Introduction to General Relativity

Preface


1. Introduction


1.1. Newton's Laws and the Galilean Transformation


1.2. Electromagnetism and the Galilean Transformation


1.3. The Lorentz Transformation


1.4. The World-Line Element


1.5. The Metric Tensor and Physical Law

Problems


2. Tensor Analysis


2.1. Coordinate Transformation of Tensors


2.2. The Metric Tensor


2.3. Parallel Translation of a Tensor


2.4. The Covariant Derivative


2.5. Tensor Products


2.6. Tensor Densities


2.7. The Riemann Curvature Tensor


2.8. The Bianchi Identities and Einstein's Tensor


2.9. Calculation of a Curvature: Example

Problems


3. Equations of Motion of Free Particles


3.1. The Equation of the Geodesic Lines


3.2. The Generation of Christoffel Symbols from the Euler Equations of the Geodesic Theorem


3.3. Gravitational Potentials and Fields


3.4. Differential Equations of the Metric Tensor in Empty Space


3.5. Field Equations in the Presence of Matter


3.6. The Matter—Energy Tensor


3.7. Deviation of Geodesic Lines in a Space with Curvature


3.8. Tangent Flat Spaces

Problems


4. Solutions of the Field Equations


4.1. The Static Spherically Symmetric Gravitation Field


4.2. The Schwarzschild Metric


4.3. Precession of Orbital Perihelion


4.4. Deflection of Light Rays in a Gravitational Field


4.5. Gravitational Red Shift of Spectral Lines


4.6. Application of the Centrosymmetric Metric in Non-Empty Space


4.7. DeSitter's Cosmological Metric


4.8. Field of a Charged Mass Point

Problems


5. Electromagnetism in General Relativity


5.1. The Electromagnetic Potential and Four-Current


5.2. The Tensor of Electromagnetic Fields and the Field Equations


5.3. The Electromagnetic Energy Tensor


5.4. Transformation of Fields: Inertial Frames


5.5. Transformation of the Field Equations: Non-inertial Frame

Problems


6. Gravitational Fields and Waves


6.1. Space and Time Measurements in Non-Inertial Frames


6.1a. Spatial Line Element of a General Metric


6.2. The Schwarzschild Gravitational Field


6.3. Gravitational Waves

Problems


7. Relativity in Cosmology


7.1. Metric Tensor of the Universe: the Robertson—Walker Metric


7.2. The Time-dependent Scale of the Universe


7.3. The Steady State Universe


7.4. Non-Expanding Model of the Universe: the Goedel Metric

Problems


8. Unified Theories and Quantized Theories of General Relativity


8.1. Unified Field Theories


8.2. "Already Unified" Field Theory


8.3. Quantization of the Gravitational Field


9. Minkowski's Coordinates and Orthogonal Transformations


9.1. Orthogonal Lorentz Transformations


9.2. World-Line Four-Vectors


9.3. Forces and Momenta in Three and Four Dimensions


9.4. Relativistic Energy of a Free Particle in an Inertial Frame


9.5. Electromagnetism in the Minkowski Frame


9.6. The Wave Propagation Vector

Problems

Appendix I

Appendix II

Appendix III

Author Index

Subject Index

Other Titles in the Series in Natural Philosophy