Table of Contents
Notations xiii
Chapter 1 Modern ideas of gravitation and cosmology - a brief essay 1
Part I Gravitation 23
Chapter 2 Fundamentals of general relativity 25
2.1 Special relativity. Minkowski geometry 26
2.1.1 Geometry 26
2.1.2 Coordinate transformations 27
2.1.3 Kinematic effects 29
2.1.4 Elements of relativistic point mechanics 31
2.2 Riemannian space-time. Coordinate systems and reference frames 33
2.2.1 Covariance, maps and atlases 34
2.2.2 Reference frames and relativity 35
2.2.3 Reference frames and chronometric invariants 36
2.2.4 Covariance and relativity 39
2.3 Riemannian space-time. Curvature 40
2.4 The gravitational field action and dynamic equations 43
2.4.1 The Einstein equations 43
2.4.2 Geodesic equations 44
2.4.3 The correspondence principle 46
2.5 Macroscopic matter and nongravitational fields in GR 46
2.5.1 Perfect fluid 47
2.5.2 Scalar fields 48
2.5.3 The electromagnetic field 48
2.6 The most symmetric spaces 50
2.6.1 Isometry groups and killing vectors 50
2.6.2 Isotropic cosmology. The dS and AdS spaces 51
Chapter 3 Spherically symmetric space-times. Black holes 55
3.1 Spherically symmetric gravitational fields 55
3.1.1 A regular centre and asymptotic flatness 59
3.2 The Reissner-Nordström-(anti-)de Sitter solution 60
3.2.1 Solution of the Einstein equations 60
3.2.2 Special cases 62
3.3 Horizons and geodesics in static, spherically symmetric space-times 65
3.3.1 The general form of geodesic equations 66
3.3.2 Horizons, geodesics and the quasiglobal coordinate 67
3.3.3 Transitions to Lemaître reference frames 70
3.3.4 Horizons, R- and T-regions 73
3.4 Schwarzschild black holes. Geodesics and a global description 74
3.4.1 R- and T-regions 74
3.4.2 Geodesics in the R-region 75
3.4.3 Particle capture by a black hole 77
3.4.4 A global description: The Kruskal metric 80
3.4.5 From Kruskal to Carter-Penrose diagram for the Schwarzschild metric 82
3.5 The global causal structure of space-times with horizons 83
3.5.1 Crossing the horizon in the general case 83
3.5.2 Construction of Carter-Penrose diagrams 86
3.6 A black hole as a result of gravitational collapse 90
3.6.1 Internal and external regions. Birkhoff's theorem 90
3.6.2 Gravitational collapse of a spherical dust cloud 93
Chapter 4 Black holes under more general conditions 97
4.1 Black holes and massless scalar fields 97
4.1.1 The general STT and the Wagoner transformations 97
4.1.2 Minimally coupled scalar fields 100
4.1.3 Conformally coupled scalar field 104
4.1.4 Anomalous (phantom) fields. The anti-Fisher solution 107
4.1.5 Cold black holes in the anti-Fisher solution 109
4.1.6 Vacuum and electrovacuum in Brans-Dicke theory 110
4.1.7 Summary for massless scalar fields 111
4.2 Scalar fields with arbitrary potentials. No-go theorems 112
4.2.1 What is the use of no-go theorems? 112
4.2.2 Basic equations 114
4.2.3 Global structure theorems 116
4.2.4 No-hair theorem 118
4.2.5 Two expressions for the mass and the properties of particle-like solutions 122
4.3 Rotating black holes 124
4.4 Black hole thermodynamics 127
4.4.1 Four laws of BH thermodynamics 127
4.4.2 Black hole evaporation 129
4.5 Regular black holes and black universes 130
4.5.1 Different kinds of regular black holes 130
4.5.2 Black universes with a minimally coupled scalar field 135
4.5.3 A black universe in a brane world 140
4.5.4 A black universe with a trapped ghost 147
Chapter 5 Wormholes 155
5.1 The notion of a wormhole 155
5.2 A wormhole as a time machine 159
5.3 Wormholes as solutions to gravitational field equations 162
5.3.1 Spherically symmetric wormholes. General properties 163
5.3.2 Wormhole construction by solving the trace of the Einstein equations 168
5.3.3 Alternative gravity and vacuum as wormhole supporters 175
5.4 Observational effects. Wormhole astrophysics 179
Chapter 6 Stability of spherically symmetric configurations 183
6.1 Preliminaries 183
6.2 Perturbation equations 185
6.2.1 General form of the field equations 186
6.2.2 Gauge δβ ≡ 0 187
6.2.3 Gauge-invariant perturbations 189
6.2.4 Regularized potential near a throat 190
6.2.5 Regular perturbations near a throat 193
6.3 Instabilities of the Fisher and anti-Fisher solutions 194
6.3.1 The static solutions 194
6.3.2 Perturbations: The Fisher solution 197
6.3.3 Perturbations: The anti-Fisher solution 197
6.4 Extensions and related problems 202
Part II Cosmology 205
Chapter 7 Stages of the Universe's evolution 207
7.1 The cosmological principle and the Einstein equations 207
7.2 De Sitter space 214
7.3 Inflation 217
7.4 Post-inflationary stages 222
7.4.1 Post-inflationary reheating of the Universe 222
7.4.2 The radiation-dominated stage 225
7.4.3 The matter-dominated stage 226
7.4.4 The modern stage of accelerated expansion (secondary inflation) 228
7.4.5 Future of the Universe: Is a Big Rip expected? 230
7.5 The scale factor in the general case 232
7.6 Why do we need an inflationary period? 234
7.6.1 The flatness problem 235
7.6.2 The initial size of the Universe 236
7.6.3 Causal connections at inflation and after it 237
7.7 Basic properties of expanding space 238
7.7.1 The redshift 238
7.7.2 The luminosity distance 240
7.7.3 The velocity of particles in FRW space-time 241
Chapter 8 Field dynamics in the inflationary period 243
8.1 Quadratic inflation 244
8.2 Quantum fluctuations during inflation 246
8.3 Hybrid inflation 255
8.4 Influence of massive fields on the process of inflation 257
8.5 Suppression of vacuum decay by virtual particles 263
Chapter 9 The large-scale structure 271
9.1 The cosmic microwave background 271
9.2 The development of density fluctuations 277
9.2.1 Density fluctuations in Minkowski space 277
9.2.2 Density perturbations in the expanding Universe 278
9.3 The baryonic asymmetry of the Universe 279
9.3.1 Baryogenesis 280
9.3.2 Large-scale fluctuations of the baryonic charge 284
9.4 Massive primordial black holes 289
9.4.1 Field fluctuations near an extremum of the potential 290
9.4.2 A specific example 292
9.4.3 Suppressed intermediate-mass black hole formation 294
9.4.4 PBH mass spectra and the scalar field dynamics 297
9.4.5 Discussion 300
Part III Extra Dimensions 303
Chapter 10 Multidimensional gravity 309
10.1 Compact extra dimensions. A brief review 309
10.1.1 A Kaluza-Klein model with a single extra dimension 311
10.1.2 Kaluza-Klein models. The general case 318
10.2 Multidimensional gravity with higher-order derivatives. Basic equations 322
10.2.1 F(R)-theory 322
10.2.2 Slow-change approximation. The Einstein frame 324
10.2.3 The first generalization: A more general form of the Lagrangian 330
10.2.4 The second generalization: Several extra factor spaces 332
10.2.5 Slow-change approximation. Reduction to d0 dimensions 333
10.3 Extra dimensions and low-energy physics 335
10.3.1 Self-stabilization of an extra space 336
10.3.2 On the influence of the number of extra dimensions on low-energy physics 338
10.3.3 Extra dimensions and inflation 340
10.3.4 Two factor spaces: Inflation and modern acceleration 343
10.3.5 Rapid particle creation in the post-inflationary period 350
10.3.6 Conclusions 358
10.4 The origin of gauge symmetries and fundamental constants 358
10.4.1 Why is the extra space symmetric? 360
10.4.2 Fundamental constants and the properties of an extra space 365
Chapter 11 The emergence of physical laws 371
11.1 Fine tuning of the universe parameters 371
11.2 Fine tuning mechanisms 377
11.2.1 Cascade birth of universes in multidimensional spaces 378
11.2.2 Simultaneous formation of space-time and the parameters of the theory 379
11.2.3 Reduction cascades 380
11.2.4 A step of the cascade in detail 382
11.3 Quadratic gravity as an explicit example 388
11.3.1 Formation of low-energy physics 391
11.3.2 Numerical computations 394
11.4 Discussion 396
Appendix 405
A.1 A controversy between adherents of multiple universes (M) and an ultimate unified theory (U) 405
A.2 Why do correct theories look elegant? 406
Bibliography 409
Index 425
