A working knowledge of Einstein's theory of general relativity is an essential tool for every physicist. This book serves as an introductory text on the subject aimed at first-year graduate students, or advanced undergraduates, in physics. Preface vii 1. Introduction 1 2. Particle on a Two-Dimensional Surface 9 2.1 Infinitesimal Displacements 10 2.2 Lagrange's Equations 12 2.3 Reciprocal Basis 14 2.4 Geodesic Motion 19 2.5 Role of Coordinates 24 3. Curvilinear Coordinate Systems 27 3.1 Line Element 27 3.2 Reciprocal Basis 29 3.3 Metric 29 3.4 Vectors 30 3.5 Tensors 31 3.6 Change of Basis 33 3.7 Transformation Law 36 3.8 Affine Connection 39 3.9 Example: Polar Coordinates 41 4. Particle on a Two-Dimensional Surface-Revisited 43 4.1 Motion in Three-Dimensional Euclidian Space 43 4.2 Parallel Displacement 46 5. Some Tensor Analysis 51 5.1 Covariant Derivative 51 5.2 The Riemann Curvature Tensor 54 5.3 Second Covariant Derivative 60 5.4 Covariant Differentiation 64 5.5 Symmetry Properties of the Riemann Tensor 67 5.6 Bianchi Identities 68 5.7 The Einstein Tensor 69 5.8 Example-Surface of a Sphere 72 5.9 Volume element 76 6. Special Relativity 87 6.1 Basic Principles 87 6.2 Four-Vectors 88 6.3 Relativistic Particle Motion 91 6.4 Lorentz Transformations 98 6.5 General Tensor Transformation Law 109 6.6 Relativistic Hydrodynamics 111 6.7 Transition to General Relativity 120 7. General Relativity 123 7.1 Einstein's Theory 123 7.2 Newtonian Limit 126 7.3 The Equivalence Principle 131 7.4 Local Freely Falling Frame (LF ) 133 7.5 Spherically Symmetric Solution to Field Equations 135 7.6 Solution in Vacuum 143 7.7 Interpretation of Schwarzschild Metric 148 7.8 A Few Applications 152 8. Precession of Perihelion 155 8.1 Lagrangian 156 8.2 Equations of Motion 159 8.3 Equilibrium Circular Orbits 163 8.4 Small Oscillations About Circular Orbits 164 8.5 Sonic Numbers and Comparison with Experiment 169 8.6 Deflection of Light 171 9. Gravitational Redshift 173 9.1 Basic Observation 174 9.2 Frequency Shift 174 9.3 Propagation of Light 176 9.4 A Cyclic Process 178 10. Neutron Stars 181 10.1 Hydrodynamics in General Relativity 181 10.2 Tolman-Oppenheimer-Volkoff (Toy) Equations 190 10.3 Relativistic Mean Field Theory of Nuclear Matter (RMFT) 201 10.4 Neutron Stars and Black Holes 210 11. Cosmology 217 11.1 Uniform Mass Background 218 11.2 Robertson-Walker Metric with k = 0 220 11.3 Solution to Einstein's Equations 225 11.4 Interpretation 229 11.5 Cosmological Redshift 233 11.6 Horizon 237 11.7 Some Comments 239 12. Gravitational Radiation 241 12.1 Linearized Theory 241 12.2 Auxiliary (Lorentz) Condition 245 12.3 Plane-Wave Solution to the Einstein Field Equations 250 12.4 Interpretation 253 12.5 Detection 257 13. Special Topics 263 13.1 Einstein-Hilbert Lagrangian 264 13.2 Cosmological Constant 267 13.3 Additional Scalar Field 268 13.4 Robertson-Walker Metric with k not = to 0 270 13.5 Inflation 275 14. Problems 281 Appendix A Reduction of g?upsilon ?R?upsilon to covariant divergences 321 Appendix B Robertson-Walker Metric with k not = to 0 325 Bibliography 327 Index 331