1.1. Schwarzschild black holes.
1.2. Poststars.
1.3. Relics of the Big Ban.
1.4. Kerr-Newman space-time and transmetagalactic otons.
1.5. Other ways of black hole formation

Before discussing the problem of otons in the Earth's physics it is necessary first to provide a definition of the term and answer, at least, two questions. The first, it is a must to find out how otons of small masses can be formed? The second, the question should to be answered: in what ways otons appear in space bodies and in the Earth? The answer to the first question is given in the first chapter, and the answer to the second one is given in the second chapter.

The term "otons" was discussing introduced in 1971 by Ya.B.Zel'dovich and I.D.Novikov in the book "The Theory of Gravitation and the Evolution of Stars" [╟х10] "as a generic name uniting all the variety of bodies with the relativistic field of gravitation which are inside the so called "horizon" or asymptotically come close to it". Black, white, grey holes and other relativistic objects predicted within the General Relativity (GR) are refered to otons. In the book the term "otons" is often used just as a synonym of the term "black holes".

1.1. Schwarzschild black holes (to the top). Equations of the modern relativistic theory of gravitation (General Relativity):

 

ааааааааааааааааааааа G_im = and (T_im - 1/2g_imT)аааааааааааааааааааааааааааааааааааа (1.1.1.)

 

were obtained by Einstein in 1915. And the next year the first exact solution of Einstein equations for the point mass was found by K.Schwarzschild. The Schwarzschild solution is written in the metric form which has served for the basis in constructing models of simplest spherical symmetric otons:

 

ааааааааа ааааааааааааааааааа а(1.1.2.)

 

where R_g = 2GMc^-2 Ч gravitational radius, Ì Ч black hole mass, c Ч speed of light, G Ч gravitational constant. A sphere drawn with the gravitational radius is for a Schwarzschild black hole a surface of infinite red shift and the event horizon. For estimating sizes of stellar mass black holes the formula is convenient:

 

аааааааааааааааааа а аааааааааааааааааааааааааааааааааааааааа (1.1.3)

where M _o Ч mass of the Sun equal to 2 а10³³ ã, R _o а Ч gravitational radius of the Sun equal to 3 а10⁵ sm. To small black holes it is possible to refer ones of masses in the bounds of 10²⁰ g < M_âí < M _o and, accordingly, of sizes in the bounds of 10^-8 sm < R_g < 3 а10⁵ sm. The bottom limit is defined by the value of the order of atomic sizes. For black holes with Ì < 10²⁰ g quantum effects of evaporation become appreciable and such black holes should be refered to micro-black holes. The black hole substance density is estimated by the formula:

 

аааааааааааааааааааааааааааааааааааааа r = (M/M _o )^-2 r _o аааааааааааааааааааааааааааааа (1.1.4.)

 

ааааааааааааааааааааааааааааааааааааа

where r _o = 1,85 а10¹⁶ g^.sm^-3 is the solar mass black hole substance density. It is evident from (1.1.4.) that the substance density of small and micro-black holes is more in many orders than substance densities of known forms of matter, and black holes themselves with respect to space bodies are with sufficient accuracy gravitational material points described by the Newtonian law of gravitation. Effects of GR, for example, in the micro-black hole case, become appreciable at the scales less than atomic ones. Nevertheless, though micro-black holes are located in atomic volumes they can give effects at the macro-level compared with ones from space bodies. So, at the distance:

 

а аааааааааааааааааа аааааааа h = (M_BH/ M _Å )^1/2 R _Å аааааааааааааааааааааааааааааааааааа (1.1.5)

 

the force of attraction caused by the black hole is equal to the gravitational force at the Earth's surface (M _Å Ч mass of the Earth, R _Å Ч radius of the Earth). The black hole with M = 1,47 а10²⁰ g will create at the distance of one kilometre the same force of gravitation as the Earth, i.e., it will cause considerable but rather localized gravitational anomalies.

In accordance with the Hawking effect black holes radiate particles like a black body with temperature [Ha01,2]: (1.1.5) during the time tа ~ 10¹⁰ (M/10¹⁵) years. At the last stage of black hole evaporation the explosion happens, in which the energy of 10³⁰ erg is extracted in 0,1 s. This is the insignificant energy in comparison with stellar energetics (luminosity of the Sun is 3,8 а10³³ erg^.s^-1), but it is rather considerable amounts for planet energetics (thermal flow from the Earth's interiors is 3,17 а10²⁰ erg^.s^-1).

Small black holes move in substance of space bodies as if in emptiness. Therefor while considering black holes in space bodies it should be involved the idea of otons as universal centres of forming all the space objects [╥Ё00,9,11]. In addition in space objects there can be not only central germinal black holes, but also others. So, for example, planetesimals, of which falling onto the proto-Earth "germ" is considered to have resulted in formation of the modern Earth, can also contain germinal black holes. In other words, not single black holes, but space bodies containing black holes are grasped. But before discussing the problem of black holes in the Earth it is necessary to find out how are black holes formed in general case, to which next sections of this chapter are devoted.

1.2. Poststars (to the top). During star's evolving the irreversible process of energy loss runs. On exhausting the stellar nuclear energy source the poststar is formed. By the term "poststar" are meant space objects being the final product of stellar evolution (white dwarves, neutron stars, black holes).

As a whole the process of increasing deepening of gravitational potential "holes" is characteristic for the substance of classical astrophysical objects. This is connected with an irreversible nature of energy loss in the radiation form by space objects, which leads to increasing the sum of connection energy of closed space system and body substance. As a result the sum of connection energy of the closed system substance does not decrease. This is a formulation of the closed system substance connection energy non-decreasing principle. Such the formulation can be assimilated to the second principle of thermodynamics.

Due to the principle of connection energy non-decreasing the black hole formation seems to be the natural and inevitable stage of evolution. The conclusion on the black hole existence in nature is so correct, as far as correct GR itself. But the formation of black holes with masses less than mass of the Sun was considered for a long time as problematic.

1.3. Relics of the Big Bang (to the top). The second version of black hole formation is connected with the idea on white holes proposed in 1964-1965 in I.D.Novikov's [═ю00,1] and Yu. NeТeman [Ne00] works, who have suggested the hypothesis of "lagged cores". According to this idea at the initial stage of Metagalactic expansion the substance expanding was retarded in some regions and the substance has not left the gravitational radius. The so called "lagged cores" have been formed.

For some reasons, the first attempt to understand a nature of white hole formation has appeared unsuccessful, but the idea of "laнgged cores" was an incitement to developing the second way of black holeТs formation. In 1967 Ya.B.Zel'dovich and I.D.Novikov [╟х10], and then in 1971. S.Hawking [Ha00] have proposed the secнond way of black holeТs formation as a result of possible inhomogeneities at early stages of cosmological expansion. Such the black holes have received the name of primordial (relic) black holes. They can have various masses, both more, and less the solar mass.

Since in the Hawking work the idea of small black holes at once was connected with some astrophysical phenomena (in particular, with the deficiency of solar neutrino), it is the work with which the wide discussion of the primordial black hole problem begins.

Because of relic otons are formed at the initial stage of the Big Bang under quite certain conditions, namely, under large density and temperatures during very short time, there are restrictions on their number and general mass. If the primordial black holes exist, the average density of their substance in the Universe is in many orders less than the critical one. Just only for this reason the presence of relic black holes with small masses in the Earth is improbable. The difficulties in an explanation of the black and white hole origins have forced resorting to the idea of transmetagalactic oton origin from other worlds.

1.4. Kerr-Newman space-time and transmetagalactic otons (to the top). The Kerr-Newman metric is the theoretical basis of transmetagalactic oton models construction (black and white holes). In the oblate quasi-spheroidal Boyer-Lindquist coordinates it is written in the following form [╠ш10] (here the geometrized units are used, c = G = 1):

 

аааааааааааааааааа а(1.4.1)

ааааааааааааааааааааааааааааааааааааа ;

ааааааааааааааааааааааааааааааааааааа ,

 

where M is the total mass of oton, Q is its charge, a is the angular momentum of rotation per unit mass.

In a general case for the Kerr-Newman metric there are several mismatched pseudo-singular surfaces. Surfaces of event horizon for the metric (1.4.1) are defined by the expression (here and further in usual units, if it is not stipulated the opposite):

 

,ааааааааааааааааааааааааа (1.4.2.)

 

where R₊ is the external event horizon, R_- is the inner one.

Surfaces of infinite shifts are deнfined as follows:

 

ааааааааа . ааааааа (1.4.3.)

 

The surface determined by r₊а is called the infinite red-shift surнface, r_-а does the infinite blue-shift one. The pseudo-singular surfaces make the structure of the extended space-time manifold (ESTM) non-trivial. In the case of a Kerr oton (Ì  0, and а0, Q = Î) the picture qualitatively does not vary. In the case of not rotating oton (a = 0, Q а0, Ì а0) the picture qualitatively changes, since from (1.4.2) and (1.4.3) r + = R +, r _ = R _, i.e., event horizon surfaces coinнcide with the corresponding infinite shifts surfaces. Thus, the condition a = 0 makes the ESTM structure more poor.

Finally, for a Schwarzschild oton there is one pseudo-singular surface: r₊ = R₊ = R_g (R_g is the gravitational radius):

 

ааааааааааааааааааааааааааааааааааааааааааааааа . ааааааааааааааааааааааааааааа (1.4.4)

 

The second peculiar surface (r_- = R_- = 0) coinнcides with the point of true singularity.

The most realistic model of an otonic white hole is associated with the Kerr ESTM, because all known astroнphysical objects possess rotation. Let us consider the Penrose diнagram for the Kerr ESTM along the symmetry axis [Õî00] (fig. 1.4.1.), that can give the qualitative representation on the global structure of ESTM.

Taking into account results of extended relativity it is possible to designate an arbitrary region of the Kerr ESTM Ì., which is separated from another by event horizons, by the general symbol [Òð03]:

а

ааааааааааааааааааааа ааааааааааааааа аM (k, P)аааааааааааааааааааааааааааааааааааааааааааааааа (1.4.5)а

 

where P = (i)^N. N is the number of event horizons sepaнrating arbitrary region Ì from on originate Ì₍₊₎, - а< k < + . Since k is not restricted, there can be the unlimited number of regions by the type of Ì. Each such region can be an independent world, which is similar to our Metagalaxy.

Anticollapsing objects in similar ESTM are formed in the results of relativistic collapse-anticollapse process from black hole matter, which flows (see fig. 1.4.1.) through wormholes from one Ì (0, +), Ì (0, i) ESTM region (otonic world) to another Ì (1, -i), Ì (1, +). The cause of transformation of collapse to anticollapse for the Kerr oton consists in rotation, which at a certain stage of contraction of oton, namely, in the region Ì (0, ' -) at R = R = a²/c²R_b, leads to expansion.

Thus, in white hole concept we should go from the Schwarzschild STM to the Kerr ESTM, which naturally explains the nature of anticollapse and leads to the notion on non-trivial ESTM structure and on worlds variety.

This is the second possible way of the white hole origin and the third way of black hole formaнtion as relics of grey holes, which were proposed in 1973-1978 within the idea of otonic worlds variety in the multi-dimensional Universe (otonic scenario) [Òð00].

On the Penrose diagram of the Kerr STM (see fig. 1.4.1.) there can be the unlimited number of regions by the type of M₍₊₎. An inнdependent otonic world corresponds to each such region, which is similar to our Metagalaxy. Though, it is necessary to notice, that any region Ì₍₊₎ of STM must be not the asymptotically flat space-time, but the curved Friedman world, possessing "holes".

If the expansion of anticollapsar stops at the event horizon the black hole will be formed. Such the black hole is the grey hole relic, which matter is originated from other regions of extended STM. The time of grey hole manifestation at the stage of anticollapse is extremely small. On stopping anticollapse they become black holes. Such otonic black holes can possess different masses and arise at any stage of Metagalactic expansion.

   

ааааааааааа

 

  Fig. 1.4.1. The Penrose diagram for the extended along the symmetry axis Kerr STM. The broken line marks the ring singularity. The stencil picture Ì₍₊₊₎ including regions Ì₍₊₊₎, Ì _(-), Ì_(i), Ì _{(' -)}, Ì_{(- ')}, and Ì^(-i) is repeated unlimitedly to the both sides. When k о ¥ we optain the complete Kerr manifold. Curves show possible geodesics (time-like), which correspond to black hole, BH; white hole, WH; grey hole, GH; dark grey hole, DGH; light grey hole, LGH.

 

 

The discovering of white (or grey) hole flares would be the confirmation of the idea on worlds variety in the multidimensional Universe. If gamma-bursts are connected with grey hole flares, bursts of gravitational radiation can be predicted to be observed synchronously with gamma-bursts.

White and grey holes from other otonic worlds, causing extreme disturbances of STM and the graviнtational field, should lead to powerful short-term bursts of gravitational radiation and electromagnetic waves. Therefore any grandiose processes in this point of the heavenly sphere after the radiation burst should not be expected, since the grey hole relic can be a single black hole. The detection of synchronism of gravitational and gamma-bursts would be decisive argument for the discovery of white and grey hole flares [Tr00].

White holes, unlike grey holes, can manifest themselves after the short-term powerful radiation burst as grandiose space explosions. In this respect it is of interest the Supernovae - 1987À, when the large burst of gravitational radiation was registered [Tro1].

1.5. Other ways of black hole formation (to the top). According to otonic worlds concept considered above, in which there are no restricнtions on the time of space object existence, black holes in the far future of Metagalaxy can reduce their masses up to any values due to quantum evaporation.

One more way of mini-black hole formation through the condensation of poorly interacting masнsive particles in neutron stars was offered in the GoldmanТs and NassinovТs work [Go00]. According to the authors, bosonic poorly interacting massive particles which have masses more than 200 GeV can be condensed on to the neutron star nuнcleus, forming the configurations being close to the gravitational radius. These configurations collapse forming mini-black holes.

Thus, there are various ways of small black hole formation and now it is necessary to disнcuss the question, how black holes appear in space bodies, in particular, in the Earth. Black holes in space bodies will be the subject of the following chapter.