Spacetime and Geometry: An Introduction to General Relativity

Cover......Page 1
Title......Page 4
Preface......Page 8
Contents......Page 12
 1.1 Prelude......Page 16
 1.2 Space and Time, Separately and Together......Page 18
 1.3 Lorentz Transformations......Page 27
 1.4 Vectors......Page 30
 1.5 Dual Vectors (One-Forms)......Page 33
 1.6 Tensors......Page 36
 1.7 Manipulating Tensors......Page 40
 1.8 Maxwell’s Equations......Page 44
 1.9 Energy and Momentum......Page 45
 1.10 Classical Field Theory......Page 52
 1.11 Exercises......Page 60
 2.1 Gravity as Geometry......Page 63
 2.2 What Is a Manifold?......Page 69
 2.3 Vectors Again......Page 78
 2.4 Tensors Again......Page 83
 2.5 The Metric......Page 86
 2.6 An Expanding Universe......Page 91
 2.7 Causality......Page 93
 2.8 Tensor Densities......Page 97
 2.9 Differential Forms......Page 99
 2.10 Integration......Page 103
 2.11 Exercises......Page 105
 3.1 Overview......Page 108
 3.2 Covariant Derivatives......Page 109
 3.3 Parallel Transport and Geodesics......Page 117
 3.4 Properties of Geodesics......Page 123
 3.5 The Expanding Universe Revisited......Page 128
 3.6 The Riemann Curvature Tensor......Page 136
 3.7 Properties of the Riemann Tensor......Page 141
 3.8 Symmetries and Killing Vectors......Page 148
 3.9 Maximally Symmetric Spaces......Page 154
 3.10 Geodesic Deviation......Page 159
 3.11 Exercises......Page 161
 4.1 Physics in Curved Spacetime......Page 166
 4.2 Einstein’s Equation......Page 170
 4.3 Lagrangian Formulation......Page 174
 4.4 Properties of Einstein’s Equation......Page 180
 4.5 The Cosmological Constant......Page 186
 4.6 Energy Conditions......Page 189
 4.7 The Equivalence Principle Revisited......Page 192
 4.8 Alternative Theories......Page 196
 4.9 Exercises......Page 205
 5.1 The Schwarzschild Metric......Page 208
 5.2 Birkhoff’s Theorem......Page 212
 5.3 Singularities......Page 219
 5.4 Geodesics of Schwarzschild......Page 220
 5.5 Experimental Tests......Page 227
 5.6 Schwarzschild Black Holes......Page 233
 5.7 The Maximally Extended Schwarzschild Solution......Page 237
 5.8 Stars and Black Holes......Page 244
 5.9 Exercises......Page 251
 6.1 The Black Hole Zoo......Page 253
 6.2 Event Horizons......Page 254
 6.3 Killing Horizons......Page 259
 6.4 Mass, Charge, and Spin......Page 263
 6.5 Charged (Reissner-Nordstrom) Black Holes......Page 269
 6.6 Rotating (Kerr) Black Holes......Page 276
 6.7 The Penrose Process and Black-Hole Thermodynamics......Page 282
 6.8 Exercises......Page 287
 7.1 Linearized Gravity and Gauge Transformations......Page 289
 7.2 Degrees of Freedom......Page 294
 7.3 Newtonian Fields and Photon Trajectories......Page 301
 7.4 Gravitational Wave Solutions......Page 308
 7.5 Production of Gravitational Waves......Page 315
 7.6 Energy Loss Due to Gravitational Radiation......Page 322
 7.7 Detection of Gravitational Waves......Page 330
 7.8 Exercises......Page 335
 8.1 Maximally Symmetric Universes......Page 338
 8.2 Robertson-Walker Metrics......Page 344
 8.3 The Friedmann Equation......Page 348
 8.4 Evolution of the Scale Factor......Page 353
 8.5 Redshifts and Distances......Page 359
 8.6 Gravitational Lensing......Page 364
 8.7 Our Universe......Page 370
 8.8 Inflation......Page 380
 8.9 Exercises......Page 389
 9.1 Introduction......Page 391
 9.2 Quantum Mechanics......Page 393
 9.3 Quantum Field Theory in Flat Spacetime......Page 400
 9.4 Quantum Field Theory in Curved Spacetime......Page 409
 9.5 The Unruh Effect......Page 417
 9.6 The Hawking Effect and Black Hole Evaporation......Page 427
A Maps between Manifolds......Page 438
B Diffeomorphisms and Lie Derivatives......Page 444
C Submanifolds......Page 454
D Hypersurfaces......Page 458
E Stokes’s Theorem......Page 468
F Geodesic Congruences......Page 474
G Conformal Transformations......Page 482
H Conformal Diagrams......Page 486
I The Parallel Propagator......Page 494
J Noncoordinate Bases......Page 498
Bibliography......Page 510
Index......Page 516