Федосин С.Г. О метрике ковариантной теории гравитации
внутри тела в релятивистской однородной модели // Научно-технические ведомости СПбГПУ. Физико-математические науки. Т. 14. № 3. С. 168-184 (2021).

St. Petersburg Polytechnical
State University Journal. Physics and Mathematics, Vol. 14, No. 3, pp.168-184 (2021). http://dx.doi.org/10.18721/JPM.14313.

**The relativistic
uniform model: the metric of the covariant theory of gravitation inside a body**

**Sergey G. Fedosin**

PO box 614088, Sviazeva str. 22-79, Perm, Perm Krai, Russia

E-mail: fedosin@hotmail.com

It is shown
that the sum of stress-energy tensors of the electromagnetic and gravitational
fields, the acceleration field and the pressure field inside a stationary
uniform spherical body within the framework of relativistic uniform model
vanishes. This fact significantly simplifies solution of equation for the
metric in covariant theory of gravitation (CTG). The metric tensor components
are calculated inside the body, and on its surface they are combined with the
components of external metric tensor. This also allows us to exactly determine one of the two unknown coefficients in the metric outside the body.
Comparing the CTG metric and the Reissner-Nordström
metric in general theory of relativity shows their difference, which is a
consequence of difference between equations for the metric and different
understanding of essence of cosmological constant.

**Keywords:** metrics; covariant theory
of gravitation; scalar curvature; cosmological constant; relativistic uniform system.

**1. Introduction**

In modern
physics, the spacetime metric of a certain physical system is completely
defined by the corresponding metric tensor. The metric is of particular
importance in the general theory of relativity, where the metric describes the
action of gravitation. In contrast, in the covariant theory of gravitation
(CTG), gravitation is an independent physical interaction. In this case, the
metric of CTG is required mainly to describe the additional effects, associated
with the interaction of electromagnetic waves with the gravitational field in
the processes of space-time measurements by means of these waves. Accordingly,
the form of the metric depends significantly on the theory of gravitation used.

Despite the
success of the general theory of relativity in describing various gravitational
phenomena, the theoretical foundation of this theory is still unsatisfactory. First
of all, this is due to the absence of a generally recognized energy-momentum
tensor of the gravitational field itself, the search for which continues to
this day [1-3]. Accordingly, the energy and momentum of a system becomes
ambiguous or not conserved [4-6]. Other problems include emerging
singularities, the need to interpret the cosmological constant, dark matter,
dark energy, etc. In this regard, the search for alternatives to the general
theory of relativity remains relevant, in particular, among vector-tensor
theories [7-9].

The covariant
theory of gravity (CTG) refers to vector theories and has a well-defined
energy-momentum tensor of the gravitational field. Outside the fixed spherical
body, the metric tensor components within the framework of CTG were determined
in [10]. Only the gravitational and electromagnetic fields exist outside the
body, therefore only these fields exert their influence on the spacetime metric
here. Using this metric, it was possible to calculate the Pioneer effect, which
has no explanation in the general theory of relativity [11]. CTG formulas
describing the gravitational time dilation, the gravitational redshift of the
wavelength, the signal delay in the gravitational field, lead to the same
results as the general theory of relativity [12].

Next, we will
calculate the metric of CTG inside a spherical body. In the presence of the
matter, we should take into account the pressure field, which we consider in a
covariant form as a vector field. Similarly, the concept of the vector
acceleration field [13-14] is used to calculate the energy and momentum of the
matter, and its contribution into the equation for the metric. It is the
representation of these fields in the form of vector fields that made it
possible to find a covariant expression for the Navier-Stokes equation [15]. In
contrast, in general relativity, the pressure field and the acceleration field
are almost always considered as simple scalar fields. Consequently, we can
assume that CTG more accurately represents the contribution of the fields to
the energy and momentum, as well as to the metric of the system.

In order to
simplify the solution of the problem, we will assume that the matter of the
body moves chaotically in the volume of the spherical shape, and is kept from
disruption by gravitation. The force of gravitation in such macroscopic
objects, as planets and stars, is so strong that it is sufficient to form the
spherical shape of these objects. This force is counteracted by the pressure
force in the matter and the force from the acceleration field. One of the
manifestations of the force from the acceleration field is the centrifugal
force arising from that component of the particles’ velocity, which is
perpendicular to the radius-vector of the particles. We can also take into
account the electromagnetic field and the corresponding force, which usually
leads to repulsion of the charged matter in case of the excess charge of one
sign. We will also assume that the physical system under consideration is a
relativistic uniform system, in which the mass and charge distributions are
similar to each other. This will allow us to use the expressions found earlier
for the potentials and field strengths.

The need to
determine the metric inside the matter arises as a consequence of the fact that
the comparison of expressions for the components of the metric tensor inside
and outside the matter makes it possible to unambiguously determine one of the
unknown coefficients in the external metric. As a result, we obtain a more
accurate expression for the CTG metric, suitable for solving more complex
problems and considering small gravitational effects.

**2. The
equation for the metric**

The use of the
principle of least action leads to the following equation for the metric in CTG
[14]:

. (1)

Here is the speed of
light; is the constant, which is part of
the Lagrangian in the terms with the scalar curvature and with the
cosmological constant ; is the Ricci tensor
with the mixed indices; is the unit tensor or the Kronecker
symbol; , , and are the
stress-energy tensors of the gravitational and electromagnetic fields, the
acceleration field and the pressure
field, respectively.

As was shown in
[16], all the quantities in (1) should be averaged over the volume of the
system’s typical particles, if (1) is used to find the metric inside the body.
We will further assume that such averaging has already been carried out in (1).
Another conclusion in [16] is that within the framework of the relativistic
uniform model the scalar curvature inside a stationary body with the constant relativistically invariant mass density and charge is a
certain constant quantity . In this case, in CTG the relation holds, where is the averaged cosmological constant
for the matter inside the body.

Acting as in [10],
we will use the spherical coordinates , , , , related to the Cartesian coordinates by the relations: , , . For the static metric, the standard form of the
metric tensor of the spherical uniform body is as follows:

, (2)

, (3)

where are the
functions of the radial coordinate only and do not
depend on the angular variables, and there are four nonzero components of the
metric tensor , , , .

By definition,
the Christoffel coefficients are expressed
in terms of the metric tensor and its derivatives:

. (4)

If we denote
the derivatives with respect to the radius by primes, then
the nonzero Christoffel coefficients, expressed in terms of the functions in the metric
tensor (2) and (3), are equal according to (4) to the following :

, , . (5)

With the help
of (5) we will calculate the components of the Ricci tensor with the covariant
indices using the standard formula:

.

This will give
four nonzero components:

, ,

, . (6)

Equation (1)
contains the components of the Ricci tensor with the mixed indices, which can
be found by multiplying the components of this tensor with the covariant
indices by the metric tensor using the formula: . With the help of (6) and (3), we find:

,
,

, . (7)

Using (6) and
(3), we will calculate the scalar curvature as follows:

. (8)

**3. The
field tensors**

The
stress-energy tensors of the gravitational field [17-18], the electromagnetic
field, the acceleration field and the pressure field [14], located on the
right-hand side of the equation for the metric (1), can be expressed as follows

,

.

,

. (9)

In (9) , , and are the tensors
of the gravitational and electromagnetic fields, the acceleration field and the
pressure field, respectively; is the gravitational
constant; is the electric
constant (vacuum permittivity); is the
acceleration field constant; is the pressure field constant.

The
stress-energy tensors in (9) were derived from the principle of least action
under the assumption that all the physical fields in the system under
consideration were described as vector fields that had their own
four-potentials [13]. Due to the fact that the field tensors have the same
form, it was possible to combine all the fields into a single general field [19-20].

Let us express
the four-potentials of the fields in terms of the corresponding scalar and
vector potentials of these fields: for the gravitational field, for the electromagnetic field, for the acceleration field, and for the pressure field.

The
gravitational tensor is defined as the four-curl of the four-potential [17].
Similarly, the electromagnetic tensor, the acceleration tensor and the pressure
field tensor [14] are calculated:

, .

, .

(10)

In the system
under consideration, the vector potentials , , and of all the
fields are close to zero because of the random motion of the matter’s
particles. This is due to the fact that the vector potentials of individual
particles are directed along the particles’ velocities, and therefore they
change each time as a result of interactions. The global vector potential of
each field inside the body is calculated as the vector sum of the corresponding
vector potentials of the particles. At each time point, most of the particles
in the system have oppositely directed velocities and vector potentials, so
that on the average the vector sum of these potentials tends to zero. The more
particles are present in the system, the more exactly the equality to zero
holds for the global vector potentials of the fields. We will not also take
into account the proper vector potentials of individual particles. As was shown
in [21], the energy of the particles’ motion arises due to all these
potentials, which is approximately equal to their kinetic energy. Thus the
inaccuracy, arising from equating the vector potentials , , and to zero, does
not exceed the inaccuracy in the case when only the rest energy is taken into
account in the system’s energy and the kinetic energy of the particles is
neglected.

As for the
scalar field potentials , , and , in the static case for a stationary spherical body
they must depend only on the current radius and must not depend on either time
or angular variables.

Assuming that and neglecting the contribution of the
vector potential , in the spherical coordinates , , , from (10) and
(3) we find the nonzero components of the gravitational tensor:

. (11)

In (11) the
quantity in the
spherical coordinates is the projection of the vector potential on the radial
component of the four-dimensional coordinate system. In this case, the quantity
is the projection of the
gravitational field strength on the radial component of the coordinate system.

The nonzero
components of the electromagnetic tensor, the acceleration tensor, and the
pressure field tensor are obtained similarly to (11):

. .

. (12)

In Minkowski
spacetime the special theory of relativity is valid, so that the potentials and
the field strengths can be calculated exactly. For the case of the relativistic
uniform model, the field strengths, which are part of the field tensors’
components inside a spherical body, in the static case have the following form [22]:

,

,

(13)

In (13) is the Lorentz factor of the typical
particles that are moving in the center of the body; and denote the
invariant mass and charge densities of the typical particles, respectively.
These mass and charge densities are obtained in the reference frames, which are
comoving with the particles. It follows from (10-13) that the field tensors
inside the body are proportional to each other:

. (14)

Let us sum up
all the stress-energy tensors in (9) and use (14):

.

(15)

As was found in
[23] from the equation of the particles’ motion and in [24] from the
generalized Poynting theorem, the following condition holds for the sum of the
field coefficients inside the body:

.
(16)

Substituting
condition (16) into (15) we find out that the sum of the stress-energy tensors
inside the body, which is in equilibrium, becomes equal to zero:

.
(17)

Will the result
of (17) change if we consider the situation in the curved spacetime? In the
physical system under consideration in the form of a spherical body, the
spacetime metric is static and depends only on the radial coordinate. Since the
vector field potentials are assumed to be zero, the tensor of each field
contains only two nonzero components, which are equal in the absolute value.
Taking into account the metric of the curved spacetime leads to the fact that
the tensors’ components of each field in (13) must be multiplied by the same
function that depends on the metric tensor
components. Just as the metric tensor components, this function will depend
only on the radial coordinate. In this case, in the flat Minkowski spacetime
this function must be equal to unity, , so that (13) is satisfied, which does not contain .

Indeed, the
equations for calculating the tensors of all the vector fields coincide with
each other in their form, according to [13], [18],
[25], so that in the relativistic uniform model at constant mass density and charge
density the field
tensors can differ from each other only by the constant coefficients.
Therefore, if we multiply the tensor of each field, found in the Minkowski
spacetime, by the same function , in order to find this tensor in the curved spacetime, relation (14) would not
change, and an additional factor would appear on the
right-hand side of (15). Since condition (16) always holds true, then in the
system under consideration the sum of the stress-energy tensors in (15) and
(17) will also be zero in the curved spacetime.

**4. Calculation of the metric inside the body**

Equation for
the metric (1) in view of (17) is significantly simplified:

.

Substituting
here (7) and (8), we obtain three equations:

. (18)

. (19)

. (20)

Substituting (20) in (18) and in (19) gives the same equation:

,
or . (21)

If we subtract equation (18) from (19), we will obtain
again (21). Equation (21) can be easily integrated, because each term
represents the derivative of the natural logarithm of the corresponding
function:

where is a certain constant.

We will now use the condition obtained in [16],
according to which the scalar curvature inside the body must be a constant
value . With the help of (8) we find:

. (23)

The sum of (23) and (18) gives the following:

.

Comparing this expression with (20), we obtain:

.

We will substitute here the value of according to (22):

.
(24)

Next we will need from (22) and the relation from (21):

Let us substitute (25) into (20):

. (26)

Equations (24) and (26) together form a system of two
differential equations for the two functions and . Direct substitution shows us that the system of these equations has
the following solution: , .

Indeed, in the weak gravitational field, when the
curved spacetime turns into the Minkowski spacetime, in the spherical
coordinates it should be , . In order to ensure that the function is not infinitely large at the
center at , the constant must be equal to zero. From the
condition it follows that , and from (22) we obtain the equality . In addition, the constant must be sufficiently small. As a
result, for the metric tensor components we can write the following:

, . (27)

The constant in (27) represents the value of
the scalar curvature, averaged over the volume of a typical particle, which is
constant inside the body, so that .

In [16] we found the relation for the value of the
cosmological constant averaged over the volume of a
typical particle:

Expanding the sine by the rule , in view of (16), we find:

.

In (28) , where is a certain constant of the order of unity; is the scalar potential of the
gravitational field on the surface of the body at ; is the scalar potential of the electric field; is the radius of the body; is the gravitational mass of the body; is the total charge of the body; is the Lorentz factor of the particles at the center of
the body; is the potential of the pressure field at the center of the
sphere; the mass and the charge are auxiliary quantities.

In the brackets on the right-hand side of (28) there
is the sum of the volumetric energy densities of the particles in the scalar
field potentials – the first and second terms are from the gravitational field,
the third term is from the acceleration field, the fourth and fifth terms are
from the electric field, and the sixth term is from the pressure field. The
third term is the greatest, it is proportional to the rest energy density of
the body. If we take into account only this term, then in the first
approximation the constant will be equal to:

.

**5. Comparison of the metric
tensor components inside and outside the body**

At the current radius reaches the surface of the
spherical body, and here the internal metric becomes equal to the external
metric. It means that at we can equate the components of the
corresponding metric tensors. According to [10], the metric tensor components
outside the body in the covariant theory of gravitation are equal to:

, ,

,
.
(29)

Comparison of (29) and (27) shows that the components and coincide both inside and outside
the body.

Equating in (27) and (29) on condition
that , taking into account (28) and the equality , we find the constant :

. (30)

According to [22], the gravitational mass of the body and the total
electric charge are determined as follows:

.

(31)

Since , it turns out that and .

We will substitute (30) into (29) into the expression
for and take into account the
equality :

(32)

In this expression and denote the scalar potentials of
the gravitational and electric fields outside the body, respectively.

We can also determine the quantities and more exactly. In [21] we found the expression for the square of the particles’ velocities at the center of the spherical
body, with the help of which we can estimate the value of the Lorentz factor in
(32):

.

According to [26], the scalar potential of the pressure field at the center of the body
is approximately equal to:

,

while the acceleration field constant and the pressure field constant are given by the formulas:

, .

In (32) we see the complex structure of the metric
tensor components, in which additional terms appear as compared to the
Minkowski spacetime metric, where in the spherical coordinates . The main addition in (32) is the term , and if we take into account (31), then this addition will become
approximately equal to .

The second important addition includes square brackets
in (32), which by the order of magnitude determines the energy of the
gravitational and electric fields, as well as the pressure field. In these
brackets, we can also use the approximate relation of the masses in expression
(31). For the metric tensor components outside the body all this leads to the
following:

On the right-hand side of (33) in the round brackets
there are quantities with the dimension of energy. For large cosmic bodies, the
main quantity here is the negative energy associated with gravitation. In this
case we can see that the third term, containing in the denominator, is distinguished by a sign from the second term, containing in the denominator.

**6. Comparison with the
metric of the general theory of relativity**

In order to compare with the metric tensor components
(29) and (33), we will consider the Reissner-Nordström
metric in the spherical coordinates, which describes the static gravitational
field around a charged spherical body in the general theory of relativity. We
will use our notation for the field potentials:

, ,

, . (34)

As we can see, the second and third terms in the
component in the Reissner-Nordström
metric (34) differ significantly from the corresponding terms in the component in the CTG metric (33) outside
the body. For example, we can see that the metric in (34) does not in any way
reflect the energy of the pressure field inside the body, whereas in (33) the
energy is associated with the pressure
field and makes its contribution to the metric. Taking into account (28), the
energy also defines the metric (27)
inside the body.

This difference in the form of the metric in both
theories is due to the difference in the equations for determining the metric.
While equation (1) is used in CTG, in the general theory of relativity the
equation for the metric with the cosmological constant , in the matter with the stress-energy tensor has the following form:

. (35)

According to the approach of the general theory of
relativity, the action of gravitation must be described by the metric tensor,
and therefore does not include the stress-energy
tensor of the gravitational field. Outside the charged body there is no matter
and no pressure field, on the right-hand side of (35) only the electromagnetic
field is left, so that we have . As a rule, in (35) the term with the cosmological constant is neglected due to its
smallness, and then the solution for the metric (34) is obtained.

Since in CTG the cosmological constant is taken into
account fully, it turns out that the solution of (27) in view of (28) for the
CTG metric inside the body and the solution of (32) outside the body are more
precise and informative than the solution of (34) in the Reissner-Nordström
metric. Besides the cosmological constant in CTG is not equal to zero and
is proportional to the potentials of all the fields acting inside the body. If
in (28) only the main term with rest energy density is taken into account, then
with the relation we can estimate the value :

.
(36)

If we substitute here the average mass density of the
cosmic space matter of the observable Universe, we will obtain the value m^{-2}. The smallness of the
cosmological constant inside cosmic bodies is associated
with the large factor in (36).

To this end we recall that the issue of the
cosmological constant in the general theory of relativity has not yet been
resolved unambiguously [27], especially with respect to correlation with vacuum
energy. Here it is implied that a very large vacuum energy for some reason
makes little contribution to the metric and to the small cosmological constant.

In CTG, the greater is the mass density in (36), the
larger is inside the body. However if we distribute the matter of all cosmic bodies
over the space, then the mass density will be very low, which leads to
insignificantly small value m^{-2}. We should also
pay attention to the fact that in CTG the cosmological constant outside the
body is assumed to be zero due to its gauging [16]. Inside the bodies, as well
as inside the observable Universe as some global body, has a certain value. In the
approximation of the relativistic uniform body model, is determined in (28).

In contrast, in the general theory of relativity, in
(35), the nonzero value of the cosmological constant outside the body is
admitted. The latter follows from the possibility of influence of the zero vacuum’s
energy on the metric through the cosmological constant.

**7. Conclusions**

In Section 3 we showed that the sum of the
stress-energy tensors of all the four fields inside the body is zero. With this
in mind, the metric tensor components were calculated in (27) as functions of
the current radius. As a result, on the surface of the body at it became possible to compare the
metric inside and outside the body and to determine the unknown coefficient in the external metric (29).

The metric tensor components and outside the fixed spherical body
in the covariant theory of gravitation (CTG), that were presented in (29), were
specified by us in (32) and (33). It turns out that these components are the
functions of the scalar potentials of all the fields, so that, for example, the
pressure field inside the body also influences the metric outside the body.
However, the main contribution to the metric is made by the scalar potential of
the gravitational field . Apparently this is due to the fact that the
expression for the scalar potential includes the gravitational mass that characterizes the source of
the field and the gravitation force. At the same time the relativistic energy
is proportional to the inertial mass , while for an external observer the mass is the rest mass and characterizes
the system with respect to the forces acting on it. Both of these masses differ
from each other by the mass-energy of the particles’ binding by means of the
fields [26]. As for the electromagnetic field, its contribution is secondary.
The body’s charge is only indirectly included in the rest mass of the body and
is not directly included in the gravitational mass. The electric field
potentials vanish in neutral bodies in (33). Thus the gravitational field is
the main factor that distinguishes the curved spacetime metric from the
Minkowski flat spacetime metric.

Our calculations allowed us to calculate the metric CTG inside the body and to refine the
metric outside the body, but in the metric tensor components there was one more
unknown adjustable coefficient . Its appearance can be due to the assumption that the coefficient has an exact value, so that the
coefficient is intended to ensure the correct
value of the metric. The value of the coefficient can be determined in the
gravitational experiments, in which the spacetime metric should be taken into
account.

**8. ****Conflict of interest**

On behalf of all authors, the
corresponding author states that there is no conflict of interest.

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