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The Gravitational Energy of a Point Mass is Finite

We argue that our equation of gravitation ( Phys.Lett. A 156 (1991) 404 ) lead in pseudo-Euclidean space-time to the finite energy of the gravitational field of a point mass.

whereẋ α = dx α /dt and g αβ is a symmetric tensor whose components are the function of ψ αβ . The field ψ µν in flat space-time is an analog of the potential A µ of the electromagnetic field. Therefore the field equations for g µν (ψ) must be invariant under the gauge transformations ψ µν → ψ µν ¿. The simplest equations of such a kind where proposed in paper [1] . The equations can be written in the form where the comma denotes the covariant derivative with respect to the metric tensor η µν in pseudo-Euclidean space-time E.
In this equations the tensor B α βγ is given by (Greek indices run from 0 to 3) , where • Γ γ αβ are the Christoffel symbols in E and Γ γ αβ are the Christoffel symbols of the Riemannian space-time V of dimenstion n , whose fundamental tensor is g αβ .
The tensor B γ αβ can be formed by replacing the odinary derivatives in Π γ αβ with the covariant ones in E. B α βγ,α satisfies also the following identities: where tensor * R γ αβδ is obtained from the Riemannian curvarure tensor by replacing the Christoffel symbols with the Thomas symbols and by replacing ordinary derivative with the covariant one in E.
The equations (2) and (6) are field equations for the tensor B α βγ . Eqs. (2) are invariant under arbitrary transformations of the tensor g αβ retaining invariant the equations of motion of a test particle , i.e.geodesic lines in V . In other words, eqs.(2) are the geodesic-invariant . Thus , the tensor field g αβ is defined up to geodesic mappings of space-time V (in the way analogous to the definding the potential A µ in electrodynamics up to gauge transformations). Therefore, additional conditions can be imposed on the tensor g αβ . In partucular, if the tensor g αβ satisfies the conditions then eqs (2) will be reduced to vacuum Einstein equations R αβ = 0 , where R αβ is the Richi tensor [1]. Unlike the g αβ or Γ γ αβ the tensor B γ αβ is invariant under the geodesic mappings of space-time V as well as strength tensor F αβ in electrodynamics is invariant under the gauge transformations.

Spherically Symmetric Field
For a spherically symmetric field the nonzero components of the tensor B γ αβ in the spherical coordinate system are: where the functions A , B and C are : and f = (r 3 + β 3 ) 1/3 and f ′ = df /dr . The constant β is not determined from the boundary conditions [1]. If we put β = α , where α = 2GM/c 2 , M is the attractive mass, G is the gravitational constant and c is the speed of light, then the solution (8) has no a physical singularity in the sense that this solution does not lead to the collapse. Some astrophysical consequences of this solution are considered in [3] .

The Gravitational Energy of a Point Mass
Eq. (2) without the nonlinear term is analogous to the electrodynamics equations F γ β,γ = 0 for the strength tensor F αβ . Since the gravitational field must be self-interacting we can suppose that eq. (2) are of the form where κ = 4πC/c 4 and is the energy-momentum tensor of gravitation field . The components of the 3-momentum density vector are equal to zero for the solution (8).
Let us find the energy of a point masses We have and ,therefore, in the used coordinates system where where is the B-function [4] . Using the equation where Γ is Γ-function and setting β = α , we obtain finally On the contrary, if we seach the constant β from the condition that gravitational energy of a point mass is given by equation (18) , then we shall arrive at the conclution that β = α. Certainly , the 4-divergency of the tensor t αβ must be equal to zero. It is likely that the 4-divergency of the tensor B γ αβ is not equal to zero identically. However , it is should be observed that the conservation law for the energymomentum tensor must not be is satisfied for an arbitrary field B γ αβ (or g αβ ), and only for the fields wich are solutions of eq. (2). For the received solution of the field equations the 4-divergency of the tensor B γ αβ is indeed equal to zero. Thererfore, the tensor t αβ for the spherical-symmetric field is a conservation value.
The gravitation energy of the point mass being finite, it is sufficient argument to suppose that the energy -momentum of gravitational field of an attracting mass is given by equation (10) up to a term, with the integral over the volume equal to zero.