Introduction to General Relativity (eBook)

Front Cover 1
Introduction to General Relativity 4
Copyright Page 5
Table of Contents 6
Preface 8
Chapter 1. Introduction 10
1.1. Newton's Laws and the Galilean Transformation 12
1.2. Electromagnetism and the Galilean Transformation 15
1.3. The Lorentz Transformation 16
1.4. The World-line Element 23
1.5. The Metric Tensor and Physical Law 25
Chapter 2. Tensor Analysis 32
2.1. Coordinate Transformation of Tensors 33
2.2. The Metric Tensor 38
2.3. Parallel Translation of a Tensor 47
2.4. The Covariant Derivative 54
2.5. Tensor Products 58
2.6. Tensor Densities 60
2.7. The Riemann Curvature Tensor 61
2.8. The Bianchi Identities and Einstein's Tensor 66
2.9. Calculation of a Curvature: Example 67
Chapter 3. Equations of Motion of Free Particles 75
3.1. The Equation of the Geodesic Lines 76
3.2. The Generation of Christoffel Symbols from the Euler Equations of the Geodesic Theorem 79
3.3. Gravitational Potentials and Fields 85
3.4. Differential Equations of the Metric Tensor in Empty Space 89
3.5. Field Equations in the Presence of Matter 93
3.6. The Matter-Energy Tensor 96
3.7. Deviation of Geodesic Lines in a Space with Curvature 99
3.8. Tangent Flat Spaces 101
Chapter 4. Solutions of the Field Equations 107
4.1. The Static Spherically Symmetric Gravitation Field 108
4.2. The Schwarzschild Metric 112
4.3. Precession of Orbital Perihelion 114
4.4. Deflection of Light Rays in a Gravitational Field 116
4.5. Gravitational Red Shift of Spectral Lines 118
4.6. Application of the Centrosymmetric Metric in Non-empty Space 120
4.7. DeSitter's Cosmological Metric 121
4.8. Field of a Charged Mass Point 124
Chapter 5. Electromagnetism in General Relativity 129
5.1. The Electromagnetic Potential and Four-current 130
5.2. The Tensor of Electromagnetic Fields and the Field Equations 131
5.3. The Electromagnetic Energy Tensor 134
5.4. Transformation of Fields: Inertial Frames 137
5.5. Transformation of the Field Equations: Non-inertial Frame 139
Chapter 6. Gravitat~onal Fields and Waves 145
6.1. Space and Time Measurements in Non-inertial Frames 145
6.1 a. Spatial Line Element of a General Metric 149
6.2. The Schwarzschild Gravitational Field 150
6.3. Gravitational Waves 160
Chapter 7. Relcitivity in Cosmology 169
7.1. Metric Tensor of the Universe: the Robertson— Walker Metric 172
7.2. The Time-dependent Scale of the Universe 184
7.3. The Steady State Universe 190
7.4. Non-expanding Model of the Universe: the Goedel Metric 193
Chapter 8. Unified Theories and Quantized Theories of General Relcitivity 197
8.1. Unified Field Theories 197
8.2. "Already Unified" Field Theory 200
8.3. Quantization of the Gravitational Field 202
Chapter 9. Minkowski's Coordinates and Orthogonal Transformations 205
9.1. Orthogonal Lorentz Transformations 205
9.2. World-line Four-vectors 209
9.3. Forces and Momenta in Three and Four Dimensions 212
9.4. Relativistic Energy of a Free Particle in an Inertial Frame 214
9.5. Electromagnetism in the Minkowski Frame 216
9.6. The Wave Propagation Vector 218
Appendix I: Numerical Data and Physical Constants 221
Appendix II 222
Appendix III: Euler's Equation of the Calculus of Variations 223
Author Index 226
Subject Index 228
Other Titles in the Series in Icitural Philosophy 232