INTRODUCTION TO GENERAL RELATIVITY RONALD ADLER Instito de Fisica Universidade Federal de Pemambuco Recife, Brazil MAURICE BAZIN Department of Physics Rutgers University MENAHEM SCHIFFER Department of Mathematics Stanford University SECOND EDITION INTERNATIONAL STUDENT EDITION McGRAW-HILL KOGAKUSHA, LTD. Tokyo Auckland Beirut Bogota Dusseldorf Johannesburg Lisbon London Lucerne Madrid Mexico New Delhi Panama Paris San Juan Sao Paulo Singapore Sydney
PREFACE TO THE SECOND EDITION PREFACE TO THE FIRST EDITION xi ziii INTRODUCTION 1 1. Physics and Geometry 1 2. The Choice of Riemannian Geometry 7 TENSOR ALGEBRA 17 1 1.1 Definition of Scalars, Contravariant Vectors, and Covariant Vectors 18 1.2 Einstein's Summation Convention 20 1.8 Definitions of Tensors 21 14 Tensor Algebra 24 l.o Decomposition of a Tensor into a Sum of Vector Products 25 1.6 Contraction of Indices 28 1.7 The Quotient Theorem 29 1.8 Lowering and Raising of Indices Associated Tensors SO 1.9 Connection with Vector Calculus in Euclidean Space 32 1.10 Connection between Bilinear Forms and Tensor Calculus 36
v i Contents VECTOR FIELDS IN AFFINE AND RIEMANN SPACE 41 2.1 2.2 2.3 2.4 Vector Transplantation and Affine Connections Parallel Displacement Christoffel Symbols Geodesies in Affine and Riemann Space Gaussian Coordinates 42 50 53 61 3 8.1 3.2 3.3 8.4 8.5 8.6 3.7 TENSOR ANALYSIS Covariant Differentiation Applications of Tensor Analysis Symmetric and Antisymmetric Tensors Closed and Exact Tensors Tensor Densities Dual Tensors Vector Fields on Curves Intrinsic Symmetries and Killing Vectors 67 67 75 80 87 90 94 98 4 4-1 4.2 4.3 44 TENSORS IN PHYSICS Maxwell's Equations in Tensor Form Proper-Time and the Equations of Motion via an Example in Relativistic Mechanics Gravity as a Metric Phenomenon The Red Shift 105 105 120 131 135 THE GRAVITATIONAL FIELD EQUATIONS IN FREE SPACE 145 5.1 Criteria for the Field Equations 14& 5.2 The Riemann Curvature Tensor 147 5.3 Symmetry Properties of the Riemann Tensor 151 5.4 The Bianchi Identities 165 5.5 Integrability and the Riemann Tensor 167
vii 5.6 Pseudo-Euclidean and Flat Spaces 162 5.7 The Einstein Field Equations for Free Space 167 5.8 The Divergenceless Form of the Einstein Field Equations 170 5.9 The Riemann Tensor and Fields of Geodesies 171 5.10 Algebraic Properties of the Riemann Tensor 176 THE SCH WAR Z S CHI LD SOLUTION AND ITS CONSEQUENCES: EXPERI- MENTAL TESTS OF GENERAL RELATIVITY 6.1 The Schwarzschild Solution 6.2 The Schwarzschild Solution in Isotropic Coordinates 6.3 The General Relativistic Kepler Problem and the Perihelic Shift of Mercury 6.4 The Sun's Quadrupole Moment and Perihelic Motion 6.5 The Trajectory of a Light Ray in a Schwarzschild Field 6.6 Travel Time of Light in a Schwarzschild Field 6.7 Null Geodesies and Fermat's Principle 6.8 The Schwarzschild Radius, Kruskal Coordinates, and the Black Hole 185 185 196 199 214 219 THE KERB SOLUTION 7.1 Eddington's Form of the Schwarzschild Solution 7.2 Einstein's Equations for Degenerate Metrics 7.3 The Order m 2 Equations 7.4 Field Equations for the Stationary Case 7.5 The Schwarzschild and Kerr Solutions 7.6 Other Coordinates 7.7 The Kerr Solution and Rotation 237 287 238 241 250 258
viii Contents 7.8 Distinguished Surfaces and the Rotating Black Hole 7.9 Effective Potentials and Black Hole Energetics 266 8 THE MATHEMATICAL STRUCTURE OF THE EINSTEIN DIFFERENTIAL SYSTEM; THE PROBLEM OF CAUCHY 275 8.1 Formulation of the Initial-Value Problem 276 8.2 Structure of Einstein's Equations 276 8.8 Separation of the Cauchy Problem into Two Parts 281 84 Characteristic Hypersurfaces of the Einstein Equation System 284 8.5 Bicharacteristics of the Einstein System 285 8.6 Uniqueness Problem for the Einstein Equations 289 8.7 The Maximum Principle for the Generalized Laplace Equation 295 CHAPT 9 ER 9.1 9.2 9.3 94 9.5 THE LINEARIZED FIELD EQUATIONS Linearization of the Field Equations The Time-independent and Spherically Symmetric Field The Weyl Solutions to the Linearized Field Equations Structure of the Linearized Equations Gravitational Waves 301 801 305 309 314 317 CHAPT 10 t ER 10.1 10.2 THE GRAVITATIONAL FIELD EQUATIONS FOR NONEMPTY SPACE The Energy-Momentum Tensor Inclusion of Forces in T" 329 330 834
i X 10.3 The Electromagnetic Field and T*' 338 104 The Field Equations in Nonempty Space 344 10.5 Classical Limit of the Gravitational Equations 345 FURTHER CONSEQUENCES OF THE FIELD EQUATIONS 351 11 11.1 The Equations of Motion 361 11.2 Conservation Laws in General Relativity: Energy-Momentum of the Gravitational Field 364 11.3 An Alternative Approach to the Conservation Laws: Energy-Momentum of the Schwarzschild Field 366 114 Variational Principles in General Relativity Theory: A Lagrangian Density for the Gravitational Field 376 11.5 The Scalar Tensor Variation of Relativity Theory 380 12 DESCRIPTIVE ASTRONOMY COSMIC 12.1 Observational Background 12.2 The Mathematical Problem in Outline 12.3 The Robertson-Walker Metric 12.4 Further Properties of the Robertson-Walker Metric 12.6 The Red Shift and the Robertson-Walker Metric: Hubble's Law 12.6 The Apparent Magnitude-Red Shift Relation 389 390 395 400 408 412 415 COSMOLOGICAL MODELS 425 13 18.1 Einstein's Equations and the Robertson- Walker Metric 4 5 13.2 Static Models of the Universe 428
13.3 Nonstatic Models of the Universe 431 18.4 The Godel Solution and Mach's Principle 487 13.5 The Steady-State Model of the Universe 448 13.6 Converse of the Apparent Magnitude-Red Shift Problem 454 14 14.1 14.2 14.8 144 THE ROLE OF RELATIVITY IN STELLAR STRUCTURE AND GRAVITATIONAL COLLAPSE ' 461 Relativistic Stellar Structure 462 A Simple Stellar Model The Interior Schwarzschild Solution 468 Stellar Models and Stability 476 Gravitational Collapse of a Dust Ball 478 15 ELECTROMAGNETIS'M AND GENERAL RELATIVITY 485 15.1 The Field of a Charged Mass Point 486 15.2 Weyl's Generalization of Riemannian Geometry ' 491 15.3 Weyl's Theory of Electromagnetism 500 154 Some Mathematical Machinery 507 15.5 The Equations of Rainich, Misner, and Wheeler 518 INDEX 585