General theory of Relativity

Table of contents

1. Cover
2. Title Page
3. Contents
4. Preface
5. Historical Perspective
6. Chapter 1: A Brief Review of Special Relativity
   1. 1.1 Introduction
   2. 1.2 Some Common Definitions
   3. 1.3 The Constancy of Velocity of Light
   4. 1.4 The Postulates of Special Relativity
   5. 1.5 Interval
   6. 1.6 Lorentz Transformation
   7. 1.7 Kinematic Consequences of Lorentz Transformation
   8. 1.8 Light Cone
   9. 1.9 Four-Vectors
   10. 1.10 Relativistic Mass
   11. 1.11 Electromagnetic Field Tensor
   12. 1.12 Covariant Form of Maxwell’s Equations
   13. 1.13 Photons and Neutrinos
   14. Problems
7. Chapter 2: Tensor Analysis and Riemannian Geomtry
   1. Part I: Line Element
      1. 2.1 Riemannian Space
      2. 2.2 Transformation of Coordinates
      3. 2.3 Contravariant and Covariant Vectors
      4. 2.4 Summation Convention
      5. 2.5 The Metric
      6. 2.6 The Metric as a Tensor
      7. 2.7 Contravariant, Covariant, and Mixed Tensors
      8. 2.8 Multiplication of Tensors—Inner and Outer Products andContraction
      9. 2.9 Quotient Law of Tensors
      10. 2.10 Fundamental Tensors: gμv, gμv, and gμv
      11. 2.11 Raising and Lowering of Indices
      12. Problems
   2. Part II: Geodesic Curves—Covariant Differentiation
      1. 2.12 Manifolds
      2. 2.13 Covariant Derivative
      3. 2.14 Christoffel’s 3-Index Symbols and Their Transformation Law
      4. 2.15 Geodesics
      5. 2.16 Covariant Differentiation of Vectors
      6. 2.17 Covariant Derivatives of Tensors
      7. Problems
   3. Part III: Curvature Tensor
      1. 2.18 Riemannian Coordinates
      2. 2.19 Riemann–Christoffel Curvature Tensor
      3. 2.20 Symmetries and Anti-Symmetries of Curvature Tensor
      4. 2.21 Number of Independent Components of the Curvature Tensor Rλμνσ
      5. 2.22 The Bianchi Identities
      6. 2.23 The Ricci Tensor
      7. 2.24 The Contracted Binachi Identities or the Einstein Tensor
      8. 2.25 Uniform Vector Field
      9. 2.26 The Condition for Flat Space–Time
      10. 2.27 Parallel Displacement and Affine Connections
      11. 2.28 Affine Connections for the Covariant Vector
      12. 2.29 Affine Connections and the Metric Tensor
      13. 2.30 Parallel Displacement and Covariant Differentiation
      14. 2.31 Energy-Momentum Tensor for a Perfect Fluid
      15. Problems
8. Chapter 3: Einstein’s Field Equations
   1. 3.1 Introduction
   2. 3.2 Principle of Equivalence
   3. 3.3 Principle of Covariance
   4. 3.4 Newtonian Equation of Motion as an Approximation of Geodesic Equations
   5. 3.5 Heuristic Derivation of Einstein’s Field Equations
   6. 3.6 Einstein’s Field Equations by Variational Technique
   7. 3.7 Fundamental Hypotheses and Postulates of General Relativity
   8. 3.8 Poisson’s Equation as Approximation of Einstein’s Field Equations. Evaluation of Constant k
   9. Problems
9. Chapter 4: Einstein’s Law of Gravitation in Empty Space—Schwarzschild Solution
   1. 4.1 Introduction
   2. 4.2 A Static Spherically Symmetric Space–Time
   3. 4.3 Schwarzschild Line-Element
   4. 4.4 Killing Vector
   5. 4.5 Particle Trajectories in Schwarzschild Space–Time
   6. 4.6 Experimental Tests of General Relativity
   7. Problems
10. Chapter 5: Einstein’s Field Equations for Non-Empty Space
    1. 5.1 Introduction
    2. 5.2 Static Line-Element with Spherical Symmetry
    3. 5.3 Schwarzschild’s Exterior Solution
    4. 5.4 Schwarzschild’s Interior Solution
    5. 5.5 Conservation Laws in Curved Space
    6. Problems
11. Chapter 6: Gravitational Waves
    1. 6.1 Introduction
    2. 6.2 Weak Gravitational Field, Linearized Field Equations
    3. 6.3 Plane Wave Solutions
    4. 6.4 Behaviour of a Massive Particle as a Gravitational Wave Passes
    5. 6.5 The Detection of Gravitational Waves
    6. 6.6 Quadrupolar Nature of Gravitational Waves
    7. 6.7 The Emission of Gravitational Waves
    8. 6.8 Experimental Support for Gravitational Waves
    9. Problems
12. Chapter 7: Black Holes
    1. 7.1 Introduction
    2. 7.2 Schwarzschild Black Holes—Singularities
    3. 7.3 Kruskal Coordinates
    4. 7.4 The Kerr Metric in Boyer–Lindquist Coordinates
    5. 7.5 Frame Dragging—Lense–Thirring Effect
    6. 7.6 Energy Extraction from a Rotating Black Hole. Penrose Process
    7. 7.7 The Reissner–Nordström Solution
    8. 7.8 Evidence for the Existence of Black Holes
    9. Problems
13. Chapter 8: Cosmology
    1. 8.1 Introduction
    2. 8.2 The Cosmological Principle and Weyl’s Postulate
    3. 8.3 A Spatial Metric Incorporating Homogeneity and Isotropy
    4. 8.4 Spaces of Positive, Negative, and Zero Curvature
    5. 8.5 Static Cosmological Models
    6. 8.6 The Robertson-Walker Metric. Friedmann Equations
    7. 8.7 Non-Static Cosmological Models. Time Evolution of Universe
    8. 8.8 Useful Terminology
    9. 8.9 The Red-Shift
    10. 8.10 Preliminaries for Early Universe
    11. 8.11 The Standard Model of Early Universe
    12. 8.12 The Age of the Universe
    13. 8.13 Cosmological Constant in Einstein’s Field Equations
    14. 8.14 Cosmic Microwave Background Radiation
    15. Problems
14. Chapter 9: Astrophysics
    1. 9.1 Introduction
    2. 9.2 Tolman—Oppenheimer–Volkoff Equation
    3. 9.3 Degeneracy of Matter
    4. 9.4 Model of a Star in Hydrostatic Equilibrium
    5. 9.5 Polytropic Stars
    6. 9.6 The Lane–Emden Equation
    7. 9.7 The Chandrashekhar Mass Limit
    8. 9.8 Formation of White Dwarfs, Neutron Stars and Black Holes ρ > 107gcm-3
    9. Problems
15. Notes
16. Epilogue
17. Bibliography
18. Copyright
19. Back Cover