I. Introduction

_{Such a curved and torsional space-time can be represented in general case by a 4-dimensional metric-affine space G4. Some particular cases of this space, the Weyl-Cartan space, the Weyl space, the Riemann-Cartan space U4, and others, were used in different non-symmetric unified theories, e.g. in the Einstein-Cartan theory [2, 5-6, 8]. The structures of all these spaces are characterized by two main geometrical parameters: the affine connections and the fundamental metric tensor [2-5, 9-12].}

_{To accomplish this, one should firstly choose the type of Lagrangian . We will use the simplest Lagrangian}

_{, , , , (1)}

_{where is the curvature tensor of the space.}

_{The field equations for G4 space can be derived from the variational principle for this Lagrangian, where the metric and affine connections are considered a priori the independent variables [10, 12, 15-16]. Therefore, we will vary action by metric and connections independently (the Palatini principle). The action is}

_{, (2)}

_{where is the 4-dimensional volume element.}

_{Variation by metric yields}

₍₃₎

_{Using the results from [2-3, 6, 10, 15-16] and varying action (2) by connections , we obtain}

_{, (4)}

_{where is the Kronecker tensor.}

_{Thus, for G4 space, not assuming any a priori correlation between and , we obtained the Einstein equations (3), as a result of variation of action (2) by , and equations (4), connecting geometrical characteristics of the space, as a result of variation of this action by . System (4) contains 64 algebraic equations with 64 unknown functions and allows one, in principle, to determine all the connections.}

_{Equations (3)-(4) can be simplified and solved for a spherically symmetric stationary field in G4 space. For such a case in 4-dimensional spherical coordinates , , , we have only four non-zero components of the metric tensor}

_{, , , , (5)}

_{where and , describing geometry of the space-time, are the unknown functions to be determined.}

_{Substituting (5) into (4) produces the following expressions for}

_{, , , ,}

_{, , , ,}

_{, , , (6)}

_{, , , ,}

_{, , , , ,}

_{where connections are undetermined, all other are equal to zero, and and denote the ordinary derivatives with respect to .}

_{The undetermined quantities appear because the number of independent equations in system (3)-(4) in a spherically symmetric stationary field, unlike the general case, is less than number of the variables.}

_,

_,

_,

_{, (7)}

_,

_,

_,

_,

_,

_{where denotes the second ordinary derivative with respect to .}

_{To solve system (7), one should make an assumption regarding undetermined connections .}

₍₈₎

_{, , , ,}

_{, , , (9)}

_{, ,}

_{where all the other 52 quantities are equal to zero.}

_{Assumption (8), leading to solution (9), can be physically explained in the following way. Different sets of are related to various degrees of space-time curvature and torsion. Formulae (9) represent the simplest set of obtained by the only possible means. Any other set of not related to condition (8), will lead to a more complex solution. Thus, solution (9) is the simplest one and corresponds to minimal curvature and torsion of the space-time, describing a pure gravitational spherically symmetric stationary field. Any other field (e. g. electromagnetic field) creates a gravitational field too. This means that curvature and torsion of the space-time related to that "mixed" field and the corresponding set of connections , must be more complex.}

_{, (10)}

_{, , (11)}

_{, ,}

_{where and are the constants determined by boundary conditions.}

_{Solution (11) in G4 space corresponds to a pure gravitational spherically symmetric stationary field. From a physical point of view, it is clear that such a field can exist either at very large or very small values. In the first case, we have extremely dispersed matter: in other words, gravitational waves propagating far away from their sources. Consequently, the space-time for such waves must be maximally uniform and simple. In the second case, we obtain extremely compressed matter: a pure gravitational ball, inside which no other forms of matter exist. Moreover, no transformations should occur inside such a ball. Any appearance of some other fields within the ball area will violate the spherical symmetry.}

_{In the transitional case (at medium values of ) regular matter must exist –various elementary particles and fields. Solution (11) is not valid for a transitional case. In order to answer the question whether or not the gravitational ball appears, we should analyze just such a transitional case. The scenario of development depends on mass, energy, temperature, pressure, and other physical parameters of the compressing matter. However, to find the appropriate solution of equations (7), one can no longer use condition (8), which is valid only for a pure gravitational field. Therefore, the transitional case for a spherically symmetric field will be described by complex equations, reduced from the condition related to the matter under consideration, and equations (7).}

_{To obtain a solution for the large from (11), one can use ordinary boundary conditions: the Newton’s expressions for metric at [13]. It yields a solution coinciding with the Schwarzschild solution for a spherically symmetric stationary pure gravitational field far away from its material sources}

_{, , (12)}

_{where is the gravitational radius.}

_{This match confirms the correctness of the general formulae (11) and our idea that the simplest solution of equations (7), or (10), and set (9) of connections , are actually related to the simplest physical field (pure gravitational), i.e. to space-time with minimum curvature and torsion.}

_{To obtain a physically reasonable solution for a gravitational ball (for a very small ), one should use formulae (11) and the boundary condition, which exclude the singularity of the metric at . Such a condition is natural for any closed physical theory. To satisfy it, we should choose from (11) the coefficients and . Thus we arrive to the following solution for a gravitational ball}

_{, (13)}

_{This result corresponds to the well-known Galilean metric. This means that inside an extremely compressed gravitational ball, there is just a plane space-time without any curvature and torsion. This can be explained in the following way: in a spherically symmetric field, the density of matter and energy is inversely related to radius due to the magnitude of internal pressure. It means that when decreases, this ball becomes more dense and uniform (e. g. plasma, nuclear matter, neutron matter, etc.). At this stage, the continuous "reconstruction" of the space-time occurs, increasing the density, homogeneity and simplicity of structure. Finally, at very small , a region appears where simplicity and homogeneity attain maximum. The space-time in this area is flat and possesses the Galilean metric. However, such a plane space-time means that even a gravitational field does not exist in this area. This is natural, since there is no gravitational field in the center of any spherically symmetric ball: attraction of the external regions leads to their mutual compensation.}

_{Now we simplify and solve equations (3)-(4) for the cosmological problem concerning the evolution of the universe. Analyzing equations for a uniform isotropic space, we will use the method described in [13]. Start with the closed isotropic model. For this model, the non-zero components of the metric tensor in 4-dimensional "spherical" coordinates are}

_{, , , , (14)}

_{where is the space curvature radius depending on the coordinate , which is related to time by the formula , and is the speed of light.}

_{Substituting (14) into (4), we obtain the expressions for affine connections}

_{, , ,}

_{, , ,}

_{, , , (15)}

_{, , ,}

_{, ,}

_{where connections are undetermined, all other are equal to zero, and denotes the ordinary derivative with respect to .}

_{The undetermined quantities appear because the number of independent equations in system (3)-(4) in the uniform isotropic field, unlike the general case, is less than the number of variables.}

_,

_,

_,

_,

_{, (16)}

_,

_,

_,

_{where denotes the second ordinary derivative with respect to .}

_{To solve system (16), one should again make an assumption regarding undetermined connections .}

_{, , ,}

_{, , , (17)}

_{, , all other 49 quantities are equal to zero.}

_{Note once again, that different sets of satisfying (15) correspond to various degrees of space-time curvature and torsion, describing different physical fields. The simplest set (17) of connections , obtained by the only possible means, describes the "simplest" physical field, a pure gravitational one.}

_{, (18)}

_{, (19)}

_{where and are the constants.}

_{, (20)}

_{where and are the constants.}

_{Thus we have obtained formulae (19) and (20), determining the curvature radius of the space G4 as a function of the "spherical time coordinate" for two models of the universe (closed and open).}

_{Formulae (19) and (20) for the space curvature radius are valid only for a pure gravitational field. However it is clear from a physical point of view (as it is for a spherically symmetric case), that such a field can exist either for very large (an extremely dispersed field, or gravitational waves) or for very small (an extremely compressed field, or gravitational "cluster").}

_{Analyzing (19) and (20) for very large , we see that the space curvature radius will increase infinitely with time for the open model and oscillate for the closed one. It means (for the open model, for example), that at the final stage of its development, the universe or, more accurately, the metagalaxy will infinitely expand and "straighten", as it should happen for a uniform isotropic pure gravitational field.}

_{For very small (at the initial stage of the metagalaxy development) for both models, we obtain a pure gravitational area with the Galilean metric, whose space curvature radius grows up with time.}

_{It is emphasized that analyzing either spherically symmetric or uniform isotropic fields in G4 space for very small , we obtained similar results: an extremely compressed pure gravitational ball, inside which space-time has the Galilean metric. This conclusion confirms the idea [15] that at the metagalaxy and the collapsing star are identical, and that generically, the process of collapse for massive gravitational systems is similar for a star, a nebula, or a metagalaxy.}

_{The first four equations (7) coincide with (10); that is, they describe a spherically symmetric gravitational field. Their solution is given by formulae (11), where the metric components are determined. All connections except are calculated in (6). The four connections should be obtained from the last six equations (7), describing a spherically symmetric distribution of matter.}

_{Equations 5, 6, and 7 from system (7) have the unique solution . The last three equations from (7) do not have the unique solution. They give various solutions for related to different material fields.}

_{In order to determine from (7) the quantities , connected to some material spherically symmetric stationary field, one can use the following technique. Compare unified field equations (7) in G4 space with equations describing the same material field with the help of the energy-momentum tensor in a more simple space, determine in this space through , and make some conclusions regarding in G4 space.}

_{Since equations (7) are anti-symmetric, the other system of equations containing the tensor , should be anti-symmetric too. Moreover, the tensor describing this spherically symmetric stationary material field should be anti-symmetric.}

_{On one hand, any stationary material field, according to Einstein [2], should be massless. On the other hand, only matter with spin has the anti-symmetric tensor . It means that only massless fluid with spin can play the role of the desired material field. However, to describe such a fluid, only the Riemann-Cartan space U4 can be used, because only the Einstein-Cartan theory analyses matter with spin.}

_{The two systems of equations (in G4 and U4 spaces) must be identical because they give different descriptions of the same phenomenon. The reasoning of Ponomarev et al. in [15] confirms the idea that these two systems are actually equivalent. Their identity allows one to determine the unknown connections in (7) by knowing in U4 space.}

_{First of all, one should obtain the system of Einstein-Cartan equations describing spherically symmetric stationary massless fluid with spin in U4 space. To accomplish this, one starts from calculations in U4 space for spherically symmetric stationary field, using the equations [10, 15-16]:}

_{, (21)}

_{where is the Cartan’s torsion tensor, the three tensors are connected by indices alteration, and are the connections in the Riemannian V4 space for a spherically symmetric stationary field [13].}

_{, , ,}

_{, , ,}

_{, , , (22)}

_,

_{where connections are undetermined, and all other are equal to zero.}

_{After that, the components of curvature tensor are calculated by formula (1). Finally, the system of the Einstein-Cartan equations in U4 space for spherically symmetric stationary massless fluid with spin is obtained:}

_,

_,

_,

_,

_{, (23)}

_,

_,

_,

_,

_,

_{where are the components of the energy-momentum tensor of massless spinning fluid in U4 space, k = 8G/ ⁴ = 2.07·10^-48 s²/g·cm, G = 6.67·10^-8 cm³/g·s² is the gravitational constant.}

_{The similarity between systems (7) and (23) allows determination of the connections in (7) in G4 space, if the connections in (23) in U4 space are calculated (i.e. expressed through tensor components). The fact that equations (7) and (23) are tantamount and describe one and the same physical field allows the assumption that the solutions (i.e. expressions for in U4 and G4 spaces) should be identical.}

_{To solve system (23) and express through , one should firstly obtain the energy-momentum tensor of spherically symmetric stationary massless spinning fluid.}

_{To calculate tensor , the following formulae from [6, 18] can be used}

₍₂₄₎

_,

_{where is the internal angular momentum (spin) tensor of massless Weyssenhoff spinning fluid, functions A(r), B(r) and C(r) are the components of spin vector [18] depending only on radius r, and is the 4-dimensional velocity for stationary field.}

_{Formulae (24) allow the obtaining of components of spherically symmetric stationary massless Weyssenhoff fluid with spin.}

₍₂₅₎

_,

_{where and are functions determined in (11); A, B and C are functions from (24).}

_{Substituting (25) into (23), we obtain equations determining connections in U4 space through the components of tensor of the massless matter with spin:}

₍₂₆₎

_{where , , A=A(r), B=B(r) and C=C(r).}

₍₂₇₎

_{Formulae (27) determine connections of a spherically symmetric stationary U4 space filled with massless Weyssenhoff fluid.}

_{We assume that the same expressions for should hold true in G4 space, since equations (7) and (23) are tantamount and describe the same phenomenon. As a result, the final solution of equations (7) in G4 space for a spherically symmetric stationary field of massless fluid with spin is given by formulae (5), (6), (11), (27).}

_{Thus, the geometry (i.e. metric and all connections ) of G4 space filled with massless Weyssenhoff fluid is determined through the spherical coordinates and the spin vector component of the massless fluid with spin.}

_{The first two equations (16) coincide with (18), describing a uniform isotropic gravitational field. Their solution is given by formulae (19)-(20), where the metric components are determined. All connections , except , are calculated in (15). The four connections should be obtained from the last six equations (16) describing uniform isotropic matter. These equations do not have a unique solution; they give different solutions for , related to different material fields.}

_{In order to determine from (16) the quantities , which are related to some material uniform isotropic field, one can use technique described in section V for a spherically symmetric field. Firstly, we obtain in U4 space the equations tantamount to equations (16) in G4 space and describing the same uniform isotropic material field. Secondly, we determine the tensor of the material field. Thirdly, we express in U4 and G4 spaces through components.}

_{Acting as in the case of a spherically symmetric stationary field (see section V), we use massless fluid with spin in uniform isotropic space. To obtain the system of the Einstein-Cartan equations describing massless Weyssenhoff fluid in uniform isotropic U4 space, one should calculate connections in this space, then determine curvature tensor , and finally receive the desired equations.}

_{These equations in U4 space are tantamount to equations (16) in G4 space, except their right-hand side contains the energy-momentum tensor of massless fluid with spin. The difference between these two systems of equations is exactly the same as the difference between equations (7) and (23) in a spherically symmetric stationary field.}

_{Tensor of massless Weyssenhoff fluid in uniform isotropic U4 space was calculated by formulae (24). Substituting the results into the corresponding Einstein-Cartan equations in U4 space, we arrived at the following equations determining connections in U4 space through parameters A, B and C of massless fluid with spin:}

_,

_,

_,

_{, (28)}

_,

_,

_{where are the 4-dimensional "spherical" coordinates; k is the constant from (23), space curvature radius ; spin vector components of Weyssenhoff fluid C=C(h ), B=B(h ), A=A(h ), and the ¢ sign denotes the ordinary derivative with respect to h .}

_{The simplest solution of (28) can be obtained if A=B=0, C(h )= and . Then:}

_{, (29)}

_{where and are the constants.}

_{Formulae (29) determine connections of uniform isotropic U4 space filled with massless Weyssenhoff fluid. We assume that the same expressions for should hold in G4 space.}

_{Thus, the geometry (i.e. metric and all connections ) of G4 space filled with massless Weyssenhoff fluid is determined through 4-dimensinal "spherical" coordinates and the spin vector component C(h ) of massless fluid with spin.}

1. The various types of curvature and torsion in G4 space are related to different physical fields. Expressions (3)-(4) are the general equations of the unified geometrical field theory. The idea confirms the hypothesis [2, 4], that "physics is the pure geometry".
2. _{The simplest set of affine connections describes the space-time with minimal curvature and torsion or, in other words, the "simplest" physical field – a pure gravitational field. More complex sets of represent more complex geometries of the space-time and therefore describe more complex material physical fields.}
3. Solutions were obtained for a pure gravitational field and massless Weyssenhoff fluid with spin in a spherically symmetric stationary space and uniform isotropic space.
4. For both models (spherically symmetric and uniform isotropic), a pure gravitational field can exist either at very large or very small distances.
5. Inside the extremely compressed spherically symmetric pure gravitational ball, the space-time is plane and has the Galilean metric. There is neither collapse, nor a black hole in such a field.
6. At the initial stage of its development, the uniform isotropic metagalaxy was an extremely compressed pure gravitational area with the Galilean metric of space-time.
7. There is no necessity to quantize equations (3)-(4). They describe the evolution of matter in the most general way, indicating that any physical field (i.e. curved and torsional space-time) is a deterministic system.

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