Jordan Journal of Physics. Vol. 8 (No. 1),
pp. 1-16 (2015).
http://journals.yu.edu.jo/jjp/Vol8No1Contents2015.html

**Relativistic energy and
mass in the weak field limit**

**Sergey
G. Fedosin**

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

E-mail: intelli@list.ru

Within the framework of the covariant theory of
gravitation (CTG) the energy is calculated for a system with continuously
distributed matter, taking into account the contribution of the gravitational
and electromagnetic fields and the contribution of the pressure and
acceleration fields. The total energy of all the fields is equal to zero, and
the system’s energy is formed from the energy of the particles, which are under
the influence of these fields. From the expression for the energy the inertial *M* and gravitational *m _{g}* masses of the system are found. These masses are compared with mass

*Keywords:**relativistic energy; mass; acceleration field; pressure field;
covariant theory of gravitation*

**1. ****Introduction**

Modern
physical theories usually describe the energy, momentum and mass of the system
in four-dimensional formalism and introduce various 4-vectors and 4-tensors
into consideration. In order to simplify comparison of the obtained expressions
it is convenient to turn to such a weak field limit, that most of the formulas
could be written in the same form as in the special theory of relativity,
without loss of accuracy. In this work it will be done for the covariant theory
of gravitation and general theory of relativity; particular attention will be
paid to the meaning acquired by the mass in these theories.

**2. ****Energy**** ****and**** ****mass**** ****in**** ****the**** ****covariant**** ****theory**** ****of**** ****gravitation**

We
will calculate the relativistic energy for the body in the form of a sphere
with the uniform density of mass and charge, moving at velocity along the
axis of the reference frame . The body under consideration
is a set of identical particles moving randomly in different directions within
the specified sphere with the radius . We will assume that all of
these particles are held together by the force of gravitation. In order to
simplify we will assume that the spaces between the particles are so small that
integration over the volume of all the particles is equivalent to integration
over the volume of the sphere. The sphere is at rest in the co-moving reference
frame , associated with the center of mass, and the velocities of particles in are equal
to and depend on the coordinates.

The
Hamiltonian for continuously distributed matter in the covariant theory of
gravitation is obtained from the Lagrangian with the help of Legendre
transformations. This Hamiltonian is equal to the relativistic energy of the
system and has the form [1-2]:

(1)

Here
is the speed of
light, is the mass
density of an arbitrary point particle in the reference frame associated with the particle, is the
scalar potential of the acceleration field, is the scalar potential of the gravitational
field, is the charge
density in the reference frame , is the scalar
potential of the electromagnetic field, is the scalar
potential of the pressure field, denotes the timelike component of the 4-velocity of the particle, includes the determinant of the metric tensor with the minus sign, is an element
of the three-dimensional volume in the reference frame , is the
gravitational constant, is the
gravitational tensor, is the vacuum
permeability, is the
electromagnetic tensor, is the
acceleration tensor, is the pressure field tensor, and are constants.

For
our purposes, it suffices to consider the expression for relativistic energy
(1) in the case when the sphere under consideration is at rest in . Then all the calculations
can be performed in the reference frame associated
with the system’s center of mass. Let us assume that the gravitational field is
small and the covariant theory of gravitation turns into the Lorentz-invariant
theory of gravitation. In this case, the metric tensor no longer depends on the coordinates and is
transformed into the metric tensor of Minkowski spacetime which is
used in the special theory of relativity. For the case of the single fixed
system the expressions for physical quantities are as follows:

, , , (2)

, , , .

where
the Lorentz factor is ,

is the
particle’s velocity in , is the
gravitational field strength, is the
torsion field vector, is the electric field strength, is the magnetic
field induction, is the vacuum permittivity, is the
pressure field strength, is the
solenoidal vector of the pressure field, is the
acceleration field strength, is the
solenoidal vector of the acceleration field.

Substituting
expressions (2) into (1) gives the following:

(3)

First
we will calculate the first integral in (3). According to [2], the Lorentz
factor for the particles inside the fixed sphere is
the function of the current radius :

, (4)

where is the Lorentz factor for velocities of the
particles in the center of the sphere, and due to the smallness of the argument
the sine can be expanded to the second-order terms.

For
the first term in the first integral in (3) with regard to (4) in spherical
coordinates we can write:

(5)

In
(5) the mass is the product of the density of the
particles’ mass by the volume of the sphere which is at rest in the
reference frame . The origin of the factor in (5) can be understood from
the following. The quantity is the mass density of the particles in the
center, which can be seen in the reference frame . Then the product gives the
mass of the particles in the sphere for the observer in in the case, as
if all the particles were in the center of the sphere. It is obvious that . In (5) it occurs that , that means that the total
energy of the particles, increased due to the internal motion of the particles,
is regarded by the Lorentz factor . The second term in (5)
appears due to the radial gradient of mean velocities of the particles inside
the sphere and it takes into account that not all the particles are located in
the center of the sphere.

The
scalar potential of the gravitational field in (3) inside the sphere, according
to [2] is equal to:

(6)

Based
on the similarity of the gravitational and electromagnetic fields, we can write
for the electric potential similarly to (6):

(7)

The
scalar potential of the pressure field inside the sphere equals:

, (8)

where
denotes the
potential of the pressure field in the center of the sphere.

Substituting
(6), (7) and (8) in (3), taking into account (4) we find:

(9)

(10)

(11)

With
regard to (5) and (9-11) the first integral in (3) will equal:

(12)

The
gravitational field strength and the torsion field inside the sphere are given
by the formulas:

, , (13)

where
is the
vector potential of the gravitational field.

The
vector potential of each particle is directed along its velocity, and due to
random directions of the particles’ velocities the total vector potential inside and
outside the sphere is zero. Consequently, the torsion field will also be zero: . Substituting the scalar
potential (6) in (13), we find the gravitational field strength:

(14)

Taking
into account (14) and the equality for the integral of the first term in the
second integral in (3) we have:

(15)

According
to [2], the potential of the gravitational field outside the sphere equals:

From
this it follows that the gravitational mass of the sphere is equal to the
quantity .

Using
(13), with we find the
field strength:

Substituting
in (3), using the equation , we find for the
gravitational field outside the sphere:

(16)

The
sum of (15) and (16) equals:

. (17)

The
calculation of the term with the electromagnetic field in (3) is done similarly
and it gives for uniformly charged particles inside the stationary sphere the
following:

, (18)

where
the charge is the product of the charge density of an arbitrary particle in the reference
frame associated with the
particle by the volume of the stationary sphere.

In Minkowski space, the 4-velocity of the stationary sphere is , and based on
the definition of the total 4-potential of the sphere’s pressure field , we find , , where denotes the density inside the stationary sphere.
In this case the scalar potential , density and pressure
inside the sphere are the functions of the current radius inside
the sphere, and the equality for the vector potential of the pressure field
in this case follows from the absence of ordered motion of particles inside the
sphere. In view of this and (8) for , the vectors and
inside the sphere are expressed as follows:

, .

In
case of uniform mass density , calculations for the vector of the pressure strength inside the sphere
give the following:

Using it we calculate the integral for the pressure:

(19)

We have to calculate one more term in the second
integral in (3). The components of vectors and for the acceleration field are found
as follows:

, ,

where the scalar potential and the vector potential are part of the
4-potential of the acceleration field , which is a covariant 4-velocity.

In the limit of special theory of relativity , where is the Lorentz
factor for the velocity of the particle’s motion. In the reference
frame the particle’s velocities inside the sphere are equal to and should be used
instead of . Then the potentials of an
arbitrary particle will be , . We need the total potentials of
the acceleration field inside the sphere, emerging due to direct interaction of
the particles with each other and due to the influence of fields. In case of
random motion of particles the velocities are directed in different directions, and
therefore inside the sphere and
. However, the total Lorentz factor
of particles is a function of the current radius, and the
total scalar potential is not equal to
zero. With
regard to
(4), for it gives the
following:

We will calculate the last integral:

(20)

Substituting
(12), (17), (18), (19) and (20) in (3), we find the relativistic energy of the
system:

(21)

In
[2] the coefficients and were calculated for the case under
consideration:

. (22)

If
we substitute (22) in (21), we will see that the field energies are canceled
completely. Only the energy of particles in corresponding fields remains:

. (23)

Equation
(22) fixes a definite relation between the pressure field, acceleration field
and gravitational and electromagnetic fields. This relation according to [2]
reveals in the fact, that the conserved integral 4-vector, which is the result
of integrating the equations of motion, is equal to zero. In this case
condition (22) appears, and within the given model the 4/3 problem is
explained.

Let
us estimate the total mass of particles in the sphere, for which, taking into
account (4), we integrate the mass density of particles in over the sphere’s volume:

(24)

Hence,
by solving the quadratic equation we obtain: . Similarly we can link the
charge with the charge of the
sphere, which is found by the observer in : . We will substitute this in
(23), given from (22):

. (25)

From
(8) we will express the scalar potential of the pressure field in the center in terms of the potential near the surface of the sphere, and will
consider the ratio :

. (26)

Similarly,
from (4) we will express in terms of
the Lorentz factor of the particles near the surface of the
sphere:

. (27)

If
we take into account (27) and (22) in (24), we can specify the relation between
and :

. (28)

Substitution
of (22), (26) and (27) in (25) gives the following:

. (29)

(29)
shows that when the covariant theory of gravitation in the weak field limit
turns into Lorentz- invariant theory of gravitation, all fields in the system, including
the acceleration field, pressure field, electromagnetic and gravitational
fields compensate each other so that the relativistic energy depends only on
the mass, the energy of gravitational and electromagnetic fields, the energy of
the surface pressure and the velocity of particles on the surface.

The
scalar potential of the pressure field near the sphere’s surface is connected
with the pressure by relation: , where and denote the pressure and the mass density near
the surface of the sphere. Using the relation , where is the average density with respect to the
sphere’s volume, we find: . For those massive bodies, in
which we can assume and
neglect the pressure on the
surface, (29) becomes a simple expression:

. (30)

From
(29) we will express the mass of the system, consisting of the matter mass and the mass of the four fields associated
with this system:

. (31)

The
mass is identical
and at rest and in motion, and it is the invariant inertial mass of the system.
Above we found from the formula for the external gravitational potential that the gravitational mass of the sphere is
the quantity . Comparison with (24) shows
that the sphere’s mass according to our assumptions is equal to the
gravitational mass . According to (31), the system’s
inertial mass increases relative to the mass by the value of mass-energy of the surface
pressure, and to a certain share of the mass-energy of the electromagnetic
field, but it decreases due to the same share of the mass-energy of the
gravitational field.

**3. ****Relations between the energies**

We
will compare the different energy components that make up the total
relativistic energy (29). We will denote by , , and the energy components of the electromagnetic
and gravitational fields, the pressure field and the acceleration field,
respectively. As the measurement unit of energy we will use the sum of the energy
components of the electromagnetic and gravitational fields from (17) and (18).
Taking into account
(24), (19), (20) and
(22) we find:

. (32)

, .

According
to (32), the energy components of the pressure field and acceleration field are
twice less than the sum of the energy
components of the gravitational and electromagnetic fields, and have a
different sign. As a result, the sum of field energy components in (21) is
equal to zero.

We
will now consider the energy components of the matter particles which are under
the influence of fields. We will denote these components by , , and , as the energy components of
the particle in the electromagnetic and gravitational fields, the pressure
field and the acceleration field, respectively. According to
(9), (10), (11), (26), (5) and (27) we have the following:

, ,

,

, (33)

.

Now
we will sum up the energy components in (32) and (33) separately for each
field:

, ,

. (34)

The
quantity denotes
the sum of the energy components of the gravitational and electromagnetic fields,
including the energy components of the fields themselves and of particles in
these fields, the definition of is given in
(32). The sum of all the energy components in (34) equals the relativistic
energy of the system (29):

. (35)

If
in (35) we neglect the product due to the small pressure on the body surface
and disregard the rest energy , then the energy value
remains, which is equal to: . In classical mechanics, in
which the rest energy is not considered, the total energy of the gravitational
and electromagnetic fields for a sphere with uniform distribution of mass and
charge is equal to:

.

According
to the virial theorem, it is considered that the internal kinetic energy should
equal half the absolute value of the energy of fields: . The total energy is composed
of the energy of fields and the internal energy:

. (36)

This
shows that the total energy in classical mechanics coincide with the relativistic energy (35), if we
exclude from the latter the rest energy and the energy of the surface pressure.
Thus the transition is performed of the covariant theory of gravitation into
the classical mechanics. However, in classical mechanics it is not determined
how the internal pressure makes contribution to the mass and energy of the
system.

We
will now specify, how in our model the virial theorem is realized, particularly
for field energies and particle energies. We have the energy of the pressure field and the energy of the acceleration field, and the sum of
these energies, according to (32), is equal to the absolute value of the sum of
energies of the gravitational and electromagnetic
fields. As a result, the sum of fields’ energies is equal to zero.

The
situation for the energies of particles in fields is different. The energy of a
particle in the field in the absence of the vector potential is defined by the
product of the mass (charge) by the scalar potential. The sum of the energies
of particles in the gravitational and electromagnetic fields according to (33)
is equal to , the energy of particles in the pressure field is , and the energy of particles
in the acceleration field is . From the energy of particles
in the pressure field we can distinguish the energy and the
energy – from the energy of particles in the acceleration field. But the sum of
these energies is times less
than the absolute value of the sum of energies of particles in the
gravitational and electromagnetic fields. At the same time, the excess energy
of particles in the gravitational and electromagnetic field, which is equal to , is compensated by the fact,
that the gravitational mass energy of the system increases from to .

**4. ****Relation with the cosmological constant**

In
[1] we obtained a relation that connects the cosmological constant with the 4-potentials of fields, which are
included in the Lagrangian:

. (37)

Let
us expand the products of 4- vectors:

, ,

, ,

here
is the mass
4-current; is the
charge (electromagnetic) 4-current; , , and denote the vector potentials of the
acceleration field, gravitational and electromagnetic fields and pressure
field, respectively; and we use the approximation of the special theory of
relativity, in which , where , is the velocity
of motion of the body’s arbitrary particle.

Let
us consider the situation in the reference frame , which is stationary relative to the body in question. In the particle velocities are equal to and the
Lorentz factor should be
used instead of . As a result, (37) can be rewritten as follows:

. (38)

In
relation (38) the cosmological constant has its own value for each particle of the
body. We intend to integrate (38) over the volume of the body in the form of a
fixed sphere, which is filled with moving particles as tightly as possible, and
which has uniform density of mass and
charge in the entire volume of the sphere. In the absence of general rotation
or directed matter flows the particles’ velocities are
directed randomly in different directions. Then, after integrating (38) the
contribution of vector products containing will be zero, and the total vector potentials , , and inside the sphere will be zero as well.
Therefore, the integral of (38) over the volume is as follows:

.

The
quantity in our opinion is the energy density of each
particle, and the integral of this density over the volume gives a certain
energy constant , which is associated with all the particles of the system. In the right side of
the equation there is the integral that we have already calculated in (12).
With this in mind, we can write:

. (39)

If
we compare (39) with (21), we see that the quantity is part of the relativistic energy of the system, and denotes the sum of energy
components of the particles under the influence of fields. The energy also includes
the energy components, associated with the fields themselves, but according to
(23) in case of a spherical body all these components cancel each other.
Therefore, we can assume that for a sphere , and .

In
(39) the mass is some
constant mass, which denotes the total mass of body particles, excluding the
contribution from the mass-energy of macroscopic fields, associated with this
body. If we divide the total body matter by particles and scatter from each
other to infinity, then for the matter at rest there will be no electromagnetic
and gravitational fields, associated with the interaction of these particles
with each other. There will be no internal pressure from the particles’
influence on each other. In this case, with regard to (38) written for a single
particle, the mass will consist of the total mass of all the
particles in view of the energy of particles’ proper fields, the energy of
their internal pressure and the internal kinetic energy. We considered such
mass in [4] as the total mass of the body parts, scattered from each other and
located at infinity at zero absolute temperature. At infinity , , and then the system’s mass turns into the mass .

From
(29-30) it follows that the system mass is less than the body mass: , and the body mass is equal to
the gravitational mass . Since the mass is constant
and is associated with the cosmological constant, and , it turns out that the
gravitational mass of the system in (29) can change, when in the
system there is a change in the energy of the pressure field or the energy of
the electromagnetic and gravitational fields. From (28) we find that , and is in the
middle between and . As a result, the ratio of
the masses is as follows:

. (40)

**5. Discussion of results**

**5.1. The masses**

According
to (40) in the weak field the inertial mass of the system
in the form of a sphere with particles, taking into account the field energies,
the internal pressure and the internal kinetic energy can be described either
by the formula (29) or by the system mass from
(39). The equality means conservation of the system’s energy,
regardless of whether the system’s parts are at infinity and do not interact
with each other, or these parts come into close contact and form a coupled
system. This is possible in case of ideal spherical collapse, when there are no
emission and matter ejections from the system at any stage of the collapse or
the matter accumulation. We discussed this question in [2] in connection with
the problem of energy in spherical supernova collapse. There we explained the
possibility of low energy emission by neutrinos based on the fact that almost
all the work of the gravitational forces during the matter compression can come
on increasing the kinetic energy of the stellar matter motion and the pressure
energy, as well as on creating the internal pressure gradients and particle’s
velocities.

Earlier
in [5] we found the expression for the masses, which differs from (40): . We can explain this by a
different accepted gauge of the cosmological constant – in this paper we use
the formulas obtained with the gauge according to [1], which differs from the
gauge in [5]. Also, we are currently using for analysis another physical system
in the form of a sphere, consisting of a set of particles moving inside the
sphere, which are held together by gravitation. In such a system inevitably
there is difference between the masses and , as a consequence of the
radial gradient of the Lorentz factor inside the sphere and as a consequence of the
difference between the density of the particles in the reference frame and the density of particles from the standpoint of the reference frame , associated with the system’s center of mass. The mass in (40) in its
meaning has technical nature, since it is determined only mathematically by
multiplying the density by the
sphere’s volume. We will note that the density is included in the system’s Lagrangian with
the 4-vector of the gravitational (mass) current density in the form . The density is also included in the equation of motion of
a point particle and in the field equations in [1].

According
to (29) and (40), the mass is greater than the mass . This means that the
gravitational mass of the system is always greater than the inertial mass of
the system, by the half the absolute value of the gravitational and
electromagnetic field energy minus the mass-energy of the surface pressure.

According
to (40) the gravitational mass is also greater than the mass of the system’s parts, scattered to infinity.
We can explain this in the following way. As we know, for a ball the absolute
value of the potential energy of the gravitational field is equal to the total
work on the matter transfer from infinity to the surface and inside the ball.
It is assumed that the ball is formed by gradual growth due to layering of
spherical shells as the matter is transferred. But beside the fact that the matter
is transferred from infinity inside the body, which results in increase of the
absolute value of the potential energy of the body’s gravitational field, the
force of gravitation performs other actions – it increases the kinetic energy
of the particles inside the body, the energy of the particles’ pressure on each
other, and creates the gradients of pressure and kinetic energy of the
particles inside the body. All these types of work of the gravitation force on
the body formation increase the body mass from to . The main contribution to the
gravitational mass increase is made by the emerging motion – at infinity the
particles were stationary, but inside the body the particles move at velocities
.

If
we consider the virial theorem, connecting half the absolute value of
gravitational and electromagnetic energies with the internal energy of the
body, then it turns out that half of the work of the gravitational and
electromagnetic fields on the body formation is transformed into the internal
energy of the body. The total energy of the body, according to (36), is negative
and with the help of it (35), (39) and (29) can be written as follows:

. (41)

Since
is equal to half the sum of the gravitational
and electrical energies, then we can see that half of the work of the
gravitational and electromagnetic fields on the body formation is transformed
into the mass increase from to the value .

From
the virial theorem the approximate equality follows between the absolute value
of the total system’s energy (36), the
internal body energy and the binding energy, if we define it in
(41) as the difference between the rest energy for the
mass and the rest energy of the initial state at
infinity . However, in usual
interpretation of the binding energy it is not so, since the binding energy is
defined as the difference between the total energy of the individual parts of
the system and the energy of the system made up of these parts into a whole.
This definition of the binding energy in this case gives us the relation: , i.e. in case of ideal
spherical collapse the system’s energy at the beginning and the end of the
process is the same and the binding energy is equal to zero. Despite the equality
of the binding energy to zero, the system does not fall apart because the
masses are always attracted. And the total energy (36) of the system remains negative.

The
invariant mass of the system is the measure of inertia of the
system as a whole and the measure of the relativistic energy of the system.
This means that the system’s acceleration under the influence of forces should
depend on the mass . The mass can be
calculated as the integral of the density over the volume of the sphere. The
gravitational mass is equal to and can be determined by means of
gravitational experiments near the body on the gravitational effect on the test
bodies. According to (31), at an infinitely large radius of the body the mass
of the spherical system becomes equal
to the gravitational mass of the body . Equation (31) can be
regarded as the quadratic equation to determine the gravitational mass depending on
the body radius , on its electrical charge and the total mass of the fixed parts of this
body , when these parts are motionless and infinitely distant from each other:

.

**5.2.
****Energies and masses in the
general theory of relativity**

In
the general theory of relativity (GTR) the system’s mass is considered to be less than the total mass
of the body’s parts [6-7]. In
GTR, there is gravitational mass of the system from the standpoint of a distant
observer, calculated as the volume integral of the sum , where is the
concentration of matter nucleons, is the mass
of one nucleon, is the
density of the body’s internal mass-energy [8]. The inertial mass of the system
is also considered, which is calculated with the volume integral of the timelike component of the stress-energy tensor, which is
then divided by the square of the speed of light and equated to the
gravitational mass based on the principle of equivalence. Accordingly, to
determine the system’s mass we need to know either the internal energy of
the body which is not precisely known, or use the stress-energy tensor, which
however does not include the gravitational field energy in principle. The
latter is due to the fact that in GTR the gravitational field is understood as
a metric field and is described by the stress-energy pseudotensor. As a result,
calculation of the relativistic energy and the system’s mass in GTR is much
more difficult and involves a number of conditions. For example, for
calculating the energy the coordinates of the reference frame at infinity
should transfer into the coordinates of Minkowski space.

The mass of the system, with regard of the
gravitational and electromagnetic fields, according to [6] and [9], in GTR in
the weak field in our notation relative to the mass, density and radius of the
body is equal to:

(42)

where is the mass tensor,
turning after multiplying by the square of the speed of light into the
stress-energy tensor of the system; the body mass ; and are the density of
mass and charge, respectively; is the kinetic
energy; is the pressure
energy per unit mass, and the case of uniform density is considered.

In [6] also the invariant mass density is used, which implies such mass density, which does not change
under the influence of the pressure or gravitational field. It is assumed that
such invariant density is part of the continuity relation in the curved spacetime: , here is the
determinant of the metric tensor, is the 4-velocity. We will note in this regard, that in the covariant
theory of gravitation the continuity relation is written not for but for [1], and can vary and depend on any factors, including
the pressure and gravitational field.

In the weak field for the fixed body in GTR may be
written:

. (43)

We will assume that , as it should be expected due to
virial theorem. If we substitute (43) into (42), we obtain the relation: , so that the mass of the system is greater than the mass . After substituting (43) into (42),
we obtain the expression for the mass-energy of the system, which to similar to those presented in [7] and [10] (in contrast
to [6], in [10] is an invariant density and denotes the mass density corresponding to our density ).

We
will assume that the mass of the system in (42) according to GTR is calculated
precisely and is equal to our mass of the system in (31):

.

(44)

From
the left side of (44) we see that in GTR the gravitational energy is included
in the equation with the increased weight relative to the electromagnetic
energy, and in the right side both energies have the same weight due to the
similarity of equations for the fields. This is due to the fact that in GTR the
gravitational field is replaced by the effect of the action of the metric field
of the metric tensor. As a result, the entire metric contains gravitation and
the electromagnetic field and pressure remain independent.

If
we neglect the contribution of to (44) and
consider this quantity as a unity, then with regard to the expression from (44) we can estimate the pressure energy
in GTR:

.

In
(42) the mass of the system due to the equivalence principle
is considered equal to the gravitational mass. This means that in GTR a charged
body increases its gravitational mass. Based on the stated above, the ratio of
masses in GTR is as follows:

, (45)

where
in the first approximation (here is the number
of nucleons in the body, is the
mass of a nucleon), ( is the
mass of the system in the form of the body and its fields, is the
internal energy in (36) ), the mass is equal to the gravitational mass , the mass is determined
by the integral over the volume of the invariant density (43), the mass is calculated by integrating over the volume
of the body density , and the mass is determined by us in (28) with the help of and has
technical nature.

If
the mass of the system decreases from the value to , then there is excess energy
of the order of . In GTR the collapsing system
must radiate this energy, so that the ideal spherical non-radiating collapse in
GTR is impossible [8].

As
we can see, relation (45) for the masses in GTR differs significantly from
relation (40) for the masses in the covariant theory of gravitation.

**6.
****Conclusion**

According
to (32), the total energy of the gravitational and electromagnetic fields
summed up with the energy of the acceleration field and the energy of the
pressure field inside the spherical body is equal to zero. During the body
formation distribution of energies of the body particles takes place in the
potentials of all the four fields. This leads to the kinetic energy of the
motion of particles, to the internal pressure and the energy of particles in
the gravitational and electromagnetic fields.

The
difference of our approach from the results of GTR is that the mass of the
system in the ideal spherical collapse does not change, . Really, if at the beginning
of the ideal collapse the spatial component of the total 4-momentum of the
particles falling on the center of mass is equal to zero due to the spherical
symmetry, the same will take place at the end of the collapse, so that the
mass-energy, which is part of the time component of 4-momentum, may be
conserved. However, the gravitational mass is greater
than the mass of the system, since the state of the
particles changes – they start moving inside the system and exert pressure on
each other, besides, the particles acquire additional energy in the internal
fields.

If
the system contains the electromagnetic field, its influence on the mass is opposite to the influence of the
gravitational field, i.e. the electromagnetic field must reduce the
gravitational mass . We can calculate, that if a
body with the mass of 1 kg and the radius of 1 meter is charged up to the
potential of about 5 megavolt, it must reduce the gravitational mass of the
body (not including
the mass of the additional charges) at weighing in the gravity field by mass fraction, which is close to the present
day accuracy of mass measurement.

**7. ****References**

1.
Fedosin S.G. About the cosmological constant, acceleration
field, pressure field and energy. vixra.org, 5 Mar 2014.

2.
Fedosin S.G. The Integral Energy-Momentum
4-Vector and Analysis of 4/3 Problem Based on the Pressure Field and
Acceleration Field.
American Journal of Modern Physics. Vol.
3, No. 4, 2014, pp. 152-167. doi:
10.11648/j.ajmp.20140304.12.

3.
Fedosin S.G. 4/3 Problem for the
Gravitational Field. Advances in
Physics Theories and Applications, 2013, Vol. 23, P. 19 – 25.

4.
Fedosin S.G. Energy, Momentum, Mass and Velocity of a
Moving Body in the Light of Gravitomagnetic Theory. Canadian Journal of Physics,
2014, Vol. 92, no. 10, P.
1074 – 1081. http://dx.doi.org/10.1139/cjp-2013-0683.

5.
Fedosin S.G. The
Hamiltonian in Covariant Theory of Gravitation. Advances in Natural Science,
2012, Vol. 5, No. 4, P. 55 – 75.

6.
Fock V. A. *The Theory
of Space, Time and Gravitation* (Pergamon
Press, London, 1959).

7.
Papapetrou A. Equations
of Motion in General Relativity. Proc. Phys. Soc. A, 64 (1951), P. 57.

8.
Abhas Mitra. Why Gravitational Contraction Must be Accompanied by
Emission of Radiation both in Newtonian and Einstein Gravity. Phys. Rev. D Vol. 74, 024010 (2006).

9.
Landau L.D. and Lifshitz E.M. The
Classical Theory of Fields (Vol. 2, 4th ed. Butterworth-Heinemann, 1975).

10.
Chandrasekhar S. The Post-Newtonian Equations of Hydrodynamics in
General Relativity. Ap. J. 142 (1965), P. 1488; Ap.
J. 158 (1969), P. 45.

Source: http://sergf.ru/reen.htm