+ 2 (or Y_ 2), without any ambiguity, the reflexion and the transmission coefficients for incident gravitational waves. And every strand of the theory must lead to these same coefficients. When we turn to the metric perturbations, their separation into an axial and a polar part is a directly manifest aspect—an aspect which is concealed in the Newman-Penrose formalism. The axial perturbations are described by a single scalar function, Z(_), which satisfies a one-dimensional Schrodinger wave-equation with a potential, V{~)—the Regge-Wheeler equation. The reflexion and the transmission coefficients one can derive by elementary methods from this equation must of course agree with what one derives from the equation governing Y+ 2. Clearly, a transformation must exist which relates the function Y+2 and Z{~\ The polar perturbations are described by three radial functions, L, N, and X, which satisfy a coupled system of three first-order linear equations. Since these equations must also inform on the reflexion and transmission of incident gravitational-waves, they must be reducible to a wave equation of the second order. The reducibility must be guaranteed by the existence of a special integral of the equations: this is the Xanthopoulos integral (126). But, if a reduction is guaranteed, it can be effected in numerous ways. There is, for example, the combination Z( + ) (given by equation (59)) which satisfies the wave equation with the potential, V(+'—theZerilli equation. However, there is another linear
STABILITY OF THE SCHWARZSCHILD BLACK-HOLE 199 combination 2£ (given by equation (154)) which satisfies the Regge-Wheeler equation. This ambiguity is really a consequence of the necessity that, whatever wave equation one derives for the polar perturbation, it must lead to the same reflexion and transmission coefficients as does the Regge-Wheeler equation; and that, therefore, its solutions must be expressible, explicitly, as linear combinations of Z(_) and its derivative. Can one, on the foregoing grounds, dismiss the wave equation governing Z(+)? It would not seem that one can. For, when we seek for the transformation relating the solutions for Y+ 2 and for Z(~', we find that it is one of a dual pair with potentials of the form (cf. equation (133)) V(±)= ±P-r~ + P2f2 + Kf (Pand kconstants), (379) dr+ where the minus sign leads to the Regge-Wheeler potential and the plus sign leads to the Zerilli potential. Besides, the potentials are manifestly of the forms which guarantee the equality of the reflexion and the transmission coefficients and provide simple reciprocal relations between the solutions for Z(+) and z<->. There is a further strand in the theory. The Newman-Penrose formalism is not, in its inception, concerned with the decomposition of the Weyl scalars Tq and 1*4 into their axial and polar parts. The decomposition becomes possible only if we express them, explicitly, in terms of the perturbed components of the Rjemann tensor. When the decomposition is made, the two parts decouple—as they must. The two parts must, then, of necessity, satisfy the same equation— as in fact they do! There is yet another facet to the perturbation theory which we shall consider in Chapter 10. But from the present vantage point, it would appear that the two central facts of the theory are the admissibility of the equation governing Tq to dual transformations and the associated decomposition of the perturbations into an axial and a polar part. 34. The stability of the Schwarzschild black-hole The stability of the Schwarzschild black-hole to external perturbations—the only kind we need consider—has been the subject of extensive discussions. The question one addresses in this context is the following: given any initial perturbation confined to a finite interval of r+ (i.e., of'compact support'), will it remain bounded, at all times, as it evolves? We have seen that the perturbations are governed by single one- dimensional wave-equations of the form d2Z , —5- + ff2Z= VZ, (380)
200 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE with smooth real potentials, independent of a, and of short range (i.e., one with a bounded integral). It would appear then that the standard theorems of the quantum theory are applicable. These theorems guarantee that the wave functions, belonging to any observable, form a complete set and that any square-integrable function that can describe a state of the system can be expanded in terms of them; and, further, that the integral of the absolute square of any state function must remain constant with time. For the problem on hand, the solutions Z (r+, a) of equation (380) for real a, satisfying the boundary conditions (159), provide the basic complete set of wave functions; and any initial perturbation that is smooth and confined to a finite interval of r+ can be expressed as an integral over the functions Z (r+, a) in the form 1 C+ °° $(o;0) Z(r„; a) do; (381) *K,0) = (2w ,1/2 and the evolution of the perturbation, at later times, is given by 1 <M»V) = We are guaranteed that (2n ,1/2 ^(ff;0)eifftZ(r+; a) da. (382) l"Mr*;0)|2drt = \4>(a;0)\2da = IWiVOfdv (383) Also the boundedness of ^(r^, t), for all t > 0, follows directly from a comparison of equations (381) and (382). A weaker result, but one of general interest, follows from the time- dependent version of equation (380), namely, d2 Z d2Z dt2 dr\ VZ. (384) (We should have derived this equation, had we not, in the analysis of §24, replaced d/dt by ia, in accordance with our assumption (16) that the time- dependence of the perturbations is represented by the factor eiat.) Multiplying equation (384) by dZ */dt, we obtain, after an integration by parts, dZ* d2Z dZ d2Z* dZ* dt dt2 dr^ dtdr^ dt dr+ = 0, (385) provided the various integrals converge. Adding to equation (385) its complex conjugate, we obtain the energy integral •+ 00 / - CO \ dZ dt 2 + dZ dr^ + V\Z\* dr+ = constant. (386)
THE QUASI-NORMAL MODES 201 The existence of this energy integral bounds the integral of \dZ/dt\2; it, therefore, excludes an exponential growth of any bounded solution of equation (383). 35. The quasi-normal modes of the Schwarzschild black-hole In this last section of this chapter we shall consider what we may describe as the pure tones of the black hole. The situations we have in mind are the following. A black hole can be perturbed in a variety of ways, other than by the incidence of gravitational waves: by an object falling into it, or by the accretion of matter surrounding it. Or, we may consider a black hole being formed by a slightly aspherical collapse of a star settling towards a final state described by the Schwarzschild solution. In all these cases, the evolution of the perturbations— if they can be considered as 'small'—can, in principle, be followed by expressing them as superpositions of the basic solutions, Z (r+; a), considered in §34. However, we may expect on general grounds that any initial perturbation will, during its last stages, decay in a manner characteristic of the black hole and independently of the original cause. In other words, we may expect that during the very last stages, the black hole will emit gravitational waves with frequencies and rates of damping, characteristic of itself, in the manner of a bell sounding its last dying pure notes. These considerations underlie the formulation of the concept of the quasi-normal modes of a black hole. Precisely, quasi-normal modes are defined as solutions of the perturbation equations, belonging to complex characteristic-frequencies and satisfying the boundary conditions appropriate for purely outgoing waves at infinity and purely ingoing waves at the horizon. The problem, then, is to seek solutions of the equations governing Z(±> which will satisfy the boundary conditions Z<±>-»,4<±>(ff)e-<<" (r,-+ooO (387) _> e + i'r. (r+_,. -oo). J This is clearly a characteristic-value problem for a; and the solutions belonging to the different characteristic values define the quasi-normal modes. First, we may observe that exponentially growing unstable quasi-normal modes, with Imff < 0, are excluded by the energy integral (386). For, if such modes should exist, solutions satisfying the boundary conditions (387) will vanish exponentially for r+ -► + oo; the integrals, over r+ in equations (386), converge and we shall be led to a contradiction. We can, therefore, have only damped stable quasi-normal modes; but the solutions belonging to them will diverge for r+ —»■ + oo—a fact which we must accept as corresponding to an implicit assumption of an 'infinite' perturbation in the remote past.
202 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE Next, we observe that the characteristic frequencies a are the same for Z(_) and Z( + ): for, if a is a characteristic frequency and Z{~](o) is the solution belonging to it, then the solution Z( + ) (a) derived from Z(_) (a) in accordance with the relation (152), will satisfy the boundary conditions (387) with (cf. equation (169)) A< + » = ,4<-V)^ (388) It will suffice, then, to consider only the equation governing Z(_). Letting Z(~> = exp( i </>drtl, (389) we find that the equation we have to solve is i<P rt + o2-<p2-yi-) = 0; (390) and the appropriate boundary conditions are <j>-* — a as r+->+oo and <p^+a as r+ -► — oo. (391) Solutions of equation (390), satisfying the boundary conditions (391), exist only when a assumes one of a discrete set of values. It is not known, in general, whether the set of characteristic values is finite or an enumerable infinity. A useful identity, which follws from integrating equation (390) and making use of the boundary conditions (391), is (cf. equation (225)) -2ia + (<r2-</>2)drt = V{')dr* = 2~M(^ + & (392) In Table IV we list the complex characteristic-frequencies a for different values of I. Table IV The complex characteristic-frequencies belonging to the quasi-normal modes of the Schwarzschild black-hole (a is expressed in the unit (2M)_1) ( 1Mb I 2Mb 2 3 0.74734 +0.17792i 0.69687 + 0.54938i 1.19889+ 0.1854U 1.16402 + 0.56231i 0.85257 + 0.74546i 4 5 6 1.61835 + 0.18832( 1.59313+ 0.56877i 1.12019 + 0.84658i 2.02458 + 0.18974i 2.42402 + 0.190531 (The entries in the different lines for 1 — 2,3, and 4 correspond to the characteristic values belonging to different modes.)
BIBLIOGRAPHICAL NOTES 203 Detailed calculations, pertaining to slightly aspherical collapse of dust clouds and to particles of finite mass falling into black holes along geodesies, do exhibit the phenomenon of the ringing of black holes with the frequencies and rates of damping of these quasi-normal modes. BIBLIOGRAPHICAL NOTES The theory of the perturbations of the Schwarzschild space-time was initiated in a classic paper by T. Regge and J. A. Wheeler: 1. T. Regge and J. A. Wheeler, Phys. Rev., 108, 1063-9 (1957). Following Regge and Wheeler, several investigations along the same lines were published during the early seventies, e.g.: 2. C. V. Vishveshwara, Phys. Rev. D, 1, 2870-9 (1970). 3. F. J. Zerilli, ibid, 2, 2141-60 (1970). 4. , Phys. Rev. Letters, 24, 737-8 (1970). 5. J. M. Bardeen and W. H. Press, J. Math. Phys., 14, 7-19 (1972). But certain essential features of the theory remained unrecognized: for example, it was not even known that the axial and the polar perturbations (rather, the 'odd' and the 'even' parity perturbations as they were then called) are characterized by the same reflexion and transmission coefficients. Many of the other relations emerged only gradually and often in the context of the Kerr black-hole. For this reason, in this chapter we bring together the results and methods scattered through the following papers to present a self-contained, coherent, and unified treatment of the theory of the perturbations of the Schwarzschild black-hole: 6. S. Chandrasekhar, Proc. Roy. Soc. {London) A, 343, 289-98 (1975). 7. and S. Detweiler, ibid, 344, 441-52 (1975). 8. , ibid, 345, 145-67 (1975). 9. S. Chandrasekhar, ibid, 358, 421-39 (1978). 10. , ibid, 441-65 (1978). 11. , ibid, 365, 453-65 (1979). 12. and B. C. Xanthopoulos, ibid, 367, 1-14 (1979). 13. , ibid, 369, 425-33 (1980). See also: 14. S. Chandrasekhar in General Relativity—An Einstein Centenary Survey, edited by S. W. Hawking and W. Israel, Cambridge, England, 1979. §24. The treatment in this section is essentially the same as in reference 6. The basic relations (36)-(39), which enable a separation of the variables, were discovered by J. L. Friedman: 15. J. L. Friedman, Proc. Roy. Soc. (London) A, 335, 163-90 (1973). §25. In this section we present a somewhat simplified version of a theory developed by: 16. B. C. Xanthopoulos, Proc. Roy. Soc. (London) A, 378, 61-71 (1981). §§26-27. The relations discussed in these sections emerged only gradually. They explicitly occur for the first time in reference 13; but several related aspects of them were partially known earlier. See references 7 and 11 and: 17. J. Heading, J. Phys. A. Math. Gen., 10, 885-97 (1977). §28. The theory of inverse scattering and its ramifications for the theory of the Korteweg-de Vries equation are far too extensive to allow an account of any depth in the brief section we have allocated to it. But the relations we have found between V{+)
204 PERTURBATIONS OF SCHWARZSCHILD BLACK-HOLE and V{~) (and similar relations we shall find in subsequent chapters in the contexts of the Reissner-Nordstrom and the Kerr black-holes) are far too reminiscent of the theory to simply ignore it. The account in this section (inclusive of §(a)) is largely based on the following paper by Deift and Trubowitz: 18. P. Deift and E. Trubowitz, Communications on Pure and Applied Math., 32, 121-251 (1979). I am grateful to Professor Deift for critical comments on this section. The analysis in §(b) follows closely: 19. L. D. Faddeev, Soviet Physics Dokl, 3, 747-51 (1958). For the explicit form of the relations of orders higher than those listed in equation (220) see: 20. R. M. Miura, C. S. Gardner, and M. D. Kruskal, J. Math. Phys., 9, 1204-9 (1968). General references to the classical theory of the solitons and of the Korteweg-de Vries equation in particular are: 21. G. B. Whitham, Linear and Non-Linear Waves, §§16.14-16.16 and 17.2-17.4, John Wiley & Sons, Inc., New York, 1974. 22. A. C. Scott, F. Y. F. Chu, and D. W. McLaughlin, Proc. IEEE, 61, 1443-83 (1973). 23. R. M. Miura, SIAM Rev., 18, 412-59 (1976). §29. In this section, the treatment in reference 9 in the context of the Kerr space-time is adapted to the Schwarzschild space-time. §30. The transformation theory was originally developed in reference 8 (and subsequent papers not listed here) in the context of the perturbations of the Kerr black- hole. It is adapted here for the Schwarzschild space-time. The basic ideas are implicit in reference 6. §31. The polar part of 4% was evaluated by J. L. Friedman (reference 15); the axial part is evaluated here to complete the analysis. §34. For alternative treatments of the stability of the Schwarzschild black-hole see: 24. V. Moncrief, Ann. Phys., 88, 323-42 (1973). 25. R. M. Wald, J. Math. Phys., 20, 1056-8 (1979). The basic theorems used in the discussion in the text are stated in: 26. L. D. Landau and E. M. Lifschitz, Quantum Mechanics (Non-Relativistic theory), §21, 60-3, Pergamon Press, Oxford, 1977. §35. The treatment of the quasi-normal modes in this section follows reference 7. In Table IV, the numerical results given in reference 7 are supplemented by some further results in: 27. D. L. Gunter, Phil. Trans. Roy. Soc. (London) A, 296, 497-526 (1980). The appearance of these quasi-normal modes during the last stages of the collapse of stellar masses is examined in: 28. C. T. Cunningham, R. H. Price, and V. Moncrief, Astrophys. J., 224, 643-67 (1978). 29. , ibid., 230, 870-92 (1979).
5 THE REISSNER-NORDSTROM SOLUTION 36. Introduction A spherically symmetric solution of the coupled equations of Einstein and of Maxwell is that of Reissner and Nordstrom. It represents a black hole with a mass M and a charge Q^. Since one does not expect any macroscopic body to possess a net charge, the consideration of a charged black-hole would appear to be outside the realm of reality. Nevertheless, the study of the Reissner-Nordstrom solution contributes to our understanding of the nature of space and time. Thus, we shall find that the scattering of incident electromagnetic and gravitational waves by the Reissner-Nordstrom black hole is described by a symmetric unitary scattering-matrix of order four and allows for the partial conversion of energy of one kind (electromagnetic or gravitational) into energy of the other kind. Besides, the Reissner-Nordstrom solution provides a more general framework for the many surprising analytical features which we encountered in the study of the Schwarzschild solution in Chapter 4. And finally, the fact that the Reissner-Nordstrbm solution has two horizons, an external event horizon and an internal "Cauchy horizon," provides a convenient bridge to the study of the Kerr solution in the subsequent chapters. 37. The Reissner-Nordstrom solution Since the Reissner-Nordstrom solution, like the Schwarzschild solution, represents a spherically symmetric space-time, we can, as in Chapter 3, write the line-element in either of the two forms (Ch. 3, equations (3) and (5)), ds2 = 4/dudu-e2"3[(d0)2 + (d<p)2sin20], (1) or ds2 = e2^[(dx°)2-(dx2)2]-e2"3[(d0)2+(d<p)2sin20], (2) where/= e2^2 and n3 are functions solely of u and v (or, equivalently, of x° and x2). Accordingly, the expressions for the non-vanishing (tetrad) components of the Riemann tensor (appropriate to the metric (2)) given in Chapter 3, equations (10), continue to be valid. However, since we are now considering, not a vacuum field, but one in which an electromagnetic field prevails, we cannot set the Ricci tensor to be zero, as we did in Chapter 3, equations (11)-(14). Instead we must now set Kt= -2(ticdFacFbd-ir,abFefF"), (3)
206 THE REISSNER NORDSTROM SOLUTION where Fab denotes the (tetrad) components of the Maxwell tensor which must, in turn, be determined by Maxwell's equations. (In equation (3), nab denotes the Minkowskian metric of the chosen ortho-normal tetrad-frame.) (a) The solution of Maxwell's equations It is clear that, under the assumed conditions of spherical symmetry, the axial components of the Maxwell tensor, namely, F01, F12, and F13, must vanish; and the equations governing the polar components, F02, F03, and F23, can be readily written down by transcribing the equations given in Chapter 2 (equations (95), (e), (g), (f), and (h)) appropriately for the metric (2). We find (e2,,1F02),2 + e^ + i"F03 cot 6 = 0, (e2"3 ^02). o + e»> + "> F23 cot 6 = 0, . (4) (^ + "3F23),2-(^ + "3Fo3))0 = 0, and (^ + "3F23))0-(^ + "3F03).2 = 0. Since none of the components of the Maxwell tensor can depend on 6 (by the assumption of spherical symmetry), it follows from the foregoing equations that ^03 = F23 = 0, (5) and (^^02).2 = (^^02).0 = 0. (6) Hence, the only non-vanishing component of the Maxwell tensor is F02 which must be of the form ^02 =-Q*e-2"\ (7) where Q% is a constant. With F03 = F23 = 0 and F02 given by equation (7), we readily find that -^44 = ^00 = ^11 = -i?22 = ^33 = e^-^3,andi?42 = 0. (8) (b) The solution of Einstein's equations With the non-vanishing components of the Ricci tensor given in equations (8), Chapter 3, equations (11)-(15), must now be replaced by 2*1212+ ^2424 =+<2^-4"3, (9) 2*1414 + ^2424= +Qle-4«>, (10) *1212+*1313+*1414= -Qle-4"', (11) *1313+*2323+*3434= -e^-4^, (12)
THE REISSNER NORDSTROM SOLUTION 207 and «2114 + «2334 = «24 = 0. (13) From equations (9) and (10), it follows that «1212+ «1010 = 0; (14) and from equations (13) and Chapter 3, equation (10b), we conclude that *i2io = 0. (15) Next, adding equations (9) and (10), we obtain «i2i2-«ioio-«2020= +e*e~4"3; (16) and rewriting equation (11) in the form *i2i2+*i3i3-*ioio=-Gie-4ft, (17) we can combine it with equation (16) to give «i3i3+«202o= -2Qle-4»>. (18) We observe that equations (14) and (15) are the same as the first pair of equations in Chapter 3, equation (19), while equations (17) and (18) replace the second pair. Treating these equations in the same fashion, we obtain the basic set of equations /Z„,„-Z„/„ = 0, (19) /Z ,,,-Z,/, = 0, (20) 01 z2 /(1-¾ +ZZU), + ZuZ, = 0, (21) (22) Equations (19)-(22) replace equations (31)-(34) of Chapter 3; and their solution can be accomplished in similar fashion. Thus, equations (19) and (20) enable us to conclude, as before, that /=yi(u)Z, and f=B{v)Z,u, (23) where A(u) and B(v) are functions only of the arguments specified. Next writing equation (21) in the alternative forms, A(u)[Z + ^-)+ZZ„ and Ql B(v)[Z + ^-) + ZZ,v - 0 (24) 0, (25)
208 THE REISSNER-NORDSTROM SOLUTION we deduce that and A(u)[ 1+||) + F{u) Z Ql\ , G(v) B(v)\ 1+^- + (26) (27) where F (u) and G (v) are further arbitrary functions of the arguments specified. Equations (23), (26), and (27) can be combined to give Ql z2 z„z -/(1+.2 + F(u)Z_v -/(1 + .2 Ql z + z G(v)Z,u z, Therefore, and we conclude that F(u) _ G(v) A(u) B(v) Thus, we obtain the solutions F{u)_A{u) G(v) B(v)' a constant = 2M (say). (28) (29) (30) and f- 2M Ql 2M O2 ■A(u)B(v)[ 1-^ + P (31) (32) It can now be verified that the solution of equations (19)-(21), expressed by equations (31) and (32), satisfies the remaining equation (22). The meaning of the solution for Z given by equations (31) becomes clearer in the form 2M , Ql dZ = - 1 Z+Z2 lA(u)du + B(v)dv1; (33) and the form of this solution again establishes, as in Chapter 3, 17(b), that a spherically symmetric solution of the coupled Einstein-Maxwell equations is necessarily static (outside the event horizon), i.e., a generalization of Birkhoff's theorem.
THE NATURE OF THE SPACE-TIME 209 Finally, we may note that the solution for the metric itself is given by ( 2M O2 \ ds2= -4(1—— + ^\A{u)B{v)dudv-Z2dil2. (34) The occurrence of the two arbitrary functions A (u) and B(v) in the solution (34) (as in the corresponding solution of Chapter 3, equation (48) for the Schwarzschild metric) is entirely consistent with the freedom of choice we have in defining, as coordinates in U2, u and v, or some function of u and some function of v. 38. The nature of the space-time In accordance with customary usage, we shall replace Z by r, with its meaning of 'luminosity distance,' and write equations (33) and (34) in the forms A dr= --1\_A(u)du + B(v)dv'] (35) and where Let ds2 = -4-rA(u)B(v)dudv-r2dCl2, (36) A=r2-2Mr + Ql. (37) r+=M + V(M2-e2)andr_ = M-V(M2-e2J, (38) be the roots of A = 0. These roots will be real and distinct if M2 > Q2,. We shall assume that this inequality obtains. Since A > 0 for r >r+ and 0 < r < r_ and A<0 for r_<r<r + , (39) it is clear that we must distinguish the three regions: A: 0 < r < r_; B: r_ < r < r + ; and C: r > r+. (40) We shall find that, while the surface r — r+ is an event horizon in the same sense that r = 2M is an event horizon in the Schwarzschild space-time, the surface r = r_ is a 'horizon' in a different sense which we shall clarify presently. We shall find it convenient to rewrite equation (35) in the form dr„= -[/l(u)du + B(u)du] (41) where * 2 2 2 Ldr = r + _^_lg|r_r+| ^—lg|r_r_ |. (42) A r,-r_ r+-r_
210 THE REISSNER-NORDSTROM SOLUTION As defined, — °° < r* < + °° f°r r+ < r < + oo and + oo > r„ > — oo for r-<r<r + . (43) To clarify the nature of the space-time described by the metric (36), we shall make different choices for the functions A(u) and B(v) in the regions A and C and in the region B. In the regions A and C we shall choose A(u)= -B(v)=l (44) and let dt = i(du + du). (45) Then, u = t — rif and v = t + r^; (46) and the metric takes the form ds2 = 3dudu-r2dQ2, (47) or, equivalently, ds2 =4(dO2-^(dr)2-r2[(d0)2 + (d<p)2sin20]. (48) rz A It is in this last form that the Reissner-Nordstrbm metric is generally written and commonly known. In the region B, we shall choose instead A(u) = B{v)=-l (49) and let dt = ±(du-dv). (50) Then u = rit, + t and v = r^ — t, (51) and ds2 = ^dudt;-r2dn2. (52) With the foregoing choices, the parts of the manifold in the three regions can be represented by the 'blocks' in Fig. 13 for constant values of the angular coordinates 6 and </>. The edges of the blocks are identified as indicated. Also, to the regions A, B, and C there correspond further regions A',B', and C obtained by applying the transformation, u-+ —u and v -»• — v, which reverses the light- cone structure. A maximal analytic extension of the Reissner-Nordstrom solution may then be obtained by piecing together copies of the six blocks in an analytic fashion so that an overlapping edge is covered by either a u(r)- or a u(r)-system of coordinates (with the exception of the corners of each block).
c o qs qs kl - o U- S :0 TJ o Z j I c ,, a o V ... u V k qs o i 211
/ \ ' Fig. 14. The maximum analytic extension of the Reissner-Nordstrom space-time. The different regions A, B, and C of Fig. 13 are pieced together contiguously; regions A', B', and C are obtained from A, B, and C by applying the transformation u -> —u and v -> — u. The full extension is obtained by piecing together copies of the six blocks so that an overlapping edge is covered by a u (r) or a i! (r)-chart. A time-like geodesic from region C crossing both horizons and a possible timelike path an observer may take, to be emancipated of the past, are shown. 212
THE NATURE OF THE SPACE-TIME 213 The resulting 'ladder' is shown in Fig. 14; it can be extended, indefinitely, in both directions. The principal reason for this extension is to make the manifold geodesically complete, i.e., to ensure that all geodesies, except those which terminate at the singularity, have infinite affine lengths both in the past and in the future directions. An analytic representation of the maximally extended space-time can be obtained as follows. In the regions A and C, we set tanl/= -e~«u= -e-*«-',) = -e-«{e+«'|p-r+ l^lr-r-l-A2} (53) and tanK= +e+av = + e+a{, + r^ = + e + "t{e+ar\r-r+\1'2\r-r-\-f'2}, (54) where a = (r+-r-)/2r2+ and P = rl /r+2 (> 0 and < 1). (55) In the region B, we set tan 1/= +e+au= + e + ^t + r^ = +e+at{e+"\r-r+ |1/2 |r-r_ |-^2} (56) and tanK= +e+clv= + e + a{-, + r^ = +e-"t{e + "\r-r+\l'2\r-r.\-fl2}. (57) By these substitutions, the metric takes the 'universal' form 4 ds2 = —2 |r —r+1 |r —r_ |cosec 2 U cosec 2 V dU dV-r2 dft2, (58) where r is implicitly defined in terms of U and V by tanl/tan^= -^1^-^+11^-^-1-^ (r > r+ and 0<r<r_) = +e2ar\r-r+\\r-r-\-tl (r-<r<r+). (59) The values of U and V, along the different edges of the blocks in Fig. 13, are indicated. The metric, as defined by equation (58), is analytic everywhere except at r = r_, since \r — r_ | is raised to a negative power in equation (59); but it is at least C2 at r = r_. The delineation of the coordinate axes by null geodesies enables us to visualize, pictorially, the nature of the underlying space-time. It is clear, for example, that the space-time external to r+ (in the region C) is very similar to the Schwarzschild space-time external to the surface r = 2M: any observer, who crosses the surface r = r+ (represented by CD in Fig. 13), following a future-directed time-like trajectory, is forever 'lost' to an external observer (in C). Besides, any radiation transmitted by such an observer, at the instant of his crossing, will be infinitely red-shifted (because dt/dx -*■ + oo; see equa-
214 THE REISSNER-NORDSTROM SOLUTION tion (70) below) to an external observer. The surface at r = r+ is, therefore, an event horizon in exactly the same sense that the surface r = 2M is, in the Schwarzschild space-time. However, for one who has crossed the event horizon in the Reissner-Nordstrom geometry, there would appear to exist an infinite range of rich possibilities for experience, that are denied to one who crosses the event horizon in the Schwarzschild geometry. For the latter person, there is, as we have seen, no alternative to being inexorably propelled towards the singularity at r = 0. In contrast, the corresponding person, in region B in the Reissner-Nordstrom geometry, has, for example, the option to follow a future-directed time-like trajectory and cross the surface r = r_ (represented by C'E) into the region A'. At the instant of such crossing, the person will witness, in a flash, a panorama of the entire history of the external world, in infinitely blue shifted (because dt/dr -* — oo for r -*■ r_ + 0; see equation (70) below) null rays arriving in the direction BCC'E. Once in the region A', the person's future is no longer determined by his (or her) past history: the region A' is outside the domain of dependence of the spatial slice y extending across the regions C and C in Fig. 14. For this reason, the surface at r = r_ is called a 'Cauchy" horizon. Alternatively, an observer in the region B has the option of following a timelike geodesic and entering the region A by crossing the surface r_ (represented by EF = E'F'). All such geodesies (except the purely radial null geodesies) skirt the singularity at r = 0 and cross the surface r = r_ (across E'G') to emerge, eventually, into a 'wonderfully' new asymptotically flat universe. It is clear that all these rich and varied prospects, for one who has crossed the event horizon at r = r+, are essentially a consequence of the time-like character of the singularity at r = 0 (in contrast to the space-like character of the singularity in the Schwarzschild geometry). But a word of warning: crossing the Cauchy horizon is fraught with danger (as we shall see in §49). 39. An alternative derivation of the Reissner-Nordstrom metric A derivation of the Reissner-Nordstrom metric, similar to the one for the Schwarzschild metric given in Chapter 3, §18, can be given. Thus, with a metric of the form Chapter 3, equation (62), we have the same expressions (Ch. 3, equation (64)) for the non-vanishing components of the Riemann tensor. And since, moreover, (cf. equations (8)) R42 = 0 and R22 ~ ^44 = 0, (60) the conclusions expressed in Chapter 3, equations (67) and (72) continue to hold in the present context. However, in place of Chapter 3, equation (73), we now have (see equation (8)) 2»2,r +^(1-^^) = ^33=% (61) r ir r
GEODESICS IN REISSNER-NORDSTROM SPACE-TIME 215 The solution of this equation is 1 + ^f r r „2v (62) and we recover the solution in its standard form given in equation (48). With the solution for fi2 and v given in equation (62), we find that the non- vanishing components of the Riemann tensor are (cf. Ch. 3, equation (75)) *<i (1)(3)(1)(3) 2Mr-Ql R 2Mr-3Ql, (2)(0)(2)(0) and (63) *<i )(2)(1)(2) R (2)(3)(2)(3) *<i (1)(0)(1)(0) R Mr-Ql (3)(0)(3)(0) where we have enclosed the indices in parentheses to emphasize that these are the tetrad components in a local inertial frame. These expressions show that r = 0 is, indeed, a genuine space-time singularity. The corresponding tensor- components are given by (cf. Ch. 3, equations (76)) )2 and ^0101 ^0303 *M212 ^2323 ^1313 Rfttm ■ + Rtipup - + Rtete = + Rr0r6 = + R<p0Ve + Rtrt, = Mr-Ql ^ ■ 2fl 5—- e^v sin 0, rM + ^e"» sin20, + ^e-» - -(2Mr-Ql)sm2er , 2Mr-3Ql I A * (64) where e2* = A/r2. 40. The geodesies in the Reissner-Nordstrom space-time Since the Reissner-Nordstrom metric differs from the Schwarzschild metric only in the definition of the 'horizon function' A, it follows that the basic equations governing geodesic motion are, with the redefinition of A, the same as those considered in Chapter 3, §§18 and 19. Thus, the equations governing the time-like geodesies are (Ch. 3, equations (85), (90), and (91)) dr , _ , , ,d./M1 + r» £2; df 'A' and ^. at L ~2' (65)
216 THE REISSNER NORDSTROM SOLUTION where A = r2-2Mr + Ql. (66) By considering r as a function of </> and replacing r by u = 1/r as the independent variable, we obtain the equation (cf. Ch. 3, equation (94)) (£)'--«*+«"'-"'(,+£)+£"-1^-'M ,sart (67) The corresponding equations governing the null geodesies are (cf. Ch. 3, equations (212)-(215)) drV ,A ,,, dt r2 J d</> L *)+L> = £: * = £a; and d7 = ^ (68) and ^Y=-e2X + 2Mu3-u2+-^=/(u) (say), (69) where L>( = L/E) is, as before, the impact parameter. Equations (67) and (69) differ from the equations considered in Chapter 3 by/(u) being a biquadratic instead of a cubic. We shall find that this fact makes an essential difference only for the orbits which cross the horizon at r = r+, i.e., for the orbits, which in the Schwarzschild geometry terminated at the singularity at r = 0; and this is in accord with the differences in the Schwarzschild and the Reissner-Nordstrom geometries interior to the respective event horizons. (a) The null geodesies As in the Schwarzschild space-time, the radial null-geodesies in the Reissner-Nordstrom geometry provide the base for constructing a null tetrad- frame for use in a Newman-Penrose formalism. The equations governing the radial null-geodesies can be obtained by setting L = 0 in equations (68); thus, 0. (70) (71) (72) where r„ is defined in equation (42). Therefore, for an in-going null-ray, the coordinate time t incrccises from — oo to -+- oo a.s v decreases from -+- oo to r+, dr dt r2 dr — ' dr A Accordingly, dr The solution of this equation is '=+'* and — = ax A + constant, d</> = d7
GEODESICS IN REISSNER NORDSTROM SPACE-TIME 217 decreases from + oo to — oo as r further decreases from r+ to r_, and increases again from — oo to a finite limit as r decreases from r _ to zero. All the while, the proper time decreases at a constant rate E (as behoves light!). From a finite distance r > r+, a co-moving observer will arrive at the singularity in a finite proper time. (We shall see, presently, that these are the only class of geodesies that terminate at the singularity: all others skirt it.) It should also be noted that as dt/dx tends to + oo for r -*■ r+ + 0 and to — oo as r -»• r_ + 0, any radiation received from infinity (in the region C) will appear infinitely red-shifted at the crossing of the event horizon and infinitely blue-shifted at the crossing of the Cauchy horizon. Turning to the consideration of equation (69) and the null geodesies in general, we first observe that the quartic equation, f(u) = 0, always allows two real roots: one negative (which has no physical significance) and one positive (which, one can show, occurs for r < r_). We shall be concerned only with cases when the two remaining roots are either both real (distinct or coincident) or a complex-conjugate pair. Let the value of the impact parameter, D, for which/(u) = 0 has a double root, be denoted by Dc. Then, for all values of D > Dc, we shall have orbits of two kinds (as in the Schwarzschild space- time): orbits of the first kind which lie entirely outside the event horizon, coming from + oo and receding again to + oo after a perihelion passage; and orbits of the second kind which have two turning points, one outside the event horizon and one inside the Cauchy horizon. For values of the impact parameter D < Dc (when/(u) = 0 allows only one real root, > 1 /r _, along the positive real u-axis) the orbit coming from + oo will cross both horizons and have a turning point for a finite value of r < r_. Thus, all these orbits skirt the singularity at r = 0. However, from our remarks in §39, it follows that orbits which have turning points inside the Cauchy horizon cannot be assumed to continue along the symmetrically reflected time-reversed orbits: they should be considered, instead, as continuing up the 'ladder' in Fig. 14: e.g., A-y B' ->C (where C is now a different asymptotically-flat universe). The value of the impact parameter for which/(u) = 0 allows a double root is determined by the equations /(ii) = ^ - u2 (Ql u2-2Mu +1) = 0 (73) and /'(u)= -2u(l-3Mu + 2{)2u2) = 0. (74) Besides u = 0, equation (74) allows the roots 3M r i wiyi2i At the larger of these two roots,/(u) has a maximum. The double root we are
218 THE REISSNER-NORDSTROM SOLUTION seeking must occur at the minimum of/(u) where 3M 4<22* 1-1 m 9M: 2 \l/2" uc (say). (76) The corresponding value of r is rc = 1.5M[l+(l-8^/9M2)"2]- (77) It is clear that at this radius rc, the geodesic equations allow an unstable circular orbit. The value of the impact parameter, Dc, associated with the double root rc follows from equation (73). We find Dc = r2/VAc, Ac = r2- 2Mrc + Q* = Mrc - Q\. When D = Dc, f{u)={u-uc)2l-Qlu2+2{M-Qluc)u + uc(M-Qluc)]; and the solution for cp is given by (78) (79) <P = ± l-Qlu2+2{M-Qluc)u + uc{M-Qluc)l du u — ur The substitution, _. £ = (u-uc) \ reduces cp to the elementary integral, 9 (-Qi+bi+ce)112' where b = 2(M-2Qluc) and c = uc(3M-4{)2uc). We thus obtain the solution 1 + <p = ■ Tc lg{2[c(-e*+^ + ^2)]1/2+2c<^ + fc} (c>0) 1 V- -sin 2c£+b (4(22 c + fc2) 2\l/2 (c<0) (80) (81) (82) (83) (84) The orbits of the two kinds are described by this same solution for values of the arguments in the following ranges: oo > r > rc, 0 < u < uc for orbits of the first kind, and rr>r>r„ and uc 1 > ^ > — oo, (85) "c< "** "max (= 1/fmin). and + oo > £ > £min( = (un (86)
GEODESICS IN REISSNER NORDSTROM SPACE-TIME 219 Fig. 15. The null geodesic for the critical impact-parameter Dc given by equation (78) for 2» = 0.8. The inner and the outer horizons are shown by the dashed circles; the dotted circle represents the unstable circular-orbit. The trajectory actually crosses the inner horizon, but only just barely, before it terminates. The unit of length along the coordinate axes is M. for orbits of the second kind where rmin is the positive root of the equation (cf. equation (79)), uc(M-Qluc)r2+2{M-Qluc)r-Ql=0. (87) Both these orbits approach the unstable circular orbit at r = rc, asymptotically from opposite sides, by spiralling around it an infinite number of times. An example of the solution derived from equation (84) is illustrated in Fig. 15 (cf. Fig. 9, (b) which illustrates the corresponding orbits in the Schwarzschild space-time). We can now readily visualize the nature of the null geodesies for other values of the impact parameter. (b) Time-like geodesies Considering first the radial geodesies, we have the governing equations 'dA2 drj £2-^ and — = E- dr A (88)
220 THE REISSNER-NORDSTROM SOLUTION Since A > 0 in the interval 0 < r < r _, it is clear that (E2 r2 — A) will vanish for some finite value of 0 < r < r _. We conclude that the trajectory will have a turning point inside the Cauchy horizon. Thus, even the radial time-like geodesies do not reach the singularity at r = 0: they skirt it only to emerge in other domains. It is also clear from equations (88) that the variation of the coordinate time, t, with r will exhibit singularities both for r -* r + (+ 0) and for r -y r_ (+ 0) consistently with the horizon character of these surfaces. The formal solutions of equations (88) can be readily written down in terms of elementary integrals. We shall not write them since they are too complicated to exhibit in any more manifest manner what can be deduced from a simple inspection of the equations. Turning next to equation (67) and to a consideration of time-like geodesies in general, we can classify and analyze the different cases, as we did in Chapter 3, §19 in the context of the Schwarzschild geometry. It is, however, clear that the essential difference between the geodesies in the Reissner-Nordstrom and the Schwarzschild geometries will be in the orbits which cross the event horizon: in the Schwarzschild geometry all such orbits must terminate in the singularity; in the Reissner-Nordstrom geometry they will formally terminate at some point inside the Cauchy horizon. The differences which arise on this account can be illustrated sufficiently by considering the orbits of the second kind associated with stable and unstable circular orbits. The conditions for the occurrence of circular orbits are /(«)= -Qlu4 + 2Mu3-(l + Ql/L2)u2 + [_2Mu-(l-E2)1/L2=0, (89) and /'(u)= -4Q%u3 +6Mu2 -2{l + Ql/L2)u + 2M/L2 = 0. (90) From these equations, it follows that the energy E and the angular momentum L of a circular orbit of radius rc = l/uc is given by , (l-2Muc + Qiu2)2 and l-3Muc+2Qlu2 M-Qlue (92) uc(l-3Muc+2<22»"c2)' These equations require, in particular, that l-3Muc+2Qlu2 >0. (93) Comparison of this inequality with equation (74) shows that the minimum radius for a time-like circular orbit is the radius of the unstable circular photon-orbit. In Fig. 16 we exhibit the dependences of E and L on the radius of the circular orbit.
GEODESICS IN REISSNER-NORDSTROM SPACE-TIME 221 L2/M2 10 12 (a) r/M r/M (b) Fig. 16. The variations of E2 (figure (a)) and L2 (figure (6)) with the radius of the circular orbit. The curves are labelled by the values of Ql to which they belong. When E and L have the values (91) and (92), appropriate for a circular orbit of radius rc = l/uc, equation (67) becomes (cf. equation (79)) du (u-uc)2l~Qlu2+2(M-Qluc)u + {M-Qluc-M/L2u?)ucl (94) Therefore, besides the circular orbit of radius rc = l/uc, equation (94) provides an orbit of the second kind determined by (cf. equation (80)) <p = + 1-Qlu2+2(M-Qluc)u + (M-Qluc-M/L2u2)uj 1-1/2 du u — uc (95) With the substitution { = (11-10-1 (96) we obtain the same solution (84) with, however, b = 2(M-2Qluc) and c = uc(3M-4Qluc-M2/L2uf). (97) An example of an orbit of the second kind associated with a stable circular orbit is illustrated in Fig. 17a. When the circular orbit is unstable, the orbits of both kinds can be derived from the same formula (84) (the case c < 0). An example of such orbits is illustrated in Fig. 17b. The minimum radius for a stable circular orbit will occur at a point of inflexion of the function/(u), i.e., we must supplement equations (89) and (90) with the further equation /"(u) = - \2Q\u2 + l2Mu-2(l+Q2JL2) = 0. (98)
-6 -4 -2 0 (a) -6 -4 -2 (b) 222
GEODESICS IN REISSNER-NORDSTROM SPACE-TIME 223 -6 -4 -2 Fig. 18. The last time-like unstable circular orbit which occurs at radius 4.89 M for g„ =0.8. The horizons are indicated by the dashed circles; and the unit of length along the coordinate axes is M. Eliminating L2 in this equation with the aid of equation (92), we obtain the equation 4Qtu*-9MQlu2c+6M2uc-M = 0, (99) or, alternatively, r3c-6Mr2+9Qlrc-4Qt/M = 0. (100) (It may be noted here that equation (100) gives rc — 6M when Q\ = 0, in agreement with the value for the Schwarzschild geometry.) When all three equations, (89), (90), and (98), are satisfied, equation (67) takes the form ^Y = (u-uc)3 (2M-3Qluc-Qlu); and the solution of this equation is (cf. Ch. 3, equation (165)) 2{M-2Qluc) U~Uc+(M-2Qlucf(<p-<p0f+Ql- An example of this orbit is illustrated in Fig. 18. (101) (102) Fig. 17. (a) Theorbit of the second kind associated with a stable time-like circular-orbit with a reciprocal radius uc = 0.15 for Qt = 0.8. (b) The case of a time-like unstable circular-orbit (with a reciprocal radius uc = 0.2225) when the orbits of the two kinds coalesce. In both cases illustrated, the orbits of the second kind cross the inner horizon and, again, only just barely. The horizons are indicated by the dashed circles; and the unit of length along the coordinate axes is M.
224 THE REISSNER-NORDSTROM SOLUTION (c) The motion of charged particles A test particle with a net charge will not, of course, describe a geodesic in the Reissner-Nordstrom geometry. Since, in this geometry, the only non- vanishing component of the vector potential isA0( — Q^/r), its motion will be determined, instead, by the Lagrangian 2JSf = "A/dA2_r^/dr dey to) (r2sin20) dcp d7 , 2<iQ* r dt d? (103)* where q denotes the charge per unit mass of the test particle. The equations of motion which follow from this Lagrangian are readily written down. They are (cf. equations (65)) A dt qQ* rz ax r constant, 2d(P , constant, and /dr\2 A {dx)+7 1+- AdA2 r^dx) E- iQ< (104) (105) and in place of equation (67) we now have 2 g2u4 + 2Mu3 + 72 (M -«&£)«■ i+Gi l-£2 L2- (l-?2)" (106) The only novel feature these equations present is that if a particle should have a turning point as it arrives at theevent horizon, then its energy, according to equation (105), will be E = qQJr, (107) and this will be negative if qQ% < 0. This fact gives rise to the possibility of extracting energy from the black hole, energy that is associated with its charge. We shall consider processes of this kind in Chapter 7. 41. The description of the Reissner-Nordstrom space-time in a Newman-Penrose formalism For constructing a null tetrad-frame for a description of the Reissner-Nordstrom space-time in a Newmann-Penrose formalism, we start * This expression for the Lagrangian follows from Lorentz's equation of motion, namely,
DESCRIPTION OF R-N SPACE-TIME IN A N-P FORMALISM 225 with the vectors (70) specified by the radial null geodesies. Precisely, we shall take /'■ = (/',r,/e,/*>) = i(r2, +A,0,0), and 1 2r (ri, rf, n6, n<<>) = ^j(r2,-A, 0,0), (108) as the two real null-vectors of the basis and adjoin to them the complex null- vector, m1' = {rri, mr, m6, mv) = —j- (0,0, 1, i cosec 6), (109) orthogonal to / and n. These vectors satisfy the required normalization conditions /n=l and m-m= -1. (110) The basis vectors, (l,n,m,m), differ from the ones defined in the Schwarzschild space-time (Ch. 3, equations (281)-(283)) only in the definition of A. The evaluation of the spin coefficients proceeds exactly as before with only minor changes resulting from the present definition of A. We now find (cf. Ch. 3, equations (287) and (288)) 0 (111) and and 1 1 cot0 A r-M Mr-Ql The fact, that the spin coefficients, k, a, X, and v vanish, confirms the type-D character of the space-time. The Weyl scalars, ^0,^,^3, and ¥4 must, therefore, vanish in the chosen basis as can, indeed, be directly verified by contracting the Riemann tensor (whose non-vanishing components are listed in equations (64)) in accordance with the definitions of the scalars. The Weyl scalar yV2 does not, however, vanish. We find (cf. Ch. 3, equation (289)) V2 = Rpqrsl'm'>nrms (i?oioiCOsec20 + i?o3O3 ~e4yR2323 -e4yR12i2cosec26), (113) 4r2 or, inserting for the components of the Riemann tensor their values given in equations (64), we find V2=-{Mr-Ql)r-\ (114)
226 THE REISSNER-NORDSTROM SOLUTION The description of the Reissner-Nordstrom geometry in the Newman-Penrose formalism is completed by noting that the value of the only non-vanishing Maxwell-scalar is given by (115) 42. The metric perturbations of the Reissner-Nordstrom solution As in the case of the Schwarzschild solution, the perturbations in the metric coefficients of the Reissner-Nordstrom solution can be analyzed by linearizing, about this solution, the Einstein and the Maxwell equations appropriate for the non-stationary axisymmetric metric, ds2 = e2v(dt)2-e2'l'(d<p-codt-q2dx2-q3dx3)2 -e2^{dx2)2-e2^{dx3)2. (116) But in contrast to the treatment of the Schwarzschild perturbations in Chapter 4, §24, we must now consider the linearization of the Maxwell equations as well. We start with Maxwell's equations. (a) The linearized Maxwell equations Maxwell's equations, listed in Chapter 2 (equations (95)) when specialized to the simpler case (116), take the forms (e^+^F12).3 + (^+"3F31),2=0, (^ + vFoi),2 + (e*+"2F12),o = 0, (^ + ^01),3 + (^ + ^13),0=0, \ (117) (^ + ^01),0 + (^ + ^12),2 + (^+^13),3 and where ^ + "3 F02 &>2 + ^ + "2 ^03 <2o3 -e* + v F23 Q 23V23 (^ + ^02),3 + (^+^03),3=0, -(^ + ^23),2+(^ + ^03),/)=0, + (^+V ^23),3+ (^+"3 ^02),0=0, (^+^02).3-(^+^03),2 + (^+^23),0 = e"' + "F01Q23+e*+«>F12Q03-e*+«>F13Q02, (118) QaB — lA,B —<lB,A and IA0 qA,o-co,A (A,B = 2,3). (119) It will be observed that the first set of equations (117) involves only quantities (or, products of quantities) which reverse their signs when </> is
METRIC PERTURBATIONS OF R-N SOLUTION 227 replaced by — <p, while the second set of equations (118) involves only quantities which are invariant to the reversal in the sign of <p. They correspond, respectively, to what we have described as axial and polar quantities in Chapter 4, §24. Besides, in each of the two groups of equations, we can dispense with the first equation since it provides only the integrability condition for the two following equations. Remembering that F02 (— — Q*r~2)ls the only non-vanishing component of the Maxwell tensor and also that the QAbS are quantities of the first order of smallness, we readily find that the linearized versions of equations (117) and (118) (dispensing with the first equation in each case) are {re" F01 sin0),r + re~v F12>0 sin0 = 0, (120) rev (F01 sin0)>e + r2 F13j0 sin0 = 0, (121) re-vF01,o + (revF12)ir+F13,e= - <2* («,2 -fe.o)sin 6, (122) ^-vF03,0 = (r^F23),r, (123) <5^02,0-% (^ + ^3),0 + -^ (F2i sin0),e = O, (124) r rsmv and SF. 02 %(<5v + <5^2) ,e + (r^F3o),r + re-*F23io = 0. (125) (b) The perturbations in the Ricci tensor In contrast to the Schwarzschild space-time, we cannot in our treatment of the Reissner-Nordstrom space-time set the perturbed components of the Ricci tensor to be equal to zero. Instead, we must set -*.,(», G. * JWr2]- (126) From this equation, we find SRoo^SRu 5R22 = 3R33 = -2^-3 F02, r2 -± F 2 r2 3> ^01=-2% F12, <5K03=+2- 5R12 =+2% F01, SR23 =+2% F03, and SR02 = SR13 = 0. (127)
228 THE REISSNER-NORDSTROM SOLUTION (c) Axial perturbations The perturbation equations, which follow from the linearization of Einstein's equations about the Reissner-Nordstrom solution, can be obtained exactly as in the treatment of the Schwarzschild perturbations in Chapter 4, §24. In particular, the axial perturbations can be treated independently of the polar perturbations. As in the context of the Schwarzschild solution, the axial perturbations of the Reissner-Nordstrom solution are characterized by the non-vanishing of a>, q2, and q3; and in place of Chapter 4, equations (11) and (12), we now have (r2e2v{223sin30),3+r4{2o2,osin30 = 2(rVsin20)<5K12 = 4e„revFolsin20, (128) and (r2e2yQ23sm30\2-r2e-2yQO3,osm36= -2{r2sm26)5R13 = 0, (129) where we have substituted for SR12 and SR13 from equations (127). Equations (128) and (129) must be supplemented by equations (120)-(122). These latter equations can be reduced to a single equation for F01 by eliminating F12 and F13 from equation (122) with the aid of equations (120) and (121). We thus obtain [>2*(re*B),r],r + -(-?^ sinB-re- B.0.0 r \sm6Je = e*K2,o-42,o,o)sin20, (130) where B = Folsin0. (131) With the substitution (cf. Ch. 4, equations (13)-(15)), Q{r, 6, t) = r2e2*Q23sm36 = A{q2t3-q3,2)sm38, (132) equations (128) and (129) take the forms ^f =-(-,-,,0).0 + ^^ (133) and A dQ r4sin30 dr + ((0,3 -,3.o),o- (134) Eliminating co from these equations and assuming (as we always do) that the perturbations have a time-dependence e'"', we obtain the equations (cf. Ch. 4,
METRIC PERTURBATIONS OF R-N SOLUTION 229 equation (18)) d {AdQ\i.3od ( 1 dQ , r A Similarly, eliminating (co2-q2,o),o from equation (130) with the aid of equation (133), we obtain the equation [e2v(revB) r] ,+ -(-¾] sin0 + ( a1re-'-&ev ]B ■ ' r\sin0/e V r3 J = -<**&§■ (136) The variables r and 6 in equations (135) and (136) can be separated by the substitutions (cf. Ch. 4, equations (19) and (20)) Q(r,e) = Q(r)Cl-+y2(6), (137) and B(r, 9) = 4^^- = 3B(r)C,-+Y2 (6), (138) sin u au where, in writing the alternative form for B(r, 9), we have made use of the recurrence relation, 1 dCv -wThF= ~2vCn-l (I39) sin0 a6 By making use of the equations satisfied by the Gegenbauer functions C,+32/2 and C,+V2, we find that the substitutions (137) and (138) in equations (135) and (136) yield the radial equations (cf. Ch. 4, equation (23)) A d /AdQ\ , A , 4CL«2 Ad7<7d7 )-^Q^Q= --^-^b (140) and [i2'(«'ll),r],p-(^ + 2)yB+fff2«-'-^«'JB= ~Q*% (141) where ^=2n = (1-1)(1 + 2). (142) Changing to the variable r„ (defined in equation (42)) and further letting //(-) Q(r) = rH{i) and re"B = —, (143) 2n
230 THE REISSNER-NORDSTROM SOLUTION we find that H2 ' and H\ > satisfy the pair of coupled equations, A A2//<-'-r5 2H(-I O2 (H2 + 2)r-3M + 4^± H<T>-3MHi2-' + 2Q,nH\-A, A2H\ (H2 + 2)r-3M + 4 Qi H<r)+2e*^H(2~)+3M//(r) where, in accordance with our standard usage, ^-sT* Equations (144) and (145) can be decoupled by the substitutions and where Z<r»= +gifl<f>+ (-qiq2)ll2H (-) 7(-) z,2 (-^)^//^ + 41^ (144) (145) (146) (147) (148) q1=3M+s/(9M2 + 4Q2n2) and q2 = 3M- j(9M2+ 4Qln2). (149) We find that Z\~) and Z2_) satisfy the one-dimensional Schrodinger-type wave-equations, A^Z}-' = K}-»Z}-» (i = 1, 2), (150) where H-» (^ + 2)1--^(1+^ (i,j=l,2ii±j) (151) qt+q2 = 6M, and -<h<?2 = 46iU2- (152) The reduction of the equations governing the axial perturbations to the pair of one-dimensional wave equations (150) was first accomplished by Moncrief and by Zerilli (though by very different procedures). It will be observed that in the limit <2„ = 0, when q1 = 6M and q2 — 0, the equation governing Z2~) reduces to the Regge-Wheeler equation (Ch. 4, equations (27) and (28)). (d) Polar perturbations Polar perturbations are characterized by non-vanishing increments in the metric functions v, fi2, ^3, and \j/. Linearizing the expressions for R02, R03, R23, Rlt, and G22 given in Chapter 4 (equations (4)) and substituting from equations (127) for the perturbations in these quantities, we find that, in place
METRIC PERTURBATIONS OF R-N SOLUTION 231 of Ch. 4, equations (31)-(35), we now have W + ^3),r+(;-V,r")W + ^3)-;^2= -«02 = 0, (153) l(dil, + dH2\e + (dil,-5n3)cotei0= -ev + x>5R03 = -2Q.-F23, (154) r 1 {5\l/ + 5v)r6 + {5\l>-5n3\rcot0+ vr— <5ve v.r +;We = -e»> + >"5R23 = -2qJ-^-F03, (155) 1 + -=- [(<ty + <5v)e e + (2(5^ + <5v - <5^3 )e cot 0 + 2fy3] rz e~2vW + ^3),0,0 = SG22 = <5K22 = 2^-SF02 (156) and »2v + ^-[^ee + ^^ecot0 + (^ + ^v-^3+^2)ecot0 + 2^3] r -e~2v^.o.o = -«11=2%5F02. (157) r These equations must be considered along with equations (123)-(125) obtained from the linearization of Maxwell's equations. The variables r and 0 in equations (123)-(125) and (153)-(157) can be separated by the substitutions <5v = N(r)P,(0); ty2 - L(r)P,(0); ty3 = [T(r)P,+ ^^:1, ^ = [T(r)P,+ K(r)Pw cot 0], re ^02 = 2g~ B°2 (r)P" F°3 = ~ 2<r B°3 (r)P'-<" and re ^23 = -io^-B23(r)Pitg, (158) (159) (160)
232 THE REISSNER-NORDSTROM SOLUTION if proper use is made of the relations in Chapter 4, equations (40)-(42). We thus obtain the following equations for the various radial functions we have introduced: ^+(7-^) [2r-/(/+l)K]-?L = 0, (161) (T-V+L) = B23, (162) (T-V+NXr-(-r-vAN-(^ + v^L=B0i, (163) ;Nr+0 + v.r)[2r-/(/+l)K]-^i + 2v,r)L and -4^e~2viV-^e-2vr+(T2e-4v[2r-/(/+l)K] = B02 (164) B03 = -^B^),, = B2Xr + -B23, (165) r r rVvB02 = 2e*[2r-/(/ + l)l/]-/(/+l)r2B23, (166) (rV'Bb3).r + rVvB02 +ffV*-2'B23 = 2^^4^- (167) We observe that two of these equations (namely, equations (162) and (166)) are algebraic. We shall find it convenient to write X = nV = $(1-1) (l + 2)V, (168) in which case (by virtue of equation (162)) 2T-l(l + l)V= -2(L + X-B23). (169) By making use of this last relation, we can rewrite equation (161) in the form (L+X-B2i\r= -(^-vA(L + X-B23)-l-L. (170) Similarly, by combining equations (162), (163) and (165) Nr-Lr=0-vrW0+vr)L + ^B23. (171) Finally, we may note that by separating the variables in equation (157) by the substitutions (158), we are led to the equation Kr>r+2(-+v>rVr + ^-(iV + L) + (T2e-4vK = 0. (172)
METRIC PERTURBATIONS OF R N SOLUTION 233 We observe that this is the same equation that we obtained in the context of the Schwarzschild perturbations (Ch. 4, equation (51)). Equations (164), (170), and (171) provide three linear first-order equations for the three functions L, N, and V(— X/n). By suitably combining these equations, we can express each of them as linear combinations of L, N, V,B23, and B03. Thus, (cf. Ch. 4, equations (52)-(55)) N,r = aN + bL + c{X-B23\ a hvr W + lfc v. L + c{X-B23)--B23, r (173) (174) 1 + v,r )N fc + J-2v„ where n + 1 -2v r 1 n L-\c + — v e-2v + vr + rv2r + (T2re-4v-2 , r r r e*„-2v (X — B23) + B03, (175) (176) 1 n , M , -e~2v + —e~2v + M2 02 ,. , ,-e"4v + (T2re-4v -^-(1 +2e2v)e~4v, r r r r rJ (177) 1 e~2v O2 - + + rv2+ff2re-4v-2-^-e-2v 1 e~2v M2 = — + + r r r O2 3 e~4v + (T2re-4v-^f (l + e2v)e~4v. (178) The following alternative form of equation (175) may be noted: 1 nr + 3M - 2 Ql -2v N + 1 , M , 1 r r r (M2-Ql)e- + aL + B03. (L + X-B23) (179) It is a remarkable fact that the system of eq uations of order five, represented by equations (165)-(167) and (173)-(175), can be reduced to two independent equations of the second order. Thus, it can be directly verified that the functions H2+) and H[+), denned in the manner (cf. Ch. 4, equation (59)) H2+) = -X--(L + X-B23), n td (180)
234 THE REISSNER-NORDSTROM SOLUTION and H\ ( + ) 1 l r2B23+2Ql-{L + X-B23) m r2B23 + 2 Qlfr where m=nr + 3M-2Ql/r, -X-H2+) satisfy the pair of equations (cf. equations (144) and (145)), \*H2+) = 4 \_UH2+) + W{- 3MH2+) + 2Q,nH{+))], A2H[+) = ~lUH[+)+W( + 2QtHH2+) + 3MH[+))l where and U = {2nr + 3M)W + {tn-nr-M)- 2nA W- A 1 — {2nr + 3M) + -{nr+M). (181) (182) (183) (184) (185) rm Equations (183) and (184) can be decoupled by the same substitutions (147) and (148) that were used to decouple equations (144) and (145) obtained in the context of the axial perturbations. Thus, the functions r<+> ,(+) 1/2H( + ) and ZY'=+qiH\-' + (-qiq2f^H 7< + > _ ( r, r, U/2 1/(+) , _ //(+) z2 =-(-^1¾) Hi +qi"2 , (186) (187) with the same definitions of qx and q2, satisfy the one-dimensional wave- equations A2Z|+)= K! + )Zp (i=l,2), (188) where V[+) = *IU+±(91-q2W] and V2+) = ^U-Hqi -q2Wl (189) These decoupled equations were first derived by Moncrief and Zerilli (though by very different methods). Also, it can be verified that in the limit <2„ = 0, the equation governing Z2 ' reduces to Zerilli's equation (Ch. 4, equations (62) and (63)). (i) The completion of the solution From our discussion in Chapter 4 (§25) of the reduction of the equations governing the polar perturbations of the Schwarzschild black-hole to Zerilli's
RELATIONS BETWEEN V\; +> AND V\C ' AND Z\ + ) AND Z\~ 235 equation, it is clear that the present reducibility of a system of equations of order five to the pair of equations (183) and (184) must be the result of the system allowing a special solution. By applying an extension of the algorism described in Chapter 4, §25, to the more general case, when the reduction of a linear system of differential equations to more than one second-order equation has been accomplished, Xanthopoulos discovered that the present system of equations (165), (167), and (173)-(175) allows the special solution (cf. Ch. 4, equation (126)) Ml Nm M--(M2-Ql+<j2r4)-2- A * r Lm = r-iev(3Mr-4Ql),Xi0) = ne"r-1, B (0) 23 2Qlr~3e\ and B(0U3' = 2Qlr~6e-y(IQl + r2 -3Mr).^ (190) The completion of the solution for the remaining radial functions with the aid of the special solution (190) is relatively straightforward. Xanthopoulos finds JV = JV<0)$ + 2n H (+) < + h m (nrffr + G.W'). + ^02v(n7-2nr-3M)-(n + l)ro](nr//2' + Q^liH\ '), (191) rw L = L^<b—j(nrHi+) + Q^H\^\ X=X(0)(D + -//< + ), B23=Bi>-%H<+», #03 where r ' r m (nrH^> + QmnHV>). $ /< + ). r< + )> (nrH^' + Q^nHY')—dr. mr (192) (193) (194) (195) (196) It may be noted here that in the limit <2„ = 0, $ as denned in equation (196) differs from <b as denned in Chapter 4, equation (70) by a factor e~v. 43. The relations between V\+) and V{i~) and Z\+) and z!_) As in the case of the Schwarzschild perturbations, the potentials V^ (i = 1,2), associated with the polar and the axial perturbations, are related in a
236 THE REISSNER-NORDSTROM SOLUTION v2 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 Kj+) (0.9)—►/ / 0.6%\—Ki"'(0.9) -10 -8 -6 -4 -2 2 4 r./M 10 Fig. 19. The potential barriers l-']*' surrounding the Reissner-Nordstrom black-hole: the full- line curves are appropriate for axial perturbations for ¢, = 0,0.4,0.6,0.8, and 0.9. The curves are labelled by the values of Qt to which they belong. The potential barriers for the polar perturbations differ very little from those for the axial perturbations: cf. the neighbouring full-line and dashed-line curves belonging to the axial and polar perturbations for Qt = 0.9. manner which guarantees the equality of the reflexion and the transmission coefficients determined by the equations governing ZJ*'. Thus, it can be verified (and as we shall later deduce in §45), the potentials are, in fact, given by (197) where k = n2 (H2 + nPi = qP and / = rJ(nzr+qj) (i,j=l,2;i±j\ (198) and the q/s have the same meanings as hitherto. The solutions, z[+) and Z\ ' of the respective equations are, therefore, related in the manner (cf. Ch. 4,
RELATIONS BETWEEN V\ + ) AND V\T > AND Z] + t AND Zj"1 237 0.4 0.3 — Vt 0.2 — 0.1 -10 - Fig. 20. The potential barriers V}- 'surrounding the Reissner-Nordstrom black-hole: the full- line curves are appropriate for axial perturbations for Qt = 0,0.4,0.6,0.8, and 0.9. The curves are labelled by the values of Qt to which they belong. The potential barriers for the polar perturbations cannot be distinguished from those for the axial perturbations in the scale of the diagrams: cf. the neighbouring full-line and dashed-line curves belonging to the axial and polar perturbations for g„ = 0.9. equations (152) and (153)) lH2{H2+2)±2iaq]^z\±) = H2(H2+2) + 2q]A ^(^r+qj) Z (+) dZ|+> (199) It is the existence of this relation which guarantees the equality of the reflexion and the transmission coefficients determined by the wave equations governing Z\+) and Z\~\ We shall return in §47 to a consideration of the implications of the solutions of these equations to the problem of reflexion and transmission of gravitational and electromagnetic waves simultaneously incident on the Reissner-Nordstrom black-hole. Meantime, we exhibit in Figs. 19 and 20 the potentials V\^ for a range of values of Q^.
238 THE REISSNER-NORDSTROM SOLUTION 44. Perturbations treated via the Newman-Penrose formalism The solution for the Weyl scalars, ^Fq and ¥4, via the Newman-Penrose formalism, in the context of the Schwarzschild perturbations, was readily possible only because four of the Bianchi identities and two of the Ricci identities (listed in Chapter 4, equations (232) and (233)) are already linearized in the sense that they are homogeneous in the quantities which vanish in the background geometry (by virtue of its type-D character). This fact meant that these equations could be considered without a knowledge of the perturbations in the basis vectors—a problem of an altogether different order of magnitude (as we shall find in Chapter 9). While this favoured situation persists with regard to the Bianchi and the Ricci identities in the Reissner-Nordstrom geometry (again, by virtue of its type-D character) it is not present in Maxwell's equations (see equations (200)-(203) below)—they include terms in the Maxwell scalar <j>1, which do not vanish in the background. But equations can be derived which are already linearized in the same sense that the Bianchi and the Ricci identities are. We begin by deriving these equations. (a) MaxwelFs equations which are already linearized Equations governing the Maxwell scalars, <f>0, <f>1, and <f>2 m the Newman-Penrose formalism are (Ch. 1, equations (330)-(333)) (D-2p)4>1-(5* + n-2<x)4>0 + K$2 = 0, (200) (<5-2t)0!-( A+n-2y)4>0 + <j<t>2 = 0, (201) (5* + 2n)<f>1 -(D-p + 2e)4>2-l<t>0=0, (202) ( A+2h)4>1-{6-t + 2P)4>2-v$0=0. (203) Since 4>i(= Qm/2r2) is non-vanishing in the background geometry, the linearized version of these equations will involve the perturbations in the basis vectors (acting as directional derivatives on 4>i) as well as of the spin coefficients p, t, n, and fi (which are also non-vanishing in the background). To rectify this situation we proceed as follows. Applying the operator (<5-2t-oc* -[S + n*) to equation (200) and the operator (D — e + e* — 2p — p*) to equation (201) and subtracting one from the other of the resulting equations, we obtain [(<5-2T-a*-/? + 7t*)(<5* + 7t-2a) -(D-e + e*-2p-p*)(A + n-2yn$0 = [(5-2x-x*-P + n*)K-(D-£ + e*-2p-p*)<j]4>2 + k54>2 -<jD<t>2 + (3D-D5)(j>y - (25p)¢^ + (2Dt)^1 -(a* + ^-7t*)D^1+2p(a*-l-^-7t*)^1 -{-z + E*-p*)d<i>l+2x(-z + z*-p*)<}>l. (204)
PERTURBATIONS TREATED VIA N-P FORMALISM 239 On the right-hand side of this equation, we replace the commutator (SD — D6) operating on <j>u by its value given in Chapter 1, equation (304) and also substitute for Dx and dp from the Ricci identities (Ch. 1, equation (310), (c)and (k), respectively). Considerable simplification results and we are left with [(<5-2T-a*-/? + 7t*)(<5* + 7t-2a) -(D-e + e*-2p-p*)(A + Ai-2y)]4>0 = [{5-2x-oi*-P + n*)K-{D-z + E*-2p-p*)o~\<}>2 + K5<}>2-oD<j)2 + 24>l\_(A-3y-y*-n + n*)K-(5*-3<x + p*--t*-n)o] + 4$lv¥l+KA4>l-<jd*4>l. (205) Similarly, by the application of the operators — (A + p* — y* + y + 2fi) and + (<5* — x* + <x + fi* + 2n) on equations (202) and (203), addition, and subsequent simplification in like manner, we obtain the further equation [ (3 * - x * + a + p * + 2 n) (6 - x + 2 /?) - (A + n* - y * + y + 2(i) (D - p + 2e)] <f>2 = [(A + H*-y* + y + 2n)X-(d*-x* + <x + p* + 2n)v]$0 + X A(j>0 —v3*4>0 + 2$l[-(D + 3e + e* - p+p*)v+(S + n*+ x - a* + 3/2)2] + 44>lv¥3-vD4>l+134>l. (206) It will be observed that equations (205) and (206) are already 'linearized' in the sense we have been using that term. Indeed, the terms in 4>i m equation (205) and in (j>0 in equation (206) are quantities of the second order of smallness and can be ignored in a linear perturbation theory. Also, the terms A(j>y and 6 * 4> i (in equation (205)) and D 4> i and <5 4> i (in equation (206)) can be replaced, respectively, by —2fi4>l, —2n<t>u 2p<pl, and 2x(pl as required by equations (200)-(203) for the background. We thus obtain the following equations appropriate for a linear perturbation theory: [(<5-2T-a*-/} + 7t*)(<5* + 7t-2a) -(D-e + e*-2p-p*)(A + H-2y)]<t>0 = 2(j)y[( A-3y-y*-2ti + n*)K-(S*-3ot. + p*-x*-2n)<j + 2y¥l] (207) and [(<5*-T* + a + /?* + 27r)(<5-T + 2/?) -(A + n*-y* + y+2n)(D-p + 2e)]$2 = 2(j)yl(d + 7t* + 2x-<x* + 3P)X-(D + 3e + e*-2p + p*)v + 2v¥3J (208)
240 THE REISSNER-NORDSTROM SOLUTION (b) The 'phantorri gauge It is clear that it would not be a simple matter to consider equations (207) and (208), as they stand, along with the other equations derived from the Bianchi and the Ricci identities. A gauge that would manifestly simplify the equations considerably is one in which 4>o = 4>2 = 0. (209) For, in this gauge, equations (207) and (208) simply become equations relating the spin coefficients, k, <j, 1, and v, with the Weyl scalars v¥l and ¥3; thus, we shall have ( A-3y-y*-2n + n*)K-(S*-3<x + P*-x*-2n)o + 2*i'l =0 (210) and (5 + n* + 2-c-<x* + 3P)l-(D + 3e + e*-2p + p*)v + 2*i>3 = 0. (211) It is a remarkable fact that equations (210) and (211) survive even in the limit 4> 1 = 0; and in the limit 4> 1 = 0, they may be said to describe a 'phantom gauge'. Indeed, we shall find that it is, in fact, the phantom gauge to which we were led in Chapter 4, §29(fc) (see equations (217) and (218) below). It follows from our discussion of the effect of tetrad rotations on the various scalars and spin coefficients in Chapter 1 (§8(3)) that a gauge in which c£0 and 4>2 vanish can be chosen. An infinitesimal rotation of class II, for example, will change the Weyl and the Maxwell scalars, to the first order in b, according to the scheme (cf Ch. 1, equations (345) and (346)) ^0-^0 + 4^1, ¥i->¥i + 3byV2, ¥2->¥2 + 2W3, ^3-^^3 + ^4, ¥4 -»*4; 4>0 -+4>o + 2b4>u 01-^1 + ^02. and <i>i^<t>2- (212) Since *¥0, *Ft, ¥3, ¥4, 4>0, and 4>2 vanish in the unperturbed Reissner-Nordstrom space-time, it is clear that *P0, *¥2, ¥3, ¥4, ¢1, and <f>2 are unaffected to the first order by the infinitesimal rotation. But v¥l and 4>0 are not: they are affected to the first order in b since ^2 and 4>i are non-vanishing in the background. It may, however, be noted, parenthetically, that while a gauge, in which v¥l and ¢1 vanish simultaneously, cannot be chosen, the combination 2y¥l<f>l -3(^0^2 is invariant to the first order (213) for the infinitesimal rotations. In like manner, a gauge in which <p2 or ¥3 is zero can be chosen. Returning to equations (210) and (211) and substituting for the non- vanishing spin-coefficients their values given in equations (112), we obtain _Ay~* 5\ 1 2? Y^_£V + _L_if2(T = 2Ti (214)
PERTURBATIONS TREATED VIA N-P FORMALISM 241 and *&\l.-(90-% =-2^3, (215) where the operators Q>„, 3s\, Z£n, and S£\ have the same meanings as in Chapter 4, equations (228) with a difference only in the definition of A. In terms of the variables (cf Ch. 4, equations (236)) k = (r2 y/2)k, a = sr, X = 2l/r, v = n y/2/r2, *! = *JrJl, and ^3 = *3 V2/r3, equations (214) and (215) take the simpler forms, (216) t 3\ $! *>2—)k + £e2s = 2— (217) r and -\n-Se\l = 2 —. (218) r / r It will be observed that equations (217) and (218) are exactly those that were introduced in Chapter 4 (equations (273) and (276)) to rectify the 'truncated symmetry' of equations Chapter 4, (237)-(242); they represent in fact the phantom gauge. (c) The basic equations The Ricci and the Bianchi identities we shall consider are the same as the ones we considered in the context of the Schwarzschild perturbations, namely, Chapter 1, equations (310), (b) and (j) and Chapter 1, equations (321), (a), (e), (d), and (h). The Ricci identities are the same in the present context as well: they are the third in each of the two groups of three equations (Ch. 4, equations (232) and (233)). But in the first pair of equations, which represent the Bianchi identities, we must include the 'Ricci terms' listed in Chapter 1, equation (339). In the chosen gauge, 4>o = 4>2 = 0' these additional terms are, respectively, -4/c^.n, (a); 4(74)^, (e); -410^?, (d); and 4v^^f (h). (219) By the inclusion of these terms, the right-hand sides of the first pair of equations in Chapter 4, equations (232) and (233), become: + k(3¥2 -40!0f )= -K(3Mr-2Ql)r-*, + (7(3y2+4,^?)= -o(3Mr-4Ql)r-\ -A(3'F2 + 4^10f)= +H3Mr-4Ql)r-\ -v(34P2-4^)1^f)= +v(3Mr-2Ql)r-\ (220)
242 THE REISSNER-NORDSTROM SOLUTION on substituting for ¥2 and 4>i their values given in equations (114) and (115), namely, y2= -{Mr-Ql)r-4 and ¢, = QJ2r2. (221) With the terms (220) on the right-hand sides, our present basic set of equations (including equations (217) and (218)) are (cf. Ch. 4, equations (237)-(242)) if2<D0-(®o+^)*i = -2fc(3A#-2^M, (222) a(®2— W + JSPt^! = + 2s HM-4^), (223) (^0+-)5-^-1^ = -, (224) M@\ — )k+^2s = 2 — (225) r I r and (a0 — W-JSP.^j = +2/(3M-4^ JZ,5** + A(si1+-J4>3=2n(3A#-2^j (227) A(V_t +- )/ + :?_!„ = *♦, (228) ^o-^V-^S' = ;«3, (229) where <D0 = ¥0 and <D4 = ¥4r4, (230) and the remaining variables have already been denned in equation (216). (d) The separation of the variables and the decoupling and reduction of the equations Considering the first group of equations (222)-(225), we find that the variables can be separated by the substitutions (cf. Ch. 4, equations (247), (278), and (283)), O>o(r,0) = K + 2(r)S + 2(0); 9^,0) = R + l(r)S + l{6);\ k(r,6) = k(r)S + l(6); s(r,8) = s(r)S + 2(8), J where the angular functions S + 2(0) and S + 1(0) are the normalized proper
PERTURBATIONS TREATED VIA N-P FORMALISM 243 solutions of the equations ^^2S + 2=-n2S + 2 and J?2J?ttS+l=-n2S+l, (232) where ,/2 = 2n = (/-1)(/+ 2). (233) Besides, the functions S+2(9) and S+1(6) are related in the manner (cf. Ch. 4, equations (281) and (282)) y2S + 2=nS+l and ^tlS+l = ~nS + 2. (234) The substitutions (231) effect a separation of the variables in equations (222)-(225) by virtue of the relations (234) among the angular functions; and we obtain the following coupled system of equations for the radial functions we have denned: //K + 2-(®o + -V+i = -2fc(3M-2^M, (235) a(91-1)R + 2-hR+1= +2s\3M-4^A, (236) ?o + -V+^ = —• (237) (238) a(®1-% + Hs = 2^-. The system of equations (235)-(238) can be decoupled to provide a pair of independent equations of the second order by considering the functions F+l = R + 2+qik/n; G+l = R+l + qls/li, j F + 2 = R + 2+q2k/n; G + 2 = R+l+q2s/n, J where qi and q2 have the same meanings as in §42 (equation (149)). Thus, by adding equation (236) to ql/^i times equation (238), we obtain A(®i-3)F+l=uR+l+2s(3M + ^p-)-qis + 2^ n— F+l =nR+l+2s 3M + ^ )-qis + 2— R+l = HR+l+q2s + -^-[R+l+^s =Ki+?-:)(K*'+7s)-(i+?j)G"' ,240) where, in the reductions, we have made use of the relations (152) among qt and q2. The relation obtained by interchanging the indices 1 and 2 in equation (240) is manifestly also true. Both these relations can be expressed in the
244 THE REISSNER-NORDSTROM SOLUTION single equation AUlA\F + . = Jl+^L\G+j (i,j = 1,2; i±j). (241) The convention that i andj take the values 1 and 2 but i ± j will be adhered to, strictly, in the rest of this chapter; it will not be restated on every occasion. By combining equations (235) and (237) in a similar fashion, we obtain the equation i0+iy+i=n(i+-^y+j. (242) Now letting 1 A2"T' ~T' r F + i = 1^Y+i and G+i = ^X + h (243) we find that equations (241) and (242) become 2lY+i = n^l+^y+j (244) and &oX+, = ^(l+^y+l. (245) Since Ad A + d -T^o = 3— +i<r = A+ and -^S>1=- «r = A_, (246) r dr,, rz dr„ we can rewrite equations (244) and (245) in the forms A-y+i = /v(!+^)*+; (247) and A + X-=4(1+^)y- (248) Finally, eliminating the X's in favour of the Fs, we obtain the pair of basic equations, A2y+i + p;A_y+i-eiy+; = o (i = 1,2), (249) where '•-£"(0 ">-<»%)• and
THE TRANSFORMATION THEORY 245 By a similar sequence of reductions, we obtain from the second group of equations, (226)-(229), the conjugate pair of equations, A2y_; + pa + y_; - fty_, = o, (252) by the substitutions <D4(r,0) = K_2(r)S_2(0), 93(r,0) = R.^S.^U n(r, 0) = ,,(^^(0), /(r,0) = /(r)S_2(0), J A F_; = r3y_i, and G_; = ^.,. (254) The angular functions, S_t(0) and S_2(0), are now defined by the 'adjoint equations' if_!if2S_2= -|i2S_2 and ^5^..^ = -//¾..^ (255) and they are related in the manner <e\S.2= -/zS_! and if^S^ = +/*S_2. (256) Thus, the Newman-Penrose equations, governing the Weyl and the Maxwell scalars describing the perturbations of the Reissner-Nordstrom black-hole, have been reduced to the same standard equation that we considered in Chapter 4 in the context of the Schwarzschild black-hole. 45. The transformation theory The pair of equations, (249) and (252), to which the Newman-Penrose equations were reduced in §44, differ from the equations considered in §30 (Ch. 4, equation (284) and its complex conjugate) only in one inconsequential respect: A2, in the definition of Ph is replaced by Dr The transformation theory developed in §30 will, therefore, apply in the present context with only very minor modifications. The problem to which we now address ourselves is the transformation of equations (249) and (252) to a pair of one-dimensional wave equations of the form A2Zf = VtZt (257) for some suitably defined potential. We shall, in the first instance, restrict our consideration to the transformation of equation (249). Also for the sake of convenience, we shall write Yt in place of y+i. The analysis appropriate to equation (252) will proceed along exactly the same lines with the only difference that the sign of a should be changed and, in particular, A_, wherever it occurs, must be replaced by A+ , and conversely.
246 THE REISSNER-NORDSTROM SOLUTION As in §30, we start with a substitution of the form (cf Ch. 4, equation (287)) Y, = /,^,+(^ + 2ia/,)A + Z„ (258) and obtain the equations (cf. Ch. 4, equations (290)-(292)) A_yf= -^/?;Zf + KfA + Z;, (259) r r8^ dr. —rA-l^(/«^)+^^, (260) and ^=/,^ + -(^ + 2^/,). (261) The compatibility of equations (249) and (257), then, leads to the equations (Ch. 4, equations (298)-(300)) j^r=(Qifi-Ri)V; (262) ^r(^R^ = ,^\.Qi^i + 2ia(Qifi-Ri^ + ph (263) and the integral r8 — Ri fiVi + Pi(Wi + 2ia f.)=K, = constant. (264) (a) The admissibility of dual transformations We shall now verify that, as in the case of the Schwarzschild perturbations, equations (260)-(264) are compatible with Pi = constant and /, = 1, (265) and allow dual transformations with two values for /?, of opposite signs. As in §30 (Ch. 4, equations (307) and (311)) the condition for existence of dual transformations is that F, = r8^ = ~(n2r + qj) (i, j = 1, 2; i + j) (266) satisfies the differential equation l/dF,V d2F, D, Pf Y\4r-)-K+SFf = J;+K> (26?) for some suitably chosen constants, Pf and Kt = K,-2iaPi. (268)
THE TRANSFORMATION THEORY 247 We find that Fh as defined in equation (266), does satisfy equation (267) with fif=qf (i,j = l,2;i^j) and *, = k = n2(n2 + 2). (269) The potentials, KJ*', associated with the dual transformations belonging to ±qj are (cf Ch. 4, equation (317)) V\±)=±qi^- + qfff + Kft (i,j=\,2;i+j\ (270) fi=!T= 3, 2 . , (U=l,2;i^). (271) where 1 Ff H(^2r + ^-) Comparison with equation (197) shows that the potentials (270) are in fact the same that occur in the wave equations which determine the axial and the polar perturbations. Also, we may note that (cf. Ch. 4, equation (312)) ^=-^/^/., (272) or, explicitly, W\^=W^)=\(r-2,M + 2^L) (273) and ^ + )=^(-)-2^. 3 A (i,j =1,2; i^j). (274) Jr3(n2r + qj) The explicit forms of the associated transformations are (cf. Ch. 4, equations (318) and (319)) Yi= KS±)zS±)+(^±) + 2i(T)A + zS±), (275) A_ Y^T^qjZ^ + QiA + Z\±\ (276) K\Vz\V=^Q,Yt~(lV\V + 2io)A-Yh (277) KP'A + ZP- + ^+^^-^-, (278) where it may be recalled that 4(1+$y1+4.)_„>.*(1+4. and KS±) = |i2(|i2 + 2)±2i(T^. (280) a-^S'+£•+& -^'+&• <2*»
248 THE REISSNER-NORDSTROM SOLUTION Making use of equation (247), we obtain from equations (276) and (277) the further relations liXj= T^.Z^ + ^^l+^A+ZP, (281) K^ZW = ^'-t I+-^^,-(^ + 21(7)^]. (282) As we have stated earlier, in the foregoing equations, we have written Yt and Xj for y+( and X+j. To obtain the corresponding equations relating y_f and X_j to ZJ*' we need only to change the sign of cr and write A + wherever A_ occurs, and conversely. (b) The asymptotic behaviours ofY±i and X±j The short range character of the potentials, V^K ensure that the solutions Zp* have the asymptotic behaviours, e±iar* both for r„ -> + oo and for rm -> — oo (cf §47 below). We can use this knowledge (as we did in a similar context in §32) to deduce the associated asymptotic behaviours of Y±i and X±j with the aid of equations (275)-(278), (281), and (282). In making these deductions, we need only to know that V^, Qhfh and A all tend to 0 for r„ -► ± oo, (283) and that 1^-.-0 for r„ -.-+00 and W\±]^ --r(Mr+-Ql) for r^ -► - oo. f + (284) Notice that the limiting value of H^'at the horizon is independent of i and of the distinguishing superscript (±). To avoid the ambiguity of " + " used both as superscripts (to distinguish the axial from the polar in Z^') and as subscripts (to distinguish Y+i from Y-t which satisfies the complex-conjugate equation) we shall explicitly write out the relations appropriate for Z{+) and suppress also the distinguishing superscript. We find: r„ -»• + oo: Z^e + ,<"\ Zf->e -'<"-., Y+i-> -4<j2e + iar*; Kfe+'<"•• Y-l~> A 2 4 ' 4a r* Ki e -'<"•. +1 ~~* a 2 4 ' 4a1 r* y_f-> -4<j2e-iar*; X+^2ioiir2e + iar*, ^w^ X^~2i^ X.j-> -2ionr2e-iar* (285)
DIRECT EVALUATION OF WEYL AND MAXWELL SCALARS 249 *-; ^...-.-. ..„',_—^KT^r; (286) ■e'ia\ Y+i^ and r„ -»• -oo: ^-^ + ^., y+f->4i(7[i(7-(Mr+ -6*)/r3+]e+ ,or., y A^ Kf(l+2gi/^r + )g+ig% H"r8+4[i<r + (Mr+-eJ)/r3+][iff + 2(r+-M)/ri]' Kfe+iar* 2nlio+\Mr+-Ql)/r3+V A^ Ki(l + 2qi/n2r+)e-i,'r' ^4[i(7-(Mr+-Q^)/r3+][i(7-2(r+-M)/r2+]' y_i^4i(7[i(7+(Mr+-e2)/r3+]e-i-., X K'e'iar' +J" 2^[i(7-(Mr+-e2)/r3+]' To obtain the relations appropriate for Z\~\ we need, in accordance with equations (280), only to replace K{ by Kf, and conversely. 46. A direct evaluation of the Weyl and the Maxwell scalars in terms of the metric perturbations As we have stated, one of the principal objectives in the study of the perturbations of the Reissner-Nordstrfim black-hole is to ascertain how incident electromagnetic and gravitational waves will be scattered and absorbed by the black hole. To apply the perturbation theory developed in the preceding sections towards this end, it is necessary that we first relate the functions denned in the theory with the amplitudes of the waves of the two kinds. The functions, ZJ*', must, in some direct way, specify the amplitudes of the incident, the reflected, and the transmitted waves since, in the limit <2„ -► 0, the equations governing Zj*' reduce to the Zerilli and the Regge-Wheeler equations. In general, we may expect that the required amplitudes are some linear combinations of the functions Z[+) and Z(2+) and, similarly, o(Z[~) and zy\ The question is: what linear combination? The answer to this question via the solution of the Newman-Penrose equations in §44 is not also straightforward (as it was in Chapter 4, §32) since they have been solved in a special gauge; and, in consequence, the functions R±2 and R + \, in terms of which the solutions have been expressed, do not have simple physical interpretations. For these reasons, we shall evaluate the Weyl scalars *Po and
250 THE REISSNER-NORDSTROM SOLUTION ¥4 and the Maxwell scalars <f>0 and <f>2, ab initio, from our knowledge of the Riemann and the Maxwell tensors. Since the Schwarzschild and the Reissner-Nordstrom metrices differ only in the definition of the horizon function A (= r2e2v in both cases), much of the analysis of §31 can be carried over. In particular, the expression for *Po given in Chapter 4, equations (334), (339), and (350) are valid, as written, in the present context. Considering the axial part, Im *P0 of *P0, we start with Chapter 4, equation (339): -2 + 2ioA 1/^^ + 9^/1- W dr \r* dr ) V \r V'r J dr /--3/2 ^1 + 2 sin2 6' (287) On the right-hand side of this equation, we replace the first term in the square brackets by (This is equation (140) in which B has been replaced by H[~) in accordance with its definition in equation (143).) Making the replacement, we find after some rearrangement of the terms icrlm^o = \io 2 dr \AJ rA 2A dr + - 2r2A Q + 1 /1 dr r2A A2' //<"» /--3/2 W + 2 sin2 8 where Now letting --vr = -l(r2-3Mr + 2<22). r rA Q = r//<2-» (289) (290) (291) as in equation (143), we find, after some further simplifications, that the terms in curly brackets on the right-hand side of equation (289) become 10 r d/f 2" A dr ^A2 + i_(r2-3Mr + 2<22t)//2-> — a //<-» L2"2 + 27A-//r,+^//<r, + i(r2-3Mr + 2e*)(r^ + //<2"))' ^ Replacing the derivatives with respect to r by derivatives with respect to r„ and
DIRECT EVALUATION OF WEYL AND MAXWELL SCALARS 251 recalling that (cf equation (273)) ^<->=_ (r2-3Mr + 2Ql), we regroup the terms in (292) in the manner ^->+2^+%£/n-> 2A dr„ rzA (293) + ,2 r2 - + - \i- , r2-3Mr+2Ql \2rA ' r3A Returning to equation (289), we can now write 2A2 H2'\ (294) -ifflmVo = —T^(H/<-» + 2i(7)A + // r<-) + - (|i2 + 2)r-6M + 4 e2* Hi-' + ^(2e.M)Hl-'}^; or, in view of equation (144) satisfied by Hj ', -frlm*0 =^((^-. + 2^,..,,,, ^ ■2z(7)A+ + A2}//<-)^. (295) (296) We shall now express H2 > as a linear combination of Z(t ' and Z2+) by solving equations (147) and (148). Rewriting these equations in the forms ZP = [4i(4i-42)]1/2(tf(rcos.A + //2->sin.H and where '(-) — \_ql(ql-q2nll2(H^cosil,-H[-)smil,)^ sin 2i/> = 2(-qiq2)112 2Q*H qi-q2 s/(9M2 + 4Qln2y we have the solutions [?i(?i-^)]1/2«(r) = 2(r)cos^-Z2-»sin^ and lqi(qi-q2)yi2H2-) = Z[-)sin\l/ + Z2-)cos\l/. Inserting for H{2) from this last equation in equation (296), we obtain -3 I 1/2 (297) (298) (299) ^[?i(?i-^)]1/2Im^0 = ^2{(H/<-) + 2i(7)A++A x(Z<2-»cosiA + Z(r»siniA)x -4^, (300) /--3/2 ^1 + 2 sin2 6
252 THE REISSNER-NORDSTROM SOLUTION or, by virtue of the equations satisfied by Z\ >, we have (cf Ch. 4, equation (345)) -3 r 2A2 ^[q1(q1-q2)]1/2Im^0=^I{[K<2-»Z<2-»+(H/(-» + 2i(7)A + Z<2-»]cos^ r-3/2 + [K<-»Z<-» + (H/<-» + 2i(T)A + Z<r»]sin^}^^-. (301) sin u Finally, by making use of equation (275) of the transformation theory, we obtain the important relation - 2ia[<h (<h - q2)] 112 Im %, = ^ (Y+ 2 cos ^ + Y+1 sin tfr) ^^- (--3/2 = (F+ 2 cos ^ + F+1 sin "/0^0- (302) Similarly, we shall find (--3/2 -2i(7[q1(q1-q2)]1/2Imr4^4 = ir3(y_2cos^+y_1sin^)^^- = |(F_2cos^ + F_lSin^)^-. (303) sin 6 An analogous calculation for Re^Po can be carried out; but it is hardly necessary since it is clear on general grounds that we shall obtain the same relation except that the factor 2i<j on the left-hand side of equation (302) will be missing (cf. Ch. 4, equations (345) and (353)). (a) The Maxwell scalars <j>0 and <j>2 By definition, ¢0 = F{pnq/"W\ (304) where we have enclosed the indices in parentheses to emphasize that they signify the components in the tetrad frame in which equations (117) and (118) are written. The basis null-vectors in the tetrad frame are given in Chapter 4, equation (333); and by contraction with these vectors, we obtain 4>o = ~7^e-v [i(F0l + F21)+ (F03 + F23)]- (305) We observe that (j>0 is again decomposed into an axial and a polar part. Considering first the axial part, — v — 2v lm<t>0 = e-j-{F0l+F2l)= € , (re*F0l sin6+ revF2l sin 6), (306) V2 (r sin 0) ^/2
DIRECT EVALUATION OF WEYL AND MAXWELL SCALARS 253 we write the term in F21, by making use of equation (120), in the manner iarevF2l sin0 = e2v(revF01 sin0)>r = (revF01 sin0)jr,. (307) Thus, lm4,0 = (i,AsLe)j2A+{reVFoiSine) = (ioAsUj2A^> (308) where, in accordance with equation (131), we have written B in place of F01 sin0. Now substituting for re"B from equations (138)and (143), we obtain r 1 dC~3/2 -(2V2),^Im0o = -A + Hr)sin2g £' (309) where, by the recurrence relations satisfied by the Gegenbauer functions, 1 dC,V2/2 sin20 d6 n2 + 2 Now substituting for H\~) from equation (299), we have 1^^-=-^^- <310) -i<T^(^2+2)[2q1(q1-q2)]1/2Im0o = ^A+(Z<r)cos^-Z2-»sin^)P;j(,, (311) where it may be noted that by equation (281) ^-A + Zr>= 2, * (nXj-qjZ\->). (312) Similarly, by considering the expression ^2 = f<P,<>(pV«> = ^[i(F01+F21)+(F03 + F23)], (313) we find -i(JH(H2 + 2)l2ql(ql -g2)]1/2Im02 = ^-A_ (Z^' cos tfr - Z<f> sin tfr)P, e. j 2r (314) where ;A-Z", = ^T^(^--^Z'<",)- (315) Considering next the polar part, Re 0O, of 0O and substituting for F03 and F23 from equations (159) and (165), we find Re^o= -^ An /,^Ar2B2i)Pue. (316)
254 THE REISSNER-NORDSTROM SOLUTION On the other hand, by equations (181), (190), and (193) 202 (317) Therefore, (2^2)Re4>0=n^A+(H[+'+2^<Aph(). (318) We may again replace the term in H[+) in the foregoing equation by ^A+«r)=^[?i(?i-^)]_1/2A+(Z< + )cos^-Zi + »sin^), (319) where by equation (281) XA+Z-,+, = ^?^)(^ + ^Zr,)- (320) 47. The problem of reflexion and transmission; the scattering matrix We now turn to the fundamental problem to which the theory of the perturbations of black holes is addressed: the manner of their interaction with incident waves of different sorts. First, we recall that the axial and the polar perturbations have each been reduced to a pair of one-dimensional wave equations with real positive potentials which have an inverse-square behaviour for r„ -» + oo and an exponentially decreasing behaviour for r* -» — oo. On account of this short- range character of the potentials, the problem with respect to each of the equations is one of barrier penetration: i.e., we have to seek solutions of the equations which satisfy the standard boundary conditions Z{±)-^+^ + /^)(^-^ (r,-» + oo), r{±)(ff)e+tor« (r.->-oo). As we have shown in §43, the forms (197), which the potentials, J7-*', have, enable us to express, as in equation (199), the solution Zj + ) as a linear combination of Z\~) and its derivative, and conversely; and from equation (199) it follows from the analysis of §27 that (cf Ch. 4, equations (168) and (169)) R\ + )(<j) = R\-\<j)eiS' and r} + )(a) =rj_)(ff), (322) where iS H2(H2 + 2)-2iaqj ,-,.-/ n ,,,¾ H2(H2 + 2) + 2i<jq} <±>f,rt»+'■"•. '- -- <" (321)
REFLEXION AND TRANSMISSION; THE SCATTERING MATRIX 255 In view of these very simple relations between the amplitudes of the reflected and the transmitted waves belonging to the two classes of perturbations, we shall, in the rest of this section, suppress the distinguishing superscripts with the understanding that the analysis which follows applies, equally, to Z\ + ) and z\-\ Now, with respect to scattering by each of the two potentials Vi (i = 1, 2; suppressing the distinguishing superscript!) we can define a scattering matrix (Ch. 4, §28) T-Xa) Rt(o) whose elements (representing the reflexion and the transmission amplitudes) satisfy the unitarity requirements (Ch. 4, equations (181)-(186)) S,= (i = 1, 2), (324) and /J,(ff)7?(ff) + r,(ff)Kf(ff) = 0, |K,(ff)| = \Rt(a)\, (325) |K,(ff)|2 + |r,(a)|2=l. (326) (We are here denoting by R(<j) what was denoted by R2(o)m §28.) By virtue of the relations (325) and (326), 5;((7)5f((7) = 1, (327) where 5f denotes the conjugate transposed-matrix of 5(. The scattering^ matrix has an important property with respect to 'time- reversibility': If Z{p and Zf are the amplitudes of the waves incident on the barrier from the right and from the left (see Fig. 21), then the amplitudes, Z{p and Zf\ of the waves reflected to the left and to the right, are given by so that by unitarity, s, *"*l z <r> 1 zv zv z 0 = _. s, zV 1 —* zv zf zv (328) (329) Our problem is now to relate the scattering matrices St and S2 to a scattering matrix (of order four) which will describe, similarly, the reflexion and transmission of electromagnetic and gravitational waves, simultaneously, incident on the Reissner-Nordstrbm black-hole. To this end, we must relate the amplitudes of the waves of the two sorts with the functions Zt and Z2. For this purpose consider the boundary conditions satisfied by the solutions for Y+i and y_f derived from the solutions for Zt satisfying the boundary conditions Zi^Z\x')e + i'"'* + Z\a')Ri((j)e'i''r* (r„-► + 00) (330)
256 THE REISSNER-NORDSTROM SOLUTION Z,!'»* Fig. 21. Incoming and outgoing waves from ±00 are reflected by the potential barrier to outgoing and ingoing waves. The scattering matrix (equation (328)) relates the amplitudes of the waves incident on the barrier to the amplitudes of the waves reflected by the barrier. Writing the asymptotic behaviours of the solutions for Y±h derived from the solution Z;, in the forms e + icr. ■Y+i^Y^l- +Y*lf r 1 r5 e + "r. A2 (r, - + 00), (r„ -► - 00), (331) »+«"■» r3Y_, -» Y^}- + Y^r3* "*"• (r„ -» + 00), -► y<ir)A2e + 'Vr. (r„-► - 00), (332) we conclude, from the correspondences in the asymptotic behaviours listed in equations (285) and (286), that y*i">= -4ff2Zj»>; K* y(in) _ Jvi y(oo) K, (333) yW = 4i(T[«<T-(Mr+ -e2)/r3 ] (r+)3Z(°°»rf((T), ^1=-^.-11 + 2q; KfZ!°°»ri((T) (r+)5 V /^2r+ ;4[i(T+ (Mr+ -Q*)/r3+] [> + 2(r+ -M)/r2 ] J (334) Considering first the fluxes of the incident and of the reflected gravitational waves, we recall that they are determined by the leading r~'-terms in the asymptotic behaviours of *P0 and ¥4 representing the incoming and the outgoing waves, respectively. From the expressions (302) and (303) for *F0 and
REFLEXION AND TRANSMISSION; THE SCATTERING MATRIX 257 ¥4, we infer (cf Ch. 4, equations (368) and (369)) and d2£<f> (S_2)2 1 dtdft In la1 J*e2f)cos i> + Y^f sin i/r |2. (336) From the expressions for 7*+"' and Y^Lep given in equations (333), it follows that the reflexion coefficient for the incident gravitational waves is given by I Z{?lR2(o) cos \j/ + Z(+°°iKi (a) sin »> \ IZ^cosiA + ZV^siniAl gr ~ ,^(=0),,^ ,, 7.(00) •_.,. 12 • (337) From this expression it is manifest that the amplitude H2 of gravitational waves is, apart from a constant of proportionality, given by H2 = Z2cosi/> + Z1 sini/f, (338) i.e., precisely the same quantity which satisfied the coupled equations (144) and (145) and (183) and (184) before their decoupling to yield the equations for Zy and Z2 (cf. equations (299)). (a) The energy-momentum tensor of the Maxwell field and the flux of electromagnetic energy The energy-momentum tensor of a Maxwell field, written out explicitly in terms of the scalars 0O, 0j, and 02, is 4nTij = {4>o4>o«i«j + 4>2<t>Vih + 20i 0* ['(i«j) + m{im3)] - 4050^,^, - 4$T$2l(imj) + 2020*™;™;} + complex-conjugate terms. (339) In the Reissner-Nordstrom background, the only non-vanishing scalar is 0i (= 6.(,/2^2). On the assumption that the perturbations in the field, represented by 0O, 02, and 34>l, vanish at least as rapidly as 1/r, one verifies that the evaluation of the perturbed Ttj does not require a knowledge of the perturbation in the basis vectors; and ignoring the terms which fall off more rapidly than r"2, we have 4nT,j = {00 0^,^+02 02^ + 2(501^:/,,^^,,^,] - 405^ n{imj)-434>^2l{imj)+ 20205»»,»»;} + complex-conjugate terms +0(r~3) (r->oo). (340) With the basis vectors given in equation (108), we find from the foregoing
258 THE REISSNER-NORDSTROM SOLUTION equation that the flux of energy, per unit time and per unit solid angle, is given by limit (^7-%) = limit—-( — i|^0|2-t-1<^2|2). (341) From the expressions for (j>0 and 4>i given in equations (311) and (314), it follows that, apart from the same constant of proportionality, for incoming waves, Z\x) exp (+ iorj, d2£(in) 1 d^dS =^(^)2(16^)|Z(r)cos^-Z<20O»sin^|2; (342) and for the reflected waves, z!0O)i?f(cr)exp( — iorj, ^j§- = ^{Pl0)2{\(>aA)\Z^R^)cosyj,-Z^R2(o)sin^2. (343) Accordingly, the reflexion coefficient, for the incident electromagnetic waves, is given by r .l^r^iWcosiA-Z^'^^siniAl2 ele IZf'cos^-Z^'sin^l2 ' l ' From this expression it is manifest that the amplitude, Hu of the electromagnetic wave is, apart from a constant of proportionality, given by Hl = Zy cos \fi — Z2 sin \fi, (345) i.e., precisely the same quantity which satisfied the coupled equations (183) and (184) before their decoupling to yield the equations for Zy and Z2. One can arrive at the same results (344) and (345) by reducing the problem to one in which the asymptotic behaviours of the solutions for X±j play the same role as the behaviours of the solutions for Y±i do in the treatment of the gravitational waves. But the arguments are somewhat less direct. (b) The scattering matrix We shall now show how the general process of scattering and absorption of electromagnetic and gravitational waves by the black hole can be described by a symmetric unitary scattering-matrix of order four exhibiting time- reversibility. As we have seen, the amplitudes, H t and H2 of the electromagnetic and gravitational (wave-like) disturbances (of some specified frequency) are related to the functions Zt and Z2 by Hi = Zy cos i/> — Z2 sin \fi ] and \ (346) H2 = Z2 cos \j/ + Zt sin \j/, J
REFLEXION AND TRANSMISSION; THE SCATTERING MATRIX 259 where it may be recalled that superscripts, ( ±), distinguishing the axial and the polar perturbations have been suppressed. Relations, inverse to (346), are Zj = Hl cos \fi + H2 sin \fi and \ (347) Z2 = H2costj/ — Hi sini/f. Now suppose that a pure electromagnetic wave of amplitude H \r) is incident on the black hole from the right. Then //(2r) = 0 and Z(C = H^cos\l/ and Z<2r) = -//fsini/r (348) Each of these incident Z -fields will give rise to reflected and transmitted amplitudes given by ZSr) = ZSr)i?i((T) and Zf = ZpT,(o); (349) and these amplitudes, recombined in accordance with equations (346), will give reflected and transmitted amplitudes in both electromagnetic and gravitational waves. Thus, H^ = HV(RlCos2\l/ + R2sm2\l/), (a) Hi" = HV (7i cos2 j, + T2 sin2 ^), (b) H(2r) = H(1r)(i?1-i?2)sini/rcosiA, (c) Hf = H(p (Ji-Tj) sin \j/ cos \j/. (d) If on the other hand, a pure gravitational wave of amplitude H(2r) is incident on the black hole from the right, then H^ = 0 and Zf) = H(jr)sin^ and Z<2r) = H^cosi//. (351) The reflexion and transmission of these incident Z-fields, when recombined, will give rise (in analogous fashion) to the following reflected and transmitted amplitudes in electromagnetic and gravitational waves: //^ = //^(^1-/^) sin \j/ cos \j/, (a) H f = H f (Ty - T2) sin * cos tfr, (b) HJ» = ^^(^^^^ + /^2 0082^). (C) ' Hf = H(2r) (Tt sin2 \j/ + T2 cos2 \j/). (d) Comparison of equations (350) (c), (d) and (352) (a), (b) shows that the conversion of incident energy of one kind, into reflected and transmitted energy of the other kind, takes place entirely symmetrically with respect to the two kinds—a consequence, clearly, of time-reversibility. Turning next to the situation described in Fig. 22, the scattering matrix which will describe the transformation (//P, Hf; HP, H2") -»(H}'», H<r»; H2", Hf) (353)
260 THE REISSNER-NORDSTROM SOLUTION w«= ///'> — Hi' —» Fig. 22. In the case of the Reissner-Nordstrom black-hole, electromagnetic and gravitational waves simultaneously incident on the potential barrier from opposite ends, from ± oo, are reflected to + oo; and a scattering matrix of order 4 is required to describe this process. can be readily written down with the aid of equations (350) and (351). Thus, 5 - HP HP //<" Hp — ^11 ^11 ^21 ^21 ^11 '11 ^21 ^21 ^12 ^12 ^22 ^22 Rl2 Tl2 R22 '22 HP HP HP Hp — Hp HP Hp HP (354) where Rn = RiCOs2\l/ + R2sin2\l/; R22 = Rt sin2i/> + /?2cos2i/>, 7*11 = 7*i cos2 \j/ + T2 sin2 \j/; T22 = 7\ sin2 \// +T2 cos2 \]/, Rl2 = R2l = (/?! — /?2)sini/>cosi/', Tl2 = T2l = (Ti—T2) sin i/> cos i/>. (355) It can be readily verified that the scattering matrix 5 denned in equation (354) is, by virtue of the unitarity of 5t and S2, unitary itself: 55= 1. In particular, the identities and Fiil2 + |Knl2+|7\2l2 + lK2il2=l |7nl2 + |Kiil2=F22l2 + IK22I2 (356) (357) required by the conservation of energy are satisfied. In Fig. 23 the dependence of the conversion factor C = 1/¾¾^2 on Q„ and a is illustrated for the case I = 2.
QUASI-NORMAL MODES OF R N BLACK-HOLE 261 I ()".=■ Fig. 23. The conversion coefficient C( +' (the figures on the left) and C(" '(the figures on the right) for the axial and polar perturbations belonging to / = 2. Each curve is labelled by the value Qt to which it belongs. 48. The quasi-normal modes of the Reissner-Nordstrom black-hole The Reissner-Nordstrom black-hole can be characterized by quasi-normal modes satisfying the same boundary conditions and with the same meanings as described in Chapter 4, §35 in the context of the Schwarzschild black-hole. There is, however, one important difference: quasi-normal modes can be defined with respect to each of the equations governing Z,(±> (i = 1, 2). Since the potentials, Kj(±), are real and positive (in the space external to the event horizon) the imaginary parts of the complex characteristic frequencies of the quasi-normal modes will be positive. Also, in view of the relation (199) between the solutions belonging to axial and polar perturbations, the characteristic frequencies will be the same for Z| + ) and Z\~\ It should also be noticed that there is no quasi-normal mode which is purely electromagnetic or purely gravitational: any quasi-normal mode of oscillation will be accompanied by the emission of both electromagnetic and gravitational radiation in accordance with equation (346). In Table V, we list the complex characteristic frequencies ol and a2
262 THE REISSNER-NORDSTROM SOLUTION Table V The complex characteristic frequencies belonging to the quasi-normal modes ofZx and Z2 (a is expressed in the unit M _1) z, z2 Q. 0 0.2 0.4 0.6 0.8 0.9 0.95 0.999 / = l 0.24828 + 0.09250/ 0.25150 + 0.09291 i 0.26194 + 0.094161 0.28276 + 0.09619/ 0.32349 + 0.09827/ 0.36082 + 0.09744/ 0.38927 + 0.09442/ 0.43031 + 0.08388/ / = 2 0.45760 + 0.09500/ 0.46296 + 0.09537/ 0.47993 + 0.09644/ 0.51201 + 0.09802/ 0.57013 + 0.09907/ 0.61939 + 0.09758/ 0.65476 + 0.09460/ 0.70310 + 0.08627/ / = 3 0.65690 + 0.09562/ 0.66437 + 0.095971 0.68728 + 0.09697/ 0.72919 + 0.09837/ 0.80284 + 0.09911/ 0.86375 + 0.09752/ 0.90668 + 0.09469/ 0.96434 + 0.08726/ / = 4 0.85310 + 0.09586/ 0.86260 + 0.09621/ 0.89100 + 0.09716/ 0.94192 + 0.09845/ 1.03039 + 0.09903/ 1.10286 + 0.09742/ 1.1535 + 0.09467/ 1.22097 + 0.08773/ / = 1 0.11252 + 0.10040/ 0.11320 + 0.10061/ 0.11537- + 0.10123/ 0.11957 + 0.10206/ 1.12712 + 0.10198/ 0.13276 + 0.09980/ 0.13540 + 0.09675/ 0.13416 + 0.09666/ / = 2 0.37367 + 0.08896/ 0.37475 + 0.08907/ 0.37844 + 0.08940/ 0.38622 + 0.08981/ 0.40122 + 0.08964/ 0.41357 + 0.08833/ 0.42169 + 0.08666/ 0.43113 + 0.08354/ / = 3 0.59944 + 0.09270/ 0.60103 + 0.09279/ 0.60705 + 0.09306/ 0.62066 0.9341/ 0.64755 + 0.09312/ 0.67002 + 0.09164/ 0.68519 + 0.08978/ 0.70387 + 0.08608/ / = 4 0.80918 + 0.09416/ 0.81134 + 0.09425/ 0.82020 + 0.09453/ 0.84056 + 0.09493/ 0.88057 + 0.09467( 0.91396 + 0.09314/ 0.93664 + 0.09117/ 0.96510 + 0.08712/ (belonging to Z'^' and Z2(±)) of the quasi-normal modes for a range of values of Q+ and /. 49. Considerations relative to the stability of the Reissner-Nordstrom space-time The considerations, relative to the stability of the Schwarzschild black-hole to external perturbations, in Chapter 4, §34 apply, quite literally, to the Reissner-Nordstrom black-hole since the only fact relevant to those considerations was that the potential barriers, external to the event horizon, are real and positive; and stability follows from this fact. But quite different circumstances prevail in the interval, r_ <r < r+, between the two horizons. The altered circumstances are the following. While the equations governing Z,-(±) remain formally unaltered, the potential barriers, Kf(±), are negative in the interval, r_ <r <r+, and in the associated range of rt, namely + oo > rt > — oo; they are in fact potential
STABILITY OF REISSNER-NORDSTROM SPACE-TIME 263 0 -5 -10 i i i i vr i i i i i i i \ 1 i M i -4 -3 -2-1 0 1 Fig. 24. The potential wells, V\ ' and V'2 ' for g„ = 0.75, governing the dispersion of perturbations in the region between the two horizons. wells rather than potential barriers (see Fig. 24). Thus, the equation now governing Z\~) is, for example (cf. equation (151)), where d2Z(_) IAI —=i— + <t2Z-(_) = - — Arl r3 (^2 +2)r -qj + 4QI z/-» (i',;'= 1, 2; 1^;'; and r_ < r < r+, - oo < r^ < + oo), (358) »■* =^+^—IgK ~r\--—lg|r-r_|, 2k+ 2k_ Kj. =—^^—, and k_ = 2r2+ 2r2 (359) (360) In view of the relation (199) between the solutions, belonging to axial and polar perturbations, it will, again, suffice to restrict our consideration to equation (358); and for convenience, we shall suppress the distinguishing superscript. An important consequence of the fact that we are now concerned with a short-range one-dimensional potential-well, is that equation (358) will allow a
264 THE REISSNER-NORDSTROM SOLUTION finite number of discrete, non-degenerate, bound states: o=±i<Jj [;'= 1,2; n= 1, 2, . . . , m (say)]. (361) Besides these bound states, we continue to have the standard problem of reflexion and transmission associated with the continuous spectrum including all real values of cr. However, in the present context, the relevant solutions must satisfy boundary conditions different from the ones we have used hitherto. The boundary conditions we must now impose are z(r*) -" A(a)e~iOT» + B(cr)e+''OT» (r->r_-0; r„->- + oo) -»e-''«"■• (r_>r++0;r„-» - oo). (362) The reason for these altered boundary conditions is that, by virtue of the light- cone structure of the region between the two horizons, one can have only waves entering the region, by crossing the event horizon at r+ (from the 'outside world'), and none leaving it, by crossing the event horizon in the reverse direction. The coefficients A(<j) and B(<j) in equation (362) are related to the reflexion and the transmission amplitudes, as we have defined them hitherto and in Chapter 4 (equations (175) and (176)) Ala) = = B(a) = —— = -^ -, 363 K) T*(a) Tx(-a) K' T*(a) T^-a) so that \A(a)\2 -|B((7)|2 = 1. (364) In Table VI we have listed the amplification factors, \A(<j) |2, for Q\ = 0.75 M 2. We observe that |/l(cr)|2, and, therefore, also |B(cr)|2, tend to finite limits as cr -► 0. This fact has its origin in the existence of bound states of zero energy in the potential wells, Vx and V2- (See Bibliographical Notes at the end of the chapter.) In analyzing the radiation arriving at the Cauchy horizon at r_, we must distinguish the edges EC and EF in the Penrose diagram (Fig. 14). For this reason, we restore the time-dependence, ewl, of the solutions; and remembering that in the interval, r_ <r <r+ (cf equations (51)), u = r^ + t and v = r^ — t, (365) we write, in place of equation (362), z(r*. t)^e-iov + \_A{o)-Y]e-iov + B{o)e+i',u. (366) If we now suppose that the flux of radiation emerging from D'C is Z (v), then 1 Z(o) = . 2?r Z(u)eiOTda (367)
STABILITY OF REISSNER-NORDSTROM SPACE-TIME 265 Table VI Amplification factors appropriate for the potential V[^ and V^ o 0.009375 0.01875 0.0375 0.05 0.075 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 for Vi 2.7746 2.7652 2.7287 2.6923 2.5962 2.4785 2.2201 1.9781 1.7764 1.6168 1.4928 1.3967 1.3219 M Ql = for v2 2.7747 2.7655 2.7299 2.6944 2.6005 2.4853 2.2312 1.9914 1.7898 1.6286 1.5022 1.4035 1.3262 0.75 M2 o 0.50 0.55 0.65 0.70 0.75 0.80 0.85 0.90 0.95 0.100 0.105 0.110 0.115 for C, 1.2631 1.2165 1.1494 1.1248 1.1044 1.0877 1.0741 1.0628 1.0534 1.0453 1.0384 1.0325 1.0276 for V2 1.2653 1.2169 1.1471 1.1220 1.1016 1.0849 1.0710 1.0596 1.0502 1.0423 1.0358 1.0304 1.0257 This flux disperses in the domain between the two horizons and at the Cauchy horizon it is determined by Z(rt,t)^X(v)+Y(u) (v -► oo; u -► oo), (368) where X(v) = and Y(u) = Z(<x) [/1((7)-1^^(7, Z(a) B(a)eiau da. (369) (370) However, our interest is not in X (v) or Y(u), per se, but rather in quantities related to them. As we have explained in §38 (p. 214), we are primarily interested in the radiation an observer receives at the instant of his (or her) crossing the Cauchy horizon. To evaluate this quantity, we consider a freely falling observer following a radial geodesic. The four-velocity, U, of the observer is given by equations (65) for L = 0; thus, l/' = -£, A W. = - E2- 1/2 and 1/^=1/^ = 0, (371) where, consistently with the time-like character of the coordinate r in the interval r_ <r<r+,we have chosen the positive square-root in the expression for V*. Also, it should be noted that we are allowed to assign negative values for E since the coordinate t is space-like in the same interval. With the prevalent radiation-field expressed in terms of Z(r+, t), a measure
266 THE REISSNER-NORDSTROM SOLUTION of the flux of radiation, #", received by the freely falling observer is given by J^= U'Z , = — 'J A A\1/2 £Z,, + (£2-3l Z, (372) We have seen that as we approach the Cauchy horizon (cf equations (369) and (370)), Z(r„t)^X(t-rJ + Y(t + rJ. (373) Accordingly, Ztt^X,-v + Y„ and Z_% - -*,_,, + >%; and the expression (372) for #" becomes X -, A\1/2" + Yt £+ £2- A\1/2 (374) ; (375) On EF, v remains finite while u -» oo; therefore, 1 +1, u -► 2r„ lg|r —r_| as r->r_ on EF. (376) Also for £ > 0, the term in Xy _ „ remains finite while the term in y„ has a divergent factor (namely, 1/A). Hence, 2r2 -EYueK-u (u-► oo on £F). (377) On EC, u remains finite while v -* oo; therefore, r -» — t, u->2r^-> lg|r —r_| as r->r_ on £C. (378) And for £ < 0, the term in Yu remains finite while the term in X _„ has the divergent factor. Hence, ^£C-+- 2r2 r+ -r_ IjEIA',-,,^-1' (u -- oo on EC). (379) We conclude from equations (377) and (399) that the divergence, or otherwise, of the received fluxes on the Cauchy horizon, at £F and EC, depend on Ri(-o). Y,.= l(J -r , \ Z(a)eiau do and X _.,= »+00 id J — 00 1 1 Z(a)e-iavd<j, (380) (381) where we have substituted for A(a) and B(<j) from equation (363). In particular,
STABILITY OF REISSNER-NORDSTROM SPACE-TIME 267 if we wish to evaluate the infinite integrals, as is naturally suggested, by contour integration, closing the contour appropriately in the upper half-plane (for determining the behaviour of Yu for u -» oo) and in the lower half-plane (for determining the behaviour of X _„ for v -* oo), then we need to specify the domains of analyticity of A(<j) and B(a), as defined in equations (363). Returning then to the definitions of A(a) and B(a), we can, in accordance with Chapter 4, equations (180), write and AW = FT^ = ^"t /i(*' - *)'■& (*• - ^> (383^ / j ( — <J) 2i(T where, for convenience, we have written x in place of r^ and/ (x, ± cr), and / (x, + a) are solutions of the one-dimensional wave equations which satisfy the boundary conditions fi(x, ±o)->e+iax x-> +00,) and (384) /(*. ±tf)-*e±lox x->-oo. i Also, as we have shown in Chapter 4, equations (194) and (198),/2(^> — 0) satisfies the integral equation f2(x, -o) = e-i»* + sin(7(x 4F(x')/2(,;_ff)dx'. (385) a The corresponding integral equation satisfied by/ (x, +<x) is >sina(x —x') /(x, Ta) = e±u"c- V(x')fx(x',T(j)Ax'. (386) Adapting a more general investigation of Hartle and Wilkins to the simpler circumstances of our present problem, we can determine the domains of analyticity of the functions,/ (x, — a) and/ (x, + a), in the complex a-plane, by solving the Volterra integral-equations (385) and (386) by successive iterations. (As is known, iterations of Volterra-type integral equations normally provide solutions whose uniform convergence is guaranteed.) Thus, considering equation (385), we may express its solution as a series in the form where / (x, -a) = e- "x + £ /<«> (X, - a), (387) n= 1 /2<"»(x, -a)= f dx1Sln<T(X~;Cl)K(x1)/2("-1)(x1,-<7). (388)
268 THE REISSNER-NORDSTROM SOLUTION By this last recurrence relation, fln)(x, dx1 dx, d*«n sin(T(xi_1 — xt) V(xt)e- (389) where x0 = x; or, after some rearrangement, /?'(*,-*) = Since (2/(7)- dxj . dx„ n ^ [e2,o,x' -*,)_ i]^U) K(x) ►constant e 2k+x (X oo), (390) (391) it is manifest that each of the multiplicands in (390) tends to zero, exponentially, for x -* — oo for all Ima > (392) Consequently, ^(n)(x, — cr) exists for each n for Imcr > — k+. And it can be shown (as Hartle and Wilkins have shown in their more general context) that the series (387) converges uniformly for all cr with Im cr > — k +. The domain of analyticity off2 (x, — a), therefore, includes the entire upper half-plane and the infinite strip of width k+ in the lower half-plane* It can be further shown that there is a sequence of poles along the imaginary axis at cr = —imK+(m = 1, 2 . . .). While we shall not show this entirely rigorously, we shall indicate the origin of the singularities. In view of the asymptotic behaviour (391) for V(x), we may, compatible with this behaviour, expect a representation of V(x) for x < 0, in the manner of a Laplace transform, by V(x) = dur(u)e"x, (393) J2k + where 'V (u) includes <5-functions at various locations, i.e., f (u) is a distribution in the technical sense. With the foregoing representation for V(x), the first iterate,^"'(x, —a), of the solution for^(x, —a), becomes (cf equation (389)) /2U)(*, ~o) = 2ia dx, e2i'o(x-x,)_ J dur(u)tfx^\ (394) 2K+ * I am grateful to Dr. Roza Trautman for pointing out to me that a result equivalent to the one stated is contained in M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Studies in Applied Mathematics, 53, 249 (1974).
STABILITY OF REISSNER-NORDSTROM SPACE-TIME 269 or, inverting the order of the integrations, we have ^-)=-2^ dixr~(n)e' f* J2k + After effecting the integration over x1, we are left with /2(1»(x, -a)=e dx1[e2&r(x-x>)-l]e''(x>-x). (395) TT(u) d/^— -^- e"x. (396) J2K+ %0*-2ia) From this last expression, it is evident that /2(1)(x, — cr) has singularities along the negative imaginary axis beginning at Imcr = — k+. Similar results obtain for the higher iterates. While these arguments do not conclusively establish the existence of cuts and singularities, for the sum £ f2{n) (x, — cr), beginning at Im cr = — k+ , they can be refined to show that this is the case; but we shall not attempt to do so. Turning now to the integral equation (386) satisfied by f (x, ±<r), we can show, in the same manner as in the context of f2(x, — cr), by solving the equation by iterations, that by virtue, now, of the asymptotic behaviour, V (x) -► constant e - 2k- x (x -► + oo), (397) the domain ofanalyticity off (x, — a) includes the entire upper half-plane and an infinite strip of width k_ in the lower half-plane; and the domain ofanalyticity of f (x, + a) includes the entire lower half-plane and an infinite strip of width k_ in the upper half-plane. In view of the domains ofanalyticity oif2 (x, — oo) and/i (x, + cr), it follows that as far as crB(cr) is concerned, its domain of analyticity, being the Wronskian of/2 (x, —cr) and/i (*, + <*)> certainly includes the strip of width k_ in the upper half-plane. (It can, in fact, be shown that/i (x, cr) is analytic in the entire complex a-plane except for singularities along the imaginary axis at a = imK_(m = 1,2,...).) In any event, by evaluating the integral Yu, by closing the contour along the real axis by extending it into the upper half-plane, we can conclude that, so long as Z(cr) is analytic in the upper half-plane, that Yu must fall off at least as rapidly as e~KU for u -* oo, i.e., Yu<e-KU, u-.-oo. (398) It now follows from equation (377) that the flux !FEF at the Cauchy horizon at EF is necessarily bounded. Turning next to oA(a), which appears in the integrand of X _„, we observe that, by virtue of what we have shown with respect to^ (x, — cr) and^ (x, — cr), its domain ofanalyticity in the lower half-plane includes the strip of width k +; and the function has cuts or singularities along the entire negative imaginary axis beginning at — K + i. Therefore, by evaluating the integral X__„, by a contour in the lower half-plane avoiding the cut along the negative imaginary
270 THE REISSNER-NORDSTROM SOLUTION axis which starts at Im a = — k +, we shall find, on the assumption that Z(a) is regular in the domain of integration, that X„-> constant e~K+" for v -* oo. (399) Consequently, by equation (379), &EC -► constant e<K- -"+)" (v -► oo). (400) Since k _ > k +, it follows that the flux, ^EC'> diverges on the Cauchy horizon at EC. This establishes that an observer, attempting to cross the Cauchy horizon, by following a time-like geodesic, and to emancipate himself (or herself) from the past, will experience the impact of an infinite flux of radiation at the instant of the crossing even if the perturbation crossing the event horizon is of compact support. 50. Some general observations on the static black-hole solutions With our consideration of the Schwarzschild and the Reissner-Nordstrdm space-times completed, it is opportune to bring to a focus their similarities and their differences—both equally striking. As for the similarities: both are of Petrov type-D; both have an event horizon which bounds an inner region incommunicable to an asymptotically flat world outside; in both cases, the event horizon is surrounded by potential barriers which determine the reflective power of the black hole to radiation incident on them; and the potential barriers all belong to a very special class which ensures the equality of their reflective power to waves of axial and of polar types; also, the perturbation equations that follow from the Newman-Penrose formalism allow, in both cases, dual transformations—a fact which, in turn, is related to the special class to which the potential barriers belong. But the variations bring their own surprises: the description of the manner of the interaction of the Reissner-NordstrOm black-hole, to the simultaneous incidence of electromagnetic and gravitational waves, requires a scattering matrix of order 4; but it is symmetric and unitary! Similarly, the decoupling of the perturbation equations of the Reissner-Nordstrom black-hole in the Newman-Penrose formalism requires the choice of a special gauge which appeared, first, as a phantom in the treatment of the Schwarzschild perturbations. And so one variation follows another. And among the differences, the principal one—from which all others derive—is the fact that the Reissner-Nordstrom space-time has an inner Cauchy horizon which makes its maximal analytic extension reveal the possibility, in principle, of exploring other worlds and emancipating oneself from one's past. But the metamorphosis of a coordinate, space-like outside the event horizon, into one which is time-like as the event horizon is crossed, brings its own danger to such prospects: radiation crossing the event horizon
BIBLIOGRAPHICAL NOTES 271 gets amplified, as it progresses towards the Cauchy horizon to an extent that impact with it may prove fatal. And this last passage provides a startling coda to the theme and variations. BIBLIOGRAPHICAL NOTES H. Reissner (1874-1967) and G. Nordstrom (1881-1923) independently derived the solution which has come to be known as the Reissner-Nordstrom solution: 1. H. Reissner, Ann. d. Physik, 50, 106-20 (1916). 2. G. Nordstrom, Proc. Kon. Ned. Akad. Wet., 20, 1238^15 (1918). §37. The derivation of the Reissner Nordstrom solution is patterned after the derivation of the Schwarzschild solution in §17. §38. Our discussion of the nature of the Reissner-Nordstrom space-time follows closely: 3. M. Simpson and R. Penrose, Internat. J. Theoret. Phys., 7, 183-97 (1973). See also: 4. J. C. Graves and D. R. Brill, Phys. Rev., 120, 1507-13 (1960). 5. S. W. Hawking and G. F. R. Ellis, The Large-Scale Structure of Space-Time, 156-9, Cambridge, England, 1973. §40. For a complete bibliography of papers on the geodesies in the Reissner-Nordstrom space-time, see: 6. N. A. Sharp, Gen. Relativity and Gravitation, 10, 659-70 (1979). Special reference may be made to: 7. A. R. Prasanna and R. K. Varma, Pramana, 8, 229^14 (1977) which is complete on the formal side. But in none of the papers on the subject are the orbits properly classified and arranged; and they do not point out the similarities and the differences with the orbits in the Schwarzschild space-time. Fig. 16 is taken from: 8. A. Armenti, Jr., Nuovo dm., 25B, 442-8 (1973). The geodesies illustrated in Figs. 15, 17, and 18 were calculated and drawn by Mr. Garret Toomey to whom I am greatly indebted. The motion of charged particles, with special emphasis on the extreme case Ql = M2, is considered in: 9. R. Ruffini in Black Holes, 497-508, edited by C. DeWitt and B. S. DeWitt, Gordon and Breach Science Publishers, New York, 1973. 10. G. Denardo and R. Ruffini, ibid., R33-44. The primary emphasis in these papers is on the possibility of extracting charge from a Reissner-Nordstrom black-hole by encounters with particles of opposite charge. §41. The fact that the spin-coefficients and the principal null-directions are the same for the Reissner-Nordstrom black-hole as they are for the Schwarzschild black-hole, except for a redefinition of the 'horizon function' A, is pointed out in: 11. S. K. Bose, J. Math. Phys., 16, 772-5 (1975). 12. D. M. Chitre, Phys. Rev. D, 13, 2713-19 (1976). 13. C. H. Lee, J. Math. Phys., 17, 1226-35 (1976). §42. The analysis in this section is based on: 14. S. Chandrasekhar and B. C. Xanthopoulos, Proc. Roy. Soc. (London) A, 367, 1-14 (1979).
272 THE REISSNER-NORDSTROM SOLUTION 15. B. C. Xanthopoulos, Proc. Roy. Soc. (London) A, 378, 73-88 (1981). For alternative treatments using different methods see: 16. V. Moncrief, Phys. Rev. D, 9, 2707-9 (1974). 17. , ibid., 10, 1057-9 (1974). 18. , ibid., 12, 1526-37 (1975). 19. F. J. Zerilli, ibid., 9, 860-8 (1974). §43. The relations in this section are given in: 20. S. Chandrasekhar, Proc. Roy. Soc. (London) A, 369, 425-33 (1980). §44. The analysis in this section follows: 21. S. Chandrasekhar, Proc. Roy. Soc. (London) A, 365, 453-65 (1979). See also: 22. A. W. C.-Lun, Nuovo Cim. Letters, 10, 681^* (1976). 23. J.Bicak, Czech. J. Phys. B, 29, 945-80 (1979). 24. R. M. Wald, Proc. Roy. Soc. (London) A, 369, 67-81 (1979). §45. See reference 21. §§46-47. Much of the discussions in these sections is new. I am grateful to Dr. R. Sorkin who suggested the introduction of the scattering matrix. The conversion of gravitational energy into electromagnetic energy in the low- frequency limit has been considered in great detail by: 25. R. A. Matzner, Phys. Rev. D, 14, 3274-80 (1976). See also: 26. D. W. Olson and W. G. Unruh, Phys. Rev. Letters, 33, 1116-8 (1974). §48. The numerical results given in Table V are taken from: 27. D. Gunter, Phil. Trans. Roy. Soc. (London), 296,457-526 (1980) and 301, 705-9 (1981). §49. The importance of examining the development of perturbations on the Cauchy horizon was first recognized and explored by Penrose (see reference 3). The subject has been discussed extensively since that time; and the principal papers are those of: 28. J. M. McNamara, Proc. Roy. Soc. (London) A, 364, 121-34 (1978). 29. , ibid., 358, 499-517 (1978). 30. Y. Gursel, V. D. Sandberg, I. D. Novikov, and A. A. Starobinsky, Phys. Rev. D, 19, 413-20 (1979). 31. Y. Gursel, I. D. Novikov, V. D. Sandberg, and A. A. Starobinsky, ibid., 20, 1260-70 (1979). 32. R. A. Matzner, N. Zamorano, and V. D. Sandberg, ibid., 19, 2821-6 (1979). I should add that I find the use of the word 'stability' in these contexts unfortunate: it does not seem to relate to what one is actually investigating, namely, how waves from the outside world, entering the domain between the two horizons via the event horizon, develop as they approach the Cauchy horizon and affect a freely-falling observer. I am indebted to Dr. S. Detweiler for the amplification factors listed in Table VI. The existence of bound states of zero-energy in the potential wells Vt and V2 is proved in paper 31. The treatment in this section departs in some essential respects from those in papers 28-32. The principal arguments are derived from: 33. J. B. Hartle and D. C. Wilkins, Commun. Math. Phys., 38, 47-63 (1974). For a fuller treatment of the finer issues, side-stepped in the account, see: 34. S. Chandrasekhar and J. B. Hartle, Proc. Roy. Soc. (London) A (in press).
6 THE KERR METRIC 51. Introduction In this chapter, we begin the study of the Kerr solution. It represents, as we have stated in the Prologue, the unique solution which the general theory of relativity provides for the description of all black holes that can occur in the astronomical universe by the gravitational collapse of stellar masses; and it is the only instance of a physical theory providing an exact description of a macroscopic object. In large measure, the preceding chapters are but preludes to the study we now begin. It has been stated that "there is no constructive analytic derivation of the [Kerr] metric that is adequate in its physical ideas, and even a check of this solution of Einstein's equations involves cumbersome calculations" (Landau and Lifshitz). Contrary to this statement, we shall find that, once the basic equations have been properly written and reduced, the derivation of the Kerr metric is really very simple and proceeds with an adequate base of physical and mathematical motivations. In this chapter, besides the derivation of the Kerr metric and the establishment of its uniqueness, we shall give a description of the space-time in a Newman-Penrose formalism which makes its type-D character manifest. 52. Equations governing vacuum space-times which are stationary and axisymmetric As we have shown in Chapter 2 (§11), a metric appropriate to stationary axisymmetric space-times can be written in the form ds2 = e^idtf-e^idip-codtf-e^idxy-e^idx3)2, (1) where v, if/, <x>, fi2, and fi3 are functions of x2 and x3 with the freedom to impose a coordinate condition on fi2 and A*3- We can now write down the components of the Riemann tensor, in a tetrad- frame with the basis one-forms, u>° = ev dt, co1 = e* (d<p - codt), u>2 = e^ dx2, and u>3 = e^ dx3, (2) by suitably specializing the expressions listed in Chapter 2, equations (75). We find that the non-vanishing components of the Riemann tensor are
274 THE KERR METRIC «1212 = +e~*~^e*~^f 2),2+e~2^,3/^,3+ie2*~2^~2V22> «1313 = +^-^-^(^-^^,3),3+^-^^,2^3,2+^-^-^00,32. «1010 = -^2ftf2v2-r2ftf3v3-^-2'(r^(ai22 + rV32). «2323 = +e-"2-"3[(^-"^2,3),3 + (^-"^3,2),2], «2020 = -^-^-^,2),2-^-^,3^,3+1^-^-^,22. «3030 = -e-"»-»(**-*V ,3),3-^2"2V,2^3,2+l^"2"3"2V«32, «1213 = +^-^-^(^-^^,2),3-^-^-^^,3^3,2+^-^-^-2^,2^,3, «1220 = +^-^-^,2(^,2-^,2) + 1^^-^^-^^,2),2 «1330 = +^-^-^3(^3-^,3) + :^-^-^,3),3 + 1^-^-^,2^3,2, «3002 = +e-'-ft(e'-»v2),j-r''»-ftvjfe)j-p-'l'-'"-*(B,,(B,3( «1230 = +^-^-^-^(0,2) ,3+^-^-^-^,3(2^,2-^3,2-V,2), «1302 = -^^^(^^^.3)2-^-^^^,2(2^3-^3-^3), «1023= -ie-ft-"'[(**->3).2-(e*-vou).3]- (3) The components which vanish are «1210 = «1310 = «1223 = «1332 = «1002 = «1003 -r -« -o ■ (4) — ^2330 — ^3220 — u- The components of the Ricci and the Einstein tensors are given by certain linear combinations of the components of the Riemann tensor (specified in Ch. 2, equations (75)); and by setting them equal to zero, we obtain the equations governing stationary axisymmetric vacuum space-times. For our present purposes it will suffice to consider the following equations. e-2"2[v,2,2+V,2(l/'+V-Ai2+/^3),2]+e-2"3[v,3,3+V,3(lA + V + |i2-/^3),3] = +^-2v[e-2^(w2)2+e-2/,3(w3)2] («00=0), (5) e-2"2[>,2,2 +^,2 ("A + V -Hz +//3).2] + e-2"3[,/,,3,3+1/, 3(./, + V + H2 -//3).3] = -i^-^-^^jf + r^la,,,)2] («u=0), (6) (^-^+^,2),2 + (^-^-^,3),3 = 0 («oi=0), (7) (>A + V),2, 3 ~ ("A + v),2/(2, 3 - ("A + v)3/(3,2 + ^,2^,3 + V,2 V, 3 = ie*-Vi<»,3 («23=0), (8) e-^Klj/ + V),3,3 + (l// + V),3 (V -//3),3 + ^.3^,3] + «_2fc[v. 2(^ + //3).2 + ^.2//3.2] -^^'[r^^jf-e-^lfflj)2] (G„=0), (9)
EQUATIONS GOVERNING VACUUM SPACE-TIMES 275 e-*"[(^ + v)>j,j + (^ + v),j(v-//j),j+^,j^,j] + e~2"3[V)3(lA + ^2),3 + f 3/^,3] = +1^-^6-^((0,^-6-^(0.,3)2] (G33 = 0). (10) Letting /» = * + v, (11) we can rewrite equations (5) and (6) in the forms (^+^-^,2),2 + (^+^^,3),3 = +^-^^-^(0,2^ + ^-^(0.3)2]. (12) (^+^-^^,2),2 + (^+^-^^,3),3 = -^-^^-^(0),2^ + ^-^(01,3)2]. (13) The sum and the difference of these equations give [^-"»(«').2] ,2+:^-^) ,3] ,3 = 0, (14) [e/» + ft-ft(^_v) 2]2 + [e/» + ft "ft(^ _v)i3] ,3 = -^-^^-^(^,2^ + ^-^(0,3)2]. (15) The addition of equations (9) and (10) yields the same equation (14), while subtraction gives 4^-^(/^3,2 + ^,2) -4^-^(0.302.3 + IA.3V.3) = 2r»{^-"(42],2-^-ft(^),3],3} _e2^-2v|-^,-^(w2)2_^2-^(W3)2-|; (16) An examination of the foregoing equations suggests that we consider them in the following sequence. First, exercising the gauge-freedom, we have to impose a coordinate condition on fi2 and fi3, we may specify ^3-^) = A(x2,x3) (17) in some convenient manner. Equation (14) then becomes an equation for P = i]/ + v. Its solution presents no difficulty. Indeed, we shall show presently that efi can be 'absorbed' as one of the coordinates (see §(fc) below). Equations (7) and (15) then become a pair of coupled equations for i/> — v and co; and their solution is the central problem in the theory of stationary axisymmetric solutions of Einstein's equation. The significance of the foregoing grouping of the metric functions, suggested by the structure of the equations, becomes clear when we write the metric (1) in the form 1 "1 ePi+P} ds2=e" x(At)2—(d<p-codt)2 Va" [(dx2)2 + A(dx3)2], (18)
276 THE KERR METRIC where A = e2^-^\ p=i// + v, and * = <r* + v. (19) As we have stated, A is at our disposal; /? can be solved for, independently of the others; the central problem is to solve for i and co; and the solution for \i2 + /½ presents no difficulty of principle once % and co are known: it follows by simple quadratures as we shall show in detail in §52, below. It is important to observe that — e2fi is the determinant of the metric of the two-dimensional space spanned by the vectors d/dt and d/d<p. One feature of the equations governing % and co is deserving of notice. Rewriting equation (7) in the form = +^-^^-^(0),2^ + ^-^(0).3)2]. (20) we can combine it with equation (15) to give {e3^-v-^ + „3|;(w2)2+2e-2^ + 2v(l/,-v)2]}2 + {e3*-»+'«'-'«>[(a»2)i3 + 2e-2* + 2'(^-v)>3]}.3 = 0, (21) or, alternatively, [eW-*-»>+»>(f-co2\2\2 + [_eW-o+^-^(x2-co2\3\3 = 0. (22) Comparison of equations (7) and (21) shows that co and i2 — co2, formally, satisfy the same equation. (a) Conjugate metrics An important feature of the equations governing stationary axisymmetric space-times is that starting from a solution represented by, say, (%, co), one can obtain other solutions. An example is provided by subjecting the metric (18) to the transformation t -* + i<p and <p -* — it. (23) Then X(dt)2--(d<p-codt)2 ->-(dt)2 +—dtd<p-X ~" (d</>)2 X til X(dt)2--(d(p-codt)2 1 (24) where co = — J- and % = —2 j. (25) X -co X -co None of the other metric coefficients are affected by this transformation. We conclude that if (%, co) represent solutions of the equations, then (%, cb), also,
EQUATIONS GOVERNING VACUUM SPACE-TIMES 277 represent solutions of the same equations. In §53 we shall explicitly verify that this is the case: it is related to the fact that co and x2 — co2 satisfy the same equation. We shall call (%, co) and (% co) conjugate solutions and the transformation (23), which yields these solutions, conjugation. (b) The Papapetrou transformation With the choice of a gauge, /*2 = /*3 = /* the metric (18) becomes and A=l, ds2 = e> X(dt)2— (d<p-codt)2 1 -e2"[(dx2)2 + (dx3)2], (26) (27) while the equation satisfied by e^ becomes (*')...„ =0 (a = 2, 3). (28) We may now take advantage of the fact, that e^ satisfies the two-dimensional Laplace's equation, to consider e^ as one of the coordinates and seek a coordinate transformation such that (31) (x2,x3)^(p,z), (29) e2"l(dx2)2 + (dx3)2] ->/(P, z) [(dp)2 + (dz)2l (30) It is clear that such a coordinate transformation can be made if (cf. Ch. 2, §11) and P,2P,3+Z,2Z,3 =0- These conditions can be satisfied by setting P,2=+z,3 and p,3 =-z,2. (32) equations which we can certainly satisfy by virtue of the equation P....=0 (P=et) (33) satisfied by p. With efi = p chosen as one of the coordinates, the metric (27) takes the form ds2 = p X(dt)2--(d(p-codt)2 -e2"[(dp)2 + (dz)2], (34) where x, <o, and fi are functions of p and z. This is the Papapetrou form of the metric.
278 THE KERR METRIC 53. The choice of gauge and the reduction of the equations to standard forms The particular choice of gauge we shall presently make implies no loss of generality; and, indeed, it is not strictly necessary to make at this stage. But it does provide a meaningful physical motivation for the form of the metric we shall seek. It is convenient to choose the polar angle 6 (with respect to the axis of symmetry) as the spatial coordinate x3. (Later in this chapter, we shall replace 9 by cos 6 as one of the independent variables.) We shall suppose that the metric allows an event horizon, which we shall define, in the present context, as a smooth two-dimensional, null surface, which is spanned by the tangent vectors, d/dt and d/dq>—the "Killing vectors" of the space-time. In conformity with the assumed stationarity and axisymmetry of the space- time, let the equation of the event horizon be N(x2,x3) = 0. (35) The condition that it be null is j^XiN, ; = 0. (36) For the chosen form of the metric, equation (36) gives ^-^(N/tlN^O, (37) where we have written x2 = r. Exercising the gauge freedom we have, we shall suppose that ^3-^) = A (r), (38) where A(r) is some function of r which we leave unspecified for the present. From equation (37) it now follows that the equation of the null surface is, in fact, given by A(r) = 0. (39) The second condition that the null-surface be spanned by d/dt and d/d<p requires that the determinant of the metric of the subspace, (t, <p), vanish on A(r) = 0: e2" = 0 on A(r) = 0. (40) Since we have left A(r) unspecified, we may, without loss of generality, suppose that e" = A1/2f(r,8), (41) where/(r,0) is some function of r and 6 which is regular on A(r) = 0 and on the axis, 6 = 0. We shall suppose, instead, that e^ has the somewhat more restricted
THE CHOICE OF GAUGE 279 form e> = A1/2/(0), (42) i.e., efi is separable in the variables r and 6. With e1'3"1'2 and efi given by equations (38) and (42), equation (15) for ep becomes [A1'2 (^),^+1/^ = 0. (43) A solution of this equation, compatible with the requirements of regularity on the axis and convexity of the horizon, is determined by A,r,r = 2 and /= sin 6; (44) and the solution for A that is appropriate is A = r2-2Mr + a2, (45) where M and a are constants.* (We shall later show that these constants signify the mass and the angular momentum per unit mass of the black hole.) Thus, with a choice of gauge that is consistent with the existence of an event horizon, we have the solution ^3 -fi = A1/2 and e" = A1'2 sin 6, (46) where A is given by equation (45). The fact that the choice of the solutions for efi and e^ "^ we have made implies no loss of generality becomes apparent when it is noted that they are consistent with the Papapetrou form of the metric for p = e» = A1'2 sin 6 and z = (r- M) cos 6. (47) With the solutions for e^"^1 and e^ given in equations (46) and by using H = cos 6, (48) (instead of 6) as the variable indicated by the index '3', we can bring equations (15) and (7), respectively to the forms [A(*-v),2],2 +[*(*- v).3].3= -^-V)[A(W,2)2 + ^K3)2], (49) and [A^-v)w,2],2 + ^^-v)W,3],3 = 0, (50) * It should be noted that, while M and a are here introduced as constants of integration, A and (M2 — a2) are invariant to the transformation (due to Xanthopolous), M-»M' = (M2-a2)l/2/P>a-*a'=4(M2-a2)1/2/p> and r-M->r'-M\ where p and q are constants and p2 + q2 — 1.
280 THE KERR METRIC where we have written 5 = l-n2 = sin26. (51) Letting % have the meaning given in equation (19), we can rewrite equations (49) and (50) in the forms -X.2 + -X.3 =^[A(«,2)2+<5(«,3)2], (52) A \ (b \ 1 X""J.2 \X""J,3 X2 >.>)..+(?"-')..-* ,53) or, alternatively, Z[(Az.2).2 +(^,3),3] = A[(Zi2)2+((B.2)2] + 5[(Z.3)2+(0.3)2]. (54) Z[(Acu.2),2 + (^,3),3] = 2A](i2w,2 +2¾3(u 3. (55) Now letting X = x + a> and y=x-a>, (56) we obtain the pair of symmetric equations i(I + y)[(Al2),2 + (W3),3] = A(I2)2+^(I3)2, (57) ^(x + Y)[(Ay2),2 + (^3),3] = A(y2)2 + ^(y3)2. (58) Equations (8) and (16) (which determine \i2 + 1*3), after some elementary reductions, become H , r-M , 2 and 7(/½ + //2).2 +-1-(^3 +//2).3 = 7^-7^2(^.2^3 + ^2*. 3) (59) 2 (r-M) (jii +H2),i + 2/((/½ +//2).3 ^(A*.aya -^,3^3)-3^^ + ^. (60) (x + n It is now manifest that once equations (57) and (58) have been solved for X and y, the solution for (n2 + //3) follows from equations (59) and (60) by simple quadratures. Equations (57)-(60) can be written in more symmetrical forms by changing r to the new variable r, = (r-M)/(M2-a2)1'2 1 when r (61) A=(M2-a2)(r,2-l). J
THE CHOICE OF GAUGE 281 Thus, we find i(X + y){[i?2-l)*a, + [(l-//*)*. J.,,} = (r,2-l)(XJ2 + (l-n2)(XJ2, (62) i(x + Y){[(^-i)yj,, + [(i-^)yj,,} = (^-1)(7,)2+ (1-M2)(y„)2, (63) 11 (^+^).,+/7^+^).^7^(^^+¾^ (64) jyvw.rjM.^^jwj.i.j;.,, (JT+Y) 2^3 + ^2),,+2^3 + ^2),,, = XxTWl(t]2 ~ X)X-"Y-" ~(1 -rtxM -^77+rr^- (65) A convenient form of equations (62) and (63) which enables one to find by inspection some special solutions is obtained by the transformation 1+F 1+G * = T37 and y=TTG- (66) We find (1-FG){[(ij2-1)F,]., + [(1-,/j)F,].,} = -2G[(,,2-1)(F,)2 + (1-^)(F,„)2], (67) and (1-FG){[(>/2-l)G,],, + [(l-^)G, „],„} = -2F[(i,2-1)(0.,)2+(1-^)(0,,,)2]. (68) The metric functions % and co are related to F and G by 1-FG J F-G * = (1-F)(1-G) aDd W = (l-F)(l-G)- (69) We may parenthetically note that equations (67) and (68) allow the simple solution F — — prj—qn and G = — pn + qn, (70) where p and q are real constants restricted by the condition p2-q2 = l. (70') (a) Some properties of the equations governing X and Y Let (X, Y) represent a solution of equations (62) and (63). Then the following are also solutions. (i) (Y, X) also represents a solution. For this solution 1 remains unchanged while co changes sign—a trivial change equivalent to letting </)-►—</).
282 THE KERR METRIC (ii) (X + c, Y— c), where c is an arbitrary constant, also represents a solution. For this solution, % again remains unchanged while co -* co + 2c—again, a trivial change equivalent to letting q> -*q> — let. (iii) (X ~1, y~1) and (Y~ \ X ~1) also represent solutions. This fact follows from the invariance of equations (67) and (68) to simultaneous changes in the signs of F and G (when X and yare replaced by their reciprocals). To ascertain the significance of these new solutions, let the metric functions derived from the solution (Y~ \ X ~1) be distinguished from the functions derived from the solution (X, Y) by a tilde. Then 1 \Y X) XY X2-co2 and _ /1 1\ X-Y 2u> 2c5 = - y X XY r-w 2 _ ,.a ■ (71) Comparison with equation (25) shows that the solutions derived from (y~ 1,X~1) are the conjugates of the solutions derived from (X, Y). Therefore, the transformation (X, Y) -* (Y~ 1,X~1) is equivalent to conjugation. (iv) By combining the results of (i), (ii), and (iii), we infer that if (X, Y) represents a solution, then so does [x/(i + cx),y/(i-cy)] where c is an arbitrary constant. But by these transformations we do not obtain any really new solutions beyond conjugation. (b) Alternative forms of the equations Returning to equations (7), (12), and (13), we find that, with our choice of gauge and the solution for e^, they become e^) +(e*+-^-) =0, (72) sin0/>r \ Asin0/e [(Asin0)vr]ir + [(sin0)v9]>9= +^^ [A(«r)2 + («>e)2], (73) [(Asin0)fr].r + [(sin0).A,ae= -jA^e ^03^)2 + (o>.9)2]- (74) It is apparent from equation (72) that co can be derived from a 'potential' Y in the manner
THE CHOICE OF GAUGE 283 so that wr= +e-**YesinO and cu„ = -e-4*yrAsin0. The integrability condition of this last equation is (e-**AYr\r+^-(e-**YesinO),e = 0; sin 0 and equation (74), expressed in terms of Y becomes (Afr),r +^(^i9 sin 0),9= -ie-"*[(y9)2+A(yr)2]. Now letting and reverting to the variables t] and fi, we have the equations (76) (77) (78) (79) ^-1½ + (i-^)^r = (ia-1)-^ 'X2 >e 1 i)(>:,)2 + (i-^2)(y,)2], (80) + (l-,2)|f = 0. (81)* The behaviours of the solutions of equations (80) and (81) for t] -* oo, compatible with the requirements of asymptotic flatness of the space-time, can be obtained as follows. For our present purposes, it will suffice to state that asymptotic flatness requires that 2M* 2J e2v->l- + 0(r-2) and co -►-^- + 0(^4) as r->oo, (82) r r where M* (not necessarily the same as M introduced in our definition of A— see footnote on p. 279) denotes the mass and J the angular momentum of the source. The foregoing behaviours are seen to be consistent with equation (73); and, as we shall see presently, they are also consistent with equations (80) and (81). In terms of the variable r], the required behaviours are 2M* J + 0(ri-2) and <o-> 1#2 ,,3/2 3 +0(q 4). 1 (M2-a2Yl2n (M2-a*y<*n (83) * Since there will be no occasion to use equations (62) and (63) and (80) and (81) at the same time, the use of X and y with different meanings, in the two pairs of equations, is not likely to cause any confusion.
284 THE KERR METRIC Since e2* + 2v = Xe2v = A(1 _|i2) = (M2 _a2) ^2 _ {) (1-^ (g4) we conclude that 2M* X^(M2-a2)(\-n2)ri 2\„2 1+- + 0(1). (85) (M2-a2)ll2n From equation (81) it now follows that consistent with this behaviour of X, Y^2Jn(3-n2) + 0(7/-^ (86) and we verify that the corresponding behaviour of co derived with the aid of equations (75) and (85) is consistent with the requirement that co -* 2Jr~3 as r-» oo. (c) The Ernst equation A still another form of the basic equations, which has played a central role in the investigations related to the finding of all stationary axisymmetric solutions of Einstein's equations for the vacuum, is due to Ernst. Besides, his equation provides the shortest and the simplest route to the Kerr metric. First, we observe that equation (53) allows co to be derived from a potential $ in the manner: S A &2=—co3 and <D, 3 = —j(B,2. (87) The potential <b will be governed by the equation ^¢.2).2 + (^.3).3 =0; (88) and, expressed in terms of ¢, equation (52) becomes 2 2 [A(lgX),2],2 + [<5(lgX),3],3 =^,3)2 ^^-^- ^ Letting ^ = y/(A5)/X, (90) we find that equations (88) and (89) can be reduced to the forms (cf. equations (54) and (55)) f[(Af2),2 + (^i3)i3] = A[pF2)2 -(<!>,2)2] + <W3)2 -(4>,3)2], (91) n(A4>,2).2 + (89 3),3] =2Ay,20>,2 +2^,3(D,3. (92) Now expressing *F and $ as the real and the imaginary parts of a complex
THE CHOICE OF GAUGE 285 funCti°n' Z = V + id>, (93) we can combine equations (91) and (92) into the single equation Re(Z)[(AZ2),2 + (<5Z3),3] = A(Z 2)2+<5(Z 3)2. (94) From the close similarity of equation (94) with the pair of equations (57) and (58) governing X and Y, it follows that properties, analogous to those enumerated in §(a) in the context of equations (57) and (58), must also exist in the context of equation (94). Thus, if Z is a solution, then so is Z~1; and, more generally, Z/(l + icZ) is also a solution where c is an arbitrary real constant. The transformation, Z -► Z/(l + icZ) is equivalent to what is generally called 'Ehler's transformation'. Ehler's transformation, unlike the corresponding transformation allowed by the equations governing X and Y, does provide significantly new solutions derived from the circumstance that an integration is involved in passing from Z to the metric functions. By the transformation, l + £ Z=-T^ (95) (analogous to the transformation (66)), equation (94) becomes (l-££*)[(A£2),2 + (<5£3),3]= -2£*[A(£2)2 + <5(£3)2]. (96) This is Ernst's equation. We have seen that associated with a solution (%, co), we have the conjugate solution, y co 1=-1 2 and &=—l 2- (97) X -co % -co* Accordingly, we may derive a conjugate Ernst-equation. Thus, by defining VJ> = V(A^=^+vX2-^2=e2v_w2e2^ (9g) S . *2 . , A *2 ^0).3=-^-0).3^,3= -^,2=-7 $,2=^2-03,3 =—c53;$3 = -^(5,2= —r&,2\ (99) and z = *+/a=-l±|, (loo) we shall have (1-££*)[(A£ 2),2 + (<5£ 3).3] = -2£*[A(£ 2)2+ 5(Et3)2y, (101) or, reverting to the variables n and n, we have (l-££*){[(^2-l)£J,, + [(l-M2)£J,,} = -2£*[(,/2-l)(£,,)2 + (l-^2)(£„)2]. (102)
286 THE KERR METRIC In terms of E, the metric functions *P and 5> are given by l-EE* - - /(£-£*) V = ReZ = - and <E = ImZ = '■ (103) l-£ l-£ 54. The derivation of the Kerr metric It is a simple matter to verify that Ernst's equation (102) allows the elementary solution (analogous to the solution (70) of equations (67) and (68)) E=-pt]-iqn, (104) where p2 + q2 = l (105) and p and q are real constants. (We have chosen to write — p and — q in the solution (104) for later convenience.) The corresponding solution for Z is 2 = ¥ + I*=_iz£!L^. (106) 1 + pt] + iqn Separating the real and the imaginary parts of Z, we have pV+«V-l p\ri2-\)-q2{\-H2) ¥ = _ l + 2pr] + p2r]2+q2n2 (pt] + 1)2 + q2 n2 ' and ft) - 2qti ~(pif + i)2 + «V or, reverting to the variable r, * A-lq2(M2-a2)/p2V \_(r-M) + (M2 - a2)l'2lpf + [q2 (M2 - a2)/p2]n (107) and ¢ = 2lg(M2-a2)/p^ l(r-M)+(M2-a2)ll2/p¥ + lq2(M2-a2)/p2ln2 ' l With the choices, p = (M2-a2)ll2/M and q = a/M, (110) consistent with the condition (105), the solutions for ty and $ simplify considerably to give <¥ = \(A-a25) and ¢=^^, (111) where p2 = r2 + aV = r2+a2cos20. (112)
DERIVATION OF THE KERR METRIC 287 Now, by equations (99), 4aMrn V2 (A-a25)2 p A p A and , 2aMi2 2 2x ¥2 _ (A-a25)2 Accordingly, p 5 p*S (113) 4aMrnA _ 2aM(r2 -a2n2)5 Ja^W and w'2 = (A^^y and the solution for cb is co 2aMr5 X2 — to2 A —a2 5 We also have (cf. equation (98)) «3 = -zr-^ ^d a>,2 = J-'*' ; (114) ^=.-^^2 = , .2.- (115) ¥ = e2*(x2-oi2) = e2*-to2e2* = \{A-a25). P2 Making use of this last equation, we can write laMrd ,, ,, laMrb (116) co = -^T,(x2-co2) = -^^6-^. (117) A — a*o pM We can now combine equations (116) and (117) to give ^^-e2^ = e2l>-co2e^ = -^(Ap4-4a2M2r2d), (118) where we have made use of our knowledge of e2i> ( = A<5). The solutions for co and e2^, which follow from equations (117) and (118), take simple forms by virtue of the following readily verifiable identities: l(r*+a2)+aJ(A5)-](jA±aj5) = p2jA±2aMrj5, (119) and Z2(A-a25) = p4A-4a2M2r25, (120) where I2 = (r2 + a2)2-a2A<5. (121) Thus, making use of the identity (120), we find from equation (118) that From equation (117) it now follows that o> = —^-. (123)
288 THE KERR METRIC Also, e2' = e2l>-2+=^, (124) and .,2 Va *=e-*+v -yfr (i25) Next, making use of the identity (119), we find from equations (123) and (125) that *-*+»-&d»;Jwu* ,126) and jA-aJd Y=x-a = ^+^-aJim-7s- (127) Finally, to complete the solution, we turn to equations (59) and (60). The reduction of these equations is facilitated by the following formulae giving the derivatives of X and Y: *,2 = lir^a^aJmVjm1^-^-2^^^^^ ^Va (128) Making use of the foregoing formulae, we find that equations (59) and (60) become |(M3+M2),2+^— (//3 +//2).3 =-^^- M)(p*+2a2l 2(r-M) (//3 +n2),2 + 2//(//3 + //2).3 = 4 ^__! _ (130) and (129) 2(r-M)2 4rM "a J2 We readily verify that the solution of these equations is given by P2 e"3+"2 = ~7A; (131)
DERIVATION OF THE KERR METRIC 289 and since e^3 fe = ^/A (by our choice of gauge), the solutions for e^3 and e^2, separately, are e1^ = p2/A and e2^ = p2. (132) We have now completed the solution for all the metric coefficients; and the metric is A I2 I2 n.2 is2 = p2^_(dt)2 _^_^ d(p_^MLdt ) sin20-^-(dr)2-p2(d0)2. (133) P_ A This is the Kerr metric. The covariant and the contravariant forms of it are (9ij) = (9li l-2Mr/p2 0 0 2aMrsin20/p2 0 -p2/A 0 0 0 0 -p2 0 2aMrsin20/p2 0 0 - [(r2 + a2) + 2a2Mrsin20/p2]sin20 (134) I2/p2A 0 0 2aMr/p2A 0 -A/p2 0 0 0 0 -1/p2 0 2aMr/p2A 0 0 -(A-a2sin20)/p2Asin20 (135) We observe that when a = 0, the Kerr metric (133) reduces to the Schwarzschild metric in its standard form. Also, from the asymptotic behaviours, »2v_ 1- — + 0(r~2\e^^ r2sin26 + 0(r),co^^ + 0(r~4), r r -2¾ 1 — 2M + 0(r~2), and e2^ -► r2 + 0(r) (r-»oo), (136) of the various metric coefficients, it is further clear that the metric approaches the Schwarzschild metric for r -* oo. We conclude that the Kerr metric is asymptotically flat and that the parameter M is to be identified with the mase of the black hole. Again, from the interpretation of co as representing the dragging of the inertial frame, we conclude from its asymptotic behaviour (co -»2aMr~3) that a is to be interpreted as the angular momentum per unit mass of black hole. Finally, we may note the following algebraic relations among the metric functions which we shall find useful: Ap2 -2 0-(^ + 0^)(0 = 0-^- and I - aa>sin2 6 = (r2 + a2) (137)
290 THE KERR METRIC (a) The tetrad components of the Riemann tensor There is, in principle, no difficulty in inserting the various metric coefficients of the Kerr solution in the expressions listed in equation (3) and obtaining the components of the Riemann tensor. The required reductions can be simplified by first noting that, besides the components listed in equation (4) which vanish and the cyclic identity, ^1230+^1302+^1023=0, (138) we have the following relations which are consequences of the vanishing of the Ricci tensor: *M213 = —^3002> ^1330 = —^1220' ^0202 = ~~ ^1313^ "1 M-JQ\ ^0303 = —^1212> ^2323 = —^1010 = ^0202 +^0303- J The reductions can be further facilitated by having at hand the following list of the derivatives of the various metric functions: P-2,e (/½ •A. 2 — + 1*3), 1 "l2 a2 sin 6 cos 6 p2 ~ ^ 2r r-M ■l p2 A ' \_2r(r2+a2)-a2(r el A*3.2 - (/^2+^3),1 — M)sin2 r 2 '■ P 9 = ■ 0]- 2a2 r T2 = sin0 P2 ■ — V cos 6 r 2+- -M A \\> e = cot 6 + a21 -j - —J ) sin 6 cos 6 = - v 3 + cot 6; \l>xe = 2a2\-r—3-+=4 [2r(r2+a2)-a2(r-M)sin20]—^Lin0cos0 = -v,2,e, a2 A cos 26 2a* A2 . "i2 2? •A.e.e = _ cosec 0 =j -4— sin 0 cosz 0 a2cos20 2a4 . 2o 2o 2o ^ , 1—2- sin 0 cos 0 = — v fl fl — cosec 0; P2 P 4a3 Mr A . n n (0,8= "• =4—sinflcosfl, 0),2= -^[(r2+a2)(3r2-a2)-a2(r2-a2)sin20]. (140) We find that the reductions involved in deriving the following formulae are
DERIVATION OF THE KERR METRIC 291 not unduly excessive: V(A<5) ^1023 — A % 2 (-^, 3^2 —-^,2^3) *P X aM cos 0 (3r2- a2 cos2 6)= -(R1230 + R1302X ^1230- ^1302 — I2sin0' 2p4 ■ 0).^2+ Hsle- o>,e(l*2 + Hih + e-*l>[eW^) sin0 + e-#(^^e sin0/.e \ Asin0/,r Asin0 . aM cos 0 „ , , -,„1 -■ ,-, ->..->„-, = -3 g (3r2-a2cos20)-^[(r2 + a2)2+a2Asin20], R 2323-^2 P2 Mr m.**"^ - 6 (r2 ~ 3a2 cos2 6) = R0202 + R0303 = - R1010, Mr, 2 12 2m3fl\/A, -(r -3<rcos 0^ v ' P° 2^1213 = ^"2" [(3"A + V),3,2 + 3l/>>3f 2 + V 3 V 2 - (3lj> + v),2/Z2.3 = -2^-(r2-3a2cos20)::^^(r2+a2)sin0, = -2- -(3"A + v), 3/^3.2] aM cos 0 2 2 2, 3WA 2 2 ^- w,2-a2cos20)^f '-2 ■ ^ (r2+a2)sin0, 3^1313-^3030 = e-2"3[3f 3.3 + V.3.3+3(f 3)2 + (v,3)2-(3lA + v),3^3.3] + e-2^(3iA+v),2^3,2 = -^(r2-3a2cos20)^[5(r2 + a2)2 + a2Asin20]. (141) P ^- And, finally, with the aid of the relations (138) and (139), we obtain aM cos 6 ^1023 = — ^1230 = — ^1302 = + P aM cos 6 (3r2 - a2 cos2 6), 1 6 (3r2 - a2 cos2 0W [(r2 + a2)2 +2a2 Asin2 0], P aM cos 0 6 ■ - (3r2 - a2 cos2 0W [2 (r2 + a2)2 + a2A sin2 0],
292 THE KERR METRIC aM COS 6 >WA, -^3002 = ^1213= -~"'~~~ (3r2-a2cos2e)—^(r2+a2)sm6, ■^1220 = ^1330= -^(r2-3a2cos20)^—(r2+a2)sin0, '^1010 = ^2323= +—6"(r _3a COS ^) = ^0202+^0303 Mr 7 Mr V6 Mr J6 Mr J6 3a VA -^1313 = ^0202= +^(r2-3a2cos20)^[2(r2 + a2)2 + a2Asin20], ^1212 = ^0303 = -^(r2-3a2cos20)^[(r2 + a2)2 + 2a2Asin20]. (142) We observe that these non-vanishing components of the Riemann tensor become singular only for 6 = n/2 and r = 0; this, then, is the only singularity of the Kerr space-time. 55. The uniqueness of the Kerr metric; the theorems of Robinson and Carter We have seen that the Kerr metric, which endows a stationary, axisymmetric, asymptotically flat space-time with a smooth convex event-horizon, is characterized by just two parameters—the mass M and the angular momentum J (= aM). The uniqueness of the Kerr metric for the description of the black holes of nature follows from the theorem of Robinson which states: stationary axisymmetric solutions of Einstein's equation for the vacuum, which have a smooth convex event-horizon, are asymptotically flat, and are non- singular outside of the horizon, are uniquely specified by the two parameters, the mass and the angular momentum, and these two parameters only. The proof of Robinson's theorem is based on an identity (see equation (156) below) derived from the equations (cf. equations (80) and (81)) where and E(X, Y) = E = 0 and F = 0. X <"J-"5f y2 2 l\ .1 + (1 -1)^2 + (.-,^ y2 ,2\ -f + (1-/^, F(X,Y) = C;2-!)^- ■x2 + (1-M2)-^- ■x2 (143) (144) (145)
THE THEOREMS OF ROBINSON AND CARTER 293 We recall that in deriving equations (80) and (81), we exercised the gauge- freedom at our disposal to specify the event-horizon as occurring at r] = 1. Besides, the expressions (144) and (145) for the operators E and F make it manifest that if (X, Y) represents a solution of the equations then so does (cX, cY) where c is an arbitrary real constant. On these accounts, there is clearly no loss of generality in supposing that the solutions for X in which we are interested all have at infinity the same asymptotic behaviour (cf. equation (85)), X^(\-n2)t]2 + 0(ti) (q-oo). (146) This restricton to solutions having the behaviour (146) is clearly equivalent to considering solutions for the same mass. The associated asymptotic behaviour of yis (cf. equation (86)), 7-2./^(3-^) + 0^-1) (>/-oo), (147) where J denotes the angular momentum. Besides the behaviours (146) and (147) required by asymptotic flatness, we shall also require that the solutions X and Vare smooth and regular on the axis (H = ±1), and on the horizon (rj = 1), and are non-singular outside (ti > 1). It would appear that, compatible with these requirements, one may very well have several solutions for the same assigned J. But Robinson's theorem is precisely to the effect that one can have no more than one solution for any given J. The proof consists in showing that if the contrary were true and we should have two solution-pairs, (Xt, Yt) and (X2, Y2), belonging to the same value of J, then they must coincide. We suppose then that there exist two solution-pairs, (X1, Y1) and (X2, Y2), belonging to the same value of J and, following Robinson, consider the functional R = j l j ^-(Y2-Y1)F(X1,Y1) + ^-(Y1-Y2)F(X2,Y2) A2 Aj +^Vt(y2 -y1)2 + (A'i-A'f)]£(A'1,y1) + jj^Y2l(Y2-Yi)2 + (X2-X22nE(X2,Y2) drjdn. (148) By the equations satisfied by (Xt, Yt) and (X2, Y2), R must identically vanish. But we shall now show that, by virtue solely of the definitions of the operators E and F and of the boundary conditions imposed on the solutions, R can be reduced to an integral with a positive-definite integrand. The reduction crucially depends on two elementary algebraic identities which we shall state as Robinson's lemmas.
294 THE KERR METRIC ROBINSON'S LEMMAS: 1 2X$ x$ I- ^Y^jlX^Y.-Y^Y.^ + X^X.X^-X^.jV + j^llXi(Y2-Yi)Y2,n + X2(X1X2,,-X2Xu,)¥ j^Y~2 (Y2-Yxf(YUtlY + 2^3-(X\ + xlux.x^-x.x,,,)2 + ^T^2(Y2-Y1)(Y2 + Y1\,(X1 X2i,-X2 Xlf,\ (149) 1 ^i X 2 1 4XlXl 1 4X7*2 -{Y2-Y1){XlX2ttl + X2X1^)f n.T^TJi\.(X2 + Xl){XlY2,,,-X2YUt,)-(Y2-Yl)(XlX2,tl + X2XUt,ft + ^Vj\-(X2-X1)(X1Y2in + X2Yu,) i X„(Y2-Y1)i{X1X2ttl + X2Xu„Y 2XlXi 1 X i x2 (Y2 - ^)(¾ - ^).,(^1^2., + ^1.,) +^1^3^^1 + ^^^1(¾.^2 + *i(ii.,)2] -^uin.,^}, 1 2 (150) and two further identities in which the derivatives with respect to t] are replaced by derivatives with respect to fi. In view of the entire symmetry of the terms in E and F (and, therefore, also in R) involving the derivatives with respect to t]—the >;-terms—and the derivatives with respect to p—the /Merms—except for (t]2 — 1) replacing (H2 — 1), we shall explicitly indicate the manner of the reductions only with respect to the >;-terms. Analogous reductions apply to the /Merms. Considering R, we integrate by parts the terms which occur with F as a factor and the terms which occur with the quantity in square brackets in the definition of £ (cf. equation (144)) as a factor. The integrated parts arising from the >;-terms are "2_1 {(Y1-Y2\,(Y2-Y1)+—^—\_(Y2-Y1)2(X2Xu, + X1X2J ■y -y )^1 X /,f/ \ X wOV V + (X2-X2)(X2XU,-X1X2J1
THE THEOREMS OF ROBINSON AND CARTER 295 1 + y2y2 (-^2 — -^ l) (X2XX „ — XXX2 ^) = -2^-1) (Y2-Ylf + (X2-Xlf xxx2 (151) Combined with a similar term arising from the integration by parts of the /*-terms, we have J -i d^fo2-!) ji dri<(l-n2) (Y2- (Y2- -Y1f + (X2- xx x2 -YX? + {X2- -x,f -x,r xxx2 tl - + 00 r\) r\ = 1 ,1=+1 (152) ,f) »«= -1 By the required smoothness of the functions X and Von the horizon and on the axis, the terms in (152) evaluated on the axis (fi= ± 1) and on the horizon (rj = 1) do not survive. And by the required asymptotic behaviours (146) and (147) of the solutions at infinity, (Y2-Ylf + (X2-Xlf X j X2 is 0(t]-3). (153)* There is, therefore, no contribution to the surface integral from this term as well; and it vanishes altogether. The f/-terms of the integrand of the volume integral which remains, after the integration by parts, are apart from a factor (t]2 — 1), 1 2X\X2 V{Y2-YX? + X\-X\-\{YXJ + 1 2xlx\{Y2-Ylf+X\-XU(Y2J2 + Yu, X\ Yz.r, Xx *i (Y2-Yx) (Y2-Yx) Xx, 1 Ji i I ZAj A2 x2;„ f i Ji. 2 (-^-^1-^2 l{Y2-Ylf+X22-XU KY.-Y.f+XJ-XU] * Actually, this is of 0(r/ 5) if account is taken of the term of 0(r/ ') in equation (146) (cf. equation (85)).
296 THE KERR METRIC l- (iS-^i)2(^,,)2 + Txi^(y2-i;)2(iS,,)2 2X\X2 v"z '" v"1'"' ' 2X\XX A j A 2 (^-^)(^-^).,(^1., + ^1^2.,) *U2 1 2 V2 ( 1 1., -^2-^2,,)(-^1-^2,, -^2-^1,,) X\X 2 + 2j^(^i-^?){^it(^i^2 + (yi.»)2]-^[(^2.,)2 + (l2.,)2]} + 2^Xl(y2_yi)2(XlX2'" + X2Xl'")2- (154) We observe that some terms on the right-hand side of (154) are common with those included on the right-hand side of Robinson's Lemma I. Eliminating these terms, we find, after some rearrangements, 2X\X\ 1 2X\X\ 1 + ^3^3^1(^-^1)12,, + ^2(^^2,,-^2^,,)]2 (X2 + X2)\_X2(X2J2 + X2(X1J2-2X1X2XU,X2^ 2XlX32 Al A2 -^2 (y2">i)(l2-li).,(^2^1., + ^l ^2.,) A j A 2 1 A j A 2 1 ;(A'2 + A'2)A'1,,A'2„-A'1A'2[(A'1,,)2 + (A'2,,)2]} + 2*1*1 (Yl ~ yi)2(*2^"+ ^1^,,)2 + 2^1(^1-^1)(^2:(^,^ + (^,^^^^(^2,,^ + (72,,)2]}. (155) We find that many of the terms in lines 3,4,6, and 8 of the foregoing expression cancel; and the surviving terms are precisely those included on the
THE THEOREMS OF ROBINSON AND CARTER 297 right-hand side of Robinson's Lemma II. Therefore, the >/-terms of the integrand, apart from the suppressed factor (t]2 — 1), are the positive-definite expressions in Robinson's Lemmas. The /^-terms will give similar contributions to the integrand. Thus, the result of the reduction is -i 4X1*1 + 2\_X1(Y2-Yi)Y2„ + X2(X1X2„-X2X1j¥ + l(X2 + X1)(X1Y2i,-X2Yli,)-(Y2-Y1)(X1X2i, + X2XliJ]2 i J fo2-lW 2^^-^., + ^(^^.,-^^.,)]3 + l(X2-X1)(X2Y1^ + X1Y2^)-(Y2-Y1)(X1X2in+X2Xu,)V + (\-H2)mX2(Y2-Y1)YUll + X1(X1X2,ll-X2X1,ll)¥ + 2lX1(Y2-Y1)Y2ill + X2(X1X2,)l-X2XUll)¥ + l(X2 + X1)(X1Y2_ll-X2Y1J-(Y2-Y1)(X1X2ill + X2XUll)r + l(X2-X1)(X2YUll + X1Y2ill) - (Y2-Y1)(X1X2-ll + X2X1,ll)y\ = 0. (156) It follows from equation (156) that each of the eight positive-definite expressions in the integrand must vanish identically. Accordingly, (Y2-Y1)(X1X2i. + X2Xlil,)=(X2 + X1)(X1Y2,l,-X2Ylil,) = (X2-X1)(X2YUx+X1Y2J, and X2(Y2-Y1)Y1,X= -X1(X1X2iX-X2XUx), *i(l2-li)l2..= -x2(x1x2iX-x2x1j. (157) (<* = ri,n). (158) The equalities of the two expressions on the right-hand side of equations (157) require that X\Y2,a = X2YUa. (159) Using this result, we find that the equality of the expressions on the first line of equations (157) gives X1X2(Y2-Y1\x-(Y2-Yl)(XlX2>a + X2XUa) = 0, (160) or, alternatively, = 0 (a = t],n). (161) Y2-Y, X\ X2
298 THE KERR METRIC Since 1^ and Y2 asymptotically approach equality for n -* oo, it follows from equation (161) that Y2 = YU (162) In view of this last equality, either of the equations (158) implies that (XJX2\« = 0 (<x = ij,//). (163) Again, from the equality of X1 and X2 for n -* oo, we conclude that X1 = X2; (164) and the uniqueness of the solution for assigned M (by virtue of the assumed normalization of the solutions) and J follows. Since the Kerr metric does satisfy the required boundary conditions of Robinson's theorem and provides a solution for a given M and J (< M2), it follows that it is the unique solutionfor the given M and J. In other words, Kerr's discovery provides proof for the existence of a solution satisfying the requirements of Robinson's theorem! Two special cases of Robinson's identity (156) are of some interest. First, we consider the case of two 'neighbouring solutions', (X, Y) and (X + SX, Y+ SY), i.e., we consider quasi-stationary axisymmetric perturbations which preserve the mass and the angular momentum. By considering SX and SY as small quantities of the first order (as behoves perturbations!) we obtain, by linearizing equation (156), the identity i J i A + (Xt,5Y+ Y„SX-XSY,)2 + (X^SY- Y,SX)2] + (1-n2K(X,M- Y„SY- XSXJ2 + (XitlSY+YtlSX-XSYll)2 + (X<ltSY-YJX)2l}=0, (165) which is equivalent to one first derived by Carter. From equation (165) it follows, as in the proof of Robinson's theorem, that SX = 0 and SY=0. (166) In other words, the Kerr solution cannot be subject to quasi-stationary axisymmetric perturbations which leave the mass and the angular momentum unchanged; or, as one says, there can be no point of bifurcation along the Kerr sequence. This is Carter's theorem proved prior to Robinson's tour de force. Next, we consider the case of non-rotating black-holes when Y= 0.
KERR SPACE-TIME IN A N P FORMALISM 299 Robinson's identity then reduces to Ji (IJ2- !)(*! ^2.,-^2^.,^ + (1 -^HXiX^-Xj*!,,,)2 xj + xl X j X 2 dn dn = 0, (167) where Xx and X2 are two solutions belonging to the same mass. From equation (167), we conclude that Aj = A2. (168) In other words, under the assumption of axisymmetry, non-rotating black- holes are uniquely specified by their mass. This result, without the assumption of axisymmetry, is Israel's theorem which states that the Schwarzschild metric is the unique solution for representing non-rotating static black-holes. 56. The description of the Kerr space-time in a Newman-Penrose formalism For a description of the Kerr space-time in a Newman-Penrose formalism, we first need to construct a null tetrad-frame. In Chapter 7 (§63(b)) we shall show that in the Kerr space-time we have a simple class of null geodesies given by the tangent vectors df dx r2 + a2 — - E —-0 dx — ' dr and dr " A*' (169) where £ is a constant. Defining the real null-vectors / and n of the Newman-Penrose formalism in terms of these geodesies and adjoining to them a complex null-vector m, orthogonal to them, we obtain the basis l' = -(r2 + a2, + A,0,a), —j(r2 + a2, -A,0,a), m = where J7* (ia sin 6,0,1, i cosecfl), p = r + ia cos 6 and p* = r — ia cosO. (170) (171) [It is unfortunate that the letter "p" is used in so many different contexts: as a spin coefficient, as p2 = r2 + a2 cos2 6, and now as p and p*. Normally, there will be no ambiguity as to which meaning is intended; but in case of overlapping usage, we shall denote the spin coefficient p by p to distinguish it from p and p*. Also, p2 will always signify p2 = r2 + a2 cos2 0.]
300 THE KERR METRIC The vector / is affinely parametrized while n and m are not: they have been arranged to satisfy the required normalization conditions /•«= 1 and mm = — 1. (172) The covariant form of the basis vectors is /, =-^(A, -p2,0, -aAsin20), 1 V 1 m, =■ (A, + p2,0, -aAsin20), (ia sin 8,0, -p2, -i(r2 + a2)sin6). (173) PV2 The spin coefficients (as defined in Ch. 1, equations (286)) with respect to the chosen basis are most conveniently evaluated via the 2-symbols (Ch. 1, equation (265)). We find that the non-vanishing 2-symbols are: ^122= -4r[(r-M)p2-rA]; ^•134 — ^132 = +^2 ^213 = ~V^ A iar sin 6 a2 sin 6 cos 6 p2p ^•324 — — 2ia cos 6 iaA cos 6 ^•334 — + (ia + r cos 6) co sec 6 w ^243 2p2p ^•341 — — " 1 (174) and the spin-coefficients derived with the aid of these 2-symbols are: k = a = X = v = 6 = 0 P = 1 p*' /* = - P = A cote ia sin 6 ia sin 6 n = 2p2p*' y = H + - (P*)V2' r-M x = 2p2 a = n-p*. (175) The fact that the spin-coefficients, k, <j, X, and v vanish, shows that the congruence of the null geodesies, / and «, are shear free. And from the shear- free character of these congruences, we can conclude, on the basis of the Goldberg-Sachs theorem, that the Kerr space-time is of Petrov type-D. The Goldberg-Sachs theorem also allows us to conclude that in the chosen basis, the Weyl scalars *P0, *Fl5 ¥3, and ¥4 vanish. These conclusions can be directly verified by contracting the non-vanishing components of the Riemann tensor listed in equations (142) with the vectors /, n, m, and lit in accordance with the definitions of these scalars.
KERR SPACE-TIME IN A N-P FORMALISM 301 Since the components of the Riemann tensor listed in equations (142) are in the tetrad-frame, it is convenient to express the basis vectors also in the same frame by subjecting them to the transformation represented by (ewi) = We find /w =^(1^ +a2), '« =fe('2+«2)> e" 0 0 0 -cue* e* 0 0 0 0 e^ 0 0 0 0 e^ e+p2 (176) I2 e*A m Xp) PJ2 a sin 6, i U2I2' e*p2 (r2 + a2) I2 sin0 p J2 ' e"\ A IP1 ' 0 0 0 ef2 f>J2 (177) where, in obtaining the vectors in these forms, use has been made of the relations (137). Using the foregoing representation of the basis vectors and making use of equations (4) and (139), we find, for example, that -^0 = RpqrJp,m"lrms = RoioM0™1 - I'm0)2 - (/½3)2] + K0303 [('°m3)2 - (^1)2] 320 300 + *202o[('2m°)2-('1m3)2]+21l133o[(/2)2m0m1-(m3)2/0/1] + 2R3002[l2l°m°m3 +l2lWm3]+2R2i01l2m3(l°m1 -I'm0) 2/1JJ + 2R2103l2l°m1m3 + 2R2013l2l1m°m - P* I p , (r2+a2)\ a2A sin2 8 _ AS2) °101 ^2 A0303 ^rj ^0202 I2 2 (r2+a2)sin0 + 4r (r2+a2)2 ^1330 ^T) +^2301+ T2 a2A sin2 6 ^2103 "I ^2 °2013 + 2aJk., , , . X, (r2+a2)sin0 R 3002 (178) Now substituting for the components of the Reimann tensor from equations (142), we find that, as required, (179) ^0=0. The vanishing of the Weyl scalars *Fl5 ¥3, ¥4 can be similarly verified.
302 THE KERR METRIC The value of the non-vanishing Weyl scalar, ¥2, can be found by contracting the Reimann tensor, with the vectors of the null basis, in accordance with its definition. Thus, we find *2 = RMJprri>rfms = ^oioi[('°'n1-'1'n0)(«0'n1-n1m0)-/2n2|m3|2] + iW['V^m3|2-/2n2|mM2] +^02.02^1^12 -/Vim3!2] + «i33o['" (m'm0 + tr^m0) - (lln° + l°n m ,3 12 ] + ^3oo2['2«°'"0'"3+/0n2m0m3+/2n1m1m3 + /1n2m1m3] + ^2301 V2™3 (n0™1 - n1™0) + n2m3 (Pm1 - /½0)] + ^2io3['2«°'n1'«3+/0n2'n1m3]+i?2013[/1n2m3m0+/2n1m3m0] - ~i +^0101 + (r2+a2) 2\2 *0 a2 A sin2 6 303 1 + 2«' '■^2301 + I2 (r2+a2)sin0 (r2 + a2) R„ laV/A/ 2 2 I2 «13 + I2 2 (r2+a2)sin0 a2 A sin2 6 ^2103 H =2 R 2013 K * I *•■*■ _1_ /i* . Mr „ ., „ „. .aMcos0 3002 6-(r2-3a2cos20)-/: (3r2-a2cos20) M (P *\3 ' (180) With the specification of the null basis, the spin-coefficients, and the Weyl scalars, we have completed the description of the Kerr space-time in a Newman-Penrose formalism which exemplifies its special algebraic character. 57. The Kerr-Schild form of the metric The derivation of the Kerr metric via the route described in §§52-54 is not the one which was followed by Kerr in his original investigation leading to his discovery. He was seeking, instead, solutions of Einstein's equation for forms of the metric appropriate to algebraically special space-times. The form of the metric he chose for his special consideration (now known as the Kerr-Schild form) is 0,} = tltj + ltlj> (181) where jj,-y is a flat metric and / is a null vector with respect to jj( •. The Kerr
THE KERR-SCHILD FORM 303 metric belongs to this class. Indeed, it will appear that it is in the Kerr-Schild form that the nature of the Kerr space-time becomes most transparent. On this account, we shall preface our consideration of the nature of the Kerr space- time showing why a space-time with the metric of the form (181) is necessarily of Petrov type-II. The following manner of presentation is due to Xanthopoulos. We begin with a few preliminaries. Consider a manifold endowed with two metrics gtj and g'if. Let V, and V[ denote the operators defining covariant differentiation with respect to the Christ offel connections derived from each of the two metrics. If ^ denotes a covariant vector defined on the manifold, then we should have a relation of the form v^v.^.-q;.^, (182) where C™ has all the properties of a symmetric connection. Thus, we should have, for example, V;-7? = V,T) - C7S Tkm + Ckm Tf , (183) where Tf is a tensor of type (1,1). With the aid of these formulae, we can relate the Riemann tensors, defined with respect to the two metrics, via the Ricci identity. Thus, -2C"kli(VnZ.-CfiuZm). (184) On further simplification (making use of the symmetry of C" in the indices i and j), we obtain the relation R'tflT = V- 2V[fC^ + 2CJpCj5.. (185) The corresponding relation between the Ricci tensors follows by contraction; thus, R'!k = R!k-2VvCZlk + 2q[;C-]n. (186) In considering the Kerr-Schild form of the metric (181), we shall identify the unprimed and the primed metrics in the foregoing formulae with lij and gij = %- + lil,; (187) where, as we have stated, /,- is a null vector with respect to t]tl, i.e., /' = !?"/, and /'7f = 0. (188) Now, with the definition, giJ = r,!'-l% (189)
304 THE KERR METRIC we verify that 0 V = fo" -'''')(»?/» + 'y W = Si (190) In other words, g'J as denned is, indeed, the contravariant form of g{-. Moreover, 9^=(^-1^)1,- = 1,. (191) Accordingly, the index of the null vector can be raised or lowered by g as well as by tj. It follows that / is null, also, with respect to g. (However, it is to be understood that, in general, the raising and the lowering of the indices is effected only by r\.) We shall now obtain an explicit formula for the connection C". Since Vj is denned in terms of the Christoffel connection derived from g{j, V,'^ = 0. (192) But we can also evaluate this quantity with the aid of equation (183). Thus, V('0,» = V,0A - C?-gmk - CTkgjm = 0. (193) On the other hand, ^ = ^ + /^) = ^(½). (194) Therefore, W*) = q}0m< + qz0jm. (195) From this last relation we obtain (in the same manner as one obtains the Christoffel connection from the vanishing of the covariant derivative of the metric tensor): Cfig* = iCW,) + V,.(U-) - V,(/,/y)]. (196) By contracting this last relation with gkn, we obtain (by virtue of the relation (190)) CI; = W - lkl")[y,(ljlk) + V,.(/,A) - V,(/,./y); (197) or, after some simplification (remembering that / is a null vector) we find Ctj = i[ W) + Vy(/,f) - V(l,lj) + lmlkVk(l,lj)l (198) Contracting equation (198) with respect to the indices n and i, we obtain C% = 0. (199) Similarly, we find l.dj = ~ WWj) = - HWh + hiyii) (200) and I'cij = ±(ri%i}.+iji%n (201) Defining X, = l%l„ (202)
THE KERR-SCHILD FORM 305 we can rewrite the relations (200) and (201) in the forms i.cij=- UiixJ + iJ.Xt) and />Q=+i(/-^+/yX-). (203) Besides, we must also have I'Xj = 0. (204) We now assume that gtj represents a solution of Einstein's equation for the vacuum. Then the Ricci tensor R';j must vanish. But R{j also vanishes by virtue of our assumption that t];i is the metric of a flat space-time. Therefore, by equation (186), 0 = -^^,+^:3, V7 Cm I V7 Cm , cn cm c Cm — vi^mfc~r' vm^.fc "T^itf^mn '■'tonWn = Vm^ik—C1mC ;„, (205) where, in the reductions, we have made use of equation (199). Now contracting equation (205) with /'7* and making use of the relations (203) and (204), we obtain o = -^^(/■crj-cr^ri + ^cLK/'crj = -ykvm(imxk+xmik)+t(imx'+1^-)^1, + i(l"Xm + lmX")(lmXn + lnXm) = -i/TV^+iY../"^/' = + \Xkrvmlk+ \X{X' = XsXl. (206) Therefore, A" is a null vector; and since it is also orthogonal to the null vector /, it must be a multiple of /. Accordingly, we may write Xj = ¢1,., (207) where (j> is some function of proportionality. Now, by the definition (202) of Xj = l%lj = <j>lj. (208) Therefore, I defines a null geodesic congruence. And finally, by combining equations (203) and (207), we obtain the relations /.C?, = -#,/, and /'C-v= +4>l"ls. (209) We now return to equation (185) and contract it with lJlm. Remembering that R!jkm vanishes (since the metric t\{j, by assumption, is that of a flat space- time), we have inmR'ijkm = i'im(s/fTk - v,qi +Q..C? - cifi). (210)
306 THE KERR METRIC We readily verify that all the terms on the right-hand side, except the first, vanish by virtue of the relations (209). We are thus left with i'imR'iJk"> = MmVjCZ = nVjiLcm-csVjU = -(/^ + ^)/,-/,, (211) where, in the reductions, we have again made use of the relations (209). Since the index of / can be raised or lowered by either of the metrics r\ or g, we can rewrite the relation we have arrived at in the form R'ijkml'lm = Hltlk where H = -(l%4> + 4>2), (212) where it may be recalled that / represents a null geodesic congruence. From equation (212), it clearly follows that JW.]W= -Hlsl[klul=0. (213) By Chapter 1, equation (385), this last relation is precisely the one which ensures that the space-time is of Petrov type-II and that / is a principal null direction. (a) Casting the Kerr metric in the Kerr-Schild form We start with the congruence of null vectors / defined in equation (169). With the choice E = 1, we have 2 2 dt = r +" <U, dr = dl, d6 = 0, and dq>=^dL (214) A A We now introduce, in place of t and q>, the new variables r2 + a2 a du = dt —dr and dq> = dq>— — dr. (215) A A In these new coordinates, the null geodesic is given by /■ = (0,1,0,0). (216) For expressing the Kerr metric in the new coordinates, it is convenient to rewrite it, first, in the form ds2 = ^[dt - (asm2 8)dq>~\2 -^-[(r2 + a2)d<p -adt~\2 P P -^(dr)2-p2(de)2. (217)
THE KERR-SCHILD FORM 307 A direct substitution of the relations (215) now gives ds2 =^(du-ad(psin20)2-^-[(r2+a2)d(p-adu]2 + 2dr(du-ad(£sin20)-p2(d0)2. The corresponding explicit form of the metric is (218) (9ij) = («) -IMrjp2 1 0 (r) 1 0 0 (0) 0 0 -P2 (<p) 2aMrsm26/p2 — a sin2 6 0 2aMrsin20/p2 -asin20 0 -l2sin20/p2 (219) By a further regrouping of the terms, we find that the metric (218) can be brought to the form ds2 = [ (du + dr)2 - (dr)2 -p2 (d6)2 - (r2 + a2) (d(p)2 sin2 0 0 \A r — 2adrd(£sin20] 2~(d" — a dip sin2 8)2; or, in view of the covariant form of / being /,-= (1,0,0, -asin20), we can also write ds2 = [ (dx0)2 - (dr)2 - p2 (d6)2 - (r2 + a2) (dyf sin2 6 -2adrd(£sin20] —liljdx'dx1', where dx° - du + dr. (220) (221) (222) (223) We shall presently show that the part of the metric in square brackets in equation (222) represents flat space-time. We have thus reduced the Kerr metric to the Kerr-Schild form; and its algebraic speciality is now a part of its definition. We can now verify that the substitutions, x = (rcosip + asin<p)s'm6, y — (r sin ¢)- a cos ¢)) sin 0, ^1 A , „2 _ /„2 L„2\-,„2p - ' - i I \11^) x2+y2 = (r2 + a2)sin20 and z = rcos0, bring the part of the metric in square brackets in equation (222) to the form of the manifestly flat metric, (dx0)2 - (dx)2 - (dy)2 - (dz)2. (225)
308 THE KERR METRIC Explicitly, the entire metric in these new coordinates is ds2 = (dx0)2 - (dx)2 - (dy)2 - (dz)2 --7 5-^-<dx°—-^ ,[r(xdx+ydy) + a(xdy-ydx)~\--zdz r* + aMzM i r +az r (226) where, according to the substitutions (224), r2 is defined, implicitly, in terms of x, y, and z by r4 - r2 (x2 + y2 + z2 - a2) - a2z2 = 0. (227) The metric (226) is clearly analytic everywhere except at x2 + y2 + z2 = a2 and z = 0; (228) in other words, it has a ring singularity in the (x, y)-plane. We shall consider the nature of this singularity in greater detail in §58 below. Here, we shall only note that it is precisely in the form (226) that Kerr originally presented his solution. 58. The nature of the Kerr space-time There are two principal features of the Kerr space-time which require to be examined and clarified: the first relates to the nature of the null surfaces and the second relates to the nature of the singularity at r = 0 and 6 = n/2. Concerning the first, we know that the null surfaces occur at the zeros of A(r) = r2 - 2Mr + a2 = 0, (229) i.e., at r = r+ =M + J(M2-a2) and r = r_ = M - J (M2-a2). (230) These zeros are real, positive, and distinct so long as a2 < M2, an inequality which we shall assume obtains. Since A > 0 for r > r+ and r <r_ and A < 0 for r_ <r <r+, (231) it is clear that, as in the case of the Reissner-Nordstrom space-time, we must distinguish the three regions: A:r<r-; B: r_ <r<r+; and C:r>r+. (232) (We shall presently see that the region A can be extended to all negative values ofr even as the region C extends to all positive values ofr.) We shall find, again as in the Reissner-Nordstrom geometry, that while the surface at r = r+ represents an event horizon, the surface at r = r_ represents a Cauchy horizon. Concerning the singularity in the Kerr space-time, we have already pointed out that the non-vanishing components of the Riemann tensor (listed in equations (142)) diverge only for r = 0 and 9 = n/2; and that this is the only singularity there is. Since the divergence at r = 0 occurs only for 6 = n/2, it is
THE NATURE OF KERR SPACE-TIME 309 clear that its nature cannot be the same as the singularity at r = 0 of the Schwarzschild and the Reissner-Nordstrom space-times. To understand the real nature of the singularity of the Kerr space-time, we must, first, eliminate the inherent ambiguity in the coordinate system, (r, 0, (p), at r = 0. This ambiguity was abolished in §58(a) by the choice of the 'Cartesian' coordinate system (x, y, z). In the metric (226), written in these coordinates, r2 is implicitly defined in terms of x, y, and z by equation (227), while (x2 + y2) = (r2 + a2) sin2 6. (233) The surfaces of constant r are confocal ellipsoids whose principal axes coincide with the coordinate axes. These ellipsoids degenerate, for r = 0, to the disc, x2+y2^ a2, z = 0. (234) The point, (r = 0, 6 = n/2), corresponds, then, to the ring, x2+y2 =a2, z = 0; (235) and the singularity along this ring is the only singularity of the Kerr space-time. The points, interior to the ring (235), correspond to r = 0 and n/2 > 6 > 0 and 0 < <p < In. The surfaces of constant 9 cross the disc and pass to the other side. On these accounts, we are entitled to extend the domain ofr to all negative values. In other words, we analytically continue the function r, defined by equation (227), to all negative values. We accomplish this continuation by attaching another chart (x', y', z') to the disc (234) and by identifying a point on the top side of the disc to a point, with the same x and y coordinates, on the bottom side of the disc in the (x', y', z')-chart. Similarly, we identify a point on the bottom side of the disc in the (x, y, z)-chart to a point on the top side of the disc in the (x', y', z')-chart. (See Fig. 25.) The enlargement of the original manifold in this manner is analogous to the enlargement of the complex plane to the Riemann surfaces for the representation of analytic functions with singularities. With the manifold enlarged, in the region A, distinguished in (232), r is allowed to take all negative values. The metric (226), extended to the enlarged manifold, has the same form in the (x, y,z')-chart as in the (x, y, z)-chart except that r is now assigned negative values. Therefore, in the (x', y', z')-chart, the terms in the curly brackets in (226) occur with the factor, + 2Mr3/(r4 + a2z2). For large negative values ofr, the space is again asymptotically flat but with a negative mass for the source. Turning to the elimination of the coordinate singularities at r+ and r_, we first realize that, on account of the principal null congruences, / and n, not being hyper-surface orthogonal, we cannot make them the basis for deriving non-singular coordinates as we were able to do in the context of the Schwarzschild and the Reissner-Nordstrom space-times. It is, therefore, useful
310 THE KERR METRIC /•= constant (r>0) 6 = constant 6 = constant ) = 0 /\ /\ 0 = 0 constant (/• <0) ingularity [jr,^]- chart [x',z']-chart Fig. 25. The ring singularity of the Kerr metric in the equatorial plane. The nature of the singularity becomes manifest when the metric is written in "Cartesian coordinates" in the form (226) when the curves of constant r become confocal spheroids and the curves of constant 8 become confocal hyperbolae. The analytic extension to all negative values of r is accomplished by attaching to the disc (x2 + y2 < a2; z = 0), another disc in a chart (x', y', z'), by the identifications described in the text and illustrated in the figure. The figure shows the sections y = 0 and y' = 0 of these planes. to restrict ourselves, in the first instance, to the two-dimensional (r, t)-manifold along the symmetry axis, 6 — 0, to clarify that the nature of the horizons is essentially the same as in the Riessner-Nordstrom space-time and that the manifold can be completed with the same basic ideas. Along the symmetry axis, 6=0, the Kerr metric degenerates to the form ds2 = dr r2 + a2 rz + az\ A By denning the null coordinates, dr )[dt + dr u = t and v = t + r„ where V2 + a2 r\+a2 dr = r-\— \g\r-r+\ -- (236) (237) -lg|r-r_|, (238) we can write ds2 = r du dv. (239) The substitutions (237) are the proper ones for the regions A and C (cf Ch. 5, equations (46) and (48)). In the region B, A is negative and the proper substitutions are and v = r^ -1, (240) u=-rt + t
THE NATURE OF KERR SPACE-TIME 311 O + ll A •-. o OQ C = V .2 >» ffv oc ^ 1 II a + c o 6 X a ii o 'So .2 v •-. on. i o
312 THE KERR METRIC Fig. 27. The maximum analytic extension of the Kerr space-time for 8 = 0 is achieved by piecing together the blocks in Fig. 26 in the manner described for the Reissner-Nordstrbm space-time in Fig. 14. when the metric takes the form ds2=^Lduda (241) r* + a* By the foregoing choices, the parts of the manifold in the three regions can be represented by the 'blocks' in Fig. 26. The edges of the blocks are to be identified as indicated. Also, to the regions A, B, and C, there correspond further regions, A', B\ and C, obtained by applying the transformation u -»• — u and v -*■ — v which reverses the light-cone structure. A maximal analytic extension of the (r, t)-manifold may then be obtained by piecing together copies of the six blocks in an analytic fashion so that an overlapping edge is covered by a u(r) or a u(r)-chart (with the exception of the corners of the blocks). The resulting 'ladder' is shown in Fig. 27; it can be extended, indefinitely, in both directions. An analytic representation of the maximally extended space-time on the axis can be obtained, exactly as in the context of the Reissner-Nordstrom space- time, as follows.
THE NATURE OF KERR SPACE-TIME 313 and and r+-r- tanK= +e+'v, tanK= +e+av, In regions A and C, we set tan I/=-e~™ and tan V = +e+'v, (242) while in the region B, we set tanl/=+e+0IU and tan V = + e+a", (243) where a = .7, "~2„ (244) 2(r2++a2) and u and v, in the substitutions (242) and (243), have, respectively, the same meanings as in equations (237) and (240). With the foregoing substitutions, the metric takes the 'universal' form 4 ds2 = - ^-|A| cosec 21/ cosec 2V dU dV, (245) where r is denned implicitly in terms of U and V by tan U tan V - -e2xr\r-r + \ |r-r_ |~" (r > r+ and r < r_) = +e2"|r-r+||r-r_r' (r_<r<r+), (246) where ri+a2 r_ /» = -j 1 = —• (247) The delineation of the coordinate axes by null geodesies in Figs. 26 and 27 enables us to visualize, pictorially, the nature of the underlying space-time. Thus, the partition at r = r+ is an event horizon and the partition at r = r_ is a Cauchy horizon in the senses we have made clear in the earlier chapters. To eliminate the coordinate singularities at r+ and r_ in general in a unified manner, and to achieve, at the same time, a maximal analytic extension of the entire manifold is more difficult. The difficulty arises from the fact that both t and </> have singular behaviours as the horizons are approached along null (or, time-like) geodesies (see Chapter 7, §61). On this account, a simple change of coordinates, which will be satisfactory at both horizons simultaneously, does not appear practicable. We shall, therefore, be content in treating the event and the Cauchy horizons separately. We consider first the substitutions appropriate for crossing the event horizon smoothly. We start with the metric in the form (cf. equation (217)) ds2 =^(dt-ad</)sin20)2-^12-[(r2 +a2)dy-adt~\2 f \w M %»\iX Jill VI -J p2 p2 -,2 -^-(dr)2-p2(d0)2; (248)
314 THE KERR METRIC and write, as before, u — t — r„, and v — t + r„, (249) where r„, is the same variable as defined in equation (238). In addition, we shall now introduce, in place of m, <p+ = <p-at/2Mr+, (250) where we have distinguished the new variable by a subscript, +, to emphasize that we are presently concerned with a smooth crossing of the event horizon. (The introduction of a new angle variable, depending on the horizon we are crossing, is an essential feature of considerations pertaining to the Kerr space- time.) With the change of variables defined, 1 P\ dt — a dm sin2 6 = —^—*—-= dt — a dm + sin2 6 where similarly, 2r2 + a2 p\ — r\ +a2cos20; (du + dv) — a dq> + sin2 6, 1 r2 - r\ We also have dr = --, ^ (dv - du). 2 r2 + a2 K (251) (252) (r2 + a2) dm - a dt = (r2 + a2) dq> + + ^ a' 2 '\ (du + dv). (253) (254) Inserting the expressions (251), (253), and (254) in equation (248), we obtain after some further simplifications, ds2 = 2P2 ■ + (r2++a2)2 (r2 + a2) dudv-p2(d0)2 sin2 6 , ,, 1 r2 — r\ , (r2 + a2) dm + + - a -j——% (du + do) a A sin Of p\ p 4p2 \r2+a2 r2 + a2/(ri.+a2)(r2 + a2) [(du)2+(dt;)2] + - aAsin20 asin2 Bdcp. —=—^ (du+dv) r+ + <r d^ + . (255) To include the four adjoining regions, I, II, I', and IF in Fig. 27 in a single
THE NATURE OF KERR SPACE-TIME 315 coordinate neighbourhood, we can define, as before tan 1/ + = -e-«+u = -e-*+te+*+r» where Then tan V+ = +e + ^v = +e + '*'e + '*r*, ' (256) 2(rUO" ^ 2 2 du = cosec2l/+dl/ + and dv = + — cosec2K+dK+. (258) <x + a + By substituting these expressions for du and dv in equation (255), we shall obtain the metric in the required form, where r is to be understood as defined implicitly in terms of 1/ + and V+ by the equation tanl/+tanK+ = -e2ot+r|r-r+ ||r-r_|-"+, (259) where /J+=r_/r + . (260) Considering next the substitutions, appropriate for crossing the Cauchy horizon at r = r _ smoothly, we write u — r^+t and v = r„, -1 (261) and <p. = (p-at/2Mr_. (262) The resulting form of the metric can be obtained from equation (255) by replacing " + " by " - " and writing du in place of dv and - dv in place of du. And the transformations analogous to those given in equations (256)-(260) are tanl/- =e + a-u = e + a-<r.+I), tan K_ = e + «-v = e + '-C.-O, 2 , 2 (263) du =—cosec2l/_dl/_, du = —cosec2K_dK_, where 2p_+a2)' and r is defined implicitly in terms l/_ and V- by tanl/_tanK_ = e2a-r|r — r_||r — r+ |-"-, (265) where /J_=r + /r_. (266) It is an easy matter to verify that the metric, as we have defined, in terms (U + ,V+) and (U-,V-), is analytic everywhere except at the curvature singularity where p2 vanishes. a- =^rrir^ (264)
316 THE KERR METRIC In Chapter 7 we shall see that with the maximal extension of the manifold, all time-like and null geodesies are of infinite affine length (both when extended to the past and when extended to the future) except those which terminate at the singularity; in other words, the extended space-time is geodesically complete. (a) The ergosphere Besides the principal difference in the nature of the singularities in the Kerr and in the Schwarzschild and in the Reissner-Nordstrbm space-times, there is a further feature which distinguishes the Kerr space-time from the other two. The distinction arises from the fact that unlike in the Schwarzschild and in the Reissner-Nordstrbm space-times, the event horizon does not coincide with the surface <7„=l-2Mr/p2=0 (267) or The surface. r2 - 2Mr + a2 cos2 6 = A - a2 sin2 0 = 0. (268) r = re(0) = M + y(M2-a2cos20), (269) external to the event horizon, on which gtt vanishes, is called the ergosphere. It coincides with the event horizon only at the poles 0 = 0 and 6 = it. The ergosphere is a stationary limit surface in the sense that it is the inner boundary of the region in which a particle with a word-line dt = 0 is time-like. This fact becomes apparent when we inquire, for example, into the implications of the obvious requirement lo^'Kl, (270) where vM = e*-v(il-co) (271) is the (^-component of the 3-velocity in the ortho-normal tetrad-frame considered in §52. In the context of the Kerr metric, the inequality (270) gives (cf equation (125)) |n-w|<*v-*=|^. (272) It follows that p\/A 2aMrj5±p2s/A "s?= w±*7~s= —z%7*—' ( ] or, by virtue of the identities (119)-(121), "Sfnx " ±l(r2+a2)±aJ(AS)U5' ( '
BIBLIOGRAPHICAL NOTES 317 In particular, we have JtL-aJb =_ A-a25 min [FT^V^7(A^J]V7 [pVA + 2flMr^]^' (275) on making use, once again, of the identities (119)-(121). Since A — a2 6 < 0 inside the ergosphere, (276) we conclude that Q.min > 0 inside the ergosphere and = 0 on r — re(6). (277) This result is in accord with our earlier statement concerning the stationary limit-surface derived from the vanishing of gtt on it. The fact that there can be no stationary observer inside the ergosphere does not, of course, preclude the existence of time-like trajectories in the region, r+<r< re(6), which can escape to infinity. This part of space is, therefore, communicable to the space outside. We shall see in §65 in Chapter 7 that the existence of this finite region of space between the event horizon and the stationary limit-surface has important consequences: it allows, for example, processes—the Penrose processes—which will result in the extraction of the rotational energy of the black hole. Finally, we may note here, for future reference, that the surface area of the event horizon is given by Surface area = o J 'sin2 6 -^-{(r2+a2)2-a2Asin20}p2 1/2 dOd<p J o Jo = 47t(r2 +a2) = 8jtMr+ = 8jtM[M + J(M2 -a2)]. (278) BIBLIOGRAPHICAL NOTES Kerr's discovery was announced very briefly in a short letter (dated 26 July 1963): 1. R. P. Kerr, Phys. Rev. Letters, 11, 237-8 (September 1, 1963) and reported at a meeting (December 16-18, 1963) a few months later: 2. R. P. Kerr in Quasistellar Sources and Gravitational Collapse, 99—103, edited by I. Robinson and E. Schuking, University of Chicago Press, Chicago, 1965. In spite of its brevity, the original announcement is surprisingly complete in enumerating the essential features of the solution. But the announcement gave not e^en an outline of the derivation; it was provided only some two years later in: 3. R. P. Kerr and A. Schild in Comitato Nazionale per le Manifestazioni Celebrative Del IV Centenario delta Nascita di Galileo Galilei, Atti del Convegno Sulla Relativita Generate: Problemi Dell'Energia E Onde Gravitazionali, 1-12, edited by G. Barbera, Florence, 1965.
318 THE KERR METRIC 4. in Proceedings of Symposia in Applied Mathematics, 17, 199-209, American Math. Soc, 1965. However, the authors warn the reader (in reference 3) that "the calculations giving these results are by no means simple." §52. The literature on stationary axisymmetric space-times is voluminous; the extent of it can be judged from: 5. W. Kinnersley in General Relativity and Gravitation, 109-35, edited by G. Shaviv and J. Rosen, John Wiley & Sons, Inc., New York, 1975. Since the foregoing report was written, the advances in the subject have been spectacular. But here we are concerned with the general theory only to the extent that we have an adequate base for a simple derivation of the Kerr metric. While the equations in this section have been derived as special cases of the ones given in Chapter 2, they are contained in essentially the same forms in: 6. S. Chandrasekhar and J. L. Friedman, Astrophys. J., 175, 379-405 (1972). What we have called the Papapetrou transformation is described in: 7. A. Papapetrou, Ann. d. Physik, (6) 12, 309-15 (1953). 8. , Ann. d'l Institut Henri Poincare Section A, 4, 83-105 (1966). §53. The treatment in this section derives largely from: 9. S. Chandrasekhar, Proc. Roy. Soc. (London) A, 358, 406-20 (1978). The Ernst equation is derived in: 10. F. J. Ernst, Phys. Rev., 167, 1175-8 (1968). 11. , ibid., 168, 1415-7 (1968). See also: 12. B. C. Xanthopoulos, Proc. Roy. Soc. (London) A, 365, 381^411, (1979). §55. The principal papers dealing with the uniqueness of the Kerr space-time are those of: 13. B. Carter, Phys. Rev. Letters, 26, 331-3 (1972). 14. D. C. Robinson, ibid, 34, 905-6 (1975). See also: 15. B. Carter in Black Holes, edited by C. DeWitt and B. S. DeWitt, Gordon and Breach Science Publishers, New York, 1973. 16. in General Relativity—An Einstein Centenary Survey, chapter 6,294-369, edited by S. W. Hawking and W. Israel, Cambridge, England, 1979. Israel's theorem (derived in this section as a special case of Robinson's theorem and assuming somewhat more restrictive conditions than in Israel's original statement) is proved in: 17. W. Israel, Phys. Rev., 164, 1776-9 (1968). §56. An important reference in this context is: 18. W. Kinnersley, J. Math. Phys., 10, 1195-1203 (1969). §57. The 'Kerr-Schild' space-times were introduced in references 3 and 4. The principal ideas in the presentation in this section (due to Xanthopoulos) are derived from: 19. B. C. Xanthopoulos, J. Math. Phys., 19, 1607-9 (1978). §58. The basic papers dealing with the nature of the Kerr space-time are those of: 20. B. Carter, Phys. Rev., 141, 1242-7 (1966). 21. R. H. Boyer and R. W. Lindquist, J. Math. Phys., 8, 265-81 (1967). 22. B. Carter, Phys. Rev., 174, 1559-70 (1968). The coordinates in which we have derived and written the Kerr metric are often called the 'Boyer-Lindquist' coordinates; they are introduced in reference 21.
7 THE GEODESICS IN THE KERR SPACE-TIME 59. Introduction This chapter is devoted to the study of the geodesies in the Kerr space-time. We are devoting an entire chapter to this study for several reasons: apart from the fact that the delineation of the geodesies exhibits the essential features of the space-time (as for example, the relevance of extending the manifold to negative values of the radial coordinate r), the separability of the Hamilton-Jacobi equation (discovered by Carter) was the first of the many properties which have endowed the Kerr metric with an aura of the miraculous. Besides, the study discloses the possibility (discovered by Penrose) of extracting energy from the Kerr black-hole, thus revealing the existence of physical processes one had not contemplated before. At the same time the study of the geodesies in the Kerr space-time directed attention to certain quite unexpected properties of space-times of Petrov type-D in general. We begin, in fact, with the establishment of one of these properties. 60. Theorems on the integrals of geodesic motion in type-D space-times We have shown in Chapter 1 (§9(c)) that in a space-time of Petrov type-D, the principal null congruences are geodesic and shear-free; and that in a null tetrad-basis (/, n, m, m), in which / and n are the principal null-directions, the spin coefficients, k, a, A, and v, and the Weyl scalars, *P0, *F„ *F3, and ¥4, vanish: K = a = X = v = 0, (1) and ^0 = ^,=^3 = ^4=0. (2) And if, as we shall suppose, that / is affinely parametrized, then the spin- coefficient 6 is also zero: e = 0. (3) Finally, the Bianchi identities (Chapter 1, equations 321)) provide for the non- vanishing Weyl-scalar *F2 the equations Dlgy2=+3p; <51gy2 = +3t, J Algy2= -3n; <5*lgy2= -3«, J l '
320 THE GEODESICS IN THE KERR SPACE-TIME where D, A, 3, and 3* are the directional derivatives along /, n, m, and in, respectively. The metric tensor, gtj, in a coordinate basis, expressed in terms of/,«, m, and in, is given by (cf. Ch. 1, equation(246)) 9ij =h"j + hni - mifhj - m^; (5) and the covariant derivatives of I, n, and m expressed in terms of the spin coefficients are (cf. Ch. 1, equation (357)) h;i = +(y + y*)ljii- («* +P)lJ'»i- (« + /»%»«« - tmy/, - T*mj/f + prnj-m, + p*mjrhl; (6) «;;, = - (V +Y*)"jh + (<** + /?K>"i + (a + P*)njm, + n*mjni + nmJni — nmjifii — H*ifij m,; (7) mj-i =+(7- y*)mjli + (a* - ^m^ + (P* - ajm^m, + 7t* /,-n, - /z* /,m( - xtijli + pnjmi. (8) (The foregoing relations are applicable only to type-D space-times, since the relations (1) and (3) have been assumed.) If k and/denote any two vectors, then, according to equation (5), k'f^uk'p = (k-l)(f-H) + (k-H)(f-l) -(*•«) (/•«)-(*•«) (/•«), (9) \k\2 = 2[(k-l)(k-n)-(k-m)(k-m)] (10) and fc'V, = (k-n)D + (k-l)A-(k-m)3-(k-m)3*. (11) We now state the principal theorem of the subject due to Walker and Penrose. THEOREM 1. If k is a null geodesic, affinely parametrized, and f is a vector orthogonal to k and parallelly propagated along it, then, in a type-D space-time, the quantity Ks = k'fJ (l,nj - /,nf - m,mj + m, m}) "V 2""3 = fc'/'(2/,n,-2m,m, -gl}YVV'3 = 2l(k'l)(f-n)-(k-m)(f-mnV^'\ (12) is conserved along the geodesic, i.e., fc'VfKs = 0. (13) A more common formulation of the theorem is that 'a type-D space-time admits a conformal Killing tensor'.
THEOREMS ON THE INTEGRALS OF GEODESIC MOTION 321 Proof. The assumptions of the theorem that k is a null geodesic, affinely parametrized, and that/is orthogonal to k and parallelly propagated along it require that fc'V,fc; = 0, fc'V,/; = 0, (14) (k ■l)(k-n)-(k-m)(k-m)^0, (15) and (k•/) (/•«) + (*•«) (/•/)-(k-m) (f-ih)-(k-m) (/-in) = 0. (16) Since k'V, {[(fc •/) (/•«)-(fc • «)(/•!»)] ^-"3} ^^^>3{kiVil(k-l)(f-n)-(k-m)(f-mn + l(k-l)(f-n)-(k-m)(f-mUkiVi\g^2113}, (17) the theorem requires us to show that fc'V,[(*-/)(/-Jl)-(*-l!l) (/•«)] = -l(k-l)(f-n)-(k-m)(f-m)WV!\g^21'3- (18) Expanding the left-hand side of equation (18) and making use of equations (14), we obtain k'V,[(* •/)(/•«)-(*•«)(/•«)] = (/-«)fc'fcJ/,;i + (*-/)fc'/Jn,., -(f-m)klkjmJ;i-(k-m)klfJmht. (19) Now substituting for /-.,-, etc., from equations (6)-(8) and simplifying, we find that we are left with fcfV,[(fc •/)(/•«)-(* -IB) (/•«)] = P[ + (/,«)(fc-«)(fc-'")-(/-«)(fc-«)(fc-'")] + T[-(/•«)(*•«)(*•/) + (/•«)(* •«)(*•/)] + /<|I-(fc-/)(fc-m)(/-ifi) + (fc-in)(fc-fn)(/-/)] + Jt[ + (fc-/)(fc-Jl)(/-IB)-(Jk-IB) (*•«)(/•/)]. (20) Considering next the right-hand side of equation (18) and making use of equations (4) and (11), we find -\_(k-l)(f-n)-(k-m)(f-m)WVAgV;113 = K (*■/)(/■«)-(*• «)(/■*)] x \_(k -n)D + (k-l)A-(k-m)5- (k-m)5*~\ lg¥2 = [(*•/) (/■«)-(*■«)(/•«)] x\_+p(k-n)-n(k-l)-T(k-ih) + n(k-m)]. (21)
322 THE GEODESICS IN THE KERR SPACE-TIME The equality (18), which the theorem asserts, requires that the terms on the right-hand side of equation (20) agree with the term in the last two lines of equation (21). We verify that this is, indeed, the case. Thus, considering the terms which occur with the spin coefficient fi as a factor on the right-hand side of equation (20), we have, by virtue of equations (15) and (16), -(k-l)(k-m)(f>m) + (k-m)(k-m)(f-l) = -(k •/)[(*•/)(/•«) + (*•«)(/•/)-(*■ «)(/■*)] + (k-m)(k •*)(/•/) = -(*./)[(*./)(/.„)-(*./w) (/•/*!)]; (22) and this expression agrees with the coefficient of fi in equation (21). The equality of the factors of the other spin coefficients in equations (20) and (21) follows in similar fashion by virtue of the same equations (15) and (16). The conservation of Ks, along a null geodesic, affinely parametrized, is thus established. THEOREM 2. If k is an affinely parametrized geodesic in a type-D space- time, then K = 2\V2\-2'3(k-l)(k-n)-Q\k\2 = 2\V2\-2l3(k-m)(k-m)-(Q-\V2\-2<3)\k\2 (23) is conserved along k if, and only if, a scalar Q exists which satisfies the equations DQ = D\V2\-2/i, Ag= A|»F2r2'3, and SQ = S*Q = 0. (24) Proof. First we observe that the equivalence of the two alternative forms of K in the enunciation of the theorem follows from equation (10). The conservation of K for geodesic motion along k requires that 2fc'Vl[(*-0(*-*)l,i'2r2/3] = l*l2*'V,fi (25) since \k\2 is conserved. By equation (11), the right-hand side of equation (25) can be rewritten in the form |*|2*'V,fi = \k\2l(k-n)DQ + (k-l)AQ -(k-m)SQ-(k-m)S*Q-\. (26) Considering the left-hand side of equation (25), we have 2fc!v,.[(fc./)(fc.«)i^2r2/3] ^2\'¥2\-2'3lk'Vi(k-l)(k-n) + (k-l)(k-n)k1V1\g\'i'2\-2/3^. (27) On the other hand, by virtue of equations (4) and the geodesic character of k,
THEOREMS ON THE INTEGRALS OF GEODESIC MOTION 323 fc'v,igi*2r2/3 = -[(k-n)(p+p*)-(k-l)(n + n*)-(k-m)(T-n*)-(k-m)(T*-n)l (28) and klV((k • /) (k •») = (k • «)k'kJ7;;i + (k • l)klkJnhi. (29) Now inserting equations (28) and (29) in equation (27) and substituting for /;;, and nj.ti from equations (6) and (7), we find that considerable simplification results and we are left with 2k'vi[(k-/)(k.«)|y2r2/3] = 2\V2\-2>3{Uk-n)(p + p*)-(k-l)(n+n*)l x[(k-m)(k-iw)-(k-/)(k-Ji)]} = \V2\-2/3\k\2l(k-l)(ti + H*)-(k->>)(P + P*n = |k|2[(k.«)D|y2r2/3 + (k./)A|y2|-2/3, (30) where, in the last step of the reductions, we have made use, once again, of equations (4). A comparison of the final result of the foregoing reduction with the right-hand side of equation (26) leads to the requirements (24) stated in the theorem. COROLLARY 1. A null geodesic, k, in any type-D space-time allows the integral of motion K0=2\*¥2\-2l3(k-m)(k'ih) = 2\*i>2\-2/3(k-l)(k-n). (31) This follows trivially from the theorem stated, since for a null geodesic | k |2 = 0 and none of conditions (24) are required for its validity. COROLLARY 2. The constant K0 of Corollary 1 is, apart from a multiplicative factor, the square of the absolute value of the complex constant Ks of Theorem 1. Proof. By equation (12), K.Kf =4|y2r2'3[(k./) (/•«)-(k-m)(f-m)-] x [(*•/) (/•«)-(*•«) (/•«)]; (32) or, alternatively, by virtue of equation (16), iKsi2 = -4i^2r2/3[(k-/)(/-«)-(k-i«) (/.«)] x [(k•«) (/•/)-(*-in) (/•«)]. (33)
324 THE GEODESICS IN THE KERR SPACE-TIME Expanding and simplifying by repeated use of the identities (15) and (16), we obtain |KS|2= -4\V2\-2l3{(k>I)(k>n)(f>n)(f>I) + (k>m)2(f'm)2 - -4\V2\-2'3{(k>m)(k>m)(f>n)(f>l) + (k>m)2(f>m)2 - (*•«)(/• IB) [(*•«)(/• IB)+ (*•«) (/. m)]} = - 4\^2\-2'3(k-m)(k- «)[(/• if) (/•/)- (/•«)(/•«)] = -41^1-^(^-^)(^-^)1/12. (34) Since/ is parallelly propagated, |/|2 remains constant along the geodesic k and the corollary stated follows. COROLLARY 3. The Kerr metric allows a conserved quantity of the form enunciated in Theorem 2. In the coordinate system adopted in Chapter 6 (cf Ch. 6, equation (180)) | ¥-, | " 2/3 = M ' 2/3p2 = M - 2/3 (r2 + fl2 CQS2 Qy (35) Accordingly, the requirements of the theorem are met by the choice Q^M~2'3r2. (36) Suppressing the factor M ~ 2/3, we conclude that in Kerr geometry K = 2p2(k-l)(k-n)-r2\k\2 and \ (37) K = 2p2 (k m) (k in) +a2 \k\2cos2 6 are integrals of geodesic motion. The two integrals (37) may be considered as independent since their equivalence derives from the constancy of \k \2 which in common parlance is equated with the conservation of the rest mass. THEOREM 3. The necessary and sufficient conditions for an integral of geodesic motion, of the form considered in Theorem 2 to exist, for a type-D space- time, are that the spin coefficients p, r, fi, and n are related in the manner p* (i* (38) Proof. The conditions obtained in Theorem 2 for the existence of an integral of the form (23), for geodesic motions in a type-D space-time, are that a scalar Q exists with the properties DQ = Df AQ = A/ and SQ = S*Q = 0, (39)
THEOREMS ON THE INTEGRALS OF GEODESIC MOTION 325 where, for convenience, we have written /=iy2r2'3. (40) In terms of/, the Bianchi identities (4) governing ¥2 become Df=-(p + p*)f; V=(n*-T)/, \ The proof of the theorem consists in applying the available commutation relations (Ch. 1, §8(c)) to Q and tracing the consequences of the requirements (39) consistently with equations (41). For a type-D space-time, compatible with the relations (1) and (3), the commutation relations are S*S-SS* = (n*-n)D + (p*-p) A + (a-/?*)<5 + (/?-a*)<5*, AD-DA = (y + y*)D-(r* + n)5 - (t + n*)6*, 3D-D5= (0L* + p-n*)D-p*5, <5 A- A<5 = (T-<x*-P)A + (ii-y-y*)8. (42) Applying these relations to Q and making use of the requirements (39), we obtain (H*-H)Df+(p*-p)Af=0, (43) (AD-DA)f=(y + y*)Df, (44) SDf=(ot.*+p-n*)Df, (45) and <5A/=(t-<x*-0)A/. (46) We now make use of the relations (41). Equation (43) directly yields the relation (H*-Ii)(p + P*) = (p*-P)(li + H*), (47) which on simplification gives PH* = p*fi or p/p* = n/n*. (48) Next, evaluating the left-hand side of equation (44), in accordance with equation (42), we obtain (T* + n)5f+(t + n*)5*f=0, (49) or, by virtue of the relations (41), (t* + n) (t - n*) + (t + n*) (t* - n) = 0. (50) On simplification, equation (50) gives tt* — 7T7r* = 0, or t/n* — n/t*. (51)
326 THE GEODESICS IN THE KERR SPACE-TIME Tracing the consequences of equations (45) and (46) is somewhat more involved. First, with the aid of equations (41), equations (45) and (46) can be reduced to give <51g(P+P*) = (<x*+/?-27t* + T) = <51gp+<51g(l+p*/P)> (52) 5lg(n+n*) = (2z-n*-<x*-P) = 5lgn + 5lg(l+n*/n). (53) Since p/p* = n/n* by equation (48), we obtain by subtracting equation (53) from equation (52) <51gp-<51gji = 2a*+2/?-7t*-T, (54) or bp-pb\g^ = (2<x.*+2fi-n*-r)p. (55) On the other hand, by the Ricci identity, Chapter 1, equation (310(k)), bp =p(x*+P+t)-tp*. (56) Eliminating bp from equation (55) with the aid of this last relation, we obtain -<5/z = (a*+0-Jt*-2T)/z + T/z*, (57) where we have, once again, made use of equation (48). But by Ricci identity, Chapter 1, equation (310(m)), -bn* = (<x*+p + n*)n*-n*n. (58) By adding equations (57) and (58), we obtain -b(n + n*) = (<x*+P-2n*-2T)n + ((x*+l]+T + n*)n*. (59) But we also have the relation (cf equation (53)) -b(n + n*)= ~(2t-n*-0L*-p)(n + n*). (60) From a comparison of equations (59) and (60), we find that tfi* — n*n = 0, or t/n* = fi/fi*. (61) Combining equations (48), (51), and (61), we obtain the relations (35) which were to be established. It appears that the relations (35) are, in fact, satisfied by most of the type-D metrics. 61. The geodesies in the equatorial plane It is clear that the geodesies in the equatorial plane can be delineated in very much the same way as we did in the Schwarzschild and in the Reissner-Nordstrom space-times: the energy and the angular-momentum integrals will suffice to reduce the problem to one of quadratures. But two essential differences must be kept in mind. First, a distinction should be made
THE GEODESICS IN THE EQUATORIAL PLANE 327 between direct and retrograde orbits whose rotations about the axis of symmetry are in the same sense or in the opposite sense to that of the black hole. And, second, the coordinate <p, like the coordinate t, is not a 'good' coordinate for describing what 'really' happens with respect to a co-moving observer: a trajectory approaching the horizon (at r+ or r_) will spiral round the black hole an infinite number of times even as it will take an infinite coordinate time t to cross the horizon; and neither will be the experience of the co-moving observer. The Lagrangian appropriate to motions in the equatorial plane (for which # = 0 and 6 = a constant = n/2) is 2<£ = 1- 2M 4aM. -t(p~—i r A (r2+a2) + - 2a2 M r and we deduce from it that the generalized momenta are given by Pt 2M\. 2aM t -\ cp = E = constant r J r Pv = 2aM. -t + and (r2 + a2)+- Pr 2a2 M cp = L = constant (62) (63) (64) (65) where we have used superior dots to denote differentiation with respect to an affine parameter t. (The constancy of pt and pv follows from the independence of the Lagrangian on t and <p which, in turn, is a manifestation of the stationary and the axisymmetric character of the Kerr geometry.) The Hamiltonian is given by 3V=pti + pv<p + prf-£e 11". 2M\ .. 2aM iz+ tcp rf2-~\r2 + a2 + 2a2M <P 2\ r ) r T 2A and from the independence of the Hamiltonian on t, we deduce that (66) 2^r= 2M_ r ■ 2aM . t + <p rz + az+- 2a2M <p- 2aM. 1 <p- = Ei— L(p — — f2 = <5j = constant (67) We may, without loss of generality, set 31 = 1 for time-like geodesies = 0 for null geodesies. (68)
328 THE GEODESICS IN THE KERR SPACE-TIME (Setting <5i = 1 for time-like geodesies requires £ to be interpreted as the specific energy or the energy per unit mass.) Solving equations (63) and (64) for ip and i, we obtain (p t = , 2M\ r 2aM ' 1 )L+ E r J r r2+a2 + 2a2M\ 2aM E L (69) (70) and inserting these solutions in the second line of equation (67), we obtain the radial equation rlfl = r2£2 + (fl£_ L)2 + (fl2£2 _ L2) _5jA> (?1) (a) The null geodesies As we have noted, <5i = 0 for null geodesies and the radial equation (71) becomes 2M , 1 (L_a£)2 r2 = E2 +^(L~aE)2 ~^(L2 ~a2E2). . (72) r r In our further considerations, it will be more convenient to distinguish the geodesies by the impact parameter D = L/E (73) rather than by L. First, we observe that geodesies with the impact parameter D = a, when L = aE, (74) play, in the present context, the same role as the radial geodesies in the Schwarzschild and in the Reissner-Nordstrom geometry. Thus, in this case, equations (69), (70), and (72) reduce to f= ±E, i = (r2 + a2)E/A, and </) = aE/A. (75) The radial coordinate is described uniformly with respect to the affine parameter while the equations governing t and <p are r2+a2 df _ dep a and ^~=±T- dr A (76) The solutions of these equations are (cf Ch. 6, equation (238)) r\ +a2 ( r \ rl+a2 / . ±t = r + ^ lg 1 lg 1 , a . ( r .\ a . ( r ±<p= lg 1 lg 1 (77)
THE GEODESICS IN THE EQUATORIAL PLANE 329 These solutions exhibit the characteristic behaviours of t and <p of tending to + oo as the horizons at r+ and r_ are approached—a fact to which we have already drawn attention. As we shall see later, the null geodesies described by the equations (76) are members of the principal null congruences that are confined to the equatorial plane. In general it is clear that we must distinguish, as in Schwarzschild's geometry, orbits with impact parameters greater than or less than a certain critical value Dc (which will in turn be different for the direct and for the retrograde orbits). For D = Dc, the geodesic equations will allow an unstable circular orbit of radius rc (say). For D > Dc, we shall have orbits of two kinds: those of the first kind which, arriving from infinity, have perihelion distances greater than rc; and those of the second kind which, having aphelion distances less than rc, terminate at the singularity at r = 0 (and 6 — n/2\). For D = Dc, the orbits of the two kinds coalesce: they both spiral, indefinitely, about the same unstable circular orbit at r = rc. For D < Dc, there are only orbits of one kind: arriving from infinity, they cross both horizons and terminate at the singularity. The equations determining the radius rc of the unstable circular 'photon- orbit' are (cf equation (72)) E2+~(L~aE)2~\(L2-a2E2) = 0 (78) rcJ rzc and ~~(L-aE)2+^(L2-a2E2) = 0. (79) ' c ' c From equation (79), we conclude that L — aE Dr — a rc = 3M- = 3M-^ . L + aE Dc + a Inserting this last relation in equation (78), we find 1 (L + aEf = E2 (Dc + af 27M2 L-aE 21M2 Dc-a ' or (Dc + af = 27 M2(DC~ a). Letting y = Dc + a, * It is a readily verifiable consequence of equations (80) and (82) that D* = 3rf + a2—a simple generalization of a result which obtains in the Schwarzschild limit, a = 0. (80) (81) (82)* (83)
330 THE GEODESICS IN THE KERR SPACE-TIME we obtain the cubic equation y3-27M2y + 54aM2 = 0. (84) We must now distinguish a > 0 and a < 0 corresponding to the direct and the retrograde orbits. For a > 0, y = -6M cos (#+120°) where cos39=a/M, Dc = y~a and rc = 3M (1 - 2a/y); and for — a = \a\ > 0, y = 6Mcos# where cos 39 = |a|/M, Dc = y+|a| and rc = 3M(1 + 2|a|/y). (85) (86) It can be directly verified that the solution for rc, expressed directly in terms of 9, is given by r, = 4Mcos23 = 2M<U+cos -cos 3 + - M (87) where the upper sign applies to retrograde orbits and the lower sign to direct orbits. From equations (85) and (86), we find for a = 0: Dc=(3 J3)M and rc = 3M, for a = M. DC = 2M for a = — M: Dc = 7M and rc = 4M (for retrograde orbits). J (88) and rc = M (for direct orbits), ^ .«„. Turning to the equations governing the orbits when the impact parameter has the critical value Dc, and the expression on the right-hand side of equation (72) allows a double root, we find that the equation can be reduced to the form u2 = ME2 (Dc - a)2 u4 (u - uc)2 (2u + uc), where 1 _, 1 Dc + a u=- and u, = — = „,, ^ -. r c rc 3M(Dc~a) (90) (91) Equation (90) can be integrated directly tq give du lE(Dc~a)jM^x=± 1 fl u2(u-uc)(2u + uc)1/2 1 = +^><-(2u+uyi2+—^-—lg V(2u + uc)-V(3uc) V(2u + uc) + V(3uc) (92)
THE GEODESICS IN THE EQUATORIAL PLANE 331 But if we wish to exhibit the orbit in the equatorial plane, we may combine it with the equation Eu2 3(a2u2^2Mu + l)uc' <M „ 2 2 o,,.. , ,— [?Dtut-2(Dc + a)u-] (93) (which follows directly from equation (69)) to obtain du Dc + a ,, „w , (u-uc)(2u + uc)112 - = -^—(a2u2-2Mu + l V- — —, (94) d<p JM y ' 3Dcuc-2{Dc+a)u' K ' or 3Dcuc-2(Dc + a)u T ~a2(Dc + a) (u — u + ) (u — u _) (u — uc) (2u + uc)112 du, (95) where u± = l/r±. The integral on the right-hand side of equation (95) is elementary and can be evaluated explicitly. The explicit expression, involving, as it does, partial fractions, is not simple and we shall not write it down. In Fig. 28, orbits derived from the solutions (92) and (95) are illustrated. They exhibit the features we have already described. The nature of the orbits, in general, can be readily visualized from the orbits with the critical impact parameters illustrated. (b) The time-like geodesies For time-like geodesies, equations (69) and (70) for <p and i remain unchanged; but equation (72) is replaced by 2M r2r2 = -A + r2£2+ (L-aE)2 - (L2 ~a2E2\ (96) where E is now to be interpreted as the energy per unit mass of the particle describing the trajectory. (i) The special case, L = aE. Time-like geodesies with L = aE, like the null geodesies with D = a, are of interest in that their behaviour as they cross the horizons is characteristic of the orbits in general. When L = aE, equation (96) becomes r2f2 = (E2-l)r2 + 2Mr-a2, (97) while the equations for (p and i are the same as for the null geodesies (cf equations (75)): </> = a£/A and i = E(r2 +a2)/A. (98)
Fig, 28. The critical null-geodesies (direct and retrograde) in the equatorial plane of a Kerr black- hole with a parameter a = 0.8. The variation of the radial coordinate r with respect to an affine parameter i is exhibited in Fig. (a) while the corresponding variations with q> of the same orbits are illustrated in Figs, (b) and (c). The radii of the unstable circular orbits and the impact parameters to which these geodesies correspond are rc = 1,811 M and Dc = 3.237 M for the direct orbit and rc = 3.819 M and Dc = 6.662 A/ for the retrograde orbit. The unit of length along the coordinate axes is M. The two horizons are shown by the dashed circles in Figs, (fc) and (c). 332
THE GEODESICS IN THE EQUATORIAL PLANE Equation (97) on integration gives [(£2-l)r2 + 2Mr-a2]1/2 333 1 T = ■ £2-l " 7(i^i) l8|[(£2 ~ 1)r2+2Mr ~fl2]1/2+r^(£2 ~l) + M 7W^) (for E2 > 1) (99) and T = [-(l-£2)r2 + 2Mr-a2]1/2 1 + M-(l-£2)r 7(T^jSm ^M2-fl2(l-£2)]^2^ (for£2<1)- (100) (The solutions, as written, are valid only for a2 < M2.) Alternatively, we may combine equations governing r and ip in the manner Ji= ±-(u-u + )(u-u_)[(£2-l)+2Mu-a2u2]1/2, d</> £ where u = 1/r and u± = l/r±, and obtain the solution 1 a(u+-u.) + 2(M-a2u + )}-lg{2£[£2£2_+2(M-a2u_)£_ -a2]1'2 (101) <P \g{2ElE2e++2(M-a2u + )i+-a2y2 + 2E2i + + 2E2S-+2(M-a2u„)} , (102) where i±^(u-u±y\ (103) An example of an orbit derived with the aid of equation (102) is illustrated in Fig. 29. (ii) The circular and associated orbits We now turn to a consideration of the radial equation (96) in general. With the reciprocal radius u( = 1/r) as the independent variable, the equation takes the form u"4u2 = ~(a2u2-2Mu+l) + E2+2M(L-aE)2u3-(L2-a2E2)u2. (104)
334 THE GEODESICS IN THE KERR SPACE-TIME Fig. 29. An example of an unbound time-like geodesic with L = aE, described in the equatorial of a Kerr black-hole with g„ = 0.8. For the orbit illustrated, E = 1.4. The unit of length along the coordinate axes is M; and the two horizons are shown by dashed circles. As in the Schwarzschild and the Reissner-Nordstrom geometries, the circular orbits play an important role in the classification of the orbits. Besides, they are useful in providing simple examples of orbits which exhibit the essential features at the same time; and this is, after all, the reason for studying the geodesies. We seek then the values of L and E which a circular orbit at some assigned radius, r = 1/u, will have. When L and E have these values, the cubic polynomial on the right-hand side of equation (104) will have a double root. The conditions for the occurrence of a double root are and (a2u2 - 2Mu + 1) + E2 + 2Mx2u3 - (x2 + 2aEx)u2 = 0 - (a2u - M) + 3Mx2u2 - (x2 + 2aEx)u = 0, where we have written x = L — aE. Equations (105) and (106) can be combined to give E2 = (l-Mu) + Mx2u3 (105) (106) (107) (108)
THE GEODESICS IN THE EQUATORIAL PLANE 335 and 2axEu = x2(3Mu-l)u-(a2u-M). (109) By eliminating E between these equations, we obtain the following quadratic equation for x: x4u2[(3Mu-l)2-4a2Mu3] -2x2u[(3Mu - l)(a2u - M) - 2a2u(Mu - 1)] + (a2u - Mf = 0. (110) Quite remarkably, the discriminant "\{b2 — 4ac)" of this equation is 4a2MA2u3 where A„ = a2u2 ~2Mu+1; (111) and the solution of equation (110) takes a particularly simple form by writing (3Mu~l)2~4a2Mui =Q + Q_, (112) where Q± = l-3Mu±2ayJ(Mui). (113) Thus, we find ^=^^^ = ^-^)- (114) On the other hand (as we may verify), K-Q+^u(aJu±jM)2. (115) The solution for x thus takes the simple form V(«fi+) It will appear presently that the upper sign in the foregoing equations applies to retrograde orbits, while the lower sign applies to direct orbits. This convention will be adhered to consistently in this section. Inserting the solution (116) for x in equation (108), we find £ = ^-[l-2Mu + aV(Mu3)]; (117) and the value of L to be associated with this value of E is L = «£ + *= + /. n {a2u2 + \±2aJ(Mu3)-\. (118) As we have explained, and as the manner of derivation makes it explicit, E and L given by equations (117) and (118) are the energy and the angular momentum, per unit mass, of a particle describing a circular orbit of reciprocal radius u. The angular velocity, £2, and the rotational velocity, v^v\ follow from
336 THE GEODESICS IN THE KERR SPACE-TIME these equations and equations (69) and (70); thus, d</> L — 2Mux (L — 2Mux)u2 ~ dT~ (r2 + a2)E-2aMxu ~ (l + a2u2)E-2aMux and „(*>> We find" and '{'^m^w^}i+aV±2aJiMu'n- ,122) We may parenthetically note here that we can recover from equation (117) the condition for the occurrence of the unstable circular null geodesic by considering the limit E-* oo, when g_ = l-3Mu + 2a^/(Mu3) = 0, (123) or, equivalently, r3'2-3Mr1'2T2ay/h4 = 0. (124) We can directly verify that the solution of this cubic equation for yjr agrees with that given in equation (87). Let L and E have the values appropriate to a circular orbit of some assigned reciprocal radius uc, i.e., L and E have the values given by equations (117) and (118) for u = uc. To avoid any ambiguity, we shall distinguish the quantities (L, E, x, etc.) evaluated for a particular uc by a subscript c. With Lc and Ec chosen in this manner, the cubic polynomial on the right-hand side of equation (104) will allow a double root at u = uc; and we find that, in consequence, the equation reduces to ., , ,T L2 — a2E2 + a2 U-4u2 = 2M(Lc-aEc)2(u-uc)2 ■■■*■■ u + 2uc-- (125) 2M(Lc-aEc)2 For Lc and Ec given by equations (117) and (118) (for u = uc), we find * We may note here the following relations which were found useful in the reductions: L-2Mux= ,\ A., (1 +a2u2)E-2aMxui = -^-[1 Ta J(Mu3)~].
THE GEODESICS IN THE EQUATORIAL PLANE 337 and L2-a2E2 + a2 A„ "c 2Mx] ~ 2(aJuc'±Mf (12?) We may accordingly rewrite equation (125) in the simpler form u2 =2Mxc2u4(u-uc)2(u-u,), (128) where U*—U< + 2{aJucljMf (129) It is clear that u„ defines the reciprocal radius of the orbit of the second kind associated with a stable circular orbit of reciprocal radius uc. The appropriate solution of equation (128) is du 1 T = J(2M) « ("-"c)("-"*) 1/2 ' (130) Alternatively, we may combine equation (128) with <p= — (Lc-2Mxcu), (131) to obtain the trajectory (Lc-2Mxcu) 1 9 ~ xca2yJ(2M) rdu. (132) (u-u + )(u- u-)(u - uc)(u - u„)1/2 Orbits of the second kind derived with the aid of equation (130) are illustrated in Fig. 30. It is clearthat the condition forthe instability of the circular orbit is u„ = uc, or by equation (129) 4uc(aJuc±jM)2 =Au< = fl2u2-2Muc+l. (133) Expanding this equation and suppressing (as unnecessary) the subscript c, we obtain the equation 3a2u2 + 6Mu±8a^f(Mu3)-l = 0; (134) or, reverting to the variable r, we have r2-6Mr + 8aV(Mr)-3a2 =0. (135) This biquadratic equation for ^/r can be reduced by standard methods to the quadratic equation r-2gVr-2MCOShfl T2fl^-M«0, (136) cosh 33 q where S^tanh-1^ and ^ = 4MC°^n sinh2fl. M cosh 3 y (137)
±T i. 0 -2 -4 t. r. 1 > direct ' i retrograde' J\ - r* \ * i i.i ±T -20- -40 - r (a) -60 i i direct retrograde i - 5 r (b) 10 Fig . 30. Examples of orbits of the second kind associated with stable circular orbits with a radii 20 M and 9.5 M. The parameters of the orbits considered are: 20 9.5 0.9780 4.7222 Direct j 0.8911 -4.9981 Retrograde J 0.9503 3.4153 Direct \ 0.9605 -4.1243 Retrograde J Fig. (a) Fig. (b) 338
THE GEODESICS IN THE EQUATORIAL PLANE 339 300 200 ±T 100 0 r Fig. 31. Marginally stable circular orbits, retrograde and direct, in a Kerr space-time with a = 0.8. The geodesic described by a particle when its energy and angular momentum are those appropriate to an unstable circular orbit can be derived from equations (130) and (132) by simply letting uc and u„ coincide. An example of such an orbit is illustrated in Fig. 31. The circular orbits one formally obtains from equations (117) and (118) for radii less than r given by equation (135) is the time-like analogues, for E2 > 1, of the unstable null geodesic considered in §(a): indeed, as we have verified, when E2 -> oo, we are led to the same limiting case. Besides the limiting case E2 -> oo, the case of the 'marginally bound' orbit with E2 = 1 is of some interest: it corresponds to the case of a particle, at rest at infinity, falling towards the black hole. By equations (108) and (116), the radius of the marginally bound circular orbit is given by \=x2u2 = -£-(aJu±jM)2, (138) or g_ = \-3Mu + 2ay/(Mu3) = u[a2u + M±2aJ{Mu)~\. (139) This last equation simplifies to the form [_au±2s/(Mu)Y= 1; (140) and we derive from this equation r = 2M±a + 2yJ(M2±aM). (141) 1 1 1 1 1 retrograde/ | r- r* 1 1 Mi i \ direct ill i i 1
340 THE GEODESICS IN THE KERR SPACE-TIME ±T -10 Fig. 32. Examples of critical marginally bound orbits (direct and retrograde) with E2 = 1 falling from rest at infinity. (In Figures 30-32, by a choice of convention, the affine parameter varies in opposite senses along the direct and the retrograde orbits.) The circular orbit of this radius is the envelope of the trajectories of particles which are at rest at infinity. An example of a marginally bound trajectory is illustrated in Fig. 32. In the foregoing discussion of the null and the time-like geodesies, we have encountered limiting circular orbits of three kinds: the unstable photon-orbit, the last stable time-like orbit, and the marginally bound orbit. The dependence of the radii of these orbits on ajM is exhibited in Fig. 33. We observe that the radii of the direct orbits, of all three kinds, tend to M as a -* M. A more careful consideration of how this limit is approached leads to behaviours shown in Table VII. The coincidence of the three limiting radii for the direct orbits as a -* M is, in fact, 'illusory' since the proper distances between them tend to non-zero limits. Thus, for a = M(l-<5) and r+ = M[l + V(2<5)] + 0(<53/2), (142) the proper distance between r = M(\ +e) and r+ is given by E2 -2(5) + 8 —T~i h lvl l& T, Jr+=M[1+^/(25)] r=M(l+e) rdr /(g<. (143) With the e's given in Table VII, we find (rph-rJproper distance -lAf lg3 [e = ^/(85/3)], (rm.b. - r + )proper distance — Mlg(l+ J2) [e = 2y/&\, ('... - r + Wr distance - M lg(25'6<r ^6) [e = J(4S)l (144)
THE GEODESICS IN THE EQUATORIAL PLANE 341 a/M Fig. 33. Radii of circular equatorial orbits around a Kerr black-hole as functions of the parameter a. Dashed and dotted curves (for direct and retrograde orbits) refer to the innermost stable (ms), innermost bound (mb), and photon (ph) orbits. Solid curves indicate the event horizon (r+) and the equatorial boundary of the ergosphere which always occurs at 2 M. The fact that these proper distances tend to non-zero limits as a -> M is a manifestation of the ambiguity in the chosen radial coordinate as a -> M. A related fact is that the energy of the last stable circular orbit does not tend to infinity for a2 -* M 2 — 0 as one might have expected from the coincidence of its radtus with that of the horizon. Thus, for a = M, the energy of a direct circular orbit with a reciprocal radius u is given by (cf. equation (117)) F la im l-2Mu + v/(Mu3) direc'1 ~M)~ \_l-3Mu + 2y/(Mui)y2' (145) Table VII The radii of the limiting circular orbits for a = 0 and a= ±M Direct a = M(t-d) Retrograde a= -M ^photon mar. bound ^stability 3M 4M 6M M[t+7(85/3)] M[l + 2^] M [1+7(45)] 4M (3 + 272)M 9M
342 THE GEODESICS IN THE KERR SPACE-TIME and this expression, for u -* M 1 + 0, tends to the limits £direc,(« = M)^±3~1'2 (u^M-'+O), (146) exhibiting a discontinuity. The limit E = 3 ~ */2 for r -> M + 0, when a = M, is important: it gives the maximum energy (per unit mass) which a stable circular orbit can have in Kerr geometry for a2 < M2. We shall return to the implications of the discontinuity exhibited by E in §66. 62. The general equations of geodesic motion and the separability of the Hamilton Jacobi equation Quite generally, geodesic motion in a stationary axisymmetric space-time will allow two integrals of motion: the energy and the angular momentum about the axis of symmetry. Besides, the norm of the four-velocity will also be conserved by virtue of its parallel propagation. These three conservation laws will not, in general, suffice to reduce the problem of solving the equations of geodesic motion to one involving quadratures only. That, nevertheless, such a reduction is in fact possible was discovered by Carter who demonstrated explicitly the separability of the Hamilton-Jacobi equation and deduced the existence of a further conserved quantity. Subsequent to Carter's discovery, Walker and Penrose showed that any type-D space-time has a conformal Killing-tensor—and, more generally, the conserved complex quantity, Ks, of Theorem 1 of §60 for null geodesies—and, further, that in Kerr geometry we have, in addition, the integral K established in Theorem 2, Corollary 3. In this section, we shall consider the reduction of equations of geodesic motion, in general, which follows from these discoveries. Starting from the general Lagrangian, „„, A 2Mr\ ., 4aMrsm26 . p2 , ... , , 2a2Mr , \ . , „ . r2 + a2 + — sin2 6 (sin2 6)q>2, (147) we deduce the energy and the angular-momentum integrals: 2Mr\. 2aMr sin2 6 P 1 2~)' + 72 <p = £ = constant (148) and 2aMrsin20 . /2 2 2a2Mr . \ pv = 2 l + [r + a + 2~ sin B )(sin G)<P = Lz = constant. (149)
THE GENERAL EQUATIONS OF GEODESIC MOTION 343 Besides, we have the integrals (cf. equations (37)) 2p2(k-l)(k-n)-r2\k\2 = K (150) and 2p2(k-m)(k-m)+(a2 cos26)\k\2 = K, (151) where /, n, m, and m are the basis null vectors. We may consider the integrals (150) and (151)as independent since it presupposes the conservation of \k\2. And, as usual, we shall set \k\2 = S1 = 1 for time-like geodesies = 0 for null geodesies. In the present context, kl = (i,tj,$). (153) Making use of our present knowledge of the null basis-vectors (Ch. 6, equation (173)), we find that in Kerr geometry the integrals (150) and (151) have the explicit forms 1 r>4 -[At-(aA sin2 0)4>]2-^-^-^^ = ^ (154) and [(a sm6)i-(r2 + a2)(sin 0)<p]2 + p402 + S^2 cos26 = K. (155) On the other hand, by equations (148) and (149)) (a sin0)t-[(r2 + a2)sin0]0 = aE sin 6 ~LZ cosecfl (156) and At -(a A sin20)q> = (r2+ a2)E-aLz. (157) Inserting these expressions in equations (154) and (155), we obtain the equations 1 o4 -[(r2 + a2)£-aLz]2-^-r2-^1r2 = K, (158) and (aE sin6- Lz cosec6)2 + p*82 + ^a2 cos2 6 = K; (159) or, alternatively, p4r2 = [(r2 + a2)£-aLz]2-A(^1r2 + K) (160) and p402= -(a£sin0-Lzcosec0)2 + (-^1a2cos20 + K). (161) Finally, equations (156) and (157) provide the complementary pair of equations p2i = j(I.2E-2aMrLz) (162) (152)
344 THE GEODESICS IN THE KERR SPACE-TIME and p2(p = \~ [2aMrE+ (p2 -2Mr)Lzcosec20]. (163) It is now clear that the problem of solving the equations of geodesic motion has been reduced to one of quadratures. An alternative form of equation (161) which we shall find useful is p40'2 = [K-(Lz-a£)2] -[a2^ -£2) + L2cosec20]cos20. (164) (a) The separability of the Hamilton-Jacobi equation and an alternative derivation of the basic equations As we have stated, the existence of a fourth quantity that is conserved along a geodesic was first discovered by Carter by explicitly demonstrating the separability of the Hamilton-Jacobi equation. At the time, it was wholly unexpected; and it suggested that the other equations of mathematical physics might be similarly separable. Indeed, they were all eventually separated as we shall see in Chapters 8, 9, and 10. As the first of the chain of remarkable properties that characterize Kerr geometry, it is useful to follow Carter's demonstration of the separability of the Hamilton-Jacobi equation. The Hamilton-Jacobi equation governing geodesic motion in a space-time with the metric tensor glj is given by dS .. dS dS 2 = g'J : r, dt dx' dx1 (165) where S denotes Hamilton's principal function. With g,J for the Kerr geometry given in Chapter 6, equation (135), equation (165) becomes dS I2 d7~^A 4 sn -i ' 4aMr dS dS A-a2sm26 [ dS p2 A dt 8cp 2 , /,o\2 p2Asin 6 \d(p dJLX p- \or J p' It is convenient to rewrite this equation in the alternative form (166) p5S :a7 1 p~2R dS dS (r2+a2)-r-+a 1 dt d<p A /3SV_J_ p2sin26 2 (asin26) dS as dt d<p (167) Assuming that the variables can be separated, we seek a solution of equation (167) of the form S = ^ t - Et + Lz(p + Sr(r) + Se(0), (168)
THE GENERAL EQUATIONS OF GEODESIC MOTION 345 where, as the notation indicates, Sr and Sg are functions only of the variable specified. For the chosen form of S, equation (167) becomes SiP^^Kr' + a^E-aL^-^i-iaEsinte-L:)2 A sin 0 dS, dr d6 (169) With the aid of the identity (a£sin20-Lz)2cosec20 = (L? cosec2 6-a2 E2) cos2 6+ (Lz-aE)2 we can rewrite equation (169) in the form (170) + dSe de + (L2cosec26-a2E2)cos26 + S1a2cos20} = 0. (171) The separability of the equation is now manifest and we infer that Al'^j =^l(r2 + a2)E-aLz¥-\_2 + (Lz-aE)2 + Sir2l (172) and dO = 2-(L2cosec28-a2E2 + d1a2)cos28. where 2 is a separation constant. With the abbreviations R^[(r2+a2)E-aLz¥-A\_2 + (Lz-aE)2 + 51r2] and 0 = ^-[a2(^1-£2) + L2cosec20]cos20, the solution for S is (173) (174) (175) S - \5xx - Et + Lzq> + y*o-) dr + dej®(6). (176) The basic equations governing the motion can be deduced from the solution (176) for the principal function S by the standard procedure of setting to zero the partial derivatives of S with respect to the different constants of the motion—2, 51, £, and Lz in this instance. Thus, we find that dS _1 d2~2 ' 1 dR , 1 7 dr + - A^/R 32 2 1 de '0 32 d0 = O (177)
346 THE GEODESICS IN THE KERR SPACE-TIME leads to the equation Similarly, we find T = 1 t==2 7* dr dr + a2 •» de 7© 0cos20 V® de, (178) (179) 1 dR 1 A^/R dE r + 2 1 d® '0 dE de and <p = = a xE + 2M 1 dR , 1 7 dr — A^/R dLz 2 r[r2£-a(Lz-a£)] 6 1 d® dr A^R' (180) '0 dL. de [(r2 + a2)£-aLz] dr + e , de (Lzcosec^0 — aE) 0 (181) where simplifications have been effected with the aid of equation (178)' It can now be verified that equations (178)-(181) are entirely equivalent to the set (160)-(163) with the identification 2. = K-(Lz-aE)2. (182) In particular, with this identification, the right-hand sides of equations (160) and (164) agree with the present definitions of R and 0 in equations (174) and (175); and the relation (178) is an immediate consequence of equations (160) and (164). Our basic equations, then, are and where and p4r2=K; p402 = 0, pip = — [2aMrE + (p2 -2Mr)Lz cosec2 0], p2i^--(L2E-2aMrLz), (183) (184) R = £2r4 + (a2£2 -L2Z - 2,)r2 + 2Mr\_2 + (Lz -aE)2] -a2£-bxr2A (185) 0 = ^+(a2E2-L2cosec2e)cos2e-61a2cos2e (186)
THE NULL GEODESICS 347 It is convenient to assemble in one place the various formulae giving the tensor and the tetrad components of the four momentum. We have 2a Mr sin2 6 and and r2 + a2 + 2a2 Mr sin2 0 (sin2 0^ = - Lz P,= 1 2Mr\ , 2aMr sin20 Pm =P„, =e-*(£-WLz) = e + y, ,«■) _ P(D= ~e ^Pr = +e + f2 p<«)=-p(6)= -e-">pe = e+i"pe, 7W = ■Pm = «V = -% = e-*L, (187) (188) 63. The null geodesies In our considerations of the general non-planar orbits, we shall concentrate on delineating the projection of the orbits on to the (r, 6) plane: the variations of t and </> along the orbits do not reveal any special features that have not already been displayed by the planar orbits on the equatorial plane. For the null geodesies 51 = 0, and it is convenient to minimize the number of parameters by letting i^LJE and q = 2. /E2, (189) and writing R and 0 in place of R/E2 and 0/£2: R^r* + (a2-i2-r])r2 + 2M\_ri + (i-a)2]r-a2ri (190) and 0 = >; + a2cos20-£2cot20. (191) The two parameters £ and t] replace the single impact parameter, D, by which we distinguished the null geodesies in the equatorial plane. The parameters £ and t] are in fact related very simply to the 'celestial coordinates' a and /? of the image as seen by an observer at infinity who receives the light ray. Making use of the expressions (188), we readily verify that a = rp {<?)' n«> = (J cosec 60 and P~(rS) - (ri +a2 cos2 60-i2cot26o)l'2^-p0o, \ P /r->oo (192)
348 THE GEODESICS IN THE KERR SPACE-TIME where 60 is the angular coordinate of the observer at infinity. Precisely, a is the apparent perpendicular distance of the image from the axis of symmetry and /? is the apparent perpendicular distance of the image from its projection on the equatorial plane. The equation governing the projection of the orbit in the (r, 0)-plane is Lt7*~h %■ where rf and 6t are certain assigned initial values of r and 6. Also, it should be noted that while the signs of ^/R and ^/0 can be chosen independently, they must be adhered to once the choice has been made. (a) The 6-motion We shall first consider the 0-motion, as specified by the integral over 0~1/2 in equation (193), since it already gives some measure of insight into the character of the orbits. Thus, the fundamental requirement, that 0 be not negative, restricts the constant £ and t] by the inequality ri + (a-t)2>0, (194) which follows from 0 written in the alternative form: 0 = n + (a - £)2 - (a sin 6 - £ cosec0)2. (195) We shall find it convenient to replace 6 by cos 6 = fi as the variable of integration and consider, instead, the integral h = f-TS- (M=cos0), (196) 0„ = ij-({2 + ij-fl2)//2-aV- (197) where Since 0„ = n when a = 0 and 6 < n/2 ~) and (198) 0^ = - £2 < 0 when n = 1 and 8=0, J it follows that we must distinguish the cases t] > 0 and t] < 0. When t] > 0, the allowed range of fi2, for 0^ ^ 0, will be between fi2 = 0 and a certain maximum /z^: 0 < n2 < i*^,ax; and when t] < 0, \i2 will be restricted to an interval 0 < n2 < /z2 < /Zj < 1 (allowing the possibility that no such interval exists). In other words, when t] > 0, the orbits will intersect the equatorial plane and oscillate symmetrically about it; and when t] < 0, the orbits cannot intersect the equatorial plane and must be confined to cones which exclude both the axis (6 = 0) and the orthogonal direction (6 = n/2). The case t] = 0 should be considered separately.
THE NULL GEODESICS 349 Explicit expressions for the integral 1^, appropriate for the cases n > 0, n — 0, and n < 0, can be readily written down: (i)i?>0: e^a^l+fKul-n2) (0 < n2 ^ n2+), where 1 2a~2 2a2 and where H2-^^2{l(i2 + ri-a2)2+4a2r,y'2 + (l;2 + r,-a2)}; k2 =n2+/(n2+ +n2-) and cosi/f = /z//z + . (ii) i, = 0: 0, = (a2 - { V - a2/ (0 < ^2 < /4ax), where fi2^=l-i2/a2 (\i\2^\a\2), and d^ 1 , 7% TW^ej sech" "max (iii)ij<0: 0„ = a2(//2-//2.)(//2-,/2) (,/2. <//2 < ,/2), where 2 1 {(M + ^-^tKM + a2-^)2-^!]1' and (199) (200) J/i- dn _ 1 F(il/,k) where k2 = (n\ ~n2-)ln\, \n\^a2, O^liKlal-VllI, and sin^ = >2+Q/2-//2-) 1/2 (201) (202) (203) (204) (205) (206) In equations (201) and (206), F(i/r, k) denotes the Jacobian elliptic integral of the first kind. (b) The principal null-congruences From equations (194) and (195) it is clear that the case i? + (a-02 = 0 (207)
350 THE GEODESICS IN THE KERR SPACE-TIME is distinguished by the fact that it is compatible with the requirement 0^ 3* 0 only for e = 60 = a constant, and £ = a sin2 60, (208) when 0^ = 0 for 6 = 60. Equation (207) now requires that ri = -a2 cos4 60. (209) Inserting the foregoing values of £ and t] in equation (190), we find that R = (r2 +a2 cos2 60)2E2, (210) where we have restored the suppressed factor E2. We now conclude from equation (183) that in this case r=±£. (211) We similarly find from equations (184) that cp = a£/A and i = (r2 + a2)E/A. (212) Accordingly, these special null geodesies are denned by dt r2 + a2^ dr „ d0 , dw a „ -=——£, ^-=±£, -,- = 0, and -p = x£. (213) dr A dr dr dr A These equations in fact define the shear-free null-congruences which we have used for constructing a null basis for a description of the Kerr space-time, in a Newman-Penrose formalism, adapted to its type-D character (cf. Ch. 6, §56). (c) The r-motion We now turn to a consideration of the radial motion as specified by the integral over R~1/2 in equation (193). As in the case of the planar orbits considered in §61, we can distinguish between the orbits which, arriving from infinity, either cross or do not cross the event horizon. In the planar case, the distinction was one of whether the impact parameter, D, was less than or greater than a certain critical value Dc, the orbit with the impact parameter Dc, itself, representing an unstable circular orbit. In the same way, in the non-planar case we are presently considering, the distinction must be one of whether the constants of the motion, £ and r], are on one side or the other of a critical locus, (£s, rjs), in the (£, >;)-plane, with the orbits, with the constants of the motion on the locus itself, representing unstable orbits with r = constant.The locus, (£s, rjs), will play the same role in the present context as the impact parameter, Dc, in the earlier context. The equations determining the unstable orbits of constant radius are R =r4+(fl2-^-i()r2 + 2M[i, + (^(i)2]r-(iVC (214)
THE NULL GEODESICS 351 and dR/dr = 4r3 + 2(a2 -£2 -ij)r+2Jt#[ij + ({ -a)2] =0. (215) Equations (214) and (215) can be combined to give 3r* + r2a2-ri(r2-a2) = r2i2 (216) and r* -a2Mr + ri(a2 -Mr) = Mr(i2 -2aZ). (217) These equations can be solved for £ and r]. Thus, eliminating t] between them, we obtain a2(r -M)i2 -2aM (r2 -a2)i -(r2 + a2)[r(r2+a2)-M(3r2-a2)] =0. (218) Quite remarkably, we find that the discriminant, "\(b2 — ac)" of this equation is a2r2A2; (219) and we find that the solution of equation (218) is given by *= t ^[M^-OtrA]. (220) a(r — M) If we choose the upper sign in the solution (220), ^(r2 + a2)/a; (221) and from equation (216), we find that the corresponding solution for r] is ij= -r4/a2. (222) With £ and r] given by equations (221) and (222), ij + (a-{)2 = 0. (223) But this relation is inconsistent with our present requirements: as we have seen in §(fc), it requires 6 to be a constant. We must, therefore, choose the lower sign in equation (220). In this manner, we find *- , ^[M^-aVA] (224) a(r — M) and (225) These equations determine, parametrically, the critical locus (£s, rjs). (See Fig. 34.) We have seen in §(a), that the character of the orbits depends, crucially, on
352 THE GEODESICS IN THE KERR SPACE-TIME Fig. 34. The locus (£s, t]s) determining the constants of the motion for three-dimensional orbits of constant radius described around a Kerr black-hole with a = 0.8. The unit of length along the abscissa is M. whether n is greater than, equal to, or less than zero. We must, accordingly, distinguish these three cases. By equation (225), n = 0 when or, equivalently, when 4a2M=r(r-3M)2 ±2ajM^r3i2-2,Mr^2. (226) (227) But this equation is the same equation (124) which determines the radii of the unstable circular photon-orbits, direct and retrograde, in the equatorial plane. Let the radii of these unstable photon-orbits be r^' and rffl. Then, it is clear that ij>0 for r<p+h>< r < r'ph' and ^=0 for r = r'pj,' and r = r(ph> n < 0 for r+ <r < r^,' and r > r^.' (228) The solutions for the 0-motion appropriate to the three cases (228) have been given in §(a). It follows from these solutions that orbits of constant radius withn < 0 are not allowed since for n2±, given by equations (205), to be positive it is necessary that O^c^a — y/\n\ and \n | < a2; and these conditions (as one can verify) cannot be met by £ and n given by equations (224) and (225).
THE NULL GEODESICS 353 It is also important to observe that for r] = 0, the orbits lie entirely in the equatorial plane. For when t] = 0, it follows from equation (216) that ^2 = 3r2 + a2;and this relation is the sameas that noted in the footnote to equation (82) since the present meaning of <!;( = Lz/a)and the former meaning of D( = L/a) are the same. Returning to the consideration of the r-motion, let the constants of the motion, £ and t], have the values £s and r]s appropriate to an orbit of constant radius rs, i.e., £s and ris have the values given by equations (224) and (225) for r = rs. For this choice of the constants of the motion, rs is a double root of R = 0; and it can be shown that R reduces to R = (r-r.)2(r2 + 2rr,-a2ti,/r2,), (229) and the integral we have to consider is dr 7i? dr (230) (i--i-.)(i-2 + 2i-i-.-a2ij./r.2)1/2' From this equation, it is apparent that the trajectory consists of two parts: a part exterior to r — rs and a part interior to r = rs. Since r]s > 0, it will terminate before r = 0; for the larger of the two roots of the equation r2 + 2rrs-aVrs2=0, (231) namely, -r.+ (r.2 + flV.2)1/2= -rs + 2V(MArs)/(rs-M), (232) is positive and, as can be shown, is less than rs provided a2 < M2. Now, letting x = (r-r.)-\ (233) and evaluating the integral (230), we obtain the equation determining the projection of the orbit on the (r, 6 = cos-1 /^)-plane. Thus, 4|-= ±^lg{2[c(l + 4rsx + cx2)]1/2 + 2cx + 4rs}, (234) where c^3r2-a2r,Jr2 (>0). (235) The integral over n on the right-hand side of the equation is given by equation (201). Examples of trajectories derived from equation (234) are illustrated in Fig. 35. While we have considered only the orbits with the constants of the motion on the critical locus, (^, r]s), it is clear how they must look when the constants of the motion are one or the other side of the locus: the two parts of the trajectory which approaches the sphere at r = rs from one side, or the other, will either separate into two disjoint parts (which we have designated as orbits
(a) (c) Fig. 35. Critical null-geodesies in the (r, 0)-plane for the three pairs of values: (a), rs = 1.85, r\s = 1.6524; (b), rs = 3.75, rjs = 4.9087; and (c), rs = 3.0 and t]s = 27.0. These orbits do not reach the singularity. The unit of length along the coordinate axes is M; and the chosen Kerr parameter a is 0.8. 354
THE NULL GEODESICS 355 Fig . 36. An example of a null geodesic with r\ = 0 in the (r, 0)-plane. These are the only orbits, not on the equatorial plane, which terminate at the singularity. The unit of length along the coordinate axes is M; and the chosen Kerr parameter a is 0.8. of the first and the second kind) or join together as a single contiguous trajectory. The case n = 0 is of some interest. As we have seen, the orbits of constant radius, r = ^,' with n = 0, lie entirely in the equatorial plane. Also, all orbits in the equatorial plane are characterized by the Carter constant being zero. But not all orbits with n = 0 lie in the equatorial plane. For, when n = 0, the r-motion is, quite generally, determined by (cf. equation (214)) J 7R }{rlr3 dr + (a2-i2)r + 2M(a-i)2^y/2' (236) while the 0-motion is determined by equation (203). The integral (236) can be reduced to a Jacobian elliptic integral; and the trajectory in the (r, 0)-plane is expressible in the form 1 7W- where £2) sech-1^-=±^-— F(iM), = (l-£2/a2)1/2, k2 = (A + B)2-r2 4AB COSl/r = A2 = r2 + (a2-i2); B2 = 3r2 + (a2 -£2), *i = -[f(a2-£2)]1/2sinhS and sinh3S = [27M 2 (A-By + r^A (A + B)r + riA' (a + i)3 1/3 (237) It is clear from the various definitions that these trajectories exist only for £ < a. For orbits in the equatorial plane, the restriction £ < a is equivalent to the restriction D ^ a (cf. the definitions of D and £ in equations (73) and (189)). The existence of three-dimensional orbits, with n = 0 and £ < a, is the
1.0- 0.5- [/ 0.5 1.0 (a) Fig. 37. Two examples of orbits with negative rj which extend into the domain of negative r: (a), t] = -0.3 and { = -0.05; and (6), r\ - -0.3 and { = -0.07. These orbits are confined between two angles, 0max and 0min. While in the (r, 0)-plane these orbits pass through zero, they do not in fact meet the singularity: in the Kerr-Schild coordinates of Fig. 25 they smoothly pass through the disc avoiding the singularity. (The chosen Kerr parameter a is 0.8.) 356
THE NULL GEODESICS 357 manifestation of the instability of the orbits in the equatorial plane with impact parameters D < a. It is also clear that all these trajectories terminate at the singularity and do not have a natural extension into the domain of negative r (see Fig. 36 where examples of trajectories derived from equation (237) are shown). Finally, we shall consider the class of orbits with n < 0. These orbits are of interest since they explore regions of negative r in a natural way and make the enlargement of the manifold to include these regions physically necessary. But the conditions for their occurrence are stringent: \r\\ < a2 and 0 < \£\ < \a\ — y/\n |. But in view of the interest in these orbits, the r- and the 0-motions were numerically integrated by Garret Toomey and the results of his integrations are shown in Fig. 37. (d) The case a = M While it has been our judgement (contrary to the commonly held view) not to give any special attention to a consideration of'what happens' when a = M, we shall briefly note the limiting forms which the formulae of the preceding section take when a = M. (We have already noticed in §61(b), in the discussion following equation (142), the inappropriateness of the coordinate r for considerations involving matters close to the event horizon when a -» M.) When a = M, equations (224) and (225) become £=^(-r2 + 2Mr+M2); ,, = -^(4M-r), (238) and we can obtain the locus n (£) explicitly. Thus, solving for r in terms of £, we obtain r(i) = M + (2M2-Mi)1/2; (239) and inserting this value in the expression for n, we obtain ^((^) = ^T[M + (2M2-M^)1/2]3[3M-(2M2-M^)1/2], (240) where it may be recalled that <x=+{cosec0o and p = (n + a2cos2 O0-Z2cot280)1/2. (241) The apparent shape of the black hole, for an observer at infinity, will be determined by the critical locus, (£, n), projected on the 'celestial sphere' even as the unstable circular photon-orbit of radius 3 M (with the impact parameter, 3M J2,) determines the shape of the Schwarzschild black-hole. In the present instance, we shall simply have to transcribe the locus (£, n) into a closed curve in the (a, /?)-plane. Thus, for an observer on the equatorial plane, viewing the
358 THE GEODESI CS IN THE KERR SPACE-TIME Fig . 38. The apparent shape of an extreme (a = M) Kerr black-hole as seen by a distant observer in the equatorial plane, if the black hole is in front of a source of illumination with an angular size larger than that of the black hole. The unit of length along the coordinate axes a and /J (defined in equation (241) is M. black hole from infinity, the apparent shape will be determined by This locus is exhibited in Fig. 38. (242) (e) The propagation of the direction of polarization along a null geodesic It is customary in general relativity to describe the propagation of light in the language of geometrical optics. The basic notion underlying this description is that of light rays defined as trajectories orthogonal to the spreading wave-fronts. Precisely, a light ray is a curve in space-time whose tangent vector along any point of it is in the direction of the average Poynting-vector. These curves, often described as the trajectories of photons, are the null geodesies of general relativity. In our considerations of the null geodesies in this and in the earlier chapters, we have adhered to this picture. Another characteristic of light, which has acquired some interest in astrophysics, in the context exactly of black holes, is that of polarization. The reason for this interest is that the radiation emerging from accretion discs surrounding black holes is expected to be polarized. And the question arises as to how one may relate the state of polarization of the radiation received by an observer at infinity with the state of polarization as it emerges from the
THE NULL GEODESICS 359 accretion disc. The question is simply answered. The state of polarization is described by the direction of the electric vector in the plane transverse to the direction of propagation, i.e., of the wave-vector It; and the electric vector, orthogonal to k, is parallelly propagated along the light ray, i.e., along the null geodesic described by k. Formulated in this fashion, it is clear that in Kerr geometry the question is answered directly by identifying the vector/, in the Walker Penrose theorem of §60, with the electric vector and taking advantage of the conservation of the complex quantity, Ks, defined in equation (12), along the null geodesic. In the Kerr space-time (cf. Ch. 6, equation 180)) y2 = -M(r-ia cos 0) ~3; (243) and the Walker Penrose conserved quantity is \_{kl){fn)-(kn){fl) -(km)(fm) + (km)(fm)](r-iacose) = K2 + iKu (say). (244) With the known expressions for the basis vectors given in Chapter 6, equation (173), we find = {k,fr-k'f,) + a(krfi>-ki>fr)sm26 = A (say), (245) and (km)(fm)-(km)(fm) = i[(r2 + a2)(/c«,/o-/co/«,)-a(/c7e-/ce/')]sin0 = 'B (say)- (246) With these definitions, the conservation law, expressed by equation (244), after multiplication by (r+iacos6), gives p2(A + iB) = (r + ia cos6) (K2 + iKx); (247) and equating the real and the imaginary parts, we obtain A=^(K2r-aK1cos6) (248) P and B = ^(K.r + aK2cosO). (249) P Besides, the orthogonality of it and/requires 2 J ^ 2 P J P .?- iff' -(^ + ^ + 2°^ sin2 e\k*f* sin2 0 = 0. (250)
360 THE GEODESICS IN THE KERR SPACE-TIME And finally, the geodesic equations (183) and (184) give p2k< = i[(r2 + a2)2 -a2Asin2 0-2aMr£], and p2k^ = -[2aMr + (p2-2Mr)£cosec20]. (251) With the components of k given by these equations, equations (248)-(250) provide three equations for determining the components of/; and the three equations will just suffice since/is determinate only modulo the null vector k. The principal interest in the foregoing equations is to relate the direction of / given by some physical theory, at some point in the neighbourhood of the black hole to the direction of/at infinity. Accordingly, it will suffice to consider the asymptotic form of the solution of equations (248)-(250). The asymptotic form of k, as one may deduce from equations (251), is given by fcr->l, fc'-»l, fce->-V, and ^-►-Icosec2^, (252) where (cf. equation (192)) 02=>; + a2cos20o-£2cot20o- (253) Since/is determinate only modulo k, there is no loss of generality in setting /' = 0. (254) The orthogonality condition (250), now, gives -r2k?fe-k'f'-(I* sin2 0o)k?F =0 (r-oo), (255) or, by equations (252), f'=-pf-tf<p. (256) Similarly, the asymptotic forms of equations (248) and (249) give r + {asin2e0)f = K2r-i (257) and [r2 (kv/O-k0f^-af0] sin 60 = K^'1. (258) Eliminating /r from equation (257) with the aid of equation (256) and substituting for ke and k^ from equations (252), we obtain -pfe -({ -a sin2 e0)/f = K2r-1 (259) and (ZcosecQ0-asmQ0)fB -(PsmQ0)f = K1r"1, (260)
THE TIME-LIKE GEODESICS 361 or, letting y = £cosec60 — asin60, we have Pfe + (y sin 60)f*= -K2r-» and yf0-(psm6o)f«> =+^,^1. Solving these equations for/6 and/'', we find 1 (261) (262) and /*sin0o = fe = r(p2 + y2) 1 (pK1+yK2) (pK.-yK,). (263) r(P2+y2V If we now denote by g ° and Sv the components of the electric vector,^, in the transverse plane which determines the state of polarization at infinity, then we may identify /"= -(rsinflo)/* and S6 = -rfe, (264) and equations (263) give 1 g<" ■ (pKi+yKi) and 1 -(pKi-yKJ. (265) /i2 + y2"""1""i' — " P2 + y We can clearly arrange that K\ + K22= 1. (266) Then, (S^g0) is a unit vector in the transverse plane which determines the state of polarization: the Stokes parameters (normalized to unit intensity), commonly used to describe the state of polarization, can be readily expressed in terms of (gv,ge). 64. The time-like geodesies For time-like geodesies, we can put <5j = 1 in equations (185) and (186); and the equation determining the projection of the geodesic on the (r, 0)-plane is, again, Cr dr _ fe d0 J 7^-J 7©' (267) but now with the definitions R = r4 + (a2 - i2 - r\)r2 + 2M[))+(^-(i)2]r- a2r\ - r2A/E2 (268) 0 = t] + a2 (1 - £-2) cos2 6-i2 cot2 0, (269) and
362 THE GEODESICS IN THE KERR SPACE-TIME where 1; = Lz/E, n = QjE2, and E is the energy per unit mass. Also, we have written R and 0 in place of R/E2 and 0/£2. (a) The 6-motion Considering first the 0-motion specified by the integral over 0~1/2, we replace, as before, 6 by fi = cos 6 as the variable of integration. The right-hand side of equation (267) then becomes f 7k- <270) where % = ti-ie + tl-a2(\-E-2)^n2-a2(\-E-2)n*. (271) It is clear that we must distinguish three cases: the bound orbits with E2 < 1, the marginally bound orbits with E2 = 1, and the unbound orbits with E2 > 1. For bound orbits with E2 < 1, it is convenient to replace a2 by a2 = a2(£T2-l)>0 (272) and write 0,, in the form From this expression for 0^ and the requirements 0^ ^ 0 and 0 =¾ \i2 =¾ 1, we conclude that negative values of n are not allowed; and that the range of fi2 is between zero and the smaller of the two roots of the equation n-(i2 + n + <x2)n2 + ot.2n* = 0 (n>0). (274) The integral (270) can now be reduced to an elliptic integral of the first kind in the usual manner. We find (^ dfi 1 (^ dn 1 Jo7^ = aJoV[(^-^)(^-^)]- = o^mfc)' (2?5) where /4 =^2{(i2 + r, + *2)±l(e + r, + z2)2-4z2r,y>2}, (276) k = n_ln+ and sini/> = ^/^_. (277) For marginally bound orbits with E2 = 1, ©„ = n-(i2 + t])n2. (278) Again, negative values of n are not allowed and
THE TIME-LIKE GEODESICS 363 For unbound orbits with E2 > 1, the discussion for the case of the null geodesies in §63(a) applies with the replacement of a2 by a2 = a2(l-£T2)>0. (280) In summary then, the bound and the marginally bound orbits must necessarily cross the equatorial plane and oscillate about it. The unbound orbits are very much like the null geodesies: orbits with n > 0 which intersect the equatorial plane, orbits with n < 0 which do not intersect the equatorial plane, are confined to cones, and extend into the regions of negative r, and orbits with n = 0. (b) The r-motion Turning next to the r-motion specified by the integral over R~112, we consider in accordance with our standard procedure, the special orbits which provide not only a basis for a classification of the orbits but also simple integrable cases which sufficiently illustrate the nature of the orbits to be expected. In the present instance, the special orbits are those for which the radial coordinate remains constant. The conditions for their occurrence are R=0 and dR/dr = 0. (281) The constraints on the constants of the motion, £, n, and E2, implied by the conditions (281), can be derived exactly as in §63(c) except that we must now allow for the additional term, — r2A/£2, in the definition of R. Following, then, the same steps, we now find that equations (216) and (217) are replaced by r2 3r4 + r2a2- n(r2 - a2) - -^-(3r2 - 4Mr + a2) = r2 £2 (282) E and r3 r4 - a2Mr + n(a2 - Mr) - -=-(r - M) = rM (£2 - 2a£); (283) E and, as before, we solve these equations for £ and n. Eliminating n from equations (282) and (283) we obtain (in place of equation (218)) a2(r- M)i2 -2aM(r2 -a2)i - \ (r2 + a2) [r{r2 + a2) - M (3r2 - a2)] - ~A2 1 = 0; (284) and the solution of this equation is (cf. equation (220)) i^K)]"*}; <«. ¢ = 1 a(r-M) M(r2-a2)±rA
364 THE GEODESICS IN THE KERR SPACE-TIME -10 Fig. 39. The (£s, t]s) locus defined by equations (285) and (286) for marginally-bound time-like orbits with E1 = 1.0. The unit of length along the abscissa is M; and the chosen Kerr parameter a is 0.8. the corresponding solution for t] is (cf. equation (225)) (see Fig. 39) r]a2(r — M) = r-M 2r3M ~ r-M' [4a2M-r(r-3M)2] + -I[r(r-2M)2-fl2M] A 1 + 1-- 1 1-- M 1/2 (286) Returning to the consideration of the r-motions, let the constants of the motion, £ and t], have the values £s and r]s appropriate to a particle with energy E, per unit mass, describing an orbit of constant radius rs, i.e., £s and jjs have the values given by equations (285) and (286) for r = rs and the assigned E2. (It is of course necessary that for the chosen value of rs and E2, ris is positive for bound and marginally bound orbits.) For this choice of the constants of the motion, rs is a double root of R = 0 and, in consequence, R can be reduced to the form R = (r-rs)2 Now, letting 1 + 2rr5 1-- 1 1-- M -^rish (287) x = (r-rs)-\ (288) the integral which we must equate with one or the other of the expressions for I: 7®» (289)
THE TIME-LIKE GEODESICS 365 (a) (c) Fig. 40. Marginally-bound, critical, time-like geodesies in the (r, 0)-plane described by equations (279) and (292) for three values of rs: (a), 2.154; (fc), 4.0; and (c), 5.44. The unit of length along the coordinate axes is M; and the chosen Kerr parameter a is 0.8. given in §(a) above is 4rJ1-fF) + ^ .2 £2 -1/2 (290)
366 THE GEODESICS IN THE KERR SPACE-TIME —an elementary integral whose values for the different cases, E2 > 0, E2 = 0, and E2 < 0, can be readily written down. Thus, with the definition F(x) = ot. + px + yx2, (291) where a, /?, and y are the coefficients of the quadratic in x which appears in the integral (290), we have \-T£=±-7:\gl2j(yF) + 2yx + ft (y > 0), 1 . , . 2yx + P = +^sinh x = T where T- D -sin- 2^±l (y>0,D> 0), (y < 0, D < 0), K-i M 2' + EJ + r? E2 (292) (293) Examples of orbits derived from the equations we have derived are illustrated in Fig. 40. As in the case of the null geodesies, orbits of constant radius r, with t] = 0, are necessarily confined to the equatorial plane. This follows from the fact that equations (282) and (283), which, for t] = 0, become 1 E2 3r2 + a2--^(3r2-4Mr + a2) = £2 and — (r-M) = M(i-a)2, (294) are the same as equations (108) and (109) which describe the circular orbits in the equatorial plane. 65. The Penrose process In the last chapter (§58(a)) we have drawn attention to the significance of the surface on which g„ vanishes and the fact that, in Kerr geometry, this surface does not coincide with the event horizon except at the poles. In the toroidal space between the two surfaces, i.e., in the ergosphere, the Killing vector d/dt becomes space-like and, likewise, the conserved component, pt, of the four- momentum. The energy of a particle in this region of space, as perceived by an observer at infinity, can be negative. This last fact has important consequences: it makes possible, as Penrose first showed, physical processes which, in effect,
THE PENROSE PROCESS 367 extract energy and angular momentum from the black hole. In this section, we shall consider the nature of these processes and the limits on the energy that can be extracted. For a consideration of the Penrose processes, it is convenient to have limits on the energy which a particle, at a specified location, can have. From equations (183) and (185) it follows that the limit is set by \_r(r2 + a2) + 2a2M]E2 -(4aMLZ)E -Lz2(r-2M)-(<V + -2/r)A = 0, (295) since this equation presupposes that there is no contribution to E from the kinetic energy derived from f2. Solving equation (295) for E and Lz, separately, we obtain £ = 1 [r(r2 + a2) + 2a2M] 2aMLz±A1/2{r2L2 + [r(r2 + a2) + 2a2M](^1r+^r"1)}1/2 (296) and L = 1 r-2M ■2aM£±A1/2[r2£2-(r-2M)(^1r + ^r"1)]1/2}, (297) where, in deriving these solutions, we have made use of the identity Ar2-4a2M2 = [r2(r2 + a2) + 2a2Mr]( 1 ). (298) The circumstances under which £, as perceived by an observer at infinity, can be negative can be inferred from equation (296). First, it is important to observe that a particle of unit mass, at rest at infinity, must, in accordance with our conventions, be assigned an energy £ = 1; and to be consistent with this requirement, we must, in the present context, choose the positive sign on the right-hand side of equation (296). With this choice of the sign, it is clearly necessary that for £ < 0, 4 < 0, (299) and 4a2M2L2> A{r2L2 + [r2(r2+a2) + 2a2Mr](<51+.2/r2)}. (300) With the aid of the identity (298), this inequality can be brought to the form [r2(r2 + a2) + 2a2MrK 1 2M L2 + A <5i + 2. <0. (301)
368 THE GEODESICS IN THE KERR SPACE-TIME It follows that E < 0, if and only if Lz < 0 and {r-2M)<-T1[51 + ± (302) We conclude that only counter-rotating particles can have negative energy; and, on the equatorial plane, it is further necessary that r < 2M, i.e., the particle be inside the ergosphere. (a) The original Penrose process The process originally conceived by Penrose to illustrate the manner in which energy may be extracted from the Kerr black-hole is the following. A particle, at rest at infinity, arrives by a geodesic in the equatorial plane, at a point r(<2M) where it has a turning point (so that f = 0). At r, it decays into two photons, one of which crosses the event horizon and is 'lost' while the other escapes to infinity. We arrange that the photon which crosses the event horizon has negative energy and the photon which escapes to infinity has an energy in excess of the particle which arrived from infinity. Let £,0)=1, L<0); E'1',^'; and £,2), L[2\ (303) denote the energies and the angular momenta of the particle arriving from infinity and of the photons which cross the event horizon and escape to infinity, respectively. Since the particle from infinity arrives at r by a time-like geodesic and has a turning point at r, its angular momentum, Lf\ can be inferred from equation (297) by setting <5j = 1, E = 1, and 2. = 0. Thus, L<0)=. \_-2aM+J(2MrA)l = <x<0) (say). (304) r — 2M Similarly, by setting St = 2L = 0 and choosing, respectively, the negative and the positive sign in equation (297), we can obtain the relations between the energies and the angular momenta of the photon which crosses the event horizon and the photon which escapes to infinity. We find L"> = (-2aM-ryA)£<1) = a<1)£<1) (say) (305) r-2M and L<2) = — (-2aM + rjA)E{2) = ot.{2)Em (say). (306) r — 2M The conservation of energy and angular momentum now requires that £(D + £(2) = £(0) = 1 (307)
THE PENROSE PROCESS 369 and L™ + £<2) = a,1)£,1) + a,2)£,2) = £<°> = a,0). (308) Solving these equations, we find a(0)_a(2) a(D_a(0) £<1, = ^a» and £<2, = ^T^' (3°9) or, substituting for a<0), a(1), and <x<2) from equations (304)-(306), we find The photon escaping to infinity has, indeed, an energy in excess of £<0) = 1 so long as r < 2M (as we have postulated). The energy, A£, that has been gained is ^-1(^-1)-■*"• (311) It is apparent from equation (311), that by the process we have described, the maximum gain in energy that can be achieved is when the particle, arriving from infinity, has a turning point at the event horizon. Therefore Since the minimum value of r+ is M (when a2 = M2) A£< 1(^/2-1) = 0.207. (313) An alternative form of the inequality (312), which we shall find instructive, is to express it in terms of the 'irreducible mass' M-m = ±ij\+a*Yi* = (Wr+)112. (314) (We shall explain in §(d) below why this mass is called 'irreducible'.) By virtue of this definition, the inequality (312) can be written in the form A more general inequality than the foregoing can be derived directly from equation (296). This equation requires [r(r2 + a2) + 2a2M~\E-2aMLz ;> 0. (316) In particular, [r+(r2++a2) + 2a2M]£-2aM£z ^0. (317) But r+ (r\ + a2) + 2a2M = 2M {r\ + a2) = 4M2r+. (318)
370 THE GEODESICS IN THE KERR SPACE-TIME Therefore, 2Mr+E-aLz^Q. (319) We shall return to this inequality in §(d) below. (b) The Wald inequality There are limits to the energy that can be extracted by the Penrose process. We shall establish two inequalities (due to Wald and to Bardeen, Press, and Teukolsky) which indicate the nature and the origin of the limitations. In the spirit of the process considered in §(a), we suppose that a particle, with a four-velocity U' and specific energy E, breaks up into fragments. Let e be the specific energy and u' the four-velocity of one of the fragments. We seek limits on 6. Choose an orthonormal tetrad-frame, e{a)', in which U' coincides with e{0)' and the remaining (space-like) basis vectors are e(a)'(a = 1, 2, 3): ¢,0,1 = U' and e(B/ (a = 1,2,3). (320) In this frame, u' = y(l/' + i/«'e(B)'), (321) where u(a) are the spatial components of the three-velocity of the fragment y = (l-MT1/2, and \v\2 = 4"h{a). (322) We now suppose that the space-time allows a time-like Killing vector 4,- = d/dx°. Let its representation in the chosen tetrad-frame be 4, = 4,0,1/, + 4,^, and {' = {«» I/'+ {<«>*,.,'. (323) From this representation of £,■, it follows that £ = 4,^ = 4,0, = 4^, = 4^, (324) and 9oo = ilii = 4,o,2 -iwiw = E2-\i\2. (325) We have thus the relation 1412 = 4(a) iix) = E2- 0oo. (326) From the representation (321) of the four-velocity of the fragment, we obtain e = 4,"' = 7(4,0, + ^4,=0) = y(E+ MI4|cos3), (327) where 3 is the angle between the three-dimensional vectors u(a) and 4,^- Making use of the relation (326), we can rewrite equation (327) in the form E = yE + y\v\(E2-g00yi2cos$. (328)
THE PENROSE PROCESS 371 This equation clearly implies the inequality y£-yM(£2-0oo)1/2 < e < yE + y\v\(E2 -g00)112. (329) This is Wald's inequality. In Kerr geometry, 0oo = 1 - 2Mr/p2 < 0 in the ergosphere; (330) and its lower bound is — 1 which it attains at the event horizon for a = M and 6 = n/2. Hence in Kerr geometry, we must under all conditions have yE-y\v\(E2 + l)1'2^e^yE + y\v\(E2 + l)U2. (331) We have seen that the maximum energy that a particle describing a stable circular orbit can have is (cf. equation (146)) £max = 3-1/2. (332) For 6 to be negative, it is therefore necessary that M>7(^TT)=i (333) In other words, the fragments must have relativistic energies before any extraction of energy by a Penrose process becomes possible. It is of interest to compare the inequality (331) with what one would have in the framework of special relativity without the advantage of an ergosphere, namely, yE-y\v\(E2 - 1)1/2 s= e < yE + y\v\(E2 - 1)1/2. (334) It is clear that no very great gain can be achieved over that possible by more conventional processes in the framework of special relativity. (c) The Bardeen-Press-Teukolsky inequality Again, in the spirit of the process considered in §(a), let two particles with specific energies, £+ and £_, following two orbits, collide at some point. We ask for the lower bound on the magnitude of the relative three-velocity, \w\, between them. Choose an orthonormal tetrad-frame, e(o/ = l/' and «<«,' (a = 1,2,3), (335) in which the two orbits cross with equal and opposite three-velocities, + u(a) and — u(a) so that |w| = txtp where M2 = »(0N«)- (336)
372 THE GEODESICS IN THE KERR SPACE-TIME The representations of the four-velocities, u + ' and u_' of the two particles in the chosen tetrad-frame, at the instant of collision, are and where u_' = y (I/'-»<•%/), y = (l-MT1/2. (337) (338) We now suppose that the space-time allows a time-like Killing vector £i( = d/dx°). Let its representation in the chosen tetrad-frame be and Then, by definition, 000 = ^ = ^,0,-^)^ = ^0,-1^, so that \i\2 = i?0)-goo- (339) (340) (341) The specific energies of the two particles at the instant of collision are given by £+=&u/ = y(£(0) + i^(a)) = y(£(o,+ Mmcos3) and E- = iiUJ ^ y(t0)-v^iM) = y(i<0)-\v\\i\cos n (342) where $ is the angle between the 3-vectors ^(a> and u(a). From the foregoing equations, it follows that 7^,o, = i(£++£-) and y|o||{|cosS = i(£+-£_). (343) Accordingly, (£+-£_)2 = 4y2M2|£|2cos2S = \v\2 (4y2if0)-4y2 g00) cos2» = M2[(£++£_)2-4y20oo]cos2£>, (344) where we have made use of the relations (341) and (343). Equation (344) clearly implies the inequality (£+ -£_)2 < M2[(£+ +£_)2-4y20oo]; or, remembering the definition of y, we have (345) (£++£_)2- 1 — l"l r0oo ^(£+-£_)2, (346)
THE PENROSE PROCESS 373 or, alternatively, -\v\4(E++E.)2 + 2\v\2(E2+ + E2.-2g00)-(E+-E.)2^0. (347) With the equality sign, the roots of equation (347) are iv If ,2i:V(£2+-goo)±V(£2--goo)]2. (348) We conclude that M >E+\E lV(£2+-goo)-V(£2--goo)l; (349) and the required lower bound on \w\ follows from equation (336); and the resulting inequality is that of Bardeen, Press, and Teukolsky. If the particle with the energy £+ is describing a stable circular orbit in the equatorial plane of a Kerr black-hole, then its maximum energy (as we have seen) is 3"1/2. With the most favourable values, g00 = — 1 and £_ = 0, the inequality (349) gives m> :7(1+^)-1:73 = 2-^3; (350) and the corresponding inequality for \w\ is M^i, (351) in agreement with the result (333) derived from Wald's inequality. The principal conclusion to be drawn from the Wald and the Bardeen-Press-Teukolsky inequalities is that to achieve substantial energy extraction by the Penrose process, one must first accelerate the particle fragments to more than half the speed of light by hydrodynamical or other forces. The specific example considered in §(a) confirms this conclusion. (d) The reversible extraction of energy We return to the inequality (319) derived in §(a). The inequality becomes an equality only if the process considered takes place at the event horizon. In general, the gain in energy 5M ( = £) and the gain in the angular momentum 3 J (= Lz), resulting from a particle with negative energy, — £, and an angular momentum, — Lz, arriving at the event horizon, are subject to the inequality 2r+MSM ^a3J. (352) If we suppose that the process takes place 'adiabatically' so that the black hole evolves along the Kerr sequence, then SJ = S(aM) = MSa + adM, (353)
374 THE GEODESICS IN THE KERR SPACE-TIME and the inequality (352) gives (2Mr+ -a2)SM = r\5M > MaSa. (354) Now, by the definition of the irreducible mass, Min, in equation (314), Mj2rr=iMr+ =|M[M + V(M2-a2)]. (355) From this equation it follows that 6M?" = 2J(M2 -a2)(r2+6M ~ Ma6a)- (356) Therefore, the inequality (354) is equivalent to the restriction SM?„>0. (357) In other words, by no continuous process can the irreducible mass of a black hole be decreased—a result which justifies the designation. Since (cf. Ch. 6, equation (278)) Surface area = 4jr(r2 + a2) = 16nM?TT, (358) we may also say that by no continuous process can the surface area of a black hole be decreased. The more general assertion, that "no interaction, whatsoever, among black holes can result in a decrease of their total surface area" is the area theorem of Hawking. The considerations we have set out, however, apply only to infinitesimal processes involving a single Kerr black-hole. Since the best that can be achieved is to keep the irreducible mass unchanged, processes in which it remains constant are said to be reversible. It may also be noted that by virtue of the definition (355), we have the relation M2 = M;2rr+J2/4M2r. (359) Since Mm is irreducible, we may interpret the second term, J 2/4M2TT, as the contribution of the rotational kinetic energy to the square of the inertial mass of the black hole; and that it is this rotational energy that is being extracted by the Penrose processes. A further result of some general interest follows from considering the effect of Penrose processes on the extreme Kerr black-hole with a = Mandr+ = M. In this case, the inequality (354) gives MSM^aSa or S(M2-a2)^0. (360) In other words, the black hole evolves down the Kerr sequence to lower values ofa2/M2 and away from equality. Thus, by infinitesimal processes of the kind we have been considering, a naked singularity with a2 > M2 cannot be created. The question whether a naked singularity can be created by non-linear 'explosive' processes is an open one. The conjecture, that no naked singularity
GEODESICS FOR a2 > M1 375 can ever be created when there are none, is the cosmic-censorship hypothesis of Penrose. 66. Geodesies for a2 > M2 Because our primary interest is in black holes, we have restricted our consideration to the Kerr space-times with a2 < M2 since, only then, does the space-time allow an event horizon and represent a black hole. If a2 > M 2, the horizons cease to exist and only the ring singularity remains. What remains, in fact, is a naked singularity. Naked singularities are believed not to exist in nature: the hypothesis of cosmic censorship forbids their coming into being in the natural course of events. Nevertheless, considerable interest attaches to knowing the sort of things space-times with naked singularities are and whether there are any essential differences in the manifestations of space- times with naked singularities and space-times with singularities concealed behind event horizons. A contrastive study of the geodesies in Kerr space- times with a2 < M2 and a2 > M2 may contribute a measure of understanding to what these differences may be. (a) The null geodesies First, we observe that there are some essential features of the null geodesies which depend on whether a2 < M 2 or a2 > M 2. Thus, we have seen in §61(a) that when a2 < M2, the null geodesies in the equatorial plane are distinguished by the critical impact parameters Dc+ (the upper sign referring to retrograde orbits and the lower sign to direct orbits): orbits arriving from infinity with impact parameters D > Dc+ recede once again to infinity after perihelion passages while orbits with impact parameters D < Dc+ terminate at the singularity. But when a2 > M2, such a distinction exists only for retrograde orbits: all direct orbits, arriving from infinity with D > a, recede again into infinity. For the distinction depends upon the existence, or otherwise, of unstable circular photon-orbits. The radii of these unstable circular photon-orbits are determined by the real positive roots, y/r, of the cubic equation (cf. equation (124)) r3/2Q_ =r>/2_3Mri/2 + 2aM1/2 = 0; (361) and for a2 > M2, this equation with the positive sign (corresponding to direct orbits) does not allow any real positive root. However, with the negative sign (corresponding to retrograde orbits) equation (361) allows the root (cf. equation (87)) V'ph.' = (2 JM) cosh $ or r$ = 4M cosh2 $ where [ (362) cosh 3£ = \a\/M.
376 THE GEODESICS IN THE KERR SPACE-TIME The corresponding impact parameter, associated with this retrograde, unstable, circular orbit is D<-) = M(6cosh £ + cosh3» ). (363) The discussion of the non-planar orbits of constant radius (rs) in §63(a) applies equally for a2 > M2. In particular, orbits with t] > 0 (for rs < r^>) cross the equatorial plane and avoid the singularity while the orbits with t] < 0 (which includes retrograde orbits with rs > rj,^' and all direct orbits) are confined to cones (not including the directions 0 = 0 and 6 = n/2) and have turning points for some negative r. (The case t] = 0 is possible only for the retrograde orbit with rs = r^'.) However, in the context of the orbits with t] < 0, there is one important difference. When a2 < M2, the orbits which have turning points in the domain of negative r cannot return to the world which they left: the interposition of the horizons will necessitate their migration to other worlds. But when a2 > M 2, there are no interposing horizons; and the trajectories which have turning points in the domain of negative r can return to the realm of positive r and recount their 'wondrous tales of travel' (cf. §(c) below). (b) The time-like geodesies Some interesting features which bear on the transition across the 'barrier' a2 = M2 emerge (as first shown by de Felice) from a consideration of the expression (cf. equation (117)) E+ =—}~[\-2Mu + aJ{Mu)3\ (364) which gives the energies of the direct (+) and the retrograde (—) circular orbits with the assigned reciprocal radius u( = 1/r). In the present context, we are interested only in the direct orbits: the indirect orbits are restricted by the requirement <2- ^ 0, i.e., by the radius of the retrograde circular photon-orbit (which, as we have seen, exists for all a2) while there is no such restriction for the direct orbits. Restricting ourselves, then, to the direct orbits, we have already noted that £< + ) for a = M has a discontinuity at r = M ± 0; thus (cf. equation (146)) £( + )-,+3-1/2 (r-M±0). (365) While the discontinuity is a consequence of the ambiguity of the chosen radial coordinate r at r = M for a2 = M 2, we are here concerned with the behaviour of £< + ) for r < M. By equation (364), £< + ) will have a zero, if the equation r3l2-2Mrll2 + aMU2 = 0 (366)
GEODESICS FOR a1 > M2 311 allows a real positive root for y/r. For a = M equation (366) allows two such roots: Jr = jM and ^ = ^(^5-1)^, (367) or r = M and r = £(3 -^5) M = 0.38197 M. (368) Therefore, £< + ) is negative in the interval i(3-,/5)M <r<M (£, + )<0). (369) In other words, stable circular orbits of negative energy exist for a2 = M 2 in the interval (369). Such stable circular orbits of negative energy continue to exist for a2 > M2 but only in the limited range (32/27)2M2> a2 > M2. (370) This follows from the fact that equation (366), for a2/M2 = 32/27, namely, r3/2 _ 2Mr112 + (32/27)1/2 M3/2 = 0, (371) allows two coincident roots at Vr = V(2M/3) or r = |M; (372) and no real positive root for \a\ > (32/27)1/2M = 1.088M. The discussion of the non-planar time-like geodesies in §64 applies equally for a2 > M2. And again, the orbits associated with n < 0 are of special interest since they return to the domain of positive r after visiting the domain of negative r—a matter to which we return presently in §(c) below. (c) Violation of causality One of the important features of space-times with singularities is that one may have domains in which causality may be violated in the sense that one may have closed time-like curves which will permit one's future to influence one's past—a feature one would describe as unphysical. In the Kerr space-time, extended to allow r to be negative, one can certainly violate causality in the domain in which gv<l> > 0 and q> become time-like. The boundary of the domain in which <p is time-like is denned by the equation I2 = (r2 + a2)2-a2Asin20 = (r2 + a2) (r2 + a2 cos2 6) + 2Ma2r sin2 0 = 0 (373) —an equation which can be satisfied only if r were allowed to be negative. Letting x = — r, to avoid ambiguity, we can rewrite equation (373) in the form fx2 + a2)2 ■it L 2_,_,., ,=sin26 (x=-r). (374) a2(xz + a^ + 2Mx)
378 THE GEODESICS IN THE KERR SPACE-TIME This equation clearly requires that (x2 + a2)2 < 1; (375) a2(x2+a2+2Mx) and this inequality bounds the range of x: 0 < x < xmm (376) where 2a .,/1., . 3M , \ -Vsinhf-sinh-1— V3j, (377) is the positive root of the equation x3 + a2x-2Ma2 = 0. (378) In particular, 0 < x < M for a2 = M2 and 0 <x< 1.2878 M for a2 = 3M2. (379) Similarly 6 is also bounded: sin20min<sin20s= 1, (380) where (x2 + a2)2 sin20min = min^7-2 2 , .,.-. (381) a1 (xz + a + 2Mx) The minimum of the expression on the right-hand side occurs at x = zM where z is the positive root of the equation (z + 1)3 -(3- a2/M 2)(z + 1) + 2(1 - a2/M 2) = 0. (382) We find, for example, that z = ^2-1, sin20min = 4(3-2V2), 6min = 55.75° (a = Jiff and (383) z =^4-1, sin20min= 0.9080, 0min = 65.25° (a = M ^3). Thus, equation (374), which defines the boundary of the region in which causality can be violated, provides rather severe bounds on the ranges of — r and0. While regions in which q> is time-like exist for all values of a2 > 0, for a2 < M 2, these regions are incommunicable to the space outside and we can afford to take a nonchalant view. But we cannot afford to be indifferent when a2 > M2 and no event horizon interposes and the region in which <p is timelike is communicable to the space outside. A question of some interest in this connection is whether causality (in some sense) can be violated by travelling
BIBLIOGRAPHICAL NOTES 379 along time-like (or null) geodesies. If such violations occur, they can only be along unbound geodesies with r] < 0 and which have turning points in the domain of negative r. Since (cf. equations (179) and (180)) fr r2 Cecos26 fr dr t = \ 7*dr + a2\ VeTde + 2M\ r[r2-^-fl)]A7i<' (384) It follows that for violation of causality, the negative contribution by the last integral derived from the part of the geodesic in the domain of negative r (which we can maximize by arranging that a(£ — a) is negative and the turning point occurs in the region in which <p is time-like) suffices to compensate for the positive contribution by all three integrals from the parts of the geodesic in the domain of positive r. The results of the calculation are inconclusive. BIBLIOGRAPHICAL NOTES It was Carter's discovery of the separability of the Hamilton-Jacobi equation that made possible a complete analytical discussion of the geodesies in the Kerr space-time: 1. B. Carter, Phys. Rev., 174, 1559-71 (1968). See also: 2. B. Carter, Commun. Math. Phys., 10, 280-310 (1968). §60. In presenting the subject, we have inverted the historical order, and preferred to begin with an account of the integral for null-geodesic motion (allowed by any type-D space-time) discovered by Walker and Penrose: 3. M. Walker and R. Penrose, Commun. Math. Phys., 18, 265-74 (1970). Walker and Penrose also derived the integral (equivalent to Carter's) for time-like geodesic motion for the particular case of the Kerr metric. The treatment in the text differs from the one adopted by Walker and Penrose and avoids the machinery of the spinor formalism. It is more direct and perhaps 'pedestrian'. However, it has the advantage of yielding simple necessary and sufficient conditions, in terms of the spin-coefficients, for a type-D space-time to allow Carter- type integrals for time-like geodesic motions. The proof of Theorem 1 was devised in collaboration with B. Xanthopoulos. It appears that the conditions equivalent to those established in Theorem 3 are included in the following papers by Hauser and Malhiot (though the author has been unable to unravel their notation): 4. I. Hauser and R. J. Malhiot, J. Math. Phys., 16, 150-2 (1975). 5. , ibid., 1625-9 (1975). 6. , ibid., 17, 1306-12 (1976). §61. The geodesies in the equatorial plane of the Kerr black-hole have been discussed extensively in the literature. For a complete listing see: 7. N. A. Sharp, General Relativity and Gravitation, 10, 659-70, (1979). In writing the account the author has consulted the following papers: 8. F. de Felice, 11 Nuovo Cim., 57 B, 351-88 (1968). 9. J. M. Bardeen, W. H. Press, and S. A. Teukolsky, Astrophys. J., 178, 347-69 (1972). 10. , ibid., 161, 103-9 (1970).
380 THE GEODESICS IN THE KERR SPACE-TIME 11. , in Black Holes, 241-89, edited by C. DeWitt and B. S. DeWitt, Gordon and Breach Science Publishers, New York, 1973. The common procedure in treating the geodesies in the Kerr space-time is to start with the general equations of motion (as they are derived in §62(a)) and then specialize them appropriately for the orbits in the equatorial plane. For the delineation of the geodesies in the equatorial plane we do not, of course, require the general equations of motion: they can be treated, ab initio, as they are in the text. The illustrations of the geodesies in this and in the following sections were prepared by Mr. Garret Toomey to whom the author is again indebted. §62. For readable accounts see: 12. C. W. Misner, K. S. Thorne, and J. W. Wheeler, Gravitation, W. H. Freeman and Co., San Francisco, 1970. 13. J. Stewart and M. Walker, Springer Tracts in Modern Physics, 69,69-115(1973). §§63 and 64. Among the many papers devoted to the geodesies in the Kerr space-time, the author found the following the most useful: 14. J. M. Bardeen in Black Holes, 215-39, edited by C. DeWitt and B. S. DeWitt, Gordon and Breach Science Publishers, New York, 1973. 15. F. de Felice and M. Calvani, II Nuovo Cim., 10 B, 447-58 (1972). 16. M. Calvani and F. de Felice, General Relativity and Gravitation, 9, 889-902 (1978). See also: 17. R. H. Boyer and R. W. Lindquist, J. Math. Phys., 8, 265-81 (1967). 18. D. C. Wilkins, Phys. Rev. D, 5, 814-22 (1972). 19. C. T. Cunningham and J. M. Bardeen, Astrophys. J., 183, 237-64, (1973). 20. , ibid., 202, 788-802 (1975). 21. , ibid., 208, 534^9 (1976). The first application of the complex integral of Walker and Penrose to the problem of tracing the changing direction of polarizations along null geodesies is due to Stark and Connors: 22. P. A. Connors and R. F. Stark, Nature, 269, 128-9 (1977). See also: 23. R. F. Stark and P. A. Connors, Nature, 266, 429-30 (1977). 24. P. A. Connors, T. Piran, and R. F. Stark, Astrophys. J., 235, 224^*4 (1980). The treatment in §63(e) is an amplification of that sketched in reference 22. §65. The existence of physical processes which, in effect, extract the rotational energy of a Kerr black-hole, was first demonstrated by: 25. R. Penrose and R. M. Floyd, Nature Phys. Sci., 229, 177-9 (1971). In this paper the 'area theorem' is also stated, albeit tentatively. More detailed considerations relative to the process conceived by Penrose are those of: 26. D. Christodoulou, 'Investigations in gravitational collapse and the physics of black holes', Ph.D. dissertation, Princeton University, Princeton, N. J., 1971; also, Phys. Rev. Lett., 25, 1596-7 (1970). 27. and R. Ruffini, Phys. Rev. D, 4, 3552-5 (1971). The notion of 'irreducible mass' is introduced in reference 26. Limits to the energy that may be extracted by Penrose processes were derived by Bardeen, Press, and Teukolsky in reference 9 and also by Wald: 28. R. M. Wald, Astrophys. J., 191, 231-3 (1974). 29. , Ann. Phys., 82, 548-56 (1974).
BIBLIOGRAPHICAL NOTES 381 See also: 30. T. Piran and J. Shaham, Phys. Rev. D, 16, 1615—35 (1977). §66. The geodesies in the Kerr space-time for a2 > M2 and the possibility of causality violations in these space-times have been considered most persistently by de Felice and his associates: 31. F. de Felice, Astron. Astrophys., 34, 15-19 (1974). 32. , ibid., 45, 65-8 (1975). 33. , M. Calvani, and L. Nobili, 11 Nuovo dm., 26 B, 1-15 (1975). 34. M. Calvani, F. de Felice, B. Muchotrzeb, and F. Salmistraro, General Relativity and Gravitation, 9, 155-63 (1977). 35. F. de Felice, Nature, 273, 429-31 (1978). 36. and M. Calvani, General Relativity and Gravitation, 10, 335^13 (1979). In the text we have attempted to summarize the principal results in the foregoing papers.
8 ELECTROMAGNETIC WAVES IN KERR GEOMETRY 67. Introduction As our studies in the contexts of the Schwarzschild and the Reissner-Nordstrom black-holes have shown, our understanding of the space- times is deepened and enriched by analyzing the manner of their reaction to external perturbations. Since external perturbations can, in general, be represented by incident waves of different sorts, their study, in essence, reduces to one of the propagation of waves of the different sorts in the space-times of the black holes and of how, in particular, they are scattered and absorbed. While the problem, as thus formulated, has a physical base, its solution reveals an analytic richness of the space-times themselves for which one is unprepared. In the case of the Kerr space-time, this richness becomes manifest almost at the outset. Since the space-time of the Kerr black-hole is stationary and axisymmetric, one naturally expects to express a general perturbation as a superposition of waves of different frequencies (o*-)* and of different periods (2mn, m = 0,1, 2, . . .) in <p. In other words, one expects to analyze the perturbation as a superposition of different modes with a time- and a (^-dependence given by where m is an integer, positive, negative, or zero. But one does not expect that the dependence of the amplitudes of these waves (with the foregoing t- and q>- dependence) on the remaining two variables, r and 6, can be separated as well. But Teukolsky showed in 1971 that this further separation can, in fact, be achieved if the equations of the massless particles—the photons, the gravitons, and the two-component neutrinos—are written in a Newman-Penrose formalism with the basis vectors chosen as in Chapter 6, §56, And as it turned out, this unexpected separability of the basic equations of mathematical physics was a sort of 'open Sesame' for unlocking further doors. In this chapter, we begin our study of the perturbations of the Kerr black- hole with Maxwell's equations and the propagation of electromagnetic waves. The theory is sufficiently simple that a complete and a self-contained account can be given; and it provides a prototype for the study of other fields. * In this and the following chapter, we shall denote the frequency by o* (instead of the customary a) in order to distinguish it from the spin-coefficient a.
DEFINITIONS AND LEMMAS 383 68. Definitions and lemmas Consistent with the formulation of the perturbation problem as we have sketched in §67 and with our treatment of the same problem in the contexts of the Schwarzschild and the Reissner-Nordstrom black-holes, we shall suppose that the perturbations can be analyzed into modes having a time- and a (^-dependence specified in equation (1). The common factor (1), in all quantities describing the perturbation, will be suppressed; and the symbols representing them will be their amplitudes. The basis vectors (l,n,m,m) given in Chapter 6, equations (170)-(173), when applied as tangent vectors to the functions with a t- and a (^-dependence specified in equation (1), become the derivative operators where and / = D = @0, «= A= -^-2 »S 3 = —T-jSfJ, and in = 5* = PV2 p\/2 ^0 M _t . iK „ r-M 2>l = dr-— + 2n—-~, A A yn = dg + Q + ncote, Set = de-Q+ n cot 8, K = (r2 + a2)&- + am, Q = actsinO+ mcosec6, r+iacos6, p* = r — iacos6, p2 = r2 + a2 cos26. (2) (3) (4) (5) It will be noticed that while 3in and 9)\ are purely radial operators, S£n and JSfJ are purely angular operators. Also, it is clear that when applied to 'background' quantities (independent of t and q>) the operators 2 and 5£ reduce to dr and de, respectively. The differential operators we have denned satisfy a number of elementary identities which we shall have occasion to use frequently in our subsequent analysis in this and in the following chapters. On this account, we shall collect them here as a series of lemmas (cf. Ch. 4, equations (227)-(229)). LEMMA 1. ^,(0)= -^: (n-6), @i = (@„)*, (sin 6)<en+l=£e„ sin 6, (sin 6)<el+1= <£ Jsin 6, A^n+1=^nA, A®I+1 = stA. (6)
384 ELECTROMAGNETIC WAVES IN KERR GEOMETRY LEMMA 2. m \ ( ,„ imasin6\ /„ ima sin0\/„ m \ 'v)r+-?H"r+-iH('v} ,7) w/iere 3> can be any Qsn or ®\. and if any S£n or if nt and m is a constant (generally an integer, positive or negative). LEMMA 3. J?n + 1 J?n + 2 . . . ifn + m (/cos 0) = (cos 0)ifn + 1 . . . J?n+mf -(msin0)ifn + 2...ifn+m/, (8) where f is any smooth function of 6 and the if„'s can be replaced by if^'s. LEMMA 4. Iff(9) and g (9) are any two smooth functions of 6, in the interval, 0 ^ 6 < n, then <7 (if J") sin 0d0 = Jo /(iftn + 10)sin0d0. (9) And finally, we may note two elementary identities which play crucial roles in many simplifications: <2(, + {2cot0 = 2ao<-cos0 and K -aQsinO = pV. (10) 69. Maxwell's equations: their reduction and their separability Maxwell's equations, appropriate to Kerr geometry, can be obtained by inserting in Chapter 1, equations (330)-(333), the spin coefficients listed in Chapter 6, equations (175) and the directional derivatives denned in equations (2). They are 1 ( rn ia sin 0\ ± ( 2 \ ± 1 / 2ia sin 6 \ , / 1 \ 1 ( _+ . iasin0\ , A ( „+ .2 \ , (11) 1 / , 2iasin0\ J A ( + 1 1 / + iasin0\ A / t 2 \ These equations take simpler and more symmetric forms when they are written in terms of the variables 4>o = 4>o, *i=*iP*JX and <t>2 = 2<p2{p*)2. (12)
MAXWELL'S EQUATIONS: REDUCTION AND SEPARABILITY 385 We find ^-^1¾ ^0 + p" ia sin 6 ¢1 = 'o + ^)*i. ,. 1*2. ^-T^-^V^ t ia sin 0\ >r I )*o- (13) (14) (15) (16) (a) 77ie reduction and the separability of the equations for $0 and <b2 It is evident that the commutativity of the operators (@0+l/p*) and (J?l+ia sin 6/p*) (by Lemma 2 of §68) enables us to eliminate ¢, from equations (13) and (16) and obtain a decoupled equation for <b0. We thus obtain *j+^X*.-^M».+£ i ¢0 = 0. (17) Similarly, the commutativity of the operators (JS?0 + ia sin 6/p*) and A(^£ + 1/p*) enables us to eliminate $j from equations (14) and (15) and obtain a decoupled equation for <t>2. Thus, ^0 + ia sin 9 + ia sin 6 \ . ( * 1 1 <D2 =0. (18) Equation (17) can be simplified with the aid of the readily verifiable identities, and the further relation K - aQ sin 6 = pV. We thus obtain [A®, 3)\ + S£l J&?! - 2io*-(r + ia cos 0)] <D0 = 0. (19) (20) (21)
386 ELECTROMAGNETIC WAVES IN KERR GEOMETRY A similar reduction of equation (18) yields \_A2>1 2>0 + £e0¥\ + 2Ut(r + ia cos 0)] <D2 = 0. (22) Equations (21) and (22) are clearly separable. Thus, with the substitutions <Do = K+1(r)S+1(0) and ¢, = ^(^,(0), (23) where R± [ (r)andS± [ (0) are, respectively, functions of rand 0 only, equations (21) and (22) separate to give the two pairs of equations, (A9x®l-2i&r)R+x =/./?+,, (24) (iflSei + 2ad'cos6)S+x = -IS+1; (25) and (A^0t^0 + 2io*-r)i?_1 =;/?_!, (26) (^o^1t-2ao*-cos0)S_, = -«_,, (27) where /. is a separation constant. (We are using I in place of our customary X to distinguish it from the spin-coefficient A.) It will be noticed that we have not distinguished the separation constants that derive from equations (21) and (22). The reason is the same as in the earlier context of Chapter 4, equations (248) and (250), which we have explained. Equations equivalent to (24)-(27) were first derived by Teukolsky. We shall call them Teukolsky's equations. Also, we may note that a comparison of equation (26) with equation (24), rewritten in the form (A@02)Z-2i<?r)AR+1=lAR+1 (28) shows that R-i and AR+1 satisfy complex-conjugate equations even as S+ j (0) and S_ ! (0) satisfy a pair of equations, one of which can be obtained from the other by replacing 0 by n — 0. 70. The Teukolsky-Starobinsky identities The decoupling of equations (13)-(16) to provide a pair of independent separable equations for $0 and $2 solves the problem only partially: for, apart from the fact the solution for ^>1 is yet to be found, the relative normalization of the solutions $0 and $2 also remains to be resolved. The resolution of the latter problem is, in some ways, the more fundamental: the solution to any linear-perturbation problem must be determinate apart from a single constant of proportionality. In the present context, the determinacy requires that the solution for $2 (for example) can have no arbitrariness—not even to the extent of a constant of proportionality—if the amplitude of the solution for <J>0 has been specified, and conversely. With $0 and $2 expressed as in equations (23), the solutions for S+! (6) and S_ ! (6) can be made unambiguous by requiring
THE TEUKOLSKY-STAROBINSKY IDENTITIES 387 that they both be normalized to unity: S2+1sm6d6=l S*1sin0d0=l. (29) But this will leave the relative normalization of the radical functions R+ j and R-1 unreso/ved. The completion of the solution of Maxwell's equations requires, as we shall see, a careful examination of equations (24)-(28) and the relationships among the solutions ./?+1 and /?_l5 and S+1 and S_l5 which the same equations imply. Such an examination by Starobinsky and Teukolsky led to some very remarkable identities which are basic to the entire theory. We shall give an account of these identities (though not in the forms in which they were originally formulated). THEOREM 1. AQ>aQ>0R_x is a constant multiple of AR+l and A^2®jA./?+1 is a constant multiple of R^x. Proof. Applying the operator 3>0 3>0 to equation (26) satisfied by R- l5 we find, successively, *0o®o*-i = 3>03)0 (ASilSioR.i) + 2i&®0®0(rR_x) = 2020(A20 -2iK)20R_ 1 + 2i&r3)0Q)QR_X + 4i&20R-x = 20A2l 20 20R. i -2i20{K@0 3)0R. x + 2r&20R_ x) + 2i<?r2i03)0R_1 + 4i<?3)0R _! = 2)0 (AS)} + 2iK)2i0 2i0R- x - 2i2i0 (K Q>aQ>aR-X)- 2i&r20 20R- x = (A3il3il-2iar)3i03i0R_1. (30) Therefore, 3>0 £%0R- i satisfies the same equation as R+x and the first part of the theorem stated follows. At the same time, the result of the foregoing reductions is equivalent to establishing the validity of the identity 2i0 2>0 (A2>1 2i0 + 2iar) = (A3>x Q>\ - 2iar) 3>0 3>0. (31) The complex conjugate of this identity, namely, 2>l 2>l(A2>0®l-2ior) = (A3>\ 3>x + 2i<jr)3>l 3>l, (32) establishes the second part of the theorem. COROLLARY. By a suitable choice of the relative normalization of the functions AR +1 and R_1 we can arrange that ASi0Si0R.1 = ^AR+1 (33)
388 ELECTROMAGNETIC WAVES IN KERR GEOMETRY and A®J®jAR+I =#*/?_,, (34) where <& is a constant (which can be complex*). We can clearly arrange for the relative normalization as prescribed since AR + , and R- , satisfy complex-conjugate equations. THEOREM 2. If the relative normalization of the functions AR + , and R-iis so arranged that equations (33) and (34) are valid, then the square of the modulus of ^ is given by \<€\2 = }} -4aV2 where a2 = a1 + (am/(f). (35) Proof. Applying the operator AQi%Qi\ to equation (33), we obtain A®j0jA®o®oR-i = *A®J®jAK+I, (36) or, by virtue of equation (34), A^S^A^o^oK., = |#|2R_,. (37) We have, therefore, the identity, A2>l®lA®0@0 = l^l2 mod A®S®0 + 2iot-r-* = 0; (38) and the stated value of |^|2 must follow from a direct evaluation of the identity. Considering then the left-hand side of equation (38), we have A2>l3)lA3)03)0 = A2>l(@0A@l + 4ir(?)3i0 = A3>l@0(A@l@0) + 4iroi-A3il3i0+4i(?A2i0 = A2l20(X- 2io*r) + 4ir&A3) J S>0 + 4i<*A®0- (39) On the other hand, A2>\2>0r = A2>l(r®0 + 1) = rA®%2>0 + A2b + A2>\ = rA2>l2)0 + 2A2)0-2iK. (40) Making use of this last relation, we can further reduce the result (39) in the manner AS>l2>lA2i0Si0 = (^+2^^,5^0-40^ = (Z+2ir(f) (%-2ir&)-4&-[(r2 + a2)(fi+am\ = %2-4a2&2-4a&m = \<€\2; (41) and establish the required identity. * We shall show in §71 that t is in fact a real constant;
THE TEUKOLSKY-STAROBINSKY IDENTITIES 389 THEOREM 3. £f0£f1S+1 is a constant multiple ofS-i and 2\2\S_ 1 is a constant multiple ofS+1. Proof. Applying the operator 2021 to equation (25) satisfied by S+15 we obtain -X2021S+1 = 2021(2l21+2a&-cos6)S+1 = 202 x2\2 XS + x +2a&{cos6)202 XS + x -Aa&(sin6)21S+l\ (42) and by successive reductions, we find 2q2 ^2 q2 i = 2 §21\2 Q — 2(£)2 ^ = 20(2\ + 2Q)2021-22021Q21 = 202\2021+220(Q2021)-220(20Q + Qcot6)21 = 2o 2\202X + 22o(Q202X)- 220(a^cos6 + mcot6cosec6)2x - 220IQ2021 + (aa<-cos 6 - mcot 0cosec6)2 x ] = 2o2\202X - 4a&-20 (cos 6)2 x = 202\2021-4a(f(cos6)2021 +^^6)2^ (43) Combining the results of the reductions (42) and (43), we obtain (202\-2a& cos 6)2021S + 1 = -X2021S+l. (44) Therefore, 20 21S+i satisfies the same eq uatio n (27) as S _ 1 and the firs t part of the theorem stated follows. At the same time, the result of the foregoing reductions is equivalent to establishing the identity 2'021(2l21+2a&-cos 6)= (202\-2a&cos6)202l. (45) The adjoint of this relation (obtained by replacing 6 by n — 6), namely, 2%2\ (20 2\ - 2a&cos 6) = {2\2X + 2a<tcos6)2l2\, (46) establishes the second part of the theorem. Theorem 2 is clearly equivalent to establishing the existence of relations of the form 2021S+1 = D1S-1 and 2%2\S-1 = D2S+1, (47) where Dx and D2 are two real constants. THEOREM 4. IfS + x (6) andS- t (6) are both normalized to unity, then in the relations (47), Dx = D2, and we shall have 2021S+1 = DS-1 and 2 l2\S-x = DS+1. (48)
390 ELECTROMAGNETIC WAVES IN KERR GEOMETRY Proof. The proof follows from two successive applications of Lemma 4 to the normalization integral (29). Thus, D\ = D\ o (JSPo^iS + iH^o^iS + Osinflde Jo (^l^\^ 0^ 1S+1)S + 1 sin OdO. (49) Jo On the other hand, £else\se0£els+l = Dlselse\s^l = DlD1s+l; (50) and equation (49) gives D\ = DXD2 S\lsm6dO = DlD1, (51) 2 0 since S +1 has also been assumed to be normalized to unity. The equality of D1 and D2 follows from this last relation. THEOREM 5. The constant D of Theorem 4 has the value D2 = ^_4aV2 = m2. (52) Proof Itisclearfromequation(50)andtheequalityofD1andD2( = D)that what is required is to establish by direct evaluation that i£\i£\^£0^£x = ;t2-4aV2 mod 2\<e ^ + 2a^cos0 + A = 0. (53) First, we observe that Se\S£0 = (^i- 2Q) {Se\ + 2Q) = SexSe\ + <\a&cos 6; (54) and that, therefore, <£\<£\<£ <><£ x = 5£\{5£ 1if0 + 4oo)-cos 6)Se j = 5£ \5£x (- X- la&cos 6) + 4ao*-(cos 6)Se \i£ x -4aoi-(sin0)J&fV (55) On the other hand, by Lemma 3 of §68, if o^i cos 0= (cos6)Sel^1-(sm6)(Sei + Se\) = (cos B)SelSex - 2 (sin &)<£x + 2Q sin 6. (56) Combining the results of the reductions (55) and (56), we obtain the desired
THE TEUKOLSKY-STAROBINSKY IDENTITIES 391 result: <e%!£\<e02>x = -(X + 2a&cos6)SelSel-'kid'Qsme = (I + 2ao*-cos 6) {X - 2ao*-cos B) - 4a<t(attsin 6 + m cosec 6) sin 6 = /2 - 4a V2 - 4adm = I1 - 4aV2 = D2 = | <€ |2. (57) We shall call (€ and D the Starobinsky constants. Returning to equations (33) and (34), we shall denote AR+1 and R-1 by P+1 and P-i when their relative normalization is compatible with these equations. Thus, we shall write A.@0 ®0P-i = ^P+1 and A^J®SP+1 = «'*P_1. (58) The first of these equations can be rewritten in the form VP + i = A%%P-1 = a(V0 + ~- )90P. i = A^S^o^-i+2iX®o^-i; (59) or, by virtue of the equation satisfied by P-i, <£P+1 = (X-2i&r)P-1+2iK2i0P_1. (60) Similarly, we find from the second of the equations (58), that <€*P_X = (X+2i&r)P+1-2iK@t,P+1. (61) Equations (60) and (61) enable us to express the derivatives of P+1 and P-1 in terms of functions themselves; thus, ~= + l£-P+1-^l(Z + 2i*r)P+1-VP-1l and (62) dP-j iK i ^=-^-+2^-2^1^^ where, it should be noted that we do not as yet know the real and the imaginary parts of # separately, though we do know its absolute value. This lacuna in our information will be rectified in §71. We find from equations (48), in similar fashion, that when S +1 and S _ i are both normalized to unity, then &\S-i = -^U-2a^cos6)S^+DS+J and (63) &iS+1 = + — [(i+2o^cos0)S+1+DS_1].
392 ELECTROMAGNETIC WAVES IN KERR GEOMETRY These equations clearly enable us to express the derivatives ofS+1 and S _ j in terms of the functions themselves. 71. The completion of the solution To complete the solution of Maxwell's equations, beyond separating the variables of $0 and $2, it is necessary to determine their relative normalization. If we choose the radial functions AR+1 and i?_ j to be P +! and P_ j (consistently with equations (58)), and let S +1 and S _ 1 be normalized to unity (consistently with equations (48)), then what remains to be ascertained is the numerical factor by which we must multiply $2 = P- xS_ 1? for example, if A$0 is chosen to be P+1S+1. This factor can be ascertained only from an equation which directly relates $0 and $2. Such an equation can be obtained by eliminating $j from equations (13) and (14) with the aid of Lemma 2 of §68. Thus, by applying the operator {S£0 + ia sin6/p*) to equation (13) and (!2>0 + 1/p*) to equation (14) and adding, we obtain ( iasm8\/ ia sin 6\^ „ 1 \/ 1 V (^ + -^-j(^~^>o = (*o+^J^o-^. (64) On simplification, this equation yields JSfo^^o =®o®o*2- (65) With the solutions for <I>0 and $2 given in equations (23), equation (65) requires that (JSfo^iS+O/S-! = (A®o®o*-i)/A*+i. (66) If we now suppose that both S +1 and S _ t are normalized to unity, then by equation (48) A^o^o^-i = DAR+1. (67) This last relation is consistent with the identification R-1 = P-1 and AR+1 = P+1, (68) if <£ = <g* = D = (A2-4aV2)1/2. (69) By equation (52), this last condition requires that # is real and that we need not distinguish between <€ and <€* (in equations (60)-(62), for example). The solutions for 4>0 and 4>2 have now become determinate and in accordance with equations (12) and (23), we may write A4>0 = P+1S+1 and 4>2 =-^-IP.1S ^. (70) 2(p*) Thus, the relation (65) has not only resolved the relative normalization of the
THE COMPLETION OF THE SOLUTION 393 solutions for <f>0 and <f>2, it has also determined the constant <€ without any ambiguity. (a) The solution for c^ We shall now complete the solution of Maxwell's equations by determining the remaining scalar $!. First, we define the functions g+1(r) = ~(r20P-i-P-^ g-i(r) = -z(r9lP+1-P+1), f + A6) = ~l(cos 9)^\S ^ +(sin 6)S ^J, f-1(6) = ^l(co&0)<?1S+1 + (sinO)S+1]. (71) It can be directly verified with the aid of the Teukolsky-Starobinsky identities that the functions g± (r) and/± (6) satisfy the differential equations &Sog+i=rP+u A®Sff_1=rP_1, and (72) ^U+i =S+1cos0; ^'0f_l = S^lcos6. Now writing equation (13) in the form A^0(P**i)= (P*&i - iasin6)A<D0, (73) and substituting for A$0 its solution P + 1S+ 1? we can rewrite it, successively, with the aid of equations (71) and of the identities established in §70, in the manner A0o(P**i) = (rP+ i)^iS+ 1-iaP+1 [(cos 0)^ + , + (sin0)S+ J = (A®o0+i)^iS+i-ifltfi\i/-i = (A®0ff f i)^iS+1-ifl(A®0®o^-i)/-i- (74) We thus obtain the relation. £20 (p*<D0 = ®0 {g+1<?1S+1 - iqf- , 90P-r). (75) In similar fashion, we find from equation (14) &o(P**i) = l(r-iacos6)90-l-\P-1S-1 = (r20P-! - P_ i)S_ ! - ia{®0P- i)(cos0)S_ x = g+1Se0&1S+1-ia(90P-1)&0f-1; (76) or, equivalently, ^o(P"**i) = ^o(ff+i^iS+i-«"/-iV-i)- (77)
394 ELECTROMAGNETIC WAVES IN KERR GEOMETRY From a comparison of equations (75) and (77), we conclude that the required solution for Q>1 is given by p*^1=g+1(r)^1S+1(d)-iaf-1(d)^0P-i(r)- (78)* By treating equations (15) and (16) in analogous fashion, we find that we have the following alternative form for the solution of ¢^ -p*^ =g-1(r)J?\S-1(d)-iaf+i(d)®lP+i(r)- (79) A comparison of the solutions (78) and (79) leads to the interesting identity ff+1JSf1S+1+ff_1J2'tS-1 = «i(/-1®oi,-i+/+i®Si,+ i> (80) Postponing the verification of this identity, we observe that combining the solutions (78) and (79), we can write the solution for (j)l more symmetrically in the form -ia{f^20P^-f+19lP+l)l (81) (b) The verification of the identity (80) By making use of equations (60), (61), and (63), we can rewrite the expression for g± [ (r) and f± i (6) as linear combinations of the functions P± [ (r) and S+ (6). We find 9+1 = 2¥K LW--2aV)p-i -irVP+i1, 9- i = j^ [ - (ir* + 2aV)i\ t + MP-1], 1 l (82^ /+1=2¥Q[(_/'COS0 + 2aV/a)S'1_^S+lC°S0]' /-1=2¥o[( + ;-cos0 + 2aV/a)S+1 + ^s-lCOS0]- With the aid of these expressions, the identity (80) can be readily verified. * Strictly, we should have added to the particular solution (78) a solution P (say) of the corresponding homogeneous equations 90(P) = y„(P) = 0. But the solution for P, namely, P = constant exp [ - io>(rt + iacos 0)] ■ cot™ (0/2), where r+ is denned by the equation (see equation (100) below) dr„ = (r2 + u2)dr/A (a2 = a2 + am/o>), is singular ai 0 = 0 and 0 = n/2; and on this account we have not included it in the solution for p*<P,.
THE COMPLETION OF THE SOLUTION 395 (c) The solution for the vector potential We shall now show how an explicit solution for the vector potential, A, can be obtained from the solutions for the Maxwell scalars <j>o,<t>i, and 4>2 we have found. We start with the expression FtJ = 8jAt-8tAj, (83) and express <p0 and 4>2 m terms of A. Thus, 40 = Ftjlim'=l,mJ(djAt-dtAj) = li5Ai-m'DAi = T-VselAi-m^oA, 1 pyjl J 1 ,„Jr2 + a2 a \ =jj2^{-irA<+A'+AA*) - -^-7- 3>0 (iaAt sin 6 + Ae + iAm cosec0), (84) PV2 and 4>2 = m'n'idjAi-5,-Xy) = m' &At-nj5*Aj m'A + nj = ~ 2-«-, u{^^l(-iaAtsme +A0-iA cosqcO) + J?0\_-AAr+(r2 + a2)At + aAvV. (85) Now, letting and AF+ j = (r2 + a2)At + AAr + aAv, G+ j = ia/lt sin 0 + Xe + iAv cosec 0, G_ j = — ia/lt sin 0 + Xe — i/1,,, cosec 9, , (86) and making use of the solutions for <p0 and <f>2 given in equations (70) and of the definitions (71), we can rewrite equations (84) and (85) in the forms -fi(2l AF+1-A^0G+1) = (r+iacos6)P+1S+1 = S+1A90g+1+iaP+1&lf+1, (87)
396 ELECTROMAGNETIC WAVES IN KERR GEOMETRY and --^-(A^0tG_1 + ifoAF_1) = These equations are readily (r+i iacosft)P.1S_1 ^g-x+iaP-x&of-x- (88) solved for F±1 and G±1. We find AF+1=(iaP+J+1+A%H+1)j2, AF_1 = (-^,^-A^J «-i)V2, G+1 = (-g+1S+1 + #$H+1)y/2, f (89) G_! =(-0-^^+^0^-1)72, where H+ j and H_ x, as introduced here, are arbitrary functions but which, we shall show presently, are not independent. With the foregoing solutions for F± y and G± j, we can solve equations (i explicitly for the components of the vector potential. We find AAr=±A(F+1-F_1) and as -j^(P+1f+1+P.1f.1) + --^(A30H+1+A3lH.1), Ae = $(G+1 + G_1) p2A e = i(G+1 + G_1) = --fi(g+1S+1+g-1S-1) + -j2(&$H+1 + Se0H-1), 4, = i[A(F+1+F_1)+ifl(G+1-G_1)sin0] = ^[(i,+ 1/+1-i,-1/-1)-(ff+1S+1-ff_1S_1)sin0] + -j^l(A@0H+1-A@$H.1) + ia(Se$H+1- and P2AV = -i[aA(F+1+F_1)sin20 + i(r2 + a2)(G+1-G_1)sii = - 4^ la2(P+ J+ r-P-rf-r) sin2 ft -(r2+a2)(g+1S+1-g.1S.1)sm6-\ - -r la(A@0H+ 1 - A2lH_ Jsin2 ft (90) (91) <j/i+i-JSPoH-i) sin 0], (92) sin0] + i(r2 + a2) (JSPJH+ - Z£0H_ t) sin 0]. (93)
TEUKOLSKY'S EQUATIONS 397 There is, as we have stated, a restriction on the choice of the functions // +1 and //_ !: it follows from evaluating (j)l in terms A and comparing it with the solution (81) we have already found. Thus, evaluating <j>l in terms of the vector potential, we find (j>l = \(lin' + fhims)Fij = \{lini + fhimi)(djAi-diAj) A ffa1 tn^ = --,.^-,^+^^,-^^ = ^[iKAr-(r2 + a2)Atir-aAip,r-QAg -i(a\e sin2 e+AvJ))cosec8-]. (94) Now substituting for the components of A from equations (90)-(93) and equating the resulting expression with the solution (81) for <j>1, we find, after some considerable simplifications, that we are left with J° {p*)2 + *> (p*)2 ®° (p*)2 ^^-(^p-0- (95) This equation defines the freedom we have in the choice of gauge for the vector potential. It is reminiscent of the Coulomb gauge. 72. The transformation of Teukolsky's equations to a standard form We shall consider in place of equations (26) and (28) the more general equations ^0,.,,,02-2(2181-1)10^ + ,,,= ^ + ,,, (96) and [A0t_|S|0o + 2(2|S|-l)K*r]/>_,s,= */>_,,,, (97)* applicable to massless fields of spin \s\. Thus, for \s\ = 1, the equations reduce to equations (26) and (28) appropriate to photons of spin 1; and we shall find in Chapters 9 and 10 that for \s\ = 2 and \s\ = 1/2 they are the equations governing the propagation of gravitational waves and of the two-component neutrinos, respectively. Since P +,,, and P _ ,s, satisfy complex-conjugate equations, it will suffice to consider the equation for P + \s\ only. Also, for convenience we shall write simply s in place of \s\ with the understanding that it is to be considered positive and allowed the values 2, 1, and 1/2. The equation we shall consider, then, is ^^-,^-2(28-1)10^+,= ^+,. (98) * While />+ |si and P- ^, in general, stand for A'R + s and R _ „ we are not here assuming that the functions have been relatively normalized in any particular way.
398 ELECTROMAGNETIC WAVES IN KERR GEOMETRY In this equation m explicitly occurs through K = (r2 + a2)<?+am (99) in the operators 3> and &. We shall now show how an explicit reference to m can be eliminated by a suitable change of variables. First, we introduce in place of r a new independent variable rm denned by d Ad 3— = — t, (100) dr„ or dr where m2 = r2 + a2 and a2 = a2 + (am/<fi). (101) In view of the admissibility of negative values of a2 (when m is negative and &■-* 0), it is clear that the r„(r)-relation, that follows from equation (100), can under certain circumstances become double-valued. We shall have to consider, in due course, how this contingency, when it arises, should be met; but, meantime, we shall consider only the formal consequences of this change of variable. One immediate consequence of the change in the independent variable to rm is the resulting simplicity of the operators 3)0 and 2J: 2 2 ®0 = -^-A+ and S^l=~\^, (102) where, in conformity with our standard usage, d A±= —±i<* (103) The simplicity arises from the fact that by virtue of the definition (101) K = roV. (104) Second, in addition to the change in the independent variable, we shall also change the dependent variable. We shall let Y= \m2\-s + 1'2P + s. (105) For integral values of s, this transformation is singular if m2 should change sign in the range of r of interest—a contingency to which we shall return presently. With the change of variables made, equation (98) becomes s-l „2 ro^A + ^A_(\m2\s-l/2Y) As -2(2s- l)io*r|ro2|s-V2y_*| ro2\s~l/2Y= 0. (106)
TEUKOLSKY'S EQUATIONS 39 On expanding this equation, we find ro2|S + 3/2 A + As-'n72 A2y+ A. 8 A 2 12s' A_y+2io*-(2s-l)r- m ro2 d m 2,5-1/2 y-[2(2s-l)iVr|ro2|s-'/2 + A|ro2|s-'/2]y=0. (107) With the definitions and r-i-y-^-Z^-Hr-** (108) m \s+ 1 1 \m* |m2|s + 3/2 dr\As-' dr ^-(2,-1)^-^-1^^) 1/2 r2A' + (2s-3)^T (109) (110) equation (107) can be reduced to the form a2y+pa_y-qy= o —a form which we have encountered already in our treatment of the perturbations of the Schwarzschild black-hole (Ch. 4, equation (284)). (a) The r^ (r)-relation Integrating equation (100), we obtain the relation 2Mr+ + (am/&), ( r \ 2Mr_ + (am/<?)/r \ / (111) When considering the external perturbations of the Kerr black-hole we are indifferent to what 'happens' inside the event horizon at r = r + . We need, therefore, be concerned only with the ri)1(r)-relation for r > r + . It is now apparent from equation (111) that the ri)1(r)-relation is single- valued for r > r+ only so long as r\ + a2 = 2Mr+ - a2 + a2 = 2Mr+ + (am/&) > 0. (112) When this inequality obtains, r* ~* + °° when r -* oo and rm -* — oo when r -► r+ + 0, (113) and the r„(r)-relation is a monotonic one. Letting (?s = — am/2Mr + (for m negative), (114)
400 ELECTROMAGNETIC WAVES IN KERR GEOMETRY and remembering that our convention with respect to & is that it is to be positive, we conclude that so long as ct> as, the rm(r)-relation is single-valued outside the event horizon and the range of rn of interest is the entire interval, (— oo, + oo). But if 0 < &■ < &-s and r2++<x2 < 0, the r^(r)-relation is double- valued: r„ -» + oo both when r -* oo and when r -* r+ + 0. In the latter case, in the neighbourhood of r = |a|, the r„(r)-relation has the behaviour r„ = r,(|a|) + -^-(r - |a|)2 + 0((r - |a|)3). (115) Also, when 0 < &■ < &s, the functions P and Q in equation (110) become singular at r = |a|. Accordingly, in these cases, the equation must be considered separately in the two branches of the ri)1(r)-relation, namely for oo > r > |a| and |a| > r > r+. We shall show in §§ 74 and 79 that in the interval, 0 < & < &s, the reflexion coefficient for incident waves of integral spins exceeds unity. This is the phenomenon of super-radiance: it is the analogue, in the domain of wave propagation, of the Penrose process in the domain of particle dynamics. While we shall consider the origins of this super-radiance in the framework of equations (96) and (97) in § 74, we may yet draw attention here to the similarity of the present inequality, 2Mr+rf< -am, (116) (equivalent to & < o^) with the inequality, Chapter 7, (352): they are the same if we identify h& with 5M and hm with SJ, where h is Planck's constant! Finally, it should be noted that a2 becomes negative already before o^: a2 < 0 for &■ < &c = - m/a; (117) and a2 = 0 when & is lco-rotational\ In the interval, &s<& < &c, while a2 is negative, r\ +a2 > 0 and the r^(r)-relation continues to be single-valued. 73. A general transformation theory and the reduction to a one-dimensional wave-equation The form (110) to which the general Teukolsky equation (98) was reduced in §72 is identical to the one to which the equations of the Newman-Penrose formalism, in the context of the perturbations of the Schwarzschild black-hole, were reduced in § 20. And as in that context, we shall seek a transformation which will bring equation (110) to a one-dimensional wave-equation of the form A2Z= VZ, (118) where V is a potential function to be determined. The theory we shall now outline differs from the one presented in § 30 only in the one respect derived from the different definitions of P (cf. equations
A GENERAL TRANSFORMATION THEORY 401 (106) and Ch. 4, (286)). It is, however, convenient to have the basic formulae written in the forms we shall need. We assume, then, that