Preprints 2017, 2017040150. http://dx.doi.org/10.20944/preprints201704.0150.v1

**Energy, Momentum, Mass and Velocity
of a Moving Body**

**Sergey G. Fedosin**

*Perm, Perm Region, Russia*

*e-mail intelli@list.ru*

*In the weak-field approximation of covariant
theory of gravitation the problem of 4/3 is formulated for internal and
external gravitational fields of a body in the form of a ball. The dependence
of the energy and the mass of the moving substance on the energy of field
accompanying the substance, as well as the dependence on the characteristic
size of the volume occupied by the substance are described. Additives in the
energy and the momentum of the body, defined by energy and momentum of the
gravitational and electromagnetic fields associated with the body are
explicitly calculated. The conclusion is made that the energy and the mass of
the body can be described by the energy of ordinary and strong gravitation, and
through the energies of electromagnetic fields of particles that compose the body.*

*Keywords:** energy; momentum; theory of
relativity; gravitation; field potentials.*

**PACS**:
03.20.+i, 03.50.x, 12.10.-g

**Introduction**

In relativistic
mechanics, there are standard formulas for the dependence of total energy and
momentum of a particle with the mass on its velocity :

, . (1)

If the energy and momentum in (1) are known the mass and the velocity of
the particle can be calculated:

, . (2)

In (1) and (2) the speed
of light is included. For a particle in rest velocity
and momentum are zero, and the total energy of the particle equals the rest
energy:

.
(3)

Equation (3) reflects the principle of proportionality
of mass and energy. In elementary particle physics the energy and the momentum
are usually measured parameters, and the mass and the velocity are found from
(2) and are secondary parameters.

Now, we shall suppose
that the measured parameters are the energy and the velocity of the particle.
In this case, from (1) we can calculate the mass and momentum:

, . (4)

The case is also possible
when the measured parameters are the momentum and the velocity of the particle
and the calculated quantities are the mass and energy:

, . (5)

If the particle velocity is given, then the mass can be found either
through the energy according to (4), or through the momentum according to (5),
in both cases, the mass should be the same.

There are also two
possible combinations of parameters when the energy and the mass, or the
momentum and the mass are known. This allows us calculating the absolute value
of the momentum and the velocity, or the energy and the velocity, respectively:

, ,

, .

From the above formulas
it is not clear whether they contain the energy and the momentum of fields,
which are inherent in the particles and the test bodies. In particular, the
test bodies always have their proper gravitational field and can also carry an
electrical charge and the corresponding electromagnetic field. In
general theory of relativity (GTR) it is considered that relativistic energy
and mass of a body decrease due to the contribution of the gravitational
energy. Although in GTR there is no unique definition of the gravitational
energy and its contribution to the total energy [1], in the weak-field
approximation the following is assumed [2]:

, . (6)

where
is the relativistic energy in the gravitational field, is the energy in the absence of the field, is the potential gravitational energy of the
body.

Since
the energy is negative, then according to GTR the mass of
the body should decrease with increasing of the field.

The
main purpose of this paper is to incorporate explicitly in the relativistic formulas
for the energy and the momentum the additives, resulting from the energy and
the momentum of fields associated with the test bodies. All subsequent
calculations will be made in the framework of the covariant theory of
gravitation (CTG) [3]. We will apply the weak-field approximation, when CTG is
transformed into the Lorentz-invariant theory of gravitation (LITG), and it
becomes possible to compare our results with the formulas of GTR (6).

**External gravitational field. Problem of 4/3**

We shall assume that (1)
– (5) are valid for a small particle and energy of its proper gravitational
field is taken into account through its mass. If there are a lot of particles
in the volume of the body, then their interaction energy leads to a significant
contribution of the field energy into the total energy of the body. In order to
simplify calculations, we shall further consider the case of a weak field. In a
weak field can be assumed that either the spacetime metric differs little from
the metric of Minkowski spacetime, or the gravitational effects of time
dilation and reducing of size are considerably smaller than the similar effects
due to the motion of the body. In this approximation the general theory of
relativity turns to gravitomagnetism, the covariant theory of gravitation turns
to the Lorentz-invariant theory of gravitation (LITG).

Gravitational potentials
of the substance unit of point size, located at in a point of space and moving along the axis
at a constant speed , according to [4] are as follows:

, , (7)

here – the
scalar potential,

– the gravitational constant,

– the
mass of substance unit,

– the propagation speed of gravitation,
which further for simplification of the calculations, we shall assume equal to
the speed of light ,

– the coordinates of the point at
which the potential is determined at the time ,

– vector potential.

According to (7), the gravitational
potential at the
time of the
point mass moving
along the axis depends on
the initial position of this
mass at . After integration of (7) over all point masses
inside the ball on the basis of the principle of superposition, the standard
formulas are obtained for the potentials of gravitational field around the
moving ball, with retardation of the gravitational interaction taken into
account:

, , (8)

where – the scalar
potential of the moving ball,

– the mass
of the ball,

– the
coordinates of the point at which the potential is
determined at the time (on the
condition that the center of the ball at was in the origin
of coordinate system),

– the vector
potential.

In (8) it is assumed that
the ball is moving along the axis at a constant speed , so that , , . With the help of the field
potentials we can calculate the field strengths around the ball by the formulas
[5]:

, , (9)

where is the gravitational acceleration,

–
the gravitational torsion in LITG (gravitomagnetic
field in gravitomagnetism).

In view of (8) and (9) we
find:

,
,

, , (10)

, .

The energy density of the
gravitational field is determined by the formula [5]:

. (11)

The total energy of the
field outside the ball at a constant velocity should not depend on time. So it
is possible to integrate the energy density of the field (11) over the external
space volume at . For this purpose we shall introduce new coordinates:

, , . (12)

The volume element is determined by the formula , where:

.

It follows that .The integral over the space of the energy
density (11) will equal:

. (13)

We shall take into
account that due to the Lorentz contraction during the motion along the axis the ball must be as an ellipsoid, the surface
equation of which at is the following:

.
(14)

After substituting (12)
in (14), it becomes apparent that the radius at the integration in (13) must change from to , and the angles and change the same way as in spherical
coordinates (from 0 to for the angle , and from 0 to for the angle ). For the energy of the
gravitational field outside the moving ball we find:

, (15)

where is the field energy around the stationary
ball.

We shall assume that the
formula (4) for connection of the mass and the energy of the particle is also
valid for the gravitational field. Then we can introduce the effective mass of
the field related to the energy:

. (16)

We shall now consider the
momentum density of the gravitational field:

,
(17)

where is the vector of energy flux density of the
gravitational field [5].

Substituting in (17) the
components of the field (10), we find:

, (18)

,

.

We can see that the
components of the momentum density of gravitational field (18) look the same as
if a liquid flowed around the ball from the axis , carrying similar density of the momentum – liquid spreads
out to the sides when meeting with the ball and merges once again on the
opposite side of the ball. Integrating the components of the momentum density
of the gravitational field (18) by volume outside the moving ball at as in (13), we obtain:

. (19)

, .

In (19) the total
momentum of the field has only the component along the axis . By analogy with (5) the coefficient
before the velocity in (19) can be interpreted as the effective
mass of the external gravitational field moving with the ball:

, (20)

where is the energy of the external static field of
the ball at rest.

Comparing (20) and (16)
gives:

.
(21)

The discrepancy between
the masses and in (21) is the essence of the so-called
problem of 4/3, according to which the mass of the field, which is
calculated through the momentum of the field, at low speeds is approximately
4/3 larger than the mass of the field, found
through the field energy. A characteristic feature of the fundamental fields,
which include the gravitational and electromagnetic fields, is the similarity
of their equations for the potentials and the field strengths. The problem of
4/3 is known for a long time for the mass of electromagnetic field of a moving
charge. Joseph John Thomson, George Francis
FitzGerald, Oliver Heaviside [6], George Frederick
Charles Searle and many others wrote about it in the late 19-th century. We
also discussed this question previously with respect to the gravitational field
of a moving ball [7]. Now we present an exact solution of the problem, not
limited to the approximation of small velocities.

**The gravitational field inside a moving ball**

According to [4]
for a ball with the density of substance (measured in the
comoving frame), which is moving along the axis , the potentials inside the ball (denoted by subscript *i *)
depend on time and are as follows:

, . (22)

In
view of (9) we can calculate the internal strengths of field:

, , ,

,, . (23)

Similarly to (11) for the
energy density of the field we find:

. (24)

According to (24) the minimum energy density inside a
moving ball is achieved on its surface, and in the center at it is zero.

The integral of (24) by
volume of the ball at in coordinates (12) with the volume element equals:

.
(25)

The moving ball looks
like an ellipsoid with the surface equation (14), and in the coordinates (12)
the radius in the integration in (25) varies from 0 to. With this in mind for the energy of
the gravitational field inside the moving ball, we have:

, (26)

where is the field energy inside a stationary ball
with radius .

The effective mass of the
field associated with energy is obtained similarly (4):

. (27)

Substituting in (17) the
components of the field strengths (23), we find the components of the vector of
momentum density of the gravitational field:

,
,
.

(28)

The vector connecting the
origin of coordinate system and center of the ball depends on the time and has
the components . From this in the point, coinciding
with the center of the ball, the components of the vector of the momentum
density of the gravitational field are always zero. At the center of the ball passes through the
origin of the coordinate system, and at the time from (28) it follows that the maximum
density of the field momentum

is achieved on the
surface of the ball on the circle of radius in the plane , which is perpendicular to the line of the ball’s motion. The same follows from
(18).

We can integrate the
components of the momentum density of gravitational field (28) over the volume
inside the moving ball at in the coordinates (12) similar to (19):

. (29)

, .

As in (19), the total
momentum of the field (29) has only the component along the axis . By analogy with (5) the coefficient
before the velocity in (29) is interpreted as the effective mass
of the gravitational field inside the ball:

, (30)

where is the field energy inside a stationary ball.

Comparing (27) and (30)
gives:

.
(31)

Connection (31) between
the masses of the field inside the ball is the same as in (21) for the masses
of the external field, so the problem of 4/3 exists inside the ball too.

**The contribution of gravitational field in energy and
momentum of a moving body**

We shall try to include in
equation (1) the relations found above for the energy and the momentum of the
gravitational field of a moving test body in the form of a ball. We shall
suppose that in static case instead of (3) there is the following relation:

,
(32)

where – the
total energy of static gravitational field inside and outside the ball with
uniform density of the substance,

–
the rest energy of the substance, found in such a way that it does not depend
on the energy of the gravitational field. To determine the energy the body’s substance
should be divided into pieces and spread to infinity.

Choosing the minus sign
before in (32) will be substantiated in the last section.
We shall continue to analyze the well-known thought experiment. We shall assume
that the substance of the ball is composed of matter and antimatter, which at
some time begin to annihilate and emit photons. We shall suppose that the
photons fly in opposite directions along the axis in the number of in each direction, so that eventually the
entire mass of the ball turns into electromagnetic radiation. In the course of
emission because of the equality of all momentums of photons, and the symmetry
of the radiation along the axis the ball remains stationary. In order that the
process does not depend on the radius of the ball, we shall assume that the
radius is constant irrespective of changes in mass. The energy of the ball (32) should be transformed into
the energy of photons:

,
(33)

where –
Planck constant,

– the frequency of photons.

We shall consider the
same situation in the frame of reference , in which the ball moves at constant
velocity along the axis and at it is in the origin of coordinate system. We
believe that the velocity of the ball does not change, despite the emission of
photons. In the frame the frequency of the photons will depend on whether
they are flying along the axis or in the opposite direction. Taking into
account the relativistic Doppler effect and (32), for the photon energy instead
of (33) we shall obtain:

(34)

On the other hand, the total
energy of the gravitational field inside and outside the ball, taking into
account (15) and (26) equals:

. (35)

For the energy of
substance and field of the moving ball, we have:

.
(36)

From (34) and (36) it
follows:

. (37)

Since the energy of the static field is negative: , then in (37) in the energy of substance of the moving ball the negative additive from field energy will
appear, and the energy does not depend on .

We shall consider now the
law of conservation of momentum. Before emission of the photons the momentum of
the moving ball consists of the momentum of the ball’s substance and the
momentum of the gravitational field, and taking into account (19) for the field
momentum outside the ball, and (29) for the momentum of the field inside the
ball, the total momentum of the field is:

.

Then for the momentum of
a moving ball we can write down:

, (38)

where is the mass of the substance of the ball as a
function of the velocity .

After the photon emission
the whole momentum of the ball and its gravitational field turns into the momentum
of photons:

, (39)

where is the energy (32) of the ball at rest, equal
to the difference of the rest energy of the substance and the energy of the
gravitational field ; also is the energy of photons according to (33).
From comparing (38) and (39) it follows:

.
(40)

We shall suppose that the mass of the moving substance
of the ball is described by the formula: ,

where is the observed mass of the body at rest, and is a function. Here we assume that the
observed mass of the body, and that the mass through which
the energy and the momentum of the gravitational field are determined, are
the same mass. Then instead of (40) we shall have:

.
(41)

But the energy of substance at rest should not depend on the
speed, as well as on the field energy of a stationary ball according to (32).
Therefore, in (41) we should have , which in view of (40) implies
the following:

, , (42)

where the mass sets the rest energy of substance in (32).

We shall substitute from (42) in (37):

.
(43)

From (43) it follows that
at the energy of the substance does not include
the field energy, but during moving in the energy of the substance an additive appears, related
with the energy of the field. The field energy also makes contribution to the mass of the moving substance in (42). The total
energy (36) of the moving substance and the field in view of (43) will equal:

,
(44)

where in the case of
uniform density of the ball substance .

Equation (44) implies that the energy of the body increases
due to the contribution of negative gravitational energy .

We shall now substitute from (42) in (38) or from (42) in (39). This gives the following:

.
(45)

Comparing (44) and (45)
with (1) shows that taking into account the gravitational field the role of the
total mass of the substance and the field is played by the quantity . We believe that , that is the total mass of the substance and the field, is
nothing but the observed mass , which also determines the gravitational field and the inert
properties of the body.

If we know the energy in (44) and the momentum (45), it follows from these relations that we
can express the mass of the substance and the velocity of the body. In case of a uniform ball with
radius in the calculation of the mass of the moving ball we can write down:

, . (46)

According to (46), the mass
of the body substance depends not only on the energy-momentum of the body, but also
depends on the average body size due to the contribution of the gravitational
field mass to the constant value of the mass .

We shall note also that the problem
of 4/3 for the gravitational field (inequality of the mass of the field, found
from the energy, and the mass of the field, calculated by the momentum of the
field) was compensated by the dependence of the energy in (37) and the mass in (42) of the moving substance on the field
energy . As a result,
the field energy in formulas (44) and (45) is included
symmetrically in both the total energy and the total momentum of the body.

**Analysis of the components of mass and energy of the
body**

Until now we have not
specified of which components the mass of the substance of a body
consists, and whether other energies except the energy of gravitational field
contribute to it. For example, what shall happens if the body is heated? From
the standpoint of kinetic theory, an increase of temperature leads first to an
increase of the average velocity of the particles that makeup the body. In this
case, according to (1) the average energy of each particle of the body would
increase, and due to the additivity of energy the total energy of the body at rest should change. For the
case of the substance and the gravitational field , and for
(44) – (45) we can write down the following:

, . (47)

Heating of the body leads to the change of in (47), and the heat as a form of energy is
distributed between the kinetic energy of substance and the energy of the gravitational field. The mass of the
uniform ball can be determined through the mass from the relations:

, . (48)

Any interaction between particles of the body with
each other or with the environment, which changes the energy of the particles, also changes the energy of the body at rest. In accordance with (48)
the mass of the substance of the ball depends not only
on , but also on the radius of the ball .

Due to the similarity of equations of electromagnetic
and gravitational fields, the energy must contain contribution from the total
energy of the electromagnetic field of the body:

. (49)

For the uniformly charged by
volume ball at rest with a charge the total energy of the electric field is:

.

The energy of the magnetic
field can make contribution to , if the ball is magnetized or if there are electric
currents. We assume that other forms of energy (e.g. heat ) can change the body
mass, but can not change the charge of the body, because it is necessary to
transfer the charged particles to the body (or from the body). This is one of
the differences between the electromagnetic and gravitational fields, in
addition to the unipolarity of gravitational charges (which are the masses) and
the bipolarity of electromagnetic charges.

The mass of the substance in (49) can be divided
into two parts, one of which is the mass of the substance at zero temperature
in Kelvin, and the other part is the additional mass from the internal
kinetic energy , which includes the kinetic energy
of motion of atoms and molecules, and the energy of turbulent motion of
substance flows [8]. If is the average velocity of particles in the
body, then the following approximate relations would hold: , . As the energy of field, we include
the energy in (49) with the negative sign:

.
(50)

For the bodies that are only under influence of their
proper gravitational and electromagnetic fields, the virial theorem is
satisfied, according to which the absolute value of the potential energy of the
field on the average is twice as much than the kinetic energy of substance:

, , (51)

here is the total energy excluding the rest energy
of the particles of the body.

Substituting (51) in (50) gives the approximate
equality:

. (52)

We shall now consider the essence of the mass related to the substance mass excluding the
contribution from the mass of the internal kinetic energy and the energy of
macroscopic fields. The contributions in mass are made by the masses of various types of
energy associated with atoms and molecules at the temperature near absolute
zero: strong interaction, binding the substance of the elementary particles and
retaining the nucleons in atomic nuclei; electromagnetic interaction of
particles; the energy of motion of electrons in atoms; rotational energy of
atoms and molecules; vibrational energy of atoms in molecules, etc. In Standard
Model it is assumed that the strong interaction arises due to the action of the
gluon field between the quarks located in the hadrons (mesons and baryons), and
the strong interaction between leptons is absent.

There is also a hypothesis that the strong interaction
is a manifestation of strong gravitation at the level of elementary particles
and atoms [9]. Since gravitation has two components, in the form of the field
of acceleration and the torsion field , the stability of nucleons in nuclei
can be described as the balance of forces from the attraction of the nucleons
to each other due to , and the repulsion of nucleons due
to the torsion field [3]. The same idea is applied to describe the structure and the
stability of a number of hadrons, considered as the composition of nucleons and
mesons [4]. Strong gravitation differs from the ordinary gravitation by
replacing of the gravitational constant by the constant of strong gravitation , and acts between all particles,
including leptons. The estimation of the quantity can be obtained from the balance of four forces
acting on the electron in the hydrogen atom: 1. The force of electric
attraction between the electron and the atomic nucleus. 2. The force of
electric repulsion of the charged substance of the electron from itself (the
electron is represented as a cloud around the nucleus). 3. The centripetal
force from the rotation of the electron around the nucleus. 4. The attraction
of the electron to the nucleus under the influence of strong gravitation. These
forces are approximately equal to each other, so the relations for the forces
of attraction from strong gravitation and the electric force are satisfied [5]:

, m^{3}∙kg^{ –1}∙c^{ –2}
, (53)

where and – the mass of proton and electron,
respectively,

– the radius of rotation of the electron
cloud,

– the elementary electric charge as the proton
charge equal to the absolute value of the negative charge of electron,

– the vacuum
permittivity.

Another way to estimate is based on the theory of similarity of matter
levels and the use of coefficients of similarity. These coefficients are
defined as follows: – coefficient of similarity by mass (the ratio of the mass of
neutron star to the proton mass); – the coefficient of similarity by size (the
ratio of the radius of neutron star to the proton radius); – the coefficient of similarity by speed (the
ratio of the characteristic speed of the particles of neutron star to the speed
of light as the typical speed of the proton substance). For strong
gravitational constant a formula is obtained: ,

where degrees of similarity coefficients correspond to
the dimension of gravitational constant according to the dimension theory.

If we understand the strong interaction as the result
of strong gravitation, the main contribution to the proton rest energy should
be made by the positive kinetic energy of its substance and the negative energy
of the strong gravitation (the electrical energy of the proton can be neglected
due to its smallness). The sum of these energies gives the total energy of the
proton, and due to the virial theorem (51) this sum of energies is
approximately equal to half of the energy of strong gravitation. Since the
energy of the strong gravitation is negative, then the total energy of the
proton is negative too. The total energy of the proton up to the sign can be
regarded as the binding energy of its substance; the binding energy equals to
the work that should be done to spread the substance to infinity so that there
total energy of the substance (potential and kinetic) should be equal to zero.
According to its meaning, the positive proton rest energy must be equal to the
binding energy or the absolute value of the total energy of the proton. This
gives the equality between the rest energy and the absolute value of half of
the energy of strong gravitation:

, (54)

where for the case if the proton was uniform density
ball with the radius.

If we substitute (53) in (54), we obtain another
equation, which allows estimating the radius of the proton:

, ,

where is the classical electron radius.

In self-consistent model of the proton [10] we find
that in (54) the radius of the proton is m, and the coefficient due to a small increase in the density of
substance in the center of the proton. At the same time, in the assumption that
positive charge is distributed over the volume of proton similar to the mass
distribution and the maximum angular frequency of the proton rotation is
limited by the condition of its integrity in the field of strong gravitation,
we can find the magnetic moment of the proton as a result of rotation of the
charged substance:

,
(55)

where J/T is the magnetic moment of the proton,

(in
the case of the uniform density and the charge of proton it should be ).

Strong gravitational constant (53) explains not only
the energy (54) and the magnetic moment (55) of the proton, but also gives an
estimate of the constant of interaction between two nucleons by the strong
gravitation:

,

where for the interaction of two nucleons, and it
tends to 1 for particles with lower density of substance, – Dirac constant, – speed of light.

For the energy of interaction of two nucleons, by
means of pseudoscalar pion-nucleon interaction, located at a distance from each other, and for the corresponding
constant of interaction at low energies, the following formulas hold:

, ,

where
is the effective charge of strong interaction,
is the mass of the pion.

As we can see, the interaction constant is close to the constant of pseudoscalar pion-nucleon strong
interaction.

The fact that the rest energy of the proton is
associated with strong gravitation, also follows from the modernized Fatio-Le
Sage theory of gravitation [11]. In this theory, based on the absorption of the
fluxes of gravitons in the substance of bodies with transfer of the momentum of
gravitons to the substance, the exact formula for Newton's gravitational force (the
law of inverse squares) is derived; the energy density of the flux of gravitons
( J/m^{3}), the cross section of their
interaction with the substance ( m^{2}) and other parameters are
deduced.

In the theory of infinite
hierarchical nesting of matter [3], [5] it is shown that at each
main level of matter the corresponding type of gravitation appears: there is a
strong gravitation at the level of elementary particles, but at the level of
stars it is the ordinary gravitation. The gravitation reaches a maximum in the
densest objects – in nucleons and in neutron
stars. In the substance of the earth's density the range of strong gravitation
is less than a meter, and at such sizes of bodies strong gravitation is
replaced by the ordinary gravitation. This corresponds to the fact that the
masses and the sizes of objects at different levels of matter increase
exponentially, and the point of replacing of the strong gravitation by the
ordinary gravitation lies near the middle of the range of masses from nucleons
to the stars on the axis of the masses on the logarithmic scale.

The fluxes of gravitons, which consist of particles similar to
neutrinos, photons and charged particles, are considered the cause of
gravitation and electric forces in the modernized
Fatio-Le Sage theory. These fluxes of gravitons generated by the
substance of lower levels of matter, control the bodies with the help of
gravitational and electromagnetic forces and create massive objects at higher
levels of matter. These objects, in turn, at certain stages of their evolution
radiate portions of neutrinos, photons and charged particles that become the
basis of other fluxes of gravitons, are already acting at higher levels of
matter. So the fields and the massive objects mutually generate each other at
different levels of matter.

In the above picture the rest energy of proton (54) is
approximately equal to the absolute value of the total energy of the proton in
its proper field of strong gravitation (for increased accuracy we should also
take into account the electromagnetic energy of the proton), and the energy in (52) consists of the rest energy of
nucleons and electrons of the substance of the body, with the addition of the
energy of their gravitational and electromagnetic interactions inside the
substance and the mechanical motion in atoms and molecules. Consequently, the
energy of the body, taking into account the virial
theorem (51) can be reduced to the half of the absolute value of the sum of the
energy of strong gravitation and electromagnetic energy of the nucleons, electrons, atoms and
molecules involved in formation of the binding energy. As a result, the total
energy of the substance and the field of the body at rest instead of (52) can
be written down as follows:

. (56)

To understand the meaning of
energy better, we shall consider the energy balance
in the process of merging of substance, with formation of elementary particles
at the beginning, passing then to confluence of the elementary particles into
atoms and finally in the formation of a body of many atoms. Initially, the
substance is motionless at infinity and its parts do not interact with each
other, so that total energy of the system is zero (we do not consider here the
rest energy of substance in its condition when it is fragmented and was not yet
included into the composition of the elementary particles). If the substance
will draw together under the influence of strong then ordinary gravitation, the
negative energy of gravitational field and the positive
kinetic energy of motion of substance will appear, and due to
the energy conservation law the total energy should not change, remaining equal
to zero. In the energy balance it is necessary to take into account the
electromagnetic energy and the energy leaving the system due to the emission of
field quanta such as photons and neutrinos:

, . (57)

In (57) the virial theorem
(51) is used for the components of the total energy of the system. According to (57), the energy of the emission that left the system equal up
to a sign to the total energy , i.e. the
energy of emission equals the binding energy of the system. By
comparing (57) and (56) we now see that the total energy of substance and of the field of the body at
rest is the same as the energy extracted from the body by different emission
during the formation of the body. As a rule in the energy only those components are taken into account
that are associated with formation of elementary particles, atoms and
macroscopic molecular substance; and the binding energies of the particles of
which the substance of elementary particles is built are not taken into account
and are assumed to be constant. Heating the body due to gravitation according
to (57) and (56) leads to an increase of body energy . This
conclusion is based on the fact that although the internal kinetic energy of
the body is part of (57) with the negative sign, but
the change of the potential energy by the virial theorem compensates the
contribution of the energy. An example
is the star, which is heated and accelerates its rotation during compression by
gravitation, and the absolute value of the gravitational energy of the star
increases.

According to (56), the total
energy of the body at rest, which is used in formulas
(47) for calculation of the energy and the momentum of the moving body,
consists mainly of the energies of two fundamental fields – gravitational and
electromagnetic, responsible for the integrity of the particles of the body and
for the composition of the body of the individual particles. In this case, the
strong interaction between the particles is taken into account by the energy of
strong gravitation and the electromagnetic energy .

As for the weak interaction it is assumed to be the
result of transformation of substance, which was for a long time under the
influence of the fundamental fields. An example is the long-term evolution of a
star massive enough to form a neutron star in a supernova outburst, when the
neutrino burst is emitted with the energy of about the total energy of the star
(the gravitational energy of the substance compression into a small-sized
neutron star is converted into the energy of neutrinos, the energy of photon
emission, the kinetic energy and the heating of the expelled shell). At the
level of elementary particles, this corresponds to the process of formation of
a neutron with the emission of neutrino.

If the body at rest in the weak interaction emits
(absorbs by the body) neutrinos, photons and other particles, it leads to a
change of the total energy of the body. In general,
the energy of the body is the
function of time and speed with which the separate particles or units of
substances are emitted from the body or absorbed by it. Due to the laws of conservation
of energy and momentum, if some particles bring into the system the energy and
momentum, then after some time they are distributed in the system and according
to virial theorem they can be taken into account through the energy and the
momentum of the fundamental fields. Therefore,
we can state that according to (56), the source of the total energy of the
body, and of its mass as the measure of inertia are the
gravitational and electromagnetic fields associated with the mass and charge
(as well as the currents) in the substance. In Fatio-Le
Sage theory of gravitation it is supposed
that the fields associated with the mass and the charges are the consequence of
the interaction of the substance and the charges with the fluxes of gravitons
and tiny charged particles that penetrate the space. If we define the total
mass of the body in the form , then (47)
is as follows:

, .
(58)

**Conclusions**

Equations (58) look exactly
the same as (1) for a small test particle. However, the body mass in (58) takes fully into account the field energies,
whereas for the mass of a small particle in (1) it is only
expected. The appearance in the mass of the contribution from the energy of fields
has occurred because we have used the energy of mutual interaction of many
small particles in a massive body. Hence, by induction, we should suppose that
not only the mass of body, but the mass of any isolated small particle should
be determined taking into account the contribution from the energy of proper
fundamental fields of the particle. The described concept
of mass in the covariant theory of gravitation (CTG) is confirmed by the
analysis of the Hamiltonian [12] and of the Lagrangian in the principle of
least action [13].

We
should note the difference between the results of CTG and general theory of
relativity (GTR) with respect to mass and energy. In CTG the mass of the
uniform spherical body at rest with the radius is expressed by
formulas (46) – (49):

, (59)

where
the mass sets the mass of the
body parts, excluding the potential energy of the fields. As a result the relativistic
and gravitational mass of the body by combining the body parts into a
whole increases due to the energy of the gravitation field , and decreases due to the electric
energy .

In
GTR the mass of the body in the weak-field approximation, according to
(6) is determined by summing up all the energies and then dividing the result
by the square of the speed of light:

. (60)

In contrast to (59) in (60) not the mass but the mass is used, which is included in the formula for the rest energy and the Lagrangian. As it was shown
in [12], for three masses associated with the body, the following relation
holds: . At the same time , which also follows from (59) and
(60). In our opinion, the reason of difference between (59) and (60) is
associated with different positions of the two theories: in CTG there is
explicit stress-energy tensor of the gravitational field, included in the
Lagrangian and contributing to the spacetime metric and the energy-momentum of
the considered system. This allows us to define all the three masses and to find their meaning, and the mass is associated with the cosmological constant in the equation for the
metric of the system. In GTR the principle of equivalence is used instead of
this, the gravitational field is reduced to the metric field, and
correspondingly, the energy and the momentum do not form tensor and can be
found only indirectly, through the spacetime metric. In GTR, only two masses, and , have the meaning.

The equality means that the relativistic gravitational mass
of the body is in the middle between the
masses and , and differs from them by the
absolute value of the potential energy of fields. This implies consistency of positions
CTG and GTR, as these theories determine the mass and energy from different
standpoints.

**References**

1.
Misner, Charles W.; Kip. S. Thorne & John A. Wheeler (1973),
Gravitation, W. H. Freeman, ISBN 0-7167-0344-0.

2.
Okun L. B.
Photons, Clocks, Gravity and the Concept of Mass. Nucl. Phys. B (Proc. Suppl.) **110**
(2002) 151–155.

3. Fedosin S.G. Fizicheskie
teorii i beskonechnaia vlozhennost’ materii.
– Perm, 2009. – 844 p. ISBN
978-5-9901951-1-0.

4.
Comments
to the book: Fedosin S.G. Fizicheskie teorii i beskonechnaia
vlozhennost’ materii. – Perm, 2009, 844 pages, Tabl. 21, Pic. 41, Ref. 289. ISBN 978-5-9901951-1-0 (in Russian).

5.
Fedosin S.G. Fizika i filosofiia podobiia: ot
preonov do metagalaktik. – Perm,
1999. – 544 p. Tabl. 66, Pic. 93, Ref. 377. ISBN
5-8131-0012-1.

6. Heaviside, Oliver (1888/1894), "Electromagnetic
waves, the propagation of potential, and the electromagnetic effects of a
moving charge", *Electrical papers*, **2**, pp. 490–499.

7. Fedosin S.G. Mass, Momentum and Energy of Gravitational Field. Journal of Vectorial Relativity, Vol. 3, No. 3, September
2008, P. 30–35.

8. Fedosin
S.G. The Principle
of Proportionality of Mass and Energy: New Version. Caspian
Journal of Applied Sciences Research, 2012, Vol. 1, No 13, P. 1–15.

9. Sivaram, C. and Sinha, K.P. Strong gravity, black holes, and hadrons.
Physical Review D, 1977, Vol. 16, Issue 6, P. 1975–1978.

10. Fedosin
S.G. The radius of
the proton in the self-consistent model. vixra.org, 03 August 2012.
Accepted by
Hadronic Journal.

11. Fedosin S.G. Model of Gravitational Interaction in the Concept of Gravitons. Journal of
Vectorial Relativity, Vol. 4, No. 1, March 2009, P. 1–24.

12. Fedosin S.G. The
Hamiltonian in covariant theory of gravitation.
Advances in Natural Science, 2012, Vol. 5, No. 4, P. 55 – 75.

13. Fedosin S.G. The
Principle of Least Action in Covariant Theory of Gravitation. Hadronic Journal, February 2012, Vol. 35, No. 1, P. 35–70.

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