INTRODUCTION TO GENERAL RELATIVITY (2nd)

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Introduction to General Relativity (Pure & Applied Physics S.)

Introduction to General Relativity (Pure & Applied Physics S.)

  • INTRODUCTION
    • 1. Physics and Geometry
    • 2. The Choice of Riemannian Geometry
  • CHAPTER 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
  • CHAPTER 2 VECTOR FIELDS IN AFFINE AND RIEMANN SPACE
    • 2.1 Vector Transplantation and Affine Connections
    • 2.2 Parallel Displacement---Christoffel Symbols
    • 2.3 Geodesics in Affine and Riemann Space
    • 2.4 Gaussian Coordinates
  • CHAPTER 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
  • CHAPTER 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
  • CHAPTER 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
  • CHAPTER 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
  • CHAPTER 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
  • CHAPTER 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
    • 8.8 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
  • CHAPTER 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
  • CHAPTER 10 THE GRAVITATIONAL FIELD EQUATIONS FOR NONEMPTY SPACE
    • 10.1 The Energy-Momentum Tensor
    • 10.2 Inclusion of Forces in T^{\mu\nu}
    • 10.3 The Electromagnetic Field and T^{\mu\nu}
    • 10.4 The Field Equations in Nonempty Space
    • 10.5 Classical Limit of the Gravitational Equations
  • CHAPTER 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
  • CHAPTER 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
    • 12.6 The Red Shift and the Robertson-Walker Metric: Hubble's Law
    • 12.6 The Apparent Magnitude-Red Shift Relation
  • CHAPTER 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 Godel Solution and Mach's Principle
    • 13.5 The Steady-State Model of the Universe
    • 13.6 Converse of the Apparent Magnitude-Red Shift Problem
  • CHAPTER 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
    • 14.8 Stellar Models and Stability
    • 14.4 Gravitational Collapse of a Dust Ball
  • CHAPTER 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