GENERAL RELATIVITY Robert M . Wald

The University of Chicago Press Chicago and London GENERA L RELATIVIT Y Robert M . Wald

The University of Chicago Press Chicago and London The painting reproduced on the cover of the paperback version of this book is Rend Magritte's Les Belles Realites, OO by ADAGP, Paris, 1984 . The transparency was furnished by Galerie Isy Brachot, Bruxelles- Paris .

The University of Chicago Press, Chicago 60637 The University of Chicago Press, Ltd . , London D 1984 by The University of Chicago All rights reserved . Published 1984 Printed in the United States of Americ a

04 03 02 0100 99 98 97 96 95 891011 1 2

Library of Congress Cataloging in Publication Dat a

Wald, Robert M . General relativity .

Bibliography: p. 473 Includes index . 1 . General relativity (Physics) I . Title . QC173 .6 .W3 5 1984 530 .1'1 83-17969 ISBN 0-226-87033-2 (pbk . )

This book is printed on acid-free paper . CONTENT S

Preface ix Notation and Conventions xi

PART I . FUNDAMENTAL S 1. Introduction 3 1 .1 Introduction 3 1 .2 Space and Time in Prerelativity Physic s 4 and in Special Relativity 1 .3 The Spacetime Metri c 6 1 .4 General Relativity 8 2. Manifold s and Tensor Fields 11 2 .1 Manifold s 11 2.2 Vector s 14 2.3 Tensors ; the Metric Tensor 18 2.4 The Abstract Index Notation 23 3. Curvature 29 3 . 1 Derivat ive Operators and Parallel Tran sport 30 3 .2 Curvature 36 3 .3 Geode sic s 41 3 . 4 Methods for Computing Curvature 47 55 4. Einstein's E quatio n 4 .1 The Geometry of Space in Prerelativity Physics ; 55 General and Special Covarianc e 59 4 .2 Special Relativity 66 4.3 General Relativity 4.4 Linearized Gravity : The Newtonian Limit and Gravitational Radiatio n 74 5. Homogeneous, Isotropic Cosmology 91 91 5.1 Homogeneity and Isotrop y 96 5.2 Dynamics of a Homogeneous, Isotropic Universe

v vi Content s

5 .3 The Cosmological Redshift; Horizons 101 107 5 .4 The Evolution of Our Universe 6 . The Schwarzschild Solution 118 6 .1 Derivation of the Schwarzschild Solutio n 119 6 .2 Interior Solution s 125 6 .3 Geodesics of Schwarzschild: Gravitational Redshift , 136 Perihelion Precession, Bending of Light, and Time Delay 6 .4 The Kruskal Extension 148

PART II . ADVANCED TOPIC S 7 . Methods for Solving Einstein's Equation 161 7 .1 Stationary, Axisymmetric Solution s 162 7.2 Spatially Homogeneous Cosmologie s 168 7.3 Algebraically Special Solution s 179 7.4 Methods for Generating Solution s 180 18 3 7.5 Perturbations 8. Causal Structure 188 8 .1 Futures and Pasts: Basic Definitions and Results 189 8 .2 Causality Condition s 195 8 .3 Domains of Dependence ; Global Hyperbolicity 200 9. Singularities 2 11 9 .1 What Is a Singularity ? 212 9 .2 Timelike and Null Geodesic Congruence s 216 9.3 Conjugate Points 223 9.4 Existence of Maximum Length Curve s 233 9.5 Singularity Theorem s 23 7 10 . The Initial Value Formulation 243 10 .1 Initial Value Formulation for Particles and Fields 244 10.2 Initial Value Formulation of General Relativity 252 11. Asymptotic Flatness 269 11 .1 Conformal Infinity 271 11 .2 Energ y 285 12. Black Holes 298 12.1 Black Holes and the Cosmic Censor Conjecture 299 12.2 General Properties of Black Hole s 308 12 .3 The Charged Kerr Black Hole s 312 12 .4 Energy Extraction from Black Hole s 324 12 .5 Black Holes and Thermodynamic s 33 0 13. Spinors 340 13 .1 Spinors in Minkowski Spacetime 342 13 .2 Spinors in Curved Spacetime 359  Contents vi i

378 14 . Quantum Effects in Strong Gravitational Fields 380 14 . 1 Quantum Gravity 389 14 .2 Quantum Field s in Curved Spacetime 14 .3 Particle Creation near Black Hole s 399 416 14 .4 Black Hole Thermodynamic s

APPENDICES A. Topological Spaces 423

B. Differential Forms, Integration , and Frobenius's Theorem 428 B . I Differential Form s 428 B . 2 Integratio n 429 B .3 Frobenius's Theorem 434 Maps of Manifolds, Lie D erivati ves, and Killing Fields C. 437 C .1 Maps of Manifold s 437 C .2 Lie Derivative s 439 C .3 Killing Vector Fields 441

D. Conformal Transformations 445 E. Lagrangian and Hamiltonian Formulations of 450 Einstein's Equation 450 E.1 Lagrangian Formulation 459 E.2 Hamiltonian Formulation 470 F . U ni ts and Dime nsions

References 473 485 Index PREFACE

This book is intended to provide a thorough introduction to the theory of general relativity . It is intended to serve as both a text for graduate students and a reference book for researchers . These two goals are somewhat contradictory, and to the