Introduction to General Relativity

INTRODUCTION
1. Physics and Geometry
2. The Choice of Riemannian Geometry

1 TENSOR ALGEBRA
1.1 Definition of Scalars, Contravariant Vectors, and Covariant Vectors
1.2 Einstein's Summation Convention
1.3 Definitions of Tensors
1.4 Tensor Algebra
1.5 Decomposition of a Tensor into a Sum of Vector Products
1.6 Contraction of Indices
1.7 The Quotient Theorem
1.8 Lowering and Raising of Indices—Associated Tensors
1.9 Connection with Vector Calculus in Euclidean Space
1.10 Connection between Bilinear Forms and Tensor Calculus

2 VECTOR FIELDS IN AFFINE AND RIEMANN SPACE
2.1 Vector Transplantation and Affine Connections
2.2 Parallel Displacement—Christoffel Symbols
2.3 Geodesies in Affine and Riemann Space
2.4 Gaussian Coordinates

3 TENSOR ANALYSIS
3.1 Covariant Differentiation
3.2 Applications of Tensor Analysis
3.3 Symmetric and Antisymmetric Tensors
3.4 Closed and Exact Tensors
3.5 Tensor Densities—Dual Tensors
3.6 Vector Fields on Curves
3.7 Intrinsic Symmetries and Killing Vectors

4 TENSORS IN PHYSICS
4.1 Maxwell's Equations in Tensor Form
4.2 Proper-Time and the Equations of Motion via an Example in Relativistic Mechanics
4.3 Gravity as a Metric Phenomenon
4.4 The Red Shift

5 THE GRAVITATIONAL FIELD EQUATIONS IN FREE SPACE
5.1 Criteria for the Field Equations
5.2 The Riemann Curvature Tensor
5.3 Symmetry Properties of the Riemann Tensor
5.4 The Bianchi Identities
5.5 Integrability and the Riemann Tensor
5.6 Pseudo-Euclidean and Flat Spaces
5.7 The Einstein Field Equations for Free Space
5.8 The Divergenceless Form of the Einstein Field Equations
5.9 The Riemann Tensor and Fields of Geodesies
5.10 Algebraic Properties of the Riemann Tensor

6 THE SCHWARZSCHILD SOLUTION AND ITS CONSEQUENCES: EXPERIMENTAL 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

7 THE KERR 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
7.8 Distinguished Surfaces and the Rotating Black Hole
7.9 Effective Potentials and Black Hole Energetics

8 THE MATHEMATICAL STRUCTURE OF THE EINSTEIN DIFFERENTIAL SYSTEM; THE PROBLEM OF CAUCHY
8.1 Formulation of the Initial-Value Problem
8.2 Structure of Einstein's Equations
8.3 Separation of the Cauchy Problem into Two Parts
8.4 Characteristic Hypersurfaces of the Einstein Equation System
8.5 Bicharacteristics of the Einstein System
8.6 Uniqueness Problem for the Einstein Equations
8.7 The Maximum Principle for the Generalized Laplace Equation

9 THE LINEARIZED FIELD EQUATIONS
9.1 Linearization of the Field Equations
9.2 The Time-independent and Spherically Symmetric Field
9.3 The Weyl Solutions to the Linearized Field Equations
9.4 Structure of the Linearized Equations
9.5 Gravitational Waves

10 THE GRAVITATIONAL FIELD EQUATIONS FOR NONEMPTY SPACE
10.1 The Energy-Momentum Tensor
10.2 Inclusion of Forces in Tμν
10.3 The Electromagnetic Field and Tμν
10.4 The Field Equations in Nonempty Space
10.5 Classical Limit of the Gravitational Equations

11 FURTHER CONSEQUENCES OF THE FIELD EQUATIONS
11.1 The Equations of Motion
11.2 Conservation Laws in General Relativity: Energy-Momentum of the Gravitational Field
11.3 An Alternative Approach to the Conservation Laws: Energy-Momentum of the Schwarzschild Field
11.4 Variational Principles in General Relativity Theory: A Lagrangian Density for the Gravitational Field
11.5 The Scalar Tensor Variation of Relativity Theory

12 DESCRIPTIVE COSMIC ASTRONOMY
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
12.5 The Red Shift and the Robertson-Walker Metric: Hubble's Law
12.6 The Apparent Magnitude-Red Shift Relation

13 COSMOLOGICAL MODELS
13.1 Einstein's Equations and the Robertson-Walker Metric
13.2 Static Models of the Universe
13.3 Nonstatic Models of the Universe
13.4 The Gödel Solution and Mach's Principle
13.5 The Steady-State Model of the Universe
13.6 Converse of the Apparent Magnitude-Red Shift Problem

14 THE ROLE OF RELATIVITY IN STELLAR STRUCTURE AND GRAVITATIONAL COLLAPSE
14.1 Relativistic Stellar Structure
14.2 A Simple Stellar Model—The Interior Schwarzschild Solution
14.3 Stellar Models and Stability
14.4 Gravitational Collapse of a Dust Ball

15 ELECTROMAGNETISM AND GENERAL RELATIVITY
15.1 The Field of a Charged Mass Point
15.2 Weyl's Generalization of Riemannian Geometry
15.3 Weyl's Theory of Electromagnetism
15.4 Some Mathematical Machinery
15.5 The Equations of Rainich, Misner, and Wheeler

INDEX