General Relativity

Introduction xiii
I Geometry 1
1 Differentiable manifolds 3
1.1 Manifolds 3
1.2 The tangent space and the derivative 8
1.3 Vector bundles 11
1.4 The tangent bundle and vector fields 14
1.5 Vector fields and flows 22
1.6 The cotangent bundle and tensor fields 29
2 Differential forms and integration 35
2.1 Differential forms 35
2.2 Classical vector calculus 40
2.3 Manifolds with boundary 43
2.4 Oriented manifolds 44
2.5 Integration of forms 47
2.6 Stokes’ theorem 49
2.7 The classical integral theorems 51
2.8 An application: conservation of mass 53
3 The metric determines the geometry 57
3.1 The metric tensor 57
3.2 Length of a curve 62
3.3 Isometries and Killing vector fields 62
4 Connections, parallel transport and geodesics 69
4.1 Connections 69
4.2 The Levi-Civita connection 71
4.3 The pullback of bundles and connections 75
4.4 Parallel transport 77
4.5 Geodesics 79
5 Curvature 83
5.1 The Riemann curvature tensor 83
5.2 The Ricci tensor and scalar curvature 87
5.3 Sectional curvature 89
5.4 Curvature and parallel transport 92
5.5 Geodesic deviation and Jacobi fields 95
5.6 Gauss’ lemma and curvature 98
II Electromagnetism and Special Relativity 101
6 Electricity and Magnetism 103
6.1 Coulomb’s and Lorentz force laws 103
6.2 Electrostatics: charges at rest 104
6.3 Electrodynamics: moving charges 106
6.4 The Law of Biot-Savart 107
6.5 There are no magnetic monopoles 109
6.6 Magnetostatics 109
6.7 Varying electric fields 112
6.8 Faraday’s law of induction 113
6.9 Conservation of energy 115
6.10 Conservation of linear momentum 117
6.11 Maxwell’s equations and waves 118
6.12 Galilean transformations and the speed of light 121
7 Special Relativity 123
7.1 The Michelson-Morley experiment 123
7.2 Lorentz tranformations 126
7.3 Minkowski spacetime 133
7.4 Motion of particles and observers in Minkowski Spacetime 136
7.5 Twins 139
7.6 Time travel and causality 141
7.7 Length contraction 143
7.8 Velocities under Lorentz Transformations 145
7.9 Bell’s spaceship paradox 147
7.10 The Doppler effect 149
7.11 Aberration of Light 150
7.12 Muon Decay: An experimental test for Special Relativity 153
7.13 Energy, Momentum and Mass 154
7.14 Electromagnetism and Special Relativity 161
III Gravity and Curvature 169
8 From Minkowski to curved spacetimes 171
8.1 Light cones and causality 171
8.2 Proper time, velocity and momentum 175
8.3 Geodesic motion and Fermi coordinates 176
8.4 Acceleration and Fermi-Walker coordinates 180
8.5 An observer moving with constant acceleration 182
8.6 A Journey to Kepler 22-b 185
8.7 Redshift and blueshift 187
8.8 Fermi-Walker transport for circular motion 188
8.9 The physical meaning of coordinates 190
8.10 The equivalence principle and tidal forces 194
8.11 Tidal forces: newtonian analysis 198
8.12 Time dilation due to acceleration 200
8.13 Gravitational time dilation and redshift 203
8.14 Comparing the two situations 207
9 The Energy-Momentum tensor 209
9.1 The equation of continuity 209
9.2 Euler’s equation 211
9.3 The momentum flux 213
9.4 Perfect fluid energy-momentum tensor 214
9.5 Conservation of energy-momentum 219
9.6 Electromagnetic energy-momentum tensor 221
9.7 The symmetry of T 224
10 The Field equation 227
10.1 Newton’s law and the Poisson equation 227
10.2 Units and dimensions 230
10.3 The Einstein tensor 232
10.4 Newtonian limit and the value of α 234
10.5 Estimate of g00 237
10.6 The cosmological constant 239
10.7 The geometric meaning of Einstein’s equation 242
10.8 The astonishing analogy: geodesic deviation and tidal forces 247
10.9 Einstein’s equation and variational principles 251
10.10 Predictions and tests 258
IV Solutions to Einstein’s Equation 261
11 The Schwarzschild solution 263
11.1 Gravitational potential of a point mass 263
11.2 Spherical symmetry and Birkhoff’s theorem 265
11.3 The Schwarzschild metric 272
11.4 Planetary motion in Newtonian gravity 275
11.5 Timelike geodesics in Schwarzschild spacetime 280
11.6 Precession of Mercury’s Perihelion 287
11.7 Lightlike geodesics in Schwarzschild spacetime 292
11.8 Gravitational bending of light 296
11.9 Conformal maps and Carter-Penrose diagrams 299
11.10 Incoming Eddington-Finkelstein and black holes 307
11.11 Outgoing Eddington-Finkelstein and white holes 309
11.12 Kruskal-Szekeres coordinates 311
11.13 Interior of a non rotating star 315
11.14 Interior of a Uniformly Dense Star 318
11.15 Geometry Inside a Spherical Empty Cavity 319
11.16 Time Machines 320
12 The FLRW metric and Cosmology 323
12.1 The FLRW metric 323
12.2 Lightlike godesics 330
12.3 Conformal flatness and Penrose diagram 333
12.4 Cosmological red shift and Hubble’s law 334
12.5 Age and diameter of the observable universe 339
V Appendices 341
A Linear algebra and tensors 343
A.1 Linear algebra and matrices 343
A.2 Tensor products 345
A.3 Tensors 347
A.4 Change of basis 348
A.5 Contraction of tensors 349
B Topology and analysis 351
B.1 The Hodge star operator 351
B.2 The Picard-Lindelof theorem 352
B.3 Lie groups 353
B.4 The linking number 356
C The future is open 363
D Other geometric results 367
D.1 The theorem of Hopf-Rinow 367
D.2 The theorem of Ambrose 372
D.3 Constant curvature and Cartan-Hadamard 374
D.4 Constant curvature and Killing-Hopf 380
Bibliography 387
Index 389