The Metric Outside a Fixed Charged Body in the Covariant Theory of Gravitation

The metric outside a charged body is calculated. As part of the given approach it is shown that the gravitational and electromagnetic fields are equally involved in the formation of the metric tensor components. And the contribution of fields in the metric is proportional to the energy of these fields. From equations for the metric it follows that the metric tensor components are determined up to two constants.


Introduction
In the theory of gravitation the metric is needed for correct representation of theoretical conclusions and experimental results. The concept of metric is most important in the general theory of relativity (GTR), in which the metric tensor field plays the role of the gravitational field. As a result, transition of GTR to the Newtonian theory of gravitation is always performed by expanding the metric tensor components to the principal expansion terms. For example, for the gravitational field potential we obtain the following expression: small the field is, in GTR first we must find 00 g , in order to evaluate then the potential  .
In contrast to this, in the covariant theory of gravitation (CTG) the metric in the weak field automatically tends to the metric of the Minkowski flat space, while the gravitational field equations do not change their form due to the covariant notation of equations. In CTG first we obtain the scalar potentials  and  of the gravitational and electromagnetic fields, the vector field potentials, the tensors, corresponding to these fields, and only after that we calculate the spacetime metric outside the body [1].

The solution of the equation for the metric
The tensor equation for the metric, which was found in [2] from the principle of least action, taking into account the cosmological constant gauge, in mixed indices has the following form: Outside the body, where the mass and charge densities are equal to zero, the scalar curvature R and the stress-energy tensors B   of the acceleration field and P   of the pressure field are also equal to zero. This is due to the fact that the acceleration field describes the motion of the body particles, and the pressure field is associated with the pressure of the body particles on each other. Both of these fields exist only in the volume of the body under consideration. The equality of the scalar curvature R to zero is associated with the gauge condition of the cosmological constant.
In the right side of (1) there are two stress-energy tensors of the gravitational U   and electromagnetic W   fields: (2) Here G is the gravitational constant, 0  is the vacuum permittivity, Φ  and F  are the gravitational and electromagnetic tensors, respectively,    is the unit tensor or the Kronecker delta. We consider that the speed of light c is the same for the propagation of electromagnetic and gravitational effects through the field. The stress-energy tensor of the gravitational field in the form of (2) was presented in [1].
With regard to (2) and (3), from (1) we obtain: According to (4), the Ricci tensor R   , associated with the spacetime curvature outside a single body, depends on the gravitational and electromagnetic field strengths of this body.
Next, we will use the notation described in [3].
Suppose there is a body with a spherical shape and the mater distribution which is symmetrical relative to the center of the sphere. In the static case, the metric depends neither on time nor on the angles of the spherical reference frame. Then the metric at an arbitrary point around the body will depend only on the radial coordinate r connecting this point with the center of the sphere. It is therefore convenient to use the spherical coordinates . The metric tensor will be found in the following form: where are the functions only of the radial coordinate r .
, it is possible to find the metric tensor with contravariant indices: According to (5) r m r m m r here  is the scalar potential of the gravitational field around the massive body with the gravitational mass M ,  is the scalar potential of the electromagnetic field around the body with the charge Q , the quantities D and A denote the vector potentials of the gravitational and electromagnetic fields.
We will assume that the vector potential D of the body is equal to zero, since that the body does not rotate, its particles move randomly in different directions and the vector potentials of particles compensate each other. Similarly, we assume that there are no directed electrical currents within the body, the magnetic moments of the substance particles are compensated and 0  A .
Using (7) and (2) If we denote the derivatives with respect to r by primes, then the non-zero Christoffel coefficients, expressed in terms of functions E K B , , in the metric tensor (5) and (6), according to (11) are equal to: Substituting (13) into (12), we find non-zero components of the Ricci tensor, and it also turns out that 22 33 RR  . We obtain the expression for the Ricci tensor in mixed indices with the help of the metric tensor: Using the obtained components R   in the left side of (4), and the tensor components U   from (9) and W   from (10) in the right side (4), we find three independent equations: , it gives the following: where 1 A is a constant.
We will substitute in (16) The solution of equation (20) is the following expression: where 2 A and 3 A are some constants.
At infinity, where the gravitational and electromagnetic fields are close to zero, the metric tensor (5)