Table of Contents
Preface to the Dover Edition xi
Preface xiii
Notation xvii
1 Special Relativity 1
The Lorentz Transformations as Viewed by Einstein 1
Minkowski Space 7
The Minkowski Norm 11
2 Clocks and Gravitational Potential 13
Gravitation, Acceleration, and the Principle of Equivalence 13
The Pseudo-Riemannian Structure of Space-Time 16
Gravitational Potential 20
Is Gravitation Governed by a Single Potential? 25
3 A Heuristic Derivation of Einstein's Equations 27
Poisson's Equation 27
The Density ρ* 29
Einstein's Equations 31
4 The Geometry of Einstein's Equations 35
Curvature in a Pseudo-Riemannian M4 35
The Einstein Tensor Gij 39
The Gauss Equations in M4 41
A Geometric Form of Einstein's Equations 42
5 The Schwarzschild Solution 46
Schwarzschild Coordinates 46
Embedding the Spatial Section 47
The Gravitational Potential and g00 50
The Schwarzschild Singularity 52
Concluding Remarks 53
6 The Classical Motion of a Continuum 56
Lie Derivatives, Interior Products, and the Variation of Integrals 56
The Cauchy Stress Tensor in Classical Mechanics 64
The Stress-Energy-Momentum Tensor 69
7 The Relativistic Equations of Motion 71
Fermi Transport and the Relative Velocity Vector 71
Vorticity, Strain, and Expansion 74
Shear and the Stress Tensor for a Viscous Fluid 78
Divergence of the Einstein Tensor: Gravitational "Force" 80
The Equations of Motion 82
Geodesies and Constants of Motion 85
Tidal Forces 88
8 Light Rays and Fermat's Principle 90
Fermat's Principle of Stationary Time 90
Geodesies in Conformally Related Metrics 93
The Deflection of Light 95
9 Electromagnetism in Three-Space and Minkowski Space 99
Twisted Forms and the Vector Product 99
E, B, and the (Heaviside-) Lorentz Force in Three-Space 100
Electromagnetism in Minkowski Space 102
Integration of Twisted Forms 103
The Charge-Current Three-Form in Minkowski Space 105
The Hodge *-Operator 106
The Laws of Gauss and Ampère-Maxwell 108
Faraday's Law and the Absence of Magnetic Monopoles 112
10 Electromagnetism in General Relativity 115
Maxwell's Equations 115
The Electromagnetic Stress-Energy-Momentum Tensor 118
The Reissner Solution 121
Conformal Invariance of Maxwell's Equations 124
Poisson's Equation in a Static Universe 125
Nonstatic Fields 127
11 The Interior Solution 128
Curvature of World Lines and Gravitational Force Potential 128
The Schwarzschild Interior Solution 130
The Oppenheimer-Volkoff Equation 134
12 Cosmology 139
The Einstein Static Universe 139
The Friedmann Cosmology: Assumptions 141
The Friedmann Cosmology: The Solution 144
The (Landau-) Raychaudhuri Equation 149
The Geometry of a Vorticity-Free Flow 152
A Generalized Poisson Equation for Vorticity Free Flows 152
General Vorticity-Free Cosmologies: The Influence of Curvature on Expansion 155
General Vorticity-Free Cosmologies: Singularities 157
General Vorticity-Free Cosmologies: Closed Spatial Universes 159
References 167
Index 169
