Lecture notes on general relativity

Contents......Page 4
Preface......Page 7
Notation and Conventions......Page 9
Part I. Fundamentals......Page 12
1.1 Introduction......Page 13
1.2 Space and Time in Prerelativity Physics and in Special Relativity......Page 14
1.3 The Spacetime Metric......Page 16
1.4 General Relativity......Page 18
2.1 Manifolds......Page 21
2.2 Vectors......Page 24
2.3 Tensors; the Metric Tensor......Page 28
2.4 The Abstract Index Notation......Page 33
3. Curvature......Page 39
3.1 Derivative Operators and Parallel Transport......Page 40
3.2 Curvature......Page 46
3.3 Geodesics......Page 51
3.4 Methods for Computing Curvature......Page 57
4.1 The Geometry of Space in Prerelativity Physics; General and Special Covariance......Page 65
4.2 Special Relativity......Page 69
4.3 General Relativity......Page 76
4.4 Linearized Gravity: The Newtonian Limit and Gravitational Radiation......Page 84
5.1 Homogeneity and Isotropy......Page 101
5.2 Dynamics of a Homogeneous, Isotropic Universe......Page 106
5.3 The Cosmological Redshift; Horizons......Page 111
5.4 The Evolution of Our Universe......Page 117
6. The Schwarzschild Solution......Page 128
6.1 Derivation of the Schwarzschild Solution......Page 129
6.2 Interior Solutions......Page 135
6.3 Geodesics of Schwarzschild: Gravitational Redshift, Perihelion Precession, Bending of Light, and Time Delay......Page 146
6.4 The Kruskal Extension......Page 158
Part II. Advanced Topics......Page 169
7. Methods for Solving Einstein\'s Equation......Page 170
7.1 Stationary, Axisymmetric Solutions......Page 171
7.2 Spatially Homogeneous Cosmologies......Page 177
7.3 Algebraically Special Solutions......Page 188
7.4 Methods for Generating Solutions......Page 189
7.5 Perturbations......Page 192
8. Causal Structure......Page 197
8.1 Futures and Pasts: Basic Definitions and Results......Page 198
8.2 Causality Conditions......Page 204
8.3 Domains of Dependence; Global Hyperbolicity......Page 209
9. Singularities......Page 220
9.1 What Is a Singularity?......Page 221
9.2 Timelike and Null Geodesic Congruences......Page 225
9.3 Conjugate Points......Page 232
9.4 Existence of Maximum Length Curves......Page 242
9.5 Singularity Theorems......Page 246
10. The Initial Value Formulation......Page 252
10.1 Initial Value Formulation for Particles and Fields......Page 253
10.2 Initial Value Formulation of General Relativity......Page 261
11. Asymptotic Flatness......Page 278
11.1 Conformal Infinity......Page 280
11.2 Energy......Page 294
12. Black Holes......Page 307
12.1 Black Holes and the Cosmic Censor Conjecture......Page 308
12.2 General Properties of Black Holes......Page 317
12.3 The Charged Kerr Black Holes......Page 321
12.4 Energy Extraction from Black Holes......Page 333
12.5 Black Holes and Thermodynamics......Page 339
13. Spinors......Page 349
13.1 Spinors in Minkowski Spacetime......Page 351
13.2 Spinors in Curved Spacetime......Page 368
14. Quantum Effects in Strong Gravitational Fields......Page 387
14.1 Quantum Gravity......Page 389
14.2 Quantum Fields in Curved Spacetime......Page 398
14.3 Particle Creation near Black Holes......Page 408
14.4 Black Hole Thermodynamics......Page 425
Appendices......Page 430
A. Topological Spaces......Page 431
B.1 Differential Forms......Page 436
B.2 Integration......Page 437
B.3 Frobenius\'s Theorem......Page 442
C.1 Maps of Manifolds......Page 445
C.2 Lie Derivatives......Page 447
C.3 Killing Vector Fields......Page 449
D. Conformal Transformations......Page 453
E.1 Lagrangian Formulation......Page 458
E.2 Hamiltonian Formulation......Page 467
F. Units and Dimensions......Page 478
References ......Page 480
Index......Page 492