Aksaray
University Journal of Science and Engineering, Vol. 2, Issue 2, pp. 127-143
(2018). http://dx.doi.org/10.29002/asujse.433947.

**Energy and
metric gauging in the covariant theory of gravitation**

**Sergey G. Fedosin**

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

E-mail: fedosin@hotmail.com

Relations for the
relativistic energy and metric are analyzed inside and outside the body in the
framework of the covariant theory of gravitation.
The methods of optimal energy gauging and equations for the metric are chosen.
It is shown that for the matter inside the body a procedure is required to
average the physical quantities, including the cosmological constant and the
scalar curvature. For the case of the relativistic uniform system, the
cosmological constant and the scalar curvature are explicitly calculated, which
turn out to be constant values inside the body and are assumed to be equal to
zero outside the body. Comparison of the cosmological constants inside a
proton, a neutron star and in the observable Universe allows us to explain the
cosmological constant problem arising in the Lambda-CDM model.

**MSC: ****83A05**; 83D05; 32Q10.

**PACS:** 03.30.+p; 04.40.-b; 95.30.Sf.

**Keywords:** Cosmological constant, Scalar curvature, Relativistic
uniform system, Gravitational field, Acceleration field, Pressure field.

**1. Introduction**

The
relativistic energy of the physical system is part of the time component of the
four-momentum of the system and is one of the most important characteristics,
along with the momentum. In this case, the energy is determined with an
accuracy up to a constant, selected arbitrarily based on the convenience of
calculation. Thus, the problem of energy gauging arises in each theory. In the
covariant theory of gravitation, the energy is gauged based on the fact that
the value of the cosmological constant is proportional with an accuracy to a
constant multiplier to the energy density of the matter particles in the proper
fields of the system under consideration [1].

The use of the
cosmological constant for energy gauging results in certain changes in the
equation for the metric, in which the cosmological constant is present
alongside with the scalar curvature. Therefore, we will further analyze both –
the expressions for the metric and for the energy.

The purpose of
this article is to clarify the question - how the cosmological constant and the scalar curvature in the matter
inside bodies should be understood. The point is that, as a rule,
representative volumes occupied by typical particles should be selected in the
matter, and calculations should be carried out for such particles, including
solution of the equation of motion. Applying the method of typical particles
implies we need to use the appropriate averaging of the physical quantities
acting on such particles. Such quantities as the cosmological constant and the
scalar curvature are not exceptions.
Thus, they should also be considered as some averaged quantities. As a
result of our analysis for the case of the relativistic uniform system, we will
calculate the cosmological constant and the scalar curvature inside the body,
and will show that they are constant quantities. In addition, we will try to
clarify the cosmological constant problem in connection with its inconsistency
with the zero energy of the vacuum.

**2.
Equations for the metric and the energy**

The use of the
principle of least action in the framework of the covariant theory of
gravitation leads to the following relation for the metric [1]:

(1)

where is the speed of light; is the constant, which is part of
the Lagrangian in the terms with the scalar curvature and the
cosmological constant ; is the Ricci tensor; is the metric tensor;
is the mass
four-current; is the charge
four-current; , , and are the
four-potentials of the gravitational and electromagnetic fields, the
acceleration field and the pressure field, respectively; , , and are the
stress-energy tensors of these fields, respectively.

Equation (1)
can be contracted by means of multiplying by the metric tensor, taking into
account that , , , , , and in the four-dimensional spacetime :

. (2)

Substitution of
(2) into (1) gives an equation for the metric:

. (3)

Let us take the
covariant derivative of both sides
of equation (3):

. (4)

For the tensors
on the right-hand side, the relation is valid as an expression of the
equation of motion [1]. Consequently, the right-hand side of (4) vanishes.

The Ricci
tensor and the scalar curvature are part of the Einstein tensor, the covariant derivative
of which is equal to zero due to the properties of the curvature tensor and the
differential Bianchi identity:

.
(5)

From comparison
of (5) with the left-hand side of (4), which must also be equal to zero, it
follows that . It means that the covariant derivative of the scalar
curvature must be equal to zero at any point in space, both inside and outside
the system.

In addition to
relation (2), which contains the scalar curvature and the
cosmological constant , there is another relation in [1], which contains
these quantities. In particular, for the Hamiltonian and the relativistic
energy of the physical system with continuous distribution of matter we found
the following:

(6)

In (6) and denote the
invariant densities of mass and charge, respectively; , , and are the scalar
potentials of the gravitational and electromagnetic fields, the acceleration
field and the pressure field, respectively; , , and are the tensors
of these fields, respectively; is the time component
of the four-velocity of the matter unit; is the metric tensor determinant; is the product of the differentials
of space coordinates; is the gravitational
constant; is the magnetic
constant; is the
acceleration field constant; is the pressure field constant.

**3.
Gauging outside the body**

We will use (6)
to calculate the contribution into the system’s energy outside the body, where
there is no matter and there are only the gravitational and electromagnetic
fields, so it is sufficient to take into account only the second integral. In
this case, the mass and charge four-currents are equal to zero and the
condition remains in (2), where the symbol refers to the
quantities outside the body. Under this condition, the contribution into the
energy (6) outside the body will be:

. (7)

It is
convenient to assume that in (7) the cosmological constant , that is, the contribution into the relativistic
energy in the volume outside the system depends neither on the scalar curvature
nor on the cosmological constant. Then the condition implies the equality
. As a result, we obtain the equality , which follows from (4) and (5).

The fact that
both the scalar curvature and the
cosmological constant are assumed to be zero outside the
body was used in [2] to calculate the metric tensor components and to simplify
the equation for the metric (3) to the following form:

.
(8)

**4.
Gauging inside the body**

We will now
pass on to the situation inside the body, where all the stress-energy tensors on
the right-hand side of the equation for the metric (3) are non-zero. Since
according to (4) and (5) the condition must hold, where the
symbol refers to the quantities in the
matter inside the body, then after applying the covariant derivative to all the
terms in (2) the following remains:

. (9)

We will now
substitute the scalar curvature from (2) into the expression for the energy
(6):

We find the
cosmological constant in two relations
– in (9) and in (10). In (9), the covariant derivative must behave with an accuracy up to a
sign like the covariant derivative . And in (10) the term is some
additional energy density. The choice of the value of for gauging
purposes is not initially limited by anything, except that it must be invariant
with respect to the covariant transformations of coordinates and time. For
convenience we will use the simplest variant, which significantly simplifies
the expression for the energy.

Just as in [1],
we will suppose that the cosmological constant in the matter
inside the body is such that the following relation would hold:

. (11)

Then, according
to (10) and (11), the energy in the space inside the body occupied by the
matter and fields will no longer depend on the cosmological constant:

(12)

Assuming that
the cosmological constant outside the body is equal to zero because of the
absence of matter there, , we will compare the relations for the energy (7) and (12). From these
relations we can see that the general expression for the energy is (12), in
which the energy must be replaced with . The system’s energy with the right-hand side in the form
of (12) was derived by us earlier in [1].

The denser the cosmic object is, the higher is the energy
density of the particles in the field potentials on the right-hand side of
(11), and the greater is the cosmological constant inside the body. Since (11) and (2)
imply the relation , then the scalar curvature inside the bodies is not equal to
zero and varies proportionally to the cosmological constant . Consequently, in denser bodies the scalar curvature has greater value.

We will express the four-potentials of the fields in
(11) in terms of the respective scalar and vector potentials of these fields: for the gravitational field, for the electromagnetic field, for the acceleration field, for the pressure field. In the limit
of the special theory of relativity, the four-currents have the following form:
, , where is the Lorentz factor of the
particle of the moving and continuously distributed matter of the system, is the velocity of the particle’s
motion. This gives the following:

. (13)

In the limit of low velocities we can neglect the
terms containing the velocity of the particles and the vector
potentials , , and . Then in (13) the Lorentz factor is and only the terms with scalar
field potentials are left. For the particles scattered at infinity in cosmic
space we can assume that these potentials arise only from the particles’ proper
fields and are the potentials averaged over the volume of particles. In this
case, according to [3], , where is the Lorentz factor for the matter
at the center of the particles. Denoting the cosmological constant for
individual particles in cosmic space by we can write:

From (11) and (14) it follows that in the absence of
matter the cosmological constant vanishes. This is consistent with the fact
that in Section 3 we assumed that the cosmological constant outside the body is
equal to zero.

According to (14), is defined by the rest energy
density of the particles with a certain addition from the energy density of the
particles in the gravitational and electromagnetic fields and in the pressure
field in the matter. Now we can average over the entire space, as well as the mass density and the charge density , without changing the values of the field potentials. To do this, we will
take into account that in the first approximation the product is the ratio of the particle’s
mass to the particle’s volume as some average density. Averaging over the
entire space will take place if we distribute the particle’s mass over the
entire volume, which, on the average, can be attributed to one particle in
cosmic space. In this case is changed to and to . Leaving in (14) only the rest energy as the basic term due to its
value, we can approximately write:

.

Using the definition in the following form: , where is a constant of the order of unity, we find the averaged value . Substituting instead of the estimate of the cosmological
constant m^{–2} according to the
Lambda-CDM model [4], we find the corresponding density: kg/m^{3}, which is close
enough to the observed average mass density of the matter.

If we consider the proton, which is stable in all
respects, as the basic particle in cosmic space, then we can estimate the
cosmological constant for it in (14). To do this,
instead of we should use the average proton density of
the order of kg/m^{3} with its radius m, according to [5]. This gives the
value m^{–2}, which is 44 orders
of magnitude greater than the cosmological constant m^{–2} averaged over the
entire cosmic space.

The next step can be made by taking into account the
strong gravitation described in [6, 7], and assumed as the basis for describing
the strong interaction at the hadron level. If we substitute the gravitational
constant with the strong gravitational
constant m^{3}/(kgˑs^{2})
according to [8], then the corresponding cosmological constant for the proton
will equal m^{–2}. The relation for the
scalar curvature for the matter inside the proton
will be written as m^{–2}. Assuming in the first
approximation that the spacetime inside the proton has constant curvature, we
will estimate the radius of curvature, based on the expression in [9], which
relates the scalar curvature and the radius of curvature: m. The value , calculated in the field of strong
gravitation, is of the order of the proton radius.

**5. Averaging of physical
quantities inside the body**

While estimating the matter parameters the usual
procedure is to single out particles or volume elements of such sizes, that
they could characterize on the average the basic properties of the matter. For
example, in a crystalline solid body a typical element is a crystal cell, so
that the whole body can be divided into a number of such cells. If we consider
the intervals between the typical particles of matter to be small, then to such
matter in the form of liquid we can apply the approximation of continuous
medium. In this case, the particles remain independent to some extent and can
move at different velocities. However, due to the close interaction of
particles, in each stationary system certain dependences of physical quantities
on the coordinates and time are established, which characterize the system on
the average. We will assume that the typical particles of the system have
exactly such parameters, which define the average physical quantities in the
matter. Actually this means that in all equations used to describe the matter
all quantities refer to typical particles.

We will next consider a non-rotating body of a
spherical shape, which represents a physical system of closely interacting
particles and fields, held in equilibrium by gravitation, and will use a
relativistic uniform model to describe all physical quantities.

Within the framework of the special theory of
relativity, the covariant derivatives are replaced with the four-gradient, and
this relation follows from (11) and (9):

On the right-hand side of (15) we will express the products
of the four-potentials of the fields by the four-currents in the same way as it
was done in (13). For the physical system under consideration, the vector field
potentials , , and averaged over a sufficient number
of typical particles vanish due to the chaotic motion of these particles, and
the Lorentz factor of the particles is the quantity as a function of the current
radius. Consequently, in (13), in a first approximation we can neglect the
terms with the vector potentials, and the following remains in (15):

The scalar potentials of the fields inside the sphere
with the radius were found in [3], [10]:

,

,

,

, (17)

where is the current radius, is the electric constant, is the pressure field potential at the
center of the sphere, and is the Lorentz factor at the center of the sphere.

The scalar potentials (17) depend on the current
radius and give the averaged values as a consequence of interaction of the
entire set of typical particles. Before substituting these potentials into
(16), equation (16) should be averaged over the volume of a typical particle.
This also means that when using the cosmological constant and the scalar
curvature inside the body, these quantities should be considered as some
averaged quantities.

If we denote by the proper volume of a
typical particle and by the apparent volume of a moving
particle from the viewpoint of an observer, who is stationary relative to the body,
then averaging of the left-hand side of (16) over the volume of the moving
particle yields:

where is the averaged scalar curvature
inside the body at the location of the given typical particle.

For the right-hand side of (16), averaging leads to
the following:

. (19)

The volume
element in the integrals (18) and (19) is the volume
element of a moving particle from the viewpoint of an observer, who is
stationary relative to the body, so that . The value in the approximation of the special theory of
relativity is a constant value for the particle under consideration, and therefore
in (19) it was taken outside the derivative sign.

Since the whole set of particles densely fill the
sphere, for the given observer the sum of the volumes of all moving particles
should give the volume of the sphere: . Hence it follows that the volume element can also be considered as the volume
element of a fixed sphere, so that by
summing all these volume elements this observer can determine the volume of the
sphere.

On the other hand, the typical particle chosen by us
moves at a certain averaged velocity and with the corresponding Lorentz
factor . As a result, if a particle at rest has the volume , then a moving particle, from the point of view of the theory of
relativity, has the reduced volume , while , as well as , taking into account the equation . This fact was used in [11] when considering the virial theorem.

In (19), the quantities and represent the elements of mass
and charge of the particle. With this in mind, equation (19) can be rewritten
using the averaged scalar potentials (17) of the fields inside the sphere:

Expression (20) is a certain four-vector, each
component of which must be zero. The time component of this four-vector
vanishes, since the potentials in the stationary sphere do not depend on time,
just as the Lorentz factor of the particles. It remains to
consider the space components in (20), for which we will use the relation for
the field coefficients, derived in [12] from the equation of motion of matter:

. (21)

If we substitute the potentials (17) into (20) and
take into account (18) and (21), then we see that indeed for the time and space
components of the four-gradient of the averaged cosmological constant the
following relation holds true:

. (22)

**6. Cosmological constant
and scalar curvature inside the body**

Since (11) and (2) imply the relation , a similar equation must also exist for the averaged quantities. This
means that the scalar curvature inside the body must also be averaged and
transformed into , while the relations and must be satisfied. Accordingly,
equation for the metric (3) inside the body must be written for the averaged
quantities:

. (23)

We can assume that relation (22) was obtained by averaging
(11) and subsequent taking of the four-gradient. Removing the sign of the
four-gradient from (22), taking into account
(17) and (21), we find the following inside the body:

(24)

Let us now use the value of the system’s total charge , as well as the value of the system’s gravitational mass , which, according to [10], is equal to the total mass of the system’s
particles :

.

.

Expressing in (24) the corresponding cosines in terms
of the mass and the charge , and then expanding the sine according to the rule , in view of (21) we find:

(25)

In (25) the auxiliary mass and the auxiliary charge are used.

Introducing then the scalar potential of the
gravitational field and the scalar potential of the
electric field on the surface of the body at , we obtain:

Thus, the averaged cosmological constant is non-zero and is a constant
value inside the fixed body. The same is true for the averaged scalar curvature
in (23) since . In this case the required condition is met automatically.

Actually relations (25-26) for the body repeat
relation (14) for an individual particle. However, (25-26) are much more
informative. In particular, from (26) it follows that the cosmological constant
depends on the scalar potentials of
the fields. In this case, the potential of the pressure field and the
potential of the acceleration field are
taken at the center of the body, but the potentials of the gravitational field and the electric field are taken not at the center, but on
the surface of the body. The latter is associated with a special way of gauging
the energy and potentials of the gravitational and electric fields – they are
gauged so that as the distance to infinity increases, they vanish.

We can also specify the values of the quantities and so that in (26) all the quantities were determined more
precisely. In [11], the expression was found for the square of the particles’
velocities at the center of a spherical body,
with the help of which we can estimate the value of the corresponding Lorentz
factor:

.

,

and the constant of the acceleration field and the constant of the pressure field are expressed by the formulas:

, .

According to the conclusions in Section 3, outside the
body both the cosmological constant and the scalar curvature are assumed to be zero. This
leads to expression (7) for the contribution of the field energy outside the
body into the total relativistic energy of the system and to equation for the
external metric (8).

As for the situation inside the body, it is necessary to perform an
operation of averaging the physical quantities in such a way that they would
correspond to the typical particles, which most fully characterize the physical
system. The order of averaging of the physical quantities is described in
Section 5. After averaging, the scalar curvature and the cosmological constant
inside the body are connected by the relation , where they are constant quantities. In (25-26) the cosmological
constant is expressed in terms of the
potentials of all the fields existing in the system. The same is true for the
scalar curvature , in this case the potentials of the acceleration field and the pressure
field are taken at the center of the body, and the potentials of the
gravitational and electric fields are taken on the surface of the body.

For gravitationally bound bodies the second most
important value after the rest energy is the gravitational energy. It follows
from (26) that if the coefficient is negative, then as the mass
density inside the sphere with the
constant radius increases, then due to the
increase in the rest energy density both the cosmological constant and the scalar curvature increase as well. But since the
absolute value of the gravitational energy density inside the body increases
proportionally to the square of the mass density, this somewhat slows down the
increase of the and .

Taking into account we will apply (26) to estimate
the cosmological constant in the matter inside a neutron star, leaving on the
right-hand side only the rest energy density: m^{–2}. Here we used the
average mass density of the star of the order of kg/m^{3} with its radius of
12 km and the mass of a typical star of 1.35 solar masses. Passing on to the
scalar curvature with the help of the equation , we can estimate the radius of static spacetime curvature inside the
star as in a spherical Riemannian space: m.

As a rule, particles are located inside the bodies in
such a way that some gaps remain between the particles. This leads to the fact
that in massive objects the average densities of mass and energy do not exceed
the corresponding values of the densities inside the particles. As a
consequence, in the process of transition to such objects the values of the
averaged cosmological constant and the scalar curvature decrease. In addition,
the cosmological constant in each system turns out to be limited to a certain
value, which is, according to (11), proportional to the rest energy density of
this matter with regard to the proper fields, and which is a certain reference
point in gauging of the relativistic energy (12).

We can compare our calculation for the neutron star
and calculation for the proton made in Section 4. According to the theory of
infinite nesting of matter [13], these objects are analogous to each other in
many respects, while for the proton we used both the ordinary gravitational
constant and the strong gravitational constant. Both for the neutron star, and
using the strong gravitational constant for the proton, we obtained that the
radius of the spacetime curvature does not differ much in the order of magnitude
from the radius of the corresponding object. The ratio of the specified radii
is the following:

.
(27)

We will now use the coefficients of similarity between
the neutron star and the proton. The ratio of the star’s mass of to the
proton’s mass gives the coefficient of similarity in mass , the ratio of the radii of these objects gives the coefficient of
similarity in sizes , and the coefficient of similarity in speeds of same-type processes
equals . It seems that the ratio should equal , but this is not so, because the radius of curvature is derived through
the energy density and is not simply a linear dimension. In order to show this,
we will take into account that in (27) the scalar curvature inside each object
is proportional to the corresponding gravitational constant and mass density:

.

Meanwhile, according to the dimension theory, the
ratio of the gravitational constants is and the ratio of the mass
densities is .

From the results of Section 4 it follows that inside
the proton the cosmological constant, taking into account the strong
gravitational constant, should be of the order of m^{–2}. This is 83 orders of
magnitude greater than the cosmological constant m^{–2}, which follows from
the general theory of relativity as applied to the observable Universe. In this
connection we recall that in cosmology there is still unexplained problem of
the cosmological constant. The essence of this problem is that the cosmological
constant, calculated with the help of the general theory of relativity for the
cosmic space, is almost 120 orders of magnitude less than the cosmological
constant for the zero vacuum energy, according to quantum physics. The cosmological constant is a required
element in the Lambda-CDM model, and in this case it becomes unclear why the
expected large magnitude of the zero vacuum energy is transformed into such a
small cosmological constant of the cosmic space of the Universe [14].

Our explanation of the problem of the cosmological
constant is as follows. Discrepancy between the conclusions of the quantum
physics and the general theory of relativity in respect of the cosmological
constant is associated with the geometrical approach of the general theory of
relativity, which replaces the gravitation, as a force action, with the
spacetime curvature. This leads to the absence in this theory of the
stress-energy tensor of the gravitational field and to the impossibility of
calculating the energy of the internal parts of the physical system under
study, which also significantly complicates quantization of the general theory
of relativity.

In the covariant theory of gravitation there are no
such problems. We find separate components of the energy inside the body, and
with their help we determine the corresponding cosmological constant for each
body. The observed part of the Universe
can be considered as the internal part of some global body, and in this case
the cosmological constant is of the order of m^{–}^{2}, besides it is a constant value, which characterizes the entire space
filled with stars and galaxies. For the neutron star, the cosmological constant reaches the value of m^{–2}, and for the proton
it equals m^{-2} for the ordinary
gravitation and m^{–2} in the strong gravitational
field.

According to the approach to energy and metric gauging
in the covariant theory of gravitation, outside the body in the space without
matter the cosmological constant and scalar curvature of spacetime vanish.
Thus, the alleged relation between the cosmological constant and the zero
vacuum energy of the quantum physics outside the body is broken. As for the
difference between the cosmological constants inside the proton and the neutron
star on the one hand, and the cosmic space on the other hand, it is explained
by the fact that the cosmological constant of the observable Universe is the
cosmological constant of all the particles and bodies of the Universe averaged
over the space.

We should note that in the modernized Le Sage’s model
gravitation can arise as a consequence of the action of the fluxes of
relativistic particles of the vacuum field on the bodies [15, 16]. In this
case, the standard problems of the Le Sage’s model are eliminated by explaining
the way in which the vacuum particles interact with the matter [17]. In this
case, instead of searching for the quantum zero energy of the vacuum it is
possible to determine the energy density of the vacuum field particles, to
derive the gravitational constant and the electric constant through the vacuum
field parameters and to explain the effect of the Newton’s and Coulomb’s laws
for gravitational and electric forces.

The equations of the covariant theory of gravitation describe only the
consequences of interaction of the vacuum field’s particles with the matter,
which are expressed in changing of the acting forces, changing of the matter
energy, as well as in creating inertia of the bodies and imparting mass to
them. This means that the energy density of the vacuum field’s particles,
despite its largeness, is not taken into account as an essential component of
the cosmological constant. However, the energy of the vacuum field’s particles
influences indirectly the magnitude of the cosmological constant inside a
particular system through the averaged density of the rest energy of the
system’s particles.

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