General Theory of Relativity of Einstein
as Unified Field Theory

Author: B. Mordvinov


Relying on a fundamental empirical identity of heavy and inertial mass it is proposed to bring a status of general theory of relativity (GTR) of Einstein up to a level of Unified Field Theory. To do this, a thoroughgoing revision of physical interpretation of energy-momentum tensor components within GTR is required.
Investigations demonstrated that physical meaning of spatial components of energy-momentum tensor within mechanics of special theory of relativity and in GTR equations may differ substantially. Taking into account this fact and a correct wording of a local principle of correspondence of GTR to Newtonian mechanics enables to implement a unique choice of Friedmann cosmological model from a class of isotropic ones. A new cosmological model differs fundamentally from "a standard one".
Within the two below sections of the paper an attempt to pass beyond the limits of the cosmological model is performed. In terms of qualitative assessments a rather interesting picture of hierarchical structure of the real universe is obtained. A complete system of equations for numerical simulation of the hierarchical structure on the basis of curvature tensor invariants is proposed. It is the author's opinion that this would enable to perform the total geometrization of all material structures, thereby raising the GTR status up to a level of Unified Field Theory.


Contents

  • Introduction
  • Part I. Cosmological Model of Friedmann
  • Part II. Hierarchical Structure of the Universe (passing beyond the model limits)
  • Part III. On Total System of Equations of Unified Theory
  • Conclusions
  • References

  • Introduction

    An advantage and effect of gravitation, as an object of theoretical investigation, lies in its universality and informativity. Having an information about gravity potential of a particle and taking divergence of its gradient we would obtain all the necessary knowledge about the particle material structure. As for other physical fields, they cannot provide such a comprehensive information. Moreover, they cannot be simply and universally reduced to curvature of metrics of Riemannian space-time.

    This very fact suggests an idea to use GTR mechanism for constructing Unified field theory in general and describing a structure of elementary particles in particular. On a path leading to this task implementation there is a problem of energy-momentum tensor in Einstein equations. This problem is associated, in its turn, with search of Lagrangian, which enables through the known formalism to construct the energy-momentum tensor of Unified theory.

    The simplest structural unit of material world is the universe on the whole, approximated by an isotropic model of Friedmann. Being almost trivial as for mathematic description, it provides a great range for unique choice of true, cosmological model. The way of implementing such a choice is shown within the first part of the paper. At the same time we have managed to destroy a deep-rooting illusion of Big Bang. In the new conception "recession" of galaxies is replaced by deep and continuous structural evolution (towards complication) of the real universe.

    A specific mechanism of the structural evolution is proposed and discussed within the second part of the paper. Consideration is performed at a level of physical assessments mainly. Even at this simplest level some interesting and rather convincing qualitative results on hierarchical world structure are obtained as well.

    Later, within the third part of the paper a program of constructing a universal theory of matter on the basis of relativistic gravitational theory, which GTR of Einstein is, is stated. Such strong assertion may be made when analyzing a remarkable empirical equation of inertial and heavy masses, which Einstein took as a principle of GTR:

    (1)

    Originality of the relation (1) lies in the fact that its component values have principally different physical nature. Inertial mass is a measure of inertia, a quantitative characteristic of matter materiality, revealing itself in its interconnection with ambient physical objects. Heavy mass is "a gravitational charge" of matter within Newtonian universal law of gravitation. But, though there is such essential physical difference, no difference of the relation from one accurate within the extremely small value ~ is found even in very fine present-day experiments. We note that empirical accuracy of law of conservation of energy does not exceed ~ , i.e. much worse.

    It gives ground to consider the expression (1) as an identity applied to all matter forms, having positively defined energy density. In particular, it follows the conclusion about uniqueness of Einstein GTR as relativistic gravitational theory. In fact, all material objects possess "gravitational charge", therefore their space-time paths in nonhomogeneous and all over penetrating gravitational field of the universe are curved in a varying degree. Moreover, globally, in space scopes, space curvature may become so considerable that it will lead to its total closure, having changed its topology qualitatively.

    Consequently, within consequence gravitational theory there is no (even theoretically) such physical body, with which one may associate global flat 4D space of Minkowsky. That is the reason that alternative, field gravitational theories are a fortiori devoid of every physical basis. In GTR the gravitational field is completely geometrized, is reduced to change of space-time metrics. It has no local and positively defined energy density, which is obligatory for all other physical fields.

    Let us to return to discussion of the identity (1), focusing at the two statements:

    1)  According to the known relativistic relation of Einstein density of energy coincides with mass density accurate within the factor . Any physical object is characterized with local, positively defined density of energy. Consequently, all material structures may be, finally, completely described through specifying the scalar density at space-time continuum.

    2)  Charge is divergent condition of a field (electrical or gravitational). On the basis of these clear physical statements we are forming up a simple logic chain.

    In GTR the gravitational field is completely geometrized, is reduced to changes in Riemannian space-time metrics. The gravitational field divergence is presented here with components of curvature tensor of Riemannian 4D space. They, in their turn, expressing density of "gravitational charge", are proportional to density of heavy mass. Heavy mass is identically equal to inertial mass. Density of inertial mass(energy) is linearly connected with components of energy-momentum relativistic tensor.

    Omitting intermediates of this logic chain we obtain that the empirical identity (1) in mathematical aspect should be expressed in a form of agreement between components of curvature tensor and components of energy-momentum tensor. These are the very Einstein equations, in particular. Components of energy-momentum tensor in the right part of these equations are generally considered as physical values, expressing them through physical properties (for example, inertial mass density). Relying on the identity (1), there are grounds to suppose, however, that the same components may be expressed geometrically as well, in a form of algebraic constructions from components of metric tensor and its derivatives. In other words, material world assumes alternatives of its mathematic description (simulation), physical and geometrical ones. It should be noted that geometrical method of description, as one that does not include arbitrary physical properties, presents itself to be more universal in terms of constructing Unified field theory.


    Part I.   Cosmological Model of Friedmann

    Physical meaning of energy-momentum tensor within general theory of relativity (GTR) of Einstein is specified. A local principle of Einstein GTR correspondence with Newtonian mechanics is stated correctly. All these enable to perform a unique choice of a cosmological model out of a class of isotropic models.
    The new choice is principally differed from the so-called "standard" Friedmann model. A physical aspect of a closed cosmological model is discussed. Instead of common concept of "Big Bang" a more calm evolution variant is proposed where the universe evolution implies constant complication of its material structure.

    1. Einstein Equations

    The proposed investigations in a field of theory of gravitation and cosmological model are performed strictly within the frameworks of fundamental notions of Einstein GTR. Just a choice of a specified mathematical expression for energy-momentum tensor, which is in the right part of Einstein equations, is changed:

    (2)

    It is a usual case when the known expression:

    (3)

    is substituted into the right part of Einstein equations (2) for local-isotropic continuum (liquid, gas).
    Here: p -- is pressure, -- is relativistic density of energy, uk -- 4D-velocity of the element of a liquid.

    Formula (2) was constructed yet before GTR creation -- within the frameworks of mechanics of specific relativity theory (SRT). In a plane space-time of SRT the components are subject to relationship (7), which represent laws of conservation of energy and momentum for an arbitrary liquid element.

    Einstein GTR is a relativistic generalization of Newtonian law of gravitation, which is based on a large body of empirical data. For continuum the law of gravitation may be written in a differential form of the Poisson equation:

    (4)

    where is a gravitational field potential. Here a source of gravitational field is just a mass of a body, distributed within space with density . If we substitute in form (3) into the relation (2) we get relativistic generalization of Poisson equation (law of universal gravitation) in the form:

    (5)

    This equation brings theory beyond frameworks of experience since its right part, as against relationship (4), includes in addition to density , pressure p explicitly i.e. thermodynamical parameter of a substance. Dependence of gravitational field not only on density but on pressure as well is not supported empirically. At the same time solution of such an important scientific problem as a cosmological one depends greatly on mathematical expression of the right part of equation (2). Since there is no empirical basis the relation (5) should have serious theoretical ground. As the below account shows there is no such ground.

    At the same time, reasonable limitation of applicability of formula (3) together with local principle of GTR correspondence to Newtonian mechanics help to easily perform a unique choice of a cosmological model out of a class of isotropic ones.

    In order to avoid misunderstandings let us agree on terminology in advance. Friedmann model represents a geometrical construction in inhomogeneously distorted Riemannian space-time. Spatial curvature in Friedmann model depends neither on a point of observation nor on a direction. From here it is another name of the model - an isotropic one. Radius of spatial curvature in Friedmann model is an arbitrary function of just the time coordinate: .

    Mathematical formalism allows the alternative of an open and close Friedmann model. Cosmological problem is that one should uniquely choose such a model of Friedmann out of a vast class of isotropic ones that would approximate the real universe (oneness and uniqueness of the universe is considered obvious). We call this very model a cosmological one.

    Presently the cosmological problem has no satisfactory solution. The so-called standard model of Friedmann does not solve the alternative of the open and the close models. "Cold" and "hot" variants of solution within the framework of the standard model include arbitrary and semidefined constants of length dimension. In addition, availability of constants of length dimension in the isotropic model contradicts to simple consideration of dimension. This will be discussed in details in Section "Choice of Cosmological Model".

    2. Energy-Momentum Tensor in GTR

    Concerning the problem of energy-momentum tensor in GTR being discussed here, a statement of Einstein himself made by him 30 years after he had published his equations, is considered to be rather interesting and didactic. In one of his last papers (1946) he said:

    "All the attempts to present matter with energy-momentum tensor are unsatisfactory, and we wish to make our theory free from a specific choice of such tensor".

    Below we try to discover the reasons of these attempts shortness "...to present matter with energy-momentum tensor...".

    It is known that covariant divergence of energy-momentum tensor in the left part of the Einstein equation automatically turns to zero due to simplified Bianchi identities:

    From here as well as from equation (2) it follows that the covariant divergence of energy-momentum tensor equals to zero:

    (6)

    Further it is assumed that equality (6) generalize the known relationship for an energy-momentum tensor taking place in Galilean reference system of STR:

    (7)

    From here there occurred a name for i.e. an energy-momentum tensor.

    Equality to zero of common divergence in the form of relationship (7) allows to discover the physical meaning of some individual components of in relativistic mechanics of STR:

    (8)

    We remind in brief the way to do this.

    At first relations (7) are integrated by 4D-volume , limited at two sides with hypersurfaces , using four-dimensional Gauss theorem:

    Substituting here an element of hypersurface in the form of we get the equality:

    All this allows to identify the preserved value with 4D-momentum of closed mechanical system:

    (9)

    From here as well as from integral expression for 4D-momentum physical interpretation of (8) of the components and follows. It may be theoretically left for GTR as well in order to preserve succession with STR mechanics.

    To discover physical meaning of the space components we integrate the relations (7) over a certain volume of the space , using in the process the three-dimensional Gauss theorem:

    (10)
    (11)

    Contour integral in the right part of the relations (10) and (11) is taken over the surface covering three-dimensional volume. From here it follows interpretation of the values () as density of energy flow and as density of momentum flow. For a resting element of liquid or gas the components take the form:

    (12)

    Formula (3) is obtained from the relations (12) if we turn from a moving, locally-accompanying reference system to a laboratory (stationary).

    We repeat the above procedure in GTR. For this purpose we put down equation (6) in the synchronous

    Here the time and space parts of metric are separated. Then we integrate equations (6) over 4D-volume , limited at two sides with the hypersurfaces . Again we use the four-dimensional Gauss theorem:

    Substituting here the element of the hypersurface in the form of again we obtain the equality:

    In GTR it is impossible to choose a global Galilean metric where the right part of the above relation would turn to zero. Therefore 4D-momentum of the closed mechanical system in a curved space-time of GTR is, generally speaking, not preserved. Strictly speaking, a notion of a closed mechanical system in gravitation theory is not absolute by itself since gravitation has a global character.

    Further, following the known procedure, we integrate the relations (6) over a certain volume of the space , using in the process the three-dimensional Gauss theorem:

    (13)
    (14)

    The right part of equations (13) and (14), in contrast to (10) and (11), includes unremovable integrals over volume that makes impossible to strictly preserve the former interpretation of space components of energy-momentum tensor. Now change of energy in volume is specified not only by the value flow through its boundary. There is "volumetric radiant of energy" - one more summand in the right part of equation (13). Intensity of this radiant is proportional to the space components .

    The inconsistency of the former interpretation of the components reveals especially distinctly within the frameworks of the isotropic model of Friedmann where reference system is accompanying substance everywhere.

    In the accompanying reference system components, which are proportional to momentum density are equal to zero. Contour integral over surface in the relation (13) turns to zero together with them. From here it indisputably follows that in Friedmann model space components of energy-momentum tensor have no reference to momentum flow density, i.e. to pressure. Therefore, using of formula (3) in the right part of the Einstein equations, as it is the case with a choice of "standard" model of Friedmann, is groundless.

    The universe, being approximated by closed model, has no asymptotically flat ambient space at infinity. At the same time any of the universe structure elements may be approximately but with good accuracy considered to be placed into flat space, with the help of which conventional laws of conservation of energy and momentum are stated.

    We compensate the reject from absolute using of formula (3), by supplementing system of equation (2) with one more differential relationship. That is we establish some local correspondence of GTR to Newtonian mechanics. That would be enough for a unique choice of a physical model of Friedmann out of a wide mathematical class of isotropic ones.

    3. Local Correspondence Principle

    Necessity of correspondence between relativistic gravitation theory and Newtonian mechanics is obvious. GTR of Einstein is, strictly speaking, a local theory. Therefore a correspondence should be stated a local one as well.

    Newton gravitation theory is applicable when the two conditions are fulfilled. Macroscopic velocities should be small in comparison with the velocity of light, and a non-dimensional quantity of a characteristic potential in the vicinity of the considered point should be small as compared with 1:

    (15)

    The first of conditions (15) satisfies rather easily. It is enough to choose locally accompanying reference system where . To fulfill the second of conditions (15) we align an arbitrary point of a gravitating object with coordinate origin of locally inertial (locally geodesic) reference system. Physically it is equivalent to the transition to a free falling "Einstein lift", where 4D-gravity is close to zero, and a metric slightly differs from a Galilean one.

    Below we assume for the sake of simplicity that in the coordinate origin components of metric tensor take the Galilean values:

    (16)

    It does not limit the consideration generality.
    Transition to the locally inertial coordinates is made within a small vicinity of the coordinates origin by means of the following transformation:

    (17)

    Here and in below expressions of the Section resolution relations over contain derivatives of initial metric tensor calculated in the coordinates origin.
    Now we put down the metric tensor:

    in the form of decomposition:

    The unknown coefficients refer to the values and at the point .
    Because of the identity, joining a partial derivative of metric tensor and Christoffel's symbols

    ,components which are linear over vanish at metric tensor decomposition. As a result we get the following expression:

    (18)

    The matrix is symmetrical to permutation in pairs and . The coefficients has no ability to possess any other symmetry attributes because of arbitrariness of the values in decomposition (18). The following expression satisfies this condition:

    (19)

    Here is a geometrical object (not a tensor), being as well symmetrical to index permutation in pairs and . With the help of the value it is possible to put down a curvature tensor in the form that is convenient for the given consideration:

    (20)

    For expression (19) basing we use the equality of initial and transformed curvature tensors in the coordinates origin:

    (21)

    Presenting the relation (21) in an expanded form we get the following equality:

    (22)

    Therefore, expression (19) really satisfies the relation (22).

    As a result we get the final form of formula (18):

    (23)

    Now everything is ready to perform the local correspondence of GTR to Newtonian mechanics. This aim is achieved by means of comparison between an equation of a particle movement within gravitational field and an equation of its geodesic path in the curved space-time:

    (24)

    Now it is necessary to equate the right parts of equations (24), put down close to the origin of locally geodesic reference system:

    (25)

    Having calculated the values with the help of expression (23), it is possible to get the required agreement between Newtonian gravity potential and a metric of Riemannian space-time in locally geodesic reference system. However for the sake of practical aims it is convenient to transform the obtained agreement to three-dimensional scalar relation.

    Let us take three-dimensional divergence from both parts of equality (25) and use a differential form of law of universal gravitation in form (4) i.e. Poisson equation. As a result we get a three-dimensional scalar form of local correspondence principle:

    (26)

    Calculating the divergence with the help of formula (23), we put down the relation (26) in the form:

    (27)

    Generally speaking, equality (27) does not coincide with similar time equation of Einstein:

    (28)

    Excepting density out of equations (27) and (28) we get the relation:

    (29)

    It represents an additional limitation, which demand fulfillment of local Newtonian correspondence from the metric of Riemannian space-time. Equation (29) is invariant with respect to coordinates transformations, which does not affect time. Therefore it may be used for choosing a cosmological model out of a class of isotropic ones, where an accompanying global reference system is always synchronous.

    4. Isotropic Metric of Friedmann

    Substance in the universe is distributed in hierarchically inhomogeneous way. The base of microstructure is nucleons, where matter is localized with density . Nucleons are joined into stars with density of the order of unity. Further there are galaxies and galactic congestions of various hierarchical levels with decreasing density. After averaging over space zones having sizes more than 100 Mpc, substance of stars may be considered to be evenly distributed throughout the universe with average density . Indirect astrophysics data indicate the presence within interstellar space some nonluminous (invisible) matter with average density being approximately two orders higher than that of star:

    Physical characteristics of homogeneity and isotropy (according to data of "relict" radiation measurements) of an averaged universe correspond to similar characteristics of a metric of cosmological model space. That is, at every moment of universal time t all points of the space may be described with one scalar parameter i.e. with the radius of curvature . Theoretical body of mathematics admits an alternative of open and closed models. Square of an interval and a metric of a closed isotropic model have the form:

    (30)

    Here: -- are conventional angle spherical coordinates;
    is an additional "angle" coordinate, determining the distance from the coordinates origin (). A metric of an open model is obtained from (30) with the help of formal replacement: .

    Global reference system of isotropic model of Friedmann is synchronous and accompanying at the same time. The closed model presents a special interest, since it is the very model that approximates the real universe, as we show below.

    The closed Cosmological model of Friedmann presents a completely homogeneous physical body, each point of which is an equitable center of symmetry. There are no real physical objects with similar characteristics in nature. Thus, for example, a homogeneous ball, submerged into an empty flat space, has only one center of symmetry, towards which its radially inhomogeneous gravitation field is directed. Homogeneity of the ball is broken at its boundary. Homogeneity of the model of Friedmann is kept everywhere in the whole course of its evolution.

    A conventional form of Newtonian law of gravitation is not applicable to the closed cosmological model on the whole. However one may assert that at every point of the model a resultant force of all gravitation forces vanishes. Therefore, inner local gravitation forces in the model of Friedmann are absent. Such a system has no ability either to shrink or to extend with acceleration. These general considerations on a mechanical system dynamics are proved below with a specific and unique choice of a cosmological model, radius of curvature of which changes linearly in time, that is, without acceleration.

    At every moment of universal time the closed model has finite volume and mass. Any point of the closed model space corresponds to a point of an opposite pole of space being maximum removed from it:

    It is an essential peculiarity of the closed model.

    5. Cosmological Model Choice

    Foremost we dwell on simple physical considerations of dimensionality. Just they were a motive of revision a conventional method of a cosmological problem solution. Transition from the real universe to the isotropic model implies a certain procedure of averaging the density of one and all material structures inhomogeneously over the space. In a system, which has no space inhomogeneities in principle, it is not possible to choose a certain length standard (dimensional constant). Therefore, functional dependence of radius of curvature on time for an isotropic model approximating the real universe can not contain a constant of length dimensionality. Just the linear dependence satisfies such a condition:

    At the same time it is known that for the "standard" model of Friedmann the functional dependence is essential nonlinear. It contains a constant of length dimensionality, different for "hot" and "cold" evolution branches. Therefore, formula (3) really can not be substitutes into the right part of equations (2) for the model of Friedmann.

    Equations of Einstein for the isotropic model reduce to two second-order ordinary differential equations relative to three functions of time, what is not enough for the system completeness:

    (31)

    = ± 1     for the closed and the open models respectively.

    The local correspondence principle in the form of the relation (27) for the metric of Friedmann enlarges system of equations (31):

    (32)

    Eliminating and from the relations (31) and (32) we get the ordinary differential equation

    (33)

    which may be get directly from the relations (29).

    In a class of real positive values of radius of curvature we have the one-value solution of equation (33) for (the closed model):

    (34)

    Substituting solution (34) into the second equation of Einstein (31) we get the equality of a time component and a sum of space components of energy-momentum tensor:

    From here, if we follow formula (3), the equation of state

    results, which has no physical meaning. It proves once more the groundlessness of traditional physical interpretation of space components of energy-momentum tensor in GTR.

    6. Specific Character of New Cosmological Model

    The real Universe, being approximated with the closed cosmological model (34), has the finite volume and mass at every moment of universe time:

    (35)

    Usually enlargement of radius of curvature in the model of Friedmann is interpreted as expansion of the Universe due to the Big Bang. In this connection astrophysical data concerning Hubble "red shift" are used to be connected with mutual moving-off of galaxies. However, as it was mentioned, an explosion of completely homogeneous mechanical system is not possible because local gradients of pressure and gravitation potential are not present in it.

    There is another possibility -- to interpret the dependence as a result of time changes of local length standards, which determine a numerical value of a radius of curvature at a given stage of the universe structural evolution. Similarly one may explain time change of the universe mass in expression (35). From this point of view the empirical Hubble effect -- "red shift" spectral lines from distant galaxies -- implies information of the past concerning a character of evolution change of parameters of atomic structures electronic levels.

    Let us introduce a dimensionless time coordinate by means of the following differential relation:

    (36)

    Functional dependence of the radius of curvature on the new time coordinate extends over the unlimited interval:

    (37)

    The last condition closes a question about the initial moment of evolution being traditionally difficult for cosmology. For the new cosmological model (and just for it) there is no singular point during unlimited process of its evolution.

    We find a value of averaged parameters of the universe at a given stage of its evolution with use of a constant of Hubble "red shift" . Hubble law for a isotropic model takes the form:

    (38)

    ( it is the most reliable value of Hubble constant ).

    Assuming the time derivative in equality (38), we substitute into the relations (35) :

    (39)

    These are approximated values of basic parameters of the observed universe at a given stage of its evolution.


    Part II.   Hierarchical Structure of the Universe (passing beyond the model limits)

    The second part presents program for passing the cosmological model limits on the basis of an idea of particles as focusing conditions of curvature waves. The scheme of the universe hierarchical structure is discussed. The obtained parameters of the universe structural hierarchy are compared with the observed astrophysical objects. An interpretation of "relict" radiation phenomenon beyond the frames of "Big Bang" standard model is proposed. The presentation is implemented at the level of physical assessments.

    7. Fundamental "Planck" Units

    The equations of Einstein (2) implement the connection of the two, so different, at first sight, methods of the real world simulation --- both geometrical and physical. They include only the two of three fundamental world constants: c G are the two of three fundamental world constants:

    is the maximum velocity of all the interactions propagation,

    is the gravitational constant, equal for all material forms.

    The third fundamental world constant (besides the above mentioned c G) is the Planck's constant. It is not present explicitly in the GTR equations but reflects the universal quantum matter properties as well, which represent themselves, finally, in its microstructure discreteness. Without it the system of fundamental world constants is not complete:

    Other world physical constants have no status of fundamentality. Thus, for example, an electrical charge is not the mandatory attribute of all material forms. Therefore the electron charge

    is unable to replace the value in the set of three actually fundamental constants with mutually independent dimensionalities

    (40)

    The set (40) allows introducing the universal standards of length r0, time t0 and mass m0:

    (41)

    These are the so-called "Planck" units, the system of which is equivalent to the set (40). By its implication the values set {r0, t0, m0} should be the quantitative characteristic of physical parameters of the fundamental microstructure of matter.

    Namely, within the spatial region ~ r0, twenty orders of magnitude smaller than the nucleon size, for the time instant t0 the mass (energy) is concentrated (where mn is the nucleon mass). The spatial curvature here reaches the maximum, when the space curvature radius is comparable with r0. A kind of a local, extremely short and intense "curvature peak" takes place. These peaks, by virtue of their fundamentality, should underlie all the observed material structures. Per se, fundamental curvature peaks are not observed directly because of their extreme smallness.

    Later on we will try to define concretely the above general considerations on dimensionality, relying on the properties of the new cosmological model. Recall that the cosmological model of Friedmann was in the beginning (1977) obtained from simple considerations of dimensionalities. And only considerably later, after longstanding efforts, there was a success in finding the strict basis of the initial, comparatively simple method for the model choice. It is described in the first part of the paper sufficiently at length. It is supposed to implement the similar approach to the problem for the given case as well: from qualitative assessments of the world structure description to stricter mathematical simulation.

    8. Focusing States of Curvature Waves

    Passing beyond the limits of the cosmological model with the aim of more complex approximation of the universe real structure is proposed to be implemented with using the conceptions on focusing of curvature waves. In fact, if intense Planck "curvature peaks" take place, the diverging elementary curvature waves should spread from them when necessary. In this case the question is not about weak quadrupole gravity waves, resulting from mutual motion of massive cosmic bodies. Later on by curvature waves are meant radial waves of the Newtonian potential type, generated by a source of variable mass.

    Analyzing these curvature waves propagation, one should take into account the specific properties of the new model, its spatial closure and availability of unlimited number of focusing states of field waves at spatial antipoles. This very peculiarity of the model generates physical phenomena, extrinsic to STR.

    Thus, for example, local electromagnetic disturbance, which occurred in plane Minkowsky space, goes further to infinity in the form of monotone attenuating spherical wave. As a result. after some time the initial disturbance practically vanishes from the range of physical phenomena. Similarly the field lines flux of point electrical charge q and of the "gravity charge" mg vanishes at infinity, though the total number of field lines is kept.

    A different situation obtains with the space of the closed universe, approximated by the cosmological model . Here the wave divergent from the coordinate origin, on having passed the equator , is focused anew after some time at the antipole .

    To determine the focusing time, let us note that at the spherical wave front spreading with sound velocity the invariant 4D interval ds = 0:

    From here, assuming , we obtain the differential relation at the wave front:

    Integrating we find the equation of motion of the divergent wave front in the form:

    For we obtain the former exponential dependence:

    (42)

    A process of successive focusings within the model occurs with no limit in equal intervals . Relation of ordinary, dimensional instant of time between neighboring focusings is:

    (43)

    Thus, perturbation of electromagnetic field or curvature, once occurred in closed space, does not vanish. It represents itself again and again at space antipoles. It is a new physical phenomenon, peculiar just to the cosmological model . With its help one may qualitatively explain the main peculiarities of the observed universe hierarchical structure.

    Any local curvature of space inevitably spreads to all directions with sound velocity. Therefore one may suppose that the universe is filled with weak elementary curvature waves with minimum wavelength They (as assessments show) tightly cover all the space at the average and in its every point they spread uniformly in all directions.

    Elementary particles may be considered as comparatively stable in time succession of focusing states of elementary curvature waves of perturbation of metrics, having come from the universe antipole. On passing all the real universe space the elementary wave front undergoes curvatures near other particles, as a result of which the structure of following focusing state becomes considerably complicated.

    It seems that the universe evolution lies just in such continuous complication, the focusing states fragmentation. Such structural evolution leads to reducing physical standards of length, as a result of which the apparent increase of curvature (dimensions) radius of the universe and its total mass takes place.

    The idea of curvature waves defocusing, occurring on having passed the whole space of the real universe was suggested to the author of the given paper (in the form of a comment) by an academician of Academy of Sciences of the USSR Eugeni Ivanovich Zababakhin. Being not a specialist in Einstein GTR field, he, however, used to show the keen interest in the work. His remarks were always constructive and hence fruitful. That was and in this case either. Having listened to the comment and performed the assessment of defocusing value of curvature waves, the author has obtained (by the order of magnitude) the parameters of nucleon, and of the following levels of the universe hierarchy as well. Let us reproduce these assessments.

    9. Structural Hierarchy Levels. Assessments

    If we present the cosmological model parameters in "Planck" units , we obtain the simple chain of equalities:

    (44)

    Here should be considered as the fundamental parameter of microstructure of the universe. At the given stage of the structural evolution its numerical value is rather great: .

    The concept about focusing states allows obtaining the reliable mass values and the dimensions of the nucleon, the main elementary particle of the universe. It is not difficult to show that the values in the process of their evolution are associated with the fundamental parameter of microstructure (7) by the relation:

    (45)

    Let us consider i -s hierarchical level, consisting of "particles" (these may be the galaxies as well, in particular), each having the dimension and the mass . The curvature wave, corresponding to the given structural element of a hierarchical level, sweeps the universe space, passing through each of "particles", and then is focused at an antipole. At this process "dents" with the area ~ each are generated at the wave surface. Deviation of beams, regular to the wave front, specifies the angle of photons deflection near an object with the mass . On having passed the distance ~ a the beam deflection from a spot of ideal focusing is :

    (46)

    One may show that up to the moment of focusing the whole surface of the curvature wave, sweeping the universe, on the average uniformly and rather tightly is covered with "dents" of the whole totality of particles of a given hierarchical level. It leads to substantial deformation of the focusing state as compared with an ideal (point) one and give grounds to suppose that the defocusing value coincides with the dimensions of the particle itself: . Substituting this equality into (46) and taking into consideration that total mass of "particles" equals to the universe mass

    we obtain the relation, which is very important for the following assessments:

    (47)

    macroscopic section of all particles of a given hierarchical level equals to the Universe "section" (~ ). It is verified by the above assumption: .

    Thus, every focusing state, bearing a mark of the locally nonuniform Universe, generates a structure of a deeper hierarchical level with the number of particles:

    (48)

    We obtain the corresponding values from the relations:

    (49)

    which we use in the below section.

    10. On Quantum Laws of Matter

    Up to here our consideration has been conducted in approximation of geometric optics, when the curvature wave front has been supposed to be absolutely thin. In such approximation the structures fragmentation because of successive focusings may occur with no limit. In fact, however, the hierarchical scale has a limit from below because of finite width of elementary curvature wave front: . It is it that leads to wave, quantum properties of microparticles. The extremely small value is equalizes with huge dimensions of diffraction lattice, which indeed the whole our universe is.

    The basis of quantum mechanics is heisenberg uncertainty principle. As is known, between radius and mass of the nucleon uncertainty principle with good accuracy ("Planck" system of units) is performed in the relativistic form:

    (50)

    It should be considered both as an empirical fact, and as verification of essentially wave and relativistic nature of the nucleon.

    Using the relations (44), (47) (50), one may associate a number of nucleons within the universe, as well as their radius and mass with the microstructure fundamental parameter :

    (51)

    From here the expressions for unknown values follow:

    (52)

    If we then use the relations (49) and (52), we obtain particle dimensions of subnucleon hierarchical level:

    It is twenty orders of magnitude smaller than the fundamental length that is, naturally, impossible. Thus, the hierarchical scale ends in fact at the nucleon level.

    We dwell on another fundamental moment. The question is about the reality of fundamental curvature peaks, the presence of which is predicted by general consideration about dimensionality. As noted above, the front of elementary curvature wave before focusing is covered "rather tightly" with distortions from "particles" of various hierarchical levels. But at this process it may occur that locally the waves focusing takes place finally in the areas having Planck dimensions. the assessment performed as early as in [4] says in favor of such assumption.

    The system of Planck focusing states, scattered along the universe, represents at every moment of the universal time the diffraction spatial lattice with the step:

    the points of such lattice, where the angle of beams deflection ~ 1 , may be considered as as absolutely opaque. At the waves dispersion at the lattice with the parameter the characteristic diffraction angle is:

    The angle corresponds to the focusing state tailing by the value , determining minimum dimensions of stable particles of the lower level of the structural hierarchy, i.e. nucleons:

    (53)

    The relation (53) agrees with (52), but is obtained more strictly, without applying quantum-mechanical considerations. On the contrary, from the relation (53), having calculated the number of nucleons

    (54)

    and their mass

    (55)

    one may obtain the connection between the dimension and the mass of stable elementary particles:

    It is nothing but heisenberg relativistic quantum uncertainty principle.

    From here one may conclude that in successive gravics, considered as Unified Field Theory, there is no necessity to introduce quantum laws artificially. They would reveal automatically at mathematical simulation of focusing states.

    11. Tables of the Hierarchical Structure

    Let us construct a table of the universe hierarchical structure, having preliminary constructed recurrence formulas on the basis of the relations (47), (48), (49) (52).

    The recurrence formulas:

    (56)

    With the help of the relations (56) we construct Table 1 of main parameters (mass and dimension in "Planck" units) of the four hierarchical levels following the lowest one, the nucleon level.

    Table 1. Parameters of the four observed structures

    Structures Comets Stars Galaxies "Einasto cells"
    0.1 eV 10 KeV 3 MeV

    Names of structural levels of the first line are described in the following, generalizing Table 2. The last but one line of Table 1 presents the average kinetic energy (temperature) of nucleons with respect to their rest mass (for a given hierarchical structure). On the basis of the virial theorem (Laue) the average kinetic energy of each particle of the system, moving in the proper gravitational field, is comparable with absolute value of potential energy of a particle in this field. In particular, the kinetic energy of a nucleon is ~ . To obtain the temperature value, say, in electron-volts, it is suffice to multiply the table value by the nucleon rest mass, i. e. by , that is done in the last line of Table 1.

    Table 2. The universe hierarchical structure

    # i () () Structural type
    1 0 1 the universe
    2 ... ... ... ... ... ...
    3 >=2 --- --- --- *
    4 ~ 4 --- --- --- *
    5 ~ 18 --- --- --- *
    6 ~ 300 --- --- --- *
    7 "Einasto cells"
    8 galaxies
    9 protostars
    10 protocomets
    11 nucleons

    Lets us discuss further the results presented in Tables 1 and 2.

    12. Discussion of the Tables Data

    Satisfactory fit of numerical values presented in Table 2 with corresponding empirical data takes place just for the nucleon parameters only. From here, in particular, one may draw a conclusion that the main part of the so-called "hidden mass" in the universe, approximately two orders of magnitude greater than the observed stellar mass, consists just of nucleons, not of massive hypothetical neutrinos or photinos.

    In Table 2 we have yet learned neither the total number of levels, nor a number of intermediate structures, situated between giant galaxies concentrations, which are called "Einasto cells", and the top level --- the universe.

    Galaxies and "Einasto cells" have dimensions, compared with the parameters of the eighth and the seventh lines of the Table. Not bad agreement for dimensions is obtained for solar type star systems, if Oort comet cloud is included into it, which contains ~ comets according to some estimations and spreading to distances up to ~ astronomical units (a.u.) from our star, i.e. the Sun. Strong divergence is obtained for masses of the eighth and the ninth lines of Table 2.

    To explain the above mentioned divergencies we use the ideas occurred in astrophysics that the observed star substance is no more than one hundredth of the universe total mass. According to astrophysical data the greater the dimensions of galaxies concentrations are, the greater the hidden mass fraction is. Concentrations of luminous star matter are observed just near the boundaries of adjoining "Einasto cells". It would be logically to assume that the hidden mass in the form of invisible protogalaxies, consisting of cold nonluminous protostars with parameters from Table 2, is concentrated mainly inside these cells. Thus, in compliance with Table 2, the following picture of the universe at the given stage of its structural evolution is occurred.

    Cold nucleons (mainly, within the composition of hydrogen) to the number of ~ pieces are combined into protocomets with the mass of g and the dimension cm. Protostar clouds with the mass of g and the diameter of cm ( a.u.) are composed of protocomets each. then protogalaxy clouds follow with the mass of g and the dimension of cm, each of which contains protostars.

    Nonluminous inside of "Einasto cells" is composed of approximately a hundred thousand protogalaxies, moving in aggregate proper gravitational field with the average kinetic energy ~ 3 MeV. In the process of protogalaxies impact at the boundaries of neighbor cells their substance warms up to approximately the same temperatures ~ 3 MeV. For cooling of hot structures occurred in such a way (hot protostar and protocomet clouds) the leading role will belong to electromagnetic radiation processes (mainly, non-equilibrium). Being transparent with respect to gamma and hard roentgen radiation, these systems will cool rather fast down the temperatures of order ~ 10 eV, i.e. the energy of binding atoms and hydrogen molecules.

    As may be seen from Table 1, the proper gravitational field of protostars keeps particles with the energy ~ 0.1 eV. Such cloud, warmed up to ~ 10 eV, will be finally left by particles with the energy > 0.1 eV. Consequently,of the initial protostar cloud of the mass g, it will be left less than , i. e. one hundredth of particles with temperature less than 0.1 eV and mass of several . Further these systems will evolve according to ordinary star cycle.

    The proposed scheme qualitatively explains a considerable percent of binary stars (>= 60 %) and the presence of angular momentum in the Solar system --- it is the result of eccentric impact of protoclouds. Protocomet structures, which have not impacted, generate circumstellar comet systems of our Oort cloud type, where approximately one milliardth () of original protocomets has left. It is clear that protocomet substructure, exposed to impact, is quickly developed with protostar size.

    Table 2 presents just the main stages of the universe hierarchical scale. But there should exist some smaller intermediate stages as well. Thus, for example, the comet structure is obtained under the assumption of homogeneity of protostars, which generate it. In fact, protostars themselves have a substructure of protocomets. Thus, nucleons of the protocomet will be in some way decomposed into fragments with nucleons each. It is interesting to note that the mass of each this hypothetical fragments is comparable with the "Planck one": .

    The last condition is hardly accidental. "Planck" mass, being a fundamental physical parameter, should represent itself substantially in the material world macrostructure. The above mentioned nucleon fragments with "Planck" mass under the influence of interatomic electrical forces may be joined into more compact structures (with presence of electrons) in the form of particles of the size ~ 0.1 cm of hard hydrogen.

    13. On "Relict" Radiation Phenomenon

    The above stated considerations allow supposing that cold (unobserved) protocomets consist, in fact, of "Planck" ice dust particles of size ~ 0.1 cm each. On this basis it is possible to explain both the very fact of existence and striking isotropy of "relict" radiation. Let us perform simple physical estimations.

    Electromagnetic radiation of the Sun (photons with the energy <= 0.5 eV) is:

    (57)

    For the whole period of its existence

    the Sun in the form of electromagnetic radiation has lost

    From here we get the following relation:

    (58)

    We have purposely closed the chain of approximate equalities (58) with the relation of electron and nucleon masses, in order to show that some connection between electrons and "relict" radiation should exist. It is known that density of energy (mass) of "relict" radiation is also approximately one thousandth of average mass of the observed stellar material.

    Thus, all the stars for the period of their evolution ~ 10 billions of years, have emitted into the universe cosmic space photons, the total energy of which is comparable with the total energy of "relict" electromagnetic field. But the wavelength of the solar photons is approximately three orders of magnitude smaller than of "relict" ones.

    It is natural that the supposition that there is some cosmic mechanism of re-emission of solar short-wave photons into long-wave "relict" ones suggests itself. The role of intermediate emitter may be completely performed by the above mentioned "Planck" dust particles within protocomets. Absorbing of stellar photon produces electrical excitation in a dust particle. The system becomes a radio aerial, radiating, finally, radiophotons, the wavelength of which is comparable with the aerial size ~ 0.1 cm and the length of "relict" waves.

    As estimations show, the average track length of stellar photon before absorbing it by the dust particle is more than 100 Mpc. It leads to equalizing the space discontinuity of original emitters (stars) up to experimentally observed isotropy and homogeneity of background emission. From the point of view of the new interpretation the notion of background emission, instead of its alternative "relict" one, depict more adequately the essence of the phenomenon and may be used without inverted commas.


    Part III.   On Total System of Equations of Unified Theory

    In the beginning of the third part it is shown that the problem of energy-momentum tensor in GTR, being often discussed in scientific community, most likely, has no definite solution. It is proposed to pass this problem over, having constructed the total system of equations of Unified Theory on the basis of curvature tensor invariants. There are grounds to hope that the obtained in such a way system of equations would allow, finally, performing the complete geometrization of all material structures, completing thus the efforts of Einstein in creating the Unified field theory.

    14. Problem of Energy-Momentum Tensor

    It is clear that the above given estimations on the universe hierarchical structure are to be verified and improved with numerical simulation of the process of elementary waves passing over the real universe. For this it is necessary to have a system of equations, which describes these waves movements. The last one dies against the problem of energy-momentum tensor for equations of Einstein. As is shown in the first part of the paper, only the time component may be strictly compared with energy density. The rest components do not allow, properly speaking, so unique interpretation.

    When choosing the cosmological model of Friedmann out of class of isotropic ones the system of equations of Einstein was supplemented with using additional differential relationship of local principle of correspondence of Einstein GTR to Newtonian mechanics. As a result it turned out that components of energy-momentum tensor are expressed through substance density as follows:

    (59)

    Here density is dependent on time only that corresponds to an empirical fact, i.e. global space isotropy. To pass beyond the framework of the model one should find for metric tensor components the total system of differential equations, replacing one relationship (29). In approximation of local isotropy the total system should be reduced just to the equation (29).

    If we consider relativistic gravics of GTR as Unified Field theory, it is natural to demand from the Einstein equations (2) a status of the total system of equations of this theory. The left part of these equations is nonlinear differential operator at the set of metric tensor components. The energy-momentum tensor within the fully geometrized Unified theory should be, finally, also expressed through the metric tensor components and their derivatives up to the second one.

    We obtain the general expression for energy-momentum tensor in GTR, depending on the principle of least action

    (60)

    with the assumption that the Lagrangian is dependent not only on first but also on second derivatives of metric tensor:

    (61)

    Integrating in (60) is performed over the marked 4D volume , at the boundary of which --- the hypersurface --- all the variations vanish. The Lagrangian density is a scalar function, which is dependent explicitly on the metric tensor components gik and its derivatives for the 4D coordinates xl. then we follow the classical monograph of Landau [2], where variation of the action is considered as a result of varying of 4D coordinates: are small values, being functions of 4D coordinates. The respective variation of the metric tensor is expressed as follows:

    (62)

    Here and below the comma before tensorial index means partial derivative and the semicolon means covariant derivative over the respective coordinate. Calculate the variation :

    (63)

    Here we mark the divergent component and apply to it (two times) the Gauss' law of flux, which allow transforming the integral over 4D volume into the integral over the hypersurface , at which all the variations vanish.

    As a result the relation (63) modifies into:

    (64)

    where the tensor Tik in contravariant presentation has the form:

    (65)

    Substituting into (64) the expression through and applying once again the Gauss' law of flux, we obtain the relation:

    From here, as a result of arbitrariness of the four functions , it follows that the covariant divergence from the tensor is equal to zero and it is interpretated as energy-momentum tensor.

    The relation (65) is the most common expression for energy-momentum tensor in GTR. We transform it in such a way that to get rid of metric tensor determinant, being under the root: . To do this we use the following differential relations:

    (66)

    From the relation (66) one may see that the effect of partial derivative over coordinate on the scalar density is equivalent to the effect of the operator on the value :

    Let us introduce another operator

    (67)

    and the geometrical object derived from it:

    (68)

    Then the expression for (65) may be rewritten in the compact form:

    (69)

    Logically following the idea of full geometrization of all material objects, the Lagrangian of the Unified theory may be constructed just from 14 invariants of curvature tensor. They are given in the monograph of A.Z. Petrov [3]. Of the fourteen invariants of curvature tensor just the simplest of them coincident with space-time scalar curvature is suitable for constructing energy-momentum tensor. It is the sum of diagonal components of Ricci tensor:

    (70)

    The other invariants of Petrov [3] represent the constructions, which are nonlinear over curvature tensor components. When substituting them into the relation (65), we obtain the complex tensorial construction, which contain derivatives of the metric tensor components higher than a second one. It contradicts the initial assumption that the Lagrangian and the theory equations should contain just the metric tensor and its derivatives not higher than a second one. Therefore the invariants are a fortiori not suitable for constructing the total system of equation of Unified Theory with the help of formalism of energy-momentum tensor. Let us demonstrate the correctness of the above statement. Symbolic partial derivatives over the metric tensor derivatives and its derivatives have the form (see [2], 94):

    (71)

    Now we find the expression for the tensor (69), assuming that the Lagrangian of the Unified theory is proportional to the first invariant of Petrov, i. e. to scalar curvature of 4D space:

    (72)

    Relations (71) help to determine the necessary derivatives of scalar curvature:

    (73)

    Performing a number of transformations, it is easy to make sure that the expression (69) for energy-momentum tensor, in assumption of (72), accurate within the numerical coefficient , is proportional to the tensor of Einstein:

    (74)

    Substituting the expression for the energy-momentum tensor (74), constructed on the first invariant of Petrov, into the right part of the equations of Einstein (2), we obtain the simple relation:

    (75)

    The relations (75) are the equations of Einstein in vacuum, or rather in the curved space-time, devoid of the sources (of the right part). It is obvious that they can not pretend to the role of global equations of Unified theory:

    The another variant is possible as well:

    From here it follows that the tensor of Einstein may be equal to the arbitrary second-rank tensor, the covariant divergence from which is equal to zero. By that we return to the original equation of Einstein with the ill-defined right part.
    Thus, the effort of the full geometrization of the equations of Einstein, relying on the principle of least action in the form (60) has not led to the expected result.

    We try then to construct the total system of equations of Unified Theory with the help of curvature tensor invariants.

    15. Invariants of Petrov

    Recall that the equations of Einstein (2) reflect the connection of and at the level of contracted forms of the curvature tensor, i. e. of Ricci tensor and the scalar curvature. Strictly speaking, it is possible to assign the determinate physical meaning just to the first, time equation (2):

    (76)

    The connection of the rest components of energy-momentum tensor with density is beyond the analysis in the general case. The relation (76) is invariant in locally invariant reference system, which may be always (locally) chosen.

    The curvature tensor invariants in physical 4D space of the real universe should be finally expressed through the scalar function .

    Let us pass to investigation of curvature tensor invariants as a mathematical object.

    In the general case Riemannian 4D curvature fourth-rank tensor has 14 invariants. To construct the total set of these invariants according to the technique of A.Z. Petrov it is necessary to preliminary decompose the curvature tensor into the isotropic (*) and the simple (°) parts:

    (77)

    In the symmetric by the number of co- and contravariant component of the note it is supposed to omit the pair of indices lm together and put it before or after ik:

    In the denominations (77) the invariants of Petrov take the form:

    (78)

    Here: is the completely antisymmetric single fourth-rank pseudotensor: , the sign of the rest components is determined by parity of the index transfers as compared with the original combination.

    We put down the invariants (78) for the globally isotropic metrics:

    (79)

    Here:

    Now it is possible to proceed to constructing the total system of equations of Unified theory on the basis of the invariants of Petrov.

    16. Total System of Equations

    We propose to construct the total system of equations of Unified theory on the basis of the invariants of Petrov (78) as follows. At first we substitute into the relations (79) the metrics of the cosmological model of Friedmann . At this process all the values turn out to be proportional to and may be expressed through the time component of the tensor of Einstein: . Generalizing the globally isotropic metrics into the arbitrary one, we obtain the system of equation versus :

    (80)

    The relations (80) may be considered as the total system of the world equations for determining all the ten components of the metric tensor with the initial and boundary conditions absent. We have 14 differential equations for the fourteen unknown values (). The obtained system of equations should, in idea, include solutions, which determine the geometrical and, accordingly, the material structure of the universe right up to "Planck" level.

    It makes sense to connect the physical reference system with the coordinates

    with the model . Then the arbitrariness of coordinate grid choice vanishes. It would not allow decreasing the number of the metric tensor components from ten to six, as is in the general case.

    On account of equality of the first invariant to the scalar curvature , the equation of the system (80) takes the simple form:

    (81)

    Substituting here the metrics of Friedmann with the arbitrary function , we get the equation from which the cosmological model follows practically uniquely, approximating the real universe.

    The system of the fourteen nonlinear differential equations (80) relative to ten components of the world metric tensor, determined at 4D Riemannian space, is extremely complex by its structure. In idea, it contains the most complete (at the elementary "Planck" level) information of space time structure of the world. But in such case the system (80) should, in particular, contain also a wave equation, which describes the movement and focusing of elementary curvature waves. From the system (80) the only one equation (81) pretend to this role, in which the values (the second derivatives of the metric tensor) enter linearly. Breaking down this equation one should take into account the nonlinear components as well, quadratic with respect to the first derivatives of the metric tensor components. Linear wave equations, the right part of which is equal to zero, can not describe the appearance of a material object in the process of focusing.

    We expand the relation (80) through expansion in terms of small addition with respect to Galilean metric tensor:

    (82)

    For more simplification of the problem we suppose that just the one component is nonzero. For one function , the minimum total set should include relations, for instance:

    (83)

    In the following it is possible to enlarge the set of relations, increasing simultaneously a number of components and thus specifying the results. The first equality in this set should be used for simplification of the three next following.

    Let us perform the extension of Ricci tensor, using the relations (82):

    Hereinafter: gik is Galilean tensor. For the note simplification we omit nonlinear components of type, since "the potential" in a given point may always be assumed to be equal to zero (locally). In the general case, when performing global numerical calculations, it should not be done. In addition, there is an opportunity to simplify the obtained relation, imposing on "the potentials" conditions, similar to Lorentz calibration:

    Then in the first round bracket the first two components are mutually reduced with the last one (Galilean tensor may be entered under the differentiation). Only the "wave" component is left:

    And the expression for is reduced to the form:

    (84)

    The contents of square bracket of the relation (84) gives nonlinear components. As a result we obtain that the relation (80) is reduced to the wave equation with the positively defined source:

    (85)

    In the same approximation the equation (76) takes the form:

    (86)

    The relation (86) allows connecting the density and the density of the source in the wave equation (85).


    Conclusions

    Let us recollect the basic problems considered in the proposed manuscript.

    The physical interpretation of components of energy-momentum tensor in GTR of Einstein is revised. Appropriateness of using conventional formula (3) in the right part of Einstein equations for choosing a cosmological model out of a class of isotropic ones is called in question.

    The principle of local correspondence between GTR of Einstein and Newtonian mechanics is stated. The invariant differential relation (29) between components of metric tensor of arbitrary gravitating system is get. It allows enlarging in a required way the system of Einstein equations, which are written as applied to the isotropic model of Friedmann. As a result it is possible to rather easily get a unique solution of the cosmological problem i.e. to perform a choice of a physical model (a closed one) approximating the real universe out of a wide mathematical class of isotropic models of Friedmann.

    In new cosmological model (35) spatial curvature radius is linearly dependent on dimension time coordinate. Introducing the dimensionless time coordinate with the relation (36) demonstrates unboundness of evolution process of the universe approximated by the cosmological model . The last condition closes a question about the initial moment of evolution being traditionally difficult for cosmology.

    Instead of the traditional concept of Big Bang an alternative variant is proposed, that is step-by-step evolution change of the universe structural elements oriented to their complication. That is just what leads to an apparent extending of the universe.

    The physical interpretation of spatial components of energy-momentum tensor in GTR of Einstein is reviewed. The baselessness of using the conventional formula (3) for energy momentum tensor of continuum is shown.

    The principle of local correspondence of GTR of Einstein to Newtonian mechanics is formulated and constructively implemented. The invariant differential relation (27), generalizing the equation of Poisson (4) for the arbitrary gravitating system (29) is obtained. It enables to supplement the system of equations of Einstein, put down with reference to the isotropic model of Friedmann. As a result it is possible to obtain rather easily the unique solution of the cosmological model, i. e. to perform a choice of a physical model (a closed one), which approximates the real universe, out of the wide mathematical class of the isotropic models of Friedmann.

    In the new cosmological model (35) the space curvature radius is linearly dependent on dimensional time coordinate. Introducing the nondimensional time coordinate with the relation (36) shows the unboundedness of the universe evolution process, approximated by the cosmological model . The last condition closes the traditionally complex for cosmology problem on the initial moment of the universe evolution.

    Instead of the conventional concept of Big Bang we propose the alternative variant -- the gradual evolutionary change of the universe structural elements towards their complication. This very fact leads to the apparent universe expansion.

    The task of the first and the second part of the paper is to destroy one of the deepest scientific fallacy of the twentieth century --- the idea of Big Bang. In fact, "the red shift", interpreted as recession of galaxies is, evidently, a result of gradual evolutionary change of all the structural elements of the universe including subatomic. The universe itself turns to be, as a matter of fact, static, subjected to just structural evolution. It is the insight of the greatest physicist of our time that is worth applauding! Let us recollect the internal resistance with which Einstein refused the static universe and agreed with the idea of its expansion.

    The second part of the manuscript presents the considerations on the opportunity of passing beyond the limits of the cosmological model on the basis of idea of the curvature waves focusing. At this process the geometry specific character of the new cosmological model is used. The picture of the world constructed on this basis (for the time present with the help of estimations) agrees on the whole with the hierarchical structure of the real universe. Specifying of the obtained picture is possible while performing numerical simulation of focusing states.

    When implementing such not simple task (the third part of the paper) the problem on energy-momentum tensor in GTR, which can not, evidently, be completely solvable, has arisen. Nevertheless, we have managed to avoid this problem by constructing the total system of differential equations for global metrics of the real universe on the basis of the curvature tensor invariants. There are grounds to hope that this path would finally lead to transformation of Einstein GTR into the Unified field theory.


    References

    1. A. Einstein. On Particles Movement in General Theory of Relativity (1946). Proceedings of scientific papers (p. 674), Moscow, "Nauka", 1966.

    2. Landau L. D., Livshitz E. M. Field Theory. Moscow, "Nauka", 1973.

    3. Petrov A. Z. New Methods in General Theory of Relativity. Moscow, "Nauka", 1966.

    4. Mordvinov B. P. Gravitation and Structure of the Universe. Preprint 13, Chelyabinsk-70, 1991.



    Author: B. Mordvinov
    The interpreter of this paper: A. Kulkova



    Gennady M.
    Last modified: Thu Mar 18 14:34:08 PST 1999