Black Holes, Cosmology and Extra Dimensions

The following sections are included:

• Contents

• Notations

At the beginning of the 20th century, only two physical fields were known, electromagnetic and gravitational. Einstein’s theory of special relativity (SR), created in 1902–1905, described quite well the mechanical and electromagnetic phenomena at velocities up to the velocity of light, which had been impossible in the framework of Galileo and Newton’s classical approach. However, Newton’s theory of gravity, which was almost a perfect basis for celestial mechanics and all terrestrial physics, was formulated using the old notions of absolute space and time and could not be reconciled with SR where space and time were unified in Minkowski’s four-dimensional geometry…

The basic content of general relativity (GR) is very well presented in quite a number of well-known textbooks and monographs (e.g., [263, 296, 393, 413, 417]). Assuming that the reader is aware of these fundamentals, in this chapter, above all for reference purposes, we will mention only its basic facts and relations. Many geometric notions (such as those of a vector and a tensor, co- and contravariant components of vectors and tensors, contraction of tensors and so on) are supposed to be known and are used without explanations…

In this chapter, we will begin considering some questions of black-hole physics. A black hole (BH) is, by definition, a space-time region where the gravitational field is so strong that no material bodies or light rays can leave this region and escape to infinity [168].

In the opinion of the majority of astrophysicists, BHs are widespread in the Universe, a great number of BHs formed in the early Universe, they form as a result of massive star evolution, and supermassive BHs (106 to 109 solar masses) are located in the central regions of many types of galaxies, including our own Milky Way Galaxy, quasars and active galactic nuclei. BHs possess many interesting and unusual properties, both theoretical and observational ones, described in detail in the books [168, 198, 296, 413] and others as well as a lot of review papers. We here only give an elementary introduction to this complicated and rapidly developing branch of physics, and we will be mostly restricted to the simplest case of spherical symmetry. Still many of the results to be presented here (e.g., the methodology of building Carter–Penrose diagrams in the general case, exact solutions with scalar fields and others) have not yet been included in standard textbooks, and some of them, to our knowledge, have only been published in journal articles and appear in a book for the first time.

The following sections are included:

• Black holes and massless scalar fields
  • The general STT and the Wagoner transformations
  • Minimally coupled scalar fields
  • Conformally coupled scalar field
  • Anomalous (phantom) fields. The anti-Fisher solution
  • Cold black holes in the anti-Fisher solution
  • Vacuum and electrovacuum in Brans–Dicke theory
  • Summary for massless scalar fields

• Scalar fields with arbitrary potentials. No-go theorems
  • What is the use of no-go theorems?
  • Basic equations
  • Global structure theorems
  • No-hair theorem
  • Two expressions for the mass and the properties of particle-like solutions

• Rotating black holes

• Black hole thermodynamics
  • Four laws of BH thermodynamics
  • Black hole evaporation

• Regular black holes and black universes
  • Different kinds of regular black holes
  • Black universes with a minimally coupled scalar field
  • A black universe in a brane world
  • A black universe with a trapped ghost

The following sections are included:

• The notion of a wormhole

• A wormhole as a time machine

• Wormholes as solutions to gravitational field equations
  • Spherically symmetric wormholes. General properties
  • Wormhole construction by solving the trace of the Einstein equations
  • Alternative gravity and vacuum as wormhole supporters

• Observational effects. Wormhole astrophysics

The following sections are included:

• Preliminaries

• Perturbation equations
  • General form of the field equations
  • Gauge δβ ≡ 0
  • Gauge-invariant perturbations
  • Regularized potential near a throat
  • Regular perturbations near a throat

• Instabilities of the Fisher and anti-Fisher solutions
  • The static solutions
  • Perturbations: The Fisher solution
  • Perturbations: The anti-Fisher solution

• Extensions and related problems

This chapter describes the standard theory of our Universe’s evolution. The physical essence of the phenomena under study is emphasized. For understanding the material, it is sufficient to know the fundamentals of field theory, gravitation and quantum mechanics.

The inflationary paradigm has been developed for about 30 years, beginning with the papers [191, 382, 383]. It has allowed for successfully solving the basic cosmological problems, beginning with the earliest stages at which our Universe was formed and ending with the galaxy formation stage. The existence of an inflationary period in the history of our Universe seems inevitable since it explains a great number of observational facts [232, 272]. The first inflationary mechanisms [191, 383] were based on self-consistent equations of the scalar and gravitational fields. They solved in the most economic manner the key problems of the Big Bang theory: the horizon problem, the flatness problem etc. The key point is that, under certain conditions, the interacting gravitational and scalar fields lead to an exponential growth of initially small spatial regions. The scalar “inflaton” field ultimately decays into fermions, thus heating the Universe [275]…

In the first two sections of this chapter we briefly describe the modern views of galaxy formation. The subsequent sections are devoted to studying the new opportunities arising due to a correct consideration of quantum fluctuations during inflation.

In this chapter, we will describe a few classes of model universes originating in nonlinear multidimensional gravity, without invoking any fields other than the metric one in the initial action. Among them are cosmological models able to describe both inflation in the early Universe and the modern stage of accelerated expansion.

The following sections are included:

• Fine tuning of the universe parameters

• Fine tuning mechanisms
  • Cascade birth of universes in multidimensional spaces
  • Simultaneous formation of space-time and the parameters of the theory
  • Reduction cascades
  • A step of the cascade in detail

• Quadratic gravity as an explicit example
  • Formation of low-energy physics
  • Numerical computations

• Discussion

The following sections are included:

• Appendix
  • A controversy between adherents of multiple universes (M) and an ultimate unified theory (U)
  • Why do correct theories look elegant?
  • Acknowledgements

• Bibliography

• Index