INTRODUCTION TO GENERAL RELATIVITY

Transcription

1 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

2 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 Definition of Scalars, Contravariant Vectors, and Covariant Vectors Einstein's Summation Convention Definitions of Tensors Tensor Algebra 24 l.o Decomposition of a Tensor into a Sum of Vector Products Contraction of Indices The Quotient Theorem Lowering and Raising of Indices Associated Tensors SO 1.9 Connection with Vector Calculus in Euclidean Space Connection between Bilinear Forms and Tensor Calculus 36

3 v i Contents VECTOR FIELDS IN AFFINE AND RIEMANN SPACE Vector Transplantation and Affine Connections Parallel Displacement Christoffel Symbols Geodesies in Affine and Riemann Space Gaussian Coordinates 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 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 THE GRAVITATIONAL FIELD EQUATIONS IN FREE SPACE Criteria for the Field Equations 14& 5.2 The Riemann Curvature Tensor Symmetry Properties of the Riemann Tensor The Bianchi Identities Integrability and the Riemann Tensor 167

4 vii 5.6 Pseudo-Euclidean and Flat Spaces The Einstein Field Equations for Free Space The Divergenceless Form of the Einstein Field Equations The Riemann Tensor and Fields of Geodesies 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 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

5 viii Contents 7.8 Distinguished Surfaces and the Rotating Black Hole 7.9 Effective Potentials and Black Hole Energetics THE MATHEMATICAL STRUCTURE OF THE EINSTEIN DIFFERENTIAL SYSTEM; THE PROBLEM OF CAUCHY Formulation of the Initial-Value Problem Structure of Einstein's Equations Separation of the Cauchy Problem into Two Parts Characteristic Hypersurfaces of the Einstein Equation System Bicharacteristics of the Einstein System Uniqueness Problem for the Einstein Equations The Maximum Principle for the Generalized Laplace Equation 295 CHAPT 9 ER 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 CHAPT 10 t ER THE GRAVITATIONAL FIELD EQUATIONS FOR NONEMPTY SPACE The Energy-Momentum Tensor Inclusion of Forces in T"

6 i X 10.3 The Electromagnetic Field and T*' The Field Equations in Nonempty Space Classical Limit of the Gravitational Equations 345 FURTHER CONSEQUENCES OF THE FIELD EQUATIONS The Equations of Motion Conservation Laws in General Relativity: Energy-Momentum of the Gravitational Field An Alternative Approach to the Conservation Laws: Energy-Momentum of the Schwarzschild Field Variational Principles in General Relativity Theory: A Lagrangian Density for the Gravitational Field The Scalar Tensor Variation of Relativity Theory 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 COSMOLOGICAL MODELS Einstein's Equations and the Robertson- Walker Metric Static Models of the Universe 428

7 13.3 Nonstatic Models of the Universe The Godel Solution and Mach's Principle The Steady-State Model of the Universe Converse of the Apparent Magnitude-Red Shift Problem 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 ELECTROMAGNETIS'M AND GENERAL RELATIVITY The Field of a Charged Mass Point Weyl's Generalization of Riemannian Geometry ' Weyl's Theory of Electromagnetism Some Mathematical Machinery The Equations of Rainich, Misner, and Wheeler 518 INDEX 585