Black holes, cosmology and extra dimensions

Contents
Acknowledgments
Notations
Abbreviations
1. Modern ideas of gravitation and cosmology — a brief essay
	1.1 Einstein after Einstein
	1.2 The technological breakthrough
	1.3 To quantize or not?
	1.4 The zoo of theories
	1.5 Gravitation and the Universe
Part I. Gravitation
	2. Fundamentals of general relativity
		2.1 Special relativity — Minkowski geometry
			2.1.1 Geometry
			2.1.2 Coordinate transformations
			2.1.3 Kinematic effects
			2.1.4 Elements of relativistic point mechanics
		2.2 Riemannian space–time — coordinate systems and reference frames
			2.2.1 Covariance, maps and atlases
			2.2.2 Reference frames and relativity
			2.2.3 Reference frames and chronometric invariants
			2.2.4 Covariance and relativity
		2.3 Riemannian space–time — curvature
		2.4 Gravitational field action and dynamic equations
			2.4.1 The Einstein equations
			2.4.2 Geodesic equations
			2.4.3 The correspondence principle
		2.5 Macroscopic matter and nongravitational fields in GR
			2.5.1 Perfect fluids
			2.5.2 Anisotropic fluids
			2.5.3 Scalar fields
			2.5.4 The electromagnetic field
		2.6 Energy conditions
		2.7 The most symmetric spaces
			2.7.1 Isometry groups and Killing vectors
			2.7.2 Isotropic cosmology — the dS and AdS spaces
	3. Spherically symmetric space–times: Black holes
		3.1 Spherically symmetric gravitational fields
			3.1.1 A regular center and asymptotic flatness
		3.2 The Reissner–Nordström–(anti-)de Sitter solution
			3.2.1 Solution of the Einstein equations
			3.2.2 Special cases of the RNdS solution
		3.3 Horizons and geodesics in static, spherically symmetric space–times
			3.3.1 The general form of geodesic equations
			3.3.2 Horizons, geodesics and the quasiglobal coordinate
			3.3.3 Null geodesics and gravitational lensing
			3.3.4 Transitions to Lemaître reference frames
			3.3.5 Horizons, R- and T-regions
		3.4 Schwarzschild black holes — geodesics and a global description
			3.4.1 R- and T-regions
			3.4.2 Geodesics in the R-region
			3.4.3 Massive and massless particle capture by a black hole
			3.4.4 A global description — the Kruskal metric
			3.4.5 From Kruskal to Carter–Penrose diagram for the Schwarzschild metric
		3.5 Global causal structure of space–times with horizons
			3.5.1 Crossing horizons in the general case
			3.5.2 Construction of Carter–Penrose diagrams
		3.6 A black hole as a result of gravitational collapse
			3.6.1 Internal and external regions — Birkhoff’s theorem
			3.6.2 Gravitational collapse of a spherical dust cloud
	4. Spherical space–times with scalar fields
		4.1 Massless minimally coupled scalar fields in GR
			4.1.1 Solutions with a massless scalar field
			4.1.2 Phantom scalar fields — the anti-Fisher solution
			4.1.3 Cold black holes in the anti-Fisher solution
		4.2 Scalar fields in some extensions of GR
			4.2.1 The general STT, Jordan and Einstein frames
			4.2.2 F(R) gravity
			4.2.3 Hybrid metric-Palatini gravity
		4.3 Geometries with conformally coupled scalar fields
			4.3.1 Models with a canonical field (ε = 1)
			4.3.2 Models with a phantom field (ε = −1)
		4.4 Some other STT solutions
			4.4.1 Solutions with nonconformal coupling
			4.4.2 Vacuum and electrovacuum solutions in the Brans–Dicke theory
			4.4.3 Summary for massless scalar fields
		4.5 Scalar fields with arbitrary potentials — no-go theorems
			4.5.1 What is the use of no-go theorems?
			4.5.2 Basic equations
			4.5.3 Global structure theorems
			4.5.4 No-hair theorem
			4.5.5 Two expressions for the mass and the properties of particle-like solutions
	5. More about black holes
		5.1 Examples of black holes with nonzero scalar field potentials
			5.1.1 Example 1: r2(x) = x2 − a2
			5.1.2 Example 2: r2(x) = x2 + a2
			5.1.3 Extensions
		5.2 Regular black holes and black universes
			5.2.1 Different kinds of regular black holes
			5.2.2 Black universes with minimally coupled scalar fields
			5.2.3 Regular black holes in a brane world
		5.3 Rotating black holes
		5.4 Black hole thermodynamics and evaporation
			5.4.1 Four laws of BH thermodynamics
			5.4.2 Black hole evaporation
	6. Wormholes
		6.1 The notion of a wormhole
		6.2 A wormhole as a time machine
		6.3 Wormholes as solutions to gravitational field equations
			6.3.1 Spherically symmetric wormholes — general properties
			6.3.2 Wormholes in the anti-Fisher solution
			6.3.3 Wormholes with nonminimally coupled scalar fields
			6.3.4 Wormhole construction by solving the trace of the Einstein equations
			6.3.5 Alternative gravity and vacuum as wormhole supporters
			6.3.6 Wormholes with a trapped ghost
		6.4 Cylindrical wormholes
			6.4.1 Wormhole definitions and existence conditions
			6.4.2 Examples of static wormholes
			6.4.3 Examples of rotating wormholes
		6.5 Observational effects — wormhole astrophysics
	7. Stability of spherically symmetric configurations
		7.1 Preliminaries
		7.2 Perturbation equations
			7.2.1 General relations
			7.2.2 Master equation
			7.2.3 Gauge-invariant perturbations
		7.3 Throats, trapped ghosts and stability
			7.3.1 Perturbations near a throat
			7.3.2 Regular perturbations near a throat
			7.3.3 Perturbations near a transition surface Strans (where h(ϕ) = 0)
			7.3.4 Transition surfaces and throats
		7.4 The stability problem for some particular solutions
			7.4.1 Instabilities of the Fisher and anti-Fisher solutions
			7.4.2 Black universes
			7.4.3 Wormholes with a trapped ghost
		7.5 Extensions and related problems
Part II. Cosmology
	8. Stages of the Universe’s evolution
		8.1 The cosmological principle and the Einstein equations
			8.1.1 Some solutions for the scale factor
		8.2 De Sitter space
			8.2.1 Is there real expansion in de Sitter space?
		8.3 Inflation
		8.4 Post-inflationary stages
			8.4.1 Post-inflationary reheating of the Universe
			8.4.2 The radiation-dominated stage
			8.4.3 The matter-dominated stage
			8.4.4 The modern stage of accelerated expansion (secondary inflation)
			8.4.5 Future of the Universe — is a Big Rip expected?
		8.5 The scale factor in the general case
		8.6 Why do we need an inflationary period?
			8.6.1 The flatness problem
			8.6.2 The initial size of the Universe
			8.6.3 Causal connections at inflation and after it
		8.7 Basic properties of expanding space
			8.7.1 The redshift
			8.7.2 The luminosity distance
			8.7.3 The velocity of particles in FRW space–time
	9. Field dynamics in the inflationary period
		9.1 Quadratic inflation
		9.2 Quantum fluctuations during inflation
			9.2.1 A brief analysis
			9.2.2 A detailed consideration
			9.2.3 Homogeneous fluctuations
		9.3 Influence of massive fields on the process of inflation
		9.4 Suppression of vacuum decay by virtual particles
		9.5 Geometric approach to inflation
			9.5.1 f(R) models of inflation
			9.5.2 The Starobinsky model
	10. The large-scale structure
		10.1 The cosmic microwave background
			10.1.1 Classical evolution of quantum fluctuations
		10.2 The development of density fluctuations
			10.2.1 Density fluctuations in Minkowski space
			10.2.2 Density perturbations in the expanding Universe
		10.3 The baryonic asymmetry of the Universe
			10.3.1 Baryogenesis
			10.3.2 Large-scale fluctuations of the baryonic charge
		10.4 Primordial black holes as the source of large-scale inhomogeneities
			10.4.1 The basic idea
			10.4.2 Field fluctuations near an extremum of the potential
			10.4.3 Size distribution of the closed walls
			10.4.4 Wall formation and collapse
			10.4.5 PBH cluster structure — a specific example
			10.4.6 Processes inside the PBH clusters
			10.4.7 Discussion
Part III. Extra dimensions
	11. Multidimensional gravity with higher-order derivatives
		11.1 Mathematical tools for multidimensional gravity with higher derivatives
		11.2 Compact extra dimensions — a brief review
			11.2.1 Scalar fields and extra spaces
			11.2.2 A Kaluza–Klein model with a single extra dimension
			11.2.3 Kaluza–Klein models — the general case
		11.3 Multidimensional gravity as the effective theory
			11.3.1 f(R) theory
			11.3.2 Slow-change approximation — the Einstein frame
			11.3.3 The first generalization — a more general form of the Lagrangian
			11.3.4 The second generalization — several extra factor spaces
			11.3.5 Slow-change approximation — reduction to d0 dimensions
		11.4 Possible manifestations of extra dimensions in the modern world
			11.4.1 Self-stabilization of an extra space
			11.4.2 Relation of the number of extra dimensions and the cosmological constant
			11.4.3 Rapid particle creation in the post-inflationary period
			11.4.4 Multiple factor spaces and a spatially varying size of the extra dimensions
			11.4.5 Funnel solutions — the method of trial functions
		11.5 Extra dimensions and inflation
			11.5.1 The model
			11.5.2 Limits on the model parameters
			11.5.3 Slow rolling and all that
			11.5.4 Comparison with observations
		11.6 Cosmological limits on the parameters of extra dimensions
			11.6.1 Limit to the size of extra dimensions
			11.6.2 Limit on the D-dimensional Planck mass
		11.7 Conclusion
	12. The emergence of physical laws
		12.1 Fine tuning of the physical parameters
			12.1.1 The electron and the properties of the Universe
			12.1.2 The carbon level
			12.1.3 Slow reactions in stars
			12.1.4 Stellar lifetime
			12.1.5 Passage through a needle’s eye
			12.1.6 It is good time to ask a question — what should an Ultimate Theory look like?
			12.1.7 Choosing particles and interactions
		12.2 The origin of symmetries and fundamental constants
			12.2.1 Fundamental constants and the properties of an extra space
			12.2.2 Why is the extra space symmetric?
		12.3 Cascade birth of Universes in multidimensional spaces
			12.3.1 Simultaneous formation of space–time and the parameters of the theory
			12.3.2 Reduction cascades
			12.3.3 A step of the cascade in detail
			12.3.4 Quadratic gravity as an explicit example
			12.3.5 Numerical computations
		12.4 Discussion
		12.5 Final remark
Appendix
	A.1 A controversy between adherents of multiple universes (M) and an ultimate unified theory (U)
	A.2 Why do correct theories look elegant?
Bibliography
Index