Spacetime and geometry. An introduction to general relativity

Cover......Page 1
Preface......Page 6
Contents......Page 10
1.1 Prelude......Page 14
1.2 Space and Time, Separately and Together......Page 16
1.3 Lorentz Transformations......Page 25
1.4 Vectors......Page 28
1.5 Dual Vectors (One-Forms)......Page 31
1.6 Tensors......Page 34
1.7 Manipulating Tensors......Page 38
1.8 Maxwell's Equations......Page 42
1.9 Energy and Momentum......Page 43
1.10 Classical Field Theory......Page 50
1.11 Exercises......Page 58
2.1 Gravity as Geometry......Page 61
2.2 What Is a Manifold?......Page 67
2.3 Vectors Again......Page 76
2.4 Tensors Again......Page 81
2.5 The Metric......Page 84
2.6 An Expanding Universe......Page 89
2.7 Causality......Page 91
2.8 Tensor Densities......Page 95
2.9 Differential Forms......Page 97
2.10 Integration......Page 101
2.11 Exercises......Page 103
3.1 Overview......Page 106
3.2 Covariant Derivatives......Page 107
3.3 Parallel Transport and Geodesies......Page 115
3.4 Properties of Geodesies......Page 121
3.5 The Expanding Universe Revisited......Page 126
3.6 The Riemann Curvature Tensor......Page 134
3.7 Properties of the Riemann Tensor......Page 139
3.8 Symmetries and Killing Vectors......Page 146
3.9 Maximally Symmetric Spaces......Page 152
3.10 Geodesic Deviation......Page 157
3.11 Exercises......Page 159
4.1 Physics in Curved Spacetime......Page 164
4.2 Einstein's Equation......Page 168
4.3 Lagrangian Formulation......Page 172
4.4 Properties of Einstein's Equation......Page 178
4.5 The Cosmological Constant......Page 184
4.6 Energy Conditions......Page 187
4.7 The Equivalence Principle Revisited......Page 190
4.8 Alternative Theories......Page 194
4.9 Exercises......Page 203
5.1 The Schwarzschild Metric......Page 206
5.2 Birkhoff's Theorem......Page 210
5.3 Singularities......Page 217
5.4 Geodesies of Schwarzschild......Page 218
5.5 Experimental Tests......Page 225
5.6 Schwarzschild Black Holes......Page 231
5.7 The Maximally Extended Schwarzschild Solution......Page 235
5.8 Stars and Black Holes......Page 242
5.9 Exercises......Page 249
6.1 The Black Hole Zoo......Page 251
6.2 Event Horizons......Page 252
6.3 Killing Horizons......Page 257
6.4 Mass, Charge, and Spin......Page 261
6.5 Charged (Reissner-Nordstrom) Black Holes......Page 267
6.6 Rotating (Kerr) Black Holes......Page 274
6.7 The Penrose Process and Black-Hole Thermodynamics......Page 280
6.8 Exercises......Page 285
7.1 Linearized Gravity and Gauge Transformations......Page 287
7.2 Degrees of Freedom......Page 292
7.3 Newtonian Fields and Photon Trajectories......Page 299
7.4 Gravitational Wave Solutions......Page 306
7.5 Production of Gravitational Waves......Page 313
7.6 Energy Loss Due to Gravitational Radiation......Page 320
7.7 Detection of Gravitational Waves......Page 328
7.8 Exercises......Page 333
8.1 Maximally Symmetric Universes......Page 336
8.2 Robertson-Walker Metrics......Page 342
8.3 The Friedmann Equation......Page 346
8.4 Evolution of the Scale Factor......Page 351
8.5 Redshifts and Distances......Page 357
8.6 Gravitational Lensing......Page 362
8.7 Our Universe......Page 368
8.8 Inflation......Page 378
8.9 Exercises......Page 387
9.1 Introduction......Page 389
9.2 Quantum Mechanics......Page 391
9.3 Quantum Field Theory in Flat Spacetime......Page 398
9.4 Quantum Field Theory in Curved Spacetime......Page 407
9.5 The Unruh Effect......Page 415
9.6 The Hawking Effect and Black Hole Evaporation......Page 425
A Maps between Manifolds......Page 436
B Diffeomorphisms and Lie Derivatives......Page 442
C Submanifolds......Page 452
D Hypersurfaces......Page 456
E Stokes's Theorem......Page 466
F Geodesic Congruences......Page 472
G Conformal Transformations......Page 480
H Conformal Diagrams......Page 484
I The Parallel Propagator......Page 492
J Noncoordinate Bases......Page 496
Bibliography......Page 508
Index......Page 514