Bulletin of Pure and Applied Sciences, Vol.38 D (Physics), No. 2, pp.
73-80 (2019). http://dx.doi.org/10.5958/2320-3218.2019.00012.5
The mass hierarchy
in the relativistic uniform system
Sergey G. Fedosin
PO box 614088, Sviazeva str. 22-79, Perm, Perm Krai, Russia
E-mail: fedosin@hotmail.com
The analysis of
the relativistic uniform system in the framework of the covariant theory of
gravitation leads to five different masses of the physical system, in which the
particles are bound together by means of the electromagnetic and gravitational
fields, the acceleration field and the pressure field. In this case it turns out that the system’s gravitational mass is equal
to the sum of the invariant masses of all the system’s particles, and the
system’s inertial mass is less than the gravitational mass by the value of the
system’s binding mass-energy.
Keywords: relativistic uniform system; gravitational mass; inertial mass; invariant mass; energy of system.
1. Introduction
For each physical
system, we can introduce several types of masses, each of which characterizes
the system in its own way. The inertial or invariant mass, which is related to
the system’s relativistic energy by a multiplier in the form of the Lorentz
factor and the speed of light squared, is widely known. If the selected
reference frame is stationary relative to the center of momentum of the
physical system under consideration, the Lorentz factor of the system becomes
equal to unity. In such a reference frame, there still remains proper motion of
the particles and fields, and it is necessary to take into account the Lorentz
factors of individual particles. When calculating the relativistic energy, it
turns out that contribution to the inertial mass is made both by the energies
of the particles in the fields acting on them, and by the energies of the
fields themselves. In particular, the energy of the gravitational and
electromagnetic fields must be calculated both in the empty space between the
particles and even outside the system. Moreover, the matter particles,
including small particles with the size of an individual atom, contain almost
only empty space. This is due to the fact that almost the entire mass of an
atom is contained in a small atomic nucleus. Therefore, the field energy must
be calculated also inside the matter particles. In connection with this, when
calculating the field energy, as a rule, integration is performed over the
entire volume of the space of the reference frame under consideration.
In classical
mechanics, the inertial mass is not only associated with the relativistic
energy, but also determines the acceleration of bodies under the action of the
forces applied to the system. A certain exception is the gravitation force. We
can assume that the gravitational force acting from body 1 on body 2 is
proportional to the active gravitational mass of body 1 and the passive
gravitational mass of body 2. These masses are interpreted differently
depending on the theory of gravitation used. For example, in the general theory
of relativity, the active and passive gravitational masses do not differ from
each other, and the inertial mass is equal to the gravitational mass of the
system.
There is at
least one more mass, often used in astrophysics. This is the total mass of
baryons that make up the system, to which the total mass of the available
electrons is added for accuracy. Indeed, the matter of planets and stars
consists of nucleons and electrons, which can get connected with each other,
forming separate atomic nuclei, atoms and ions, as well as the matter in
different phase states. Obviously, the total mass of baryons and electrons is
not equal to the inertial mass of the system, since during formation of a bound
system, it releases the binding energy, and the relativistic energy of the
system decreases.
Next we will consider the question of how the above-mentioned masses are
interrelated in the covariant theory of gravitation. In doing so, we will use a
physical model of particles and fields, which represents the relativistic
uniform system. For the sake of simplicity, only the main four fields will be
taken into account, including the electromagnetic and gravitational fields, the
acceleration field and the pressure field. Each of these fields is assumed to be
a vector field of Maxwell-type. All these fields are considered as the
corresponding components of the general field [1], [2], and are used to
describe macroscopic systems [3].
This approach
was used earlier to estimate various masses of the system of particles and
fields of a spherical shape in [4], [5]. In the system under consideration the
particles move chaotically and so closely to each other that the approximation
of continuous matter distribution can be applied. In this case, some part of
the particles is charged, and the charge density distribution is similar to the
mass density distribution.
Using now the
properties of the relativistic uniform system [6], we will calculate the more
accurate values of the system’s masses, and will compare them by arranging in a
certain sequence. The starting point of our reasoning will be the formula for
the relativistic energy of a physical system with continuous matter
distribution [7]:
(1)
In (1) 
is the speed of light; 
and 
denote
invariant mass and charge densities, respectively; 
, 
, 
and 
are the scalar potentials of gravitational and electromagnetic
fields, acceleration field and pressure field, respectively; 
, 
, 
and 
are the tensors of these fields, respectively; 
is the time component of the
four-velocity of the element of matter; 
is the determinant of the metric tensor; 
is the product of differentials of the spatial coordinates; 
is the gravitational constant; 
is the magnetic constant; 
is the acceleration field coefficient; 
is the pressure field coefficient.
Formula (1) is
valid in the approximation, in which the potentials and field tensors, found as
superpositions of the contributions of the entire set of particles, do not
depend on the velocities of individual particles of the system. Using this
formula in [5], the energy for the equilibrium system of a spherical shape with
the continuously distributed chaotically moving matter, taking into account the
fields’ energy, was calculated explicitly with an accuracy up to the terms that
do not contain the speed of light squared in the denominators. Taking into
account the corrections made in [8], the energy is equal to [9]:
(2)
The last four
terms in (2) define the field energies, the quantity 
is the
invariant inertial mass of the system, the mass 
is the product of the mass density 
by the volume of the sphere
with the particles, 
is the Lorentz factor of the particles in the
center of the sphere, 
denotes the radius of the sphere, the charge 
is defined as the product of the charge
density by the volume of the sphere , 
is the electric constant, 
is the scalar potential of pressure field in
the center of sphere.
2. Normalization of the mass and
energy
The system’s energy
is an integral quantity, it contains contributions from the energy of particles
and fields, and in the general case requires normalization. It is convenient to
normalize the energy with the help of the cosmological constant 
, which is part of the system’s Lagrangian. Formula
(1) for the four main fields was obtained under the following condition [7]:
, (3)
where 
is the four-potential of gravitational field
in the framework of the covariant theory of gravitation,
is the mass four-current,
is the invariant mass density,
is the four-velocity of point particle,
is the electromagnetic four-potential,
is the charge four-current,
is
the invariant charge density,
is the four-potential
of acceleration field,
is the four-potential of pressure field,
,
, 
and 
are the vector potentials of gravitational and
electromagnetic fields, acceleration field and pressure field, respectively.
With the help
of (3) we simplified the equations for the metric and performed gauge of the
relativistic energy so that 
and the scalar
curvature 
disappear from
both of them.
Within the
framework of the special theory of relativity the four-currents are:
, 
, where 
is the Lorentz factor, 
is the velocity of system’s particles. We substitute this in (3):
. (4)
Within the
limit of low velocities we can neglect the terms,
containing the particle velocity 
and the vector
potentials. Then in (4) the Lorentz factor is 
and only the
terms with the scalar potentials of the fields are left. For the particles scattered to
infinity in cosmic space we can assume that these potentials arise only from
the proper fields of the particles and are the potentials averaged with respect
to the volume of the particles. In this case 
and we can write:
. (5)
The scalar
potentials of the fields inside the sphere were found in [10]. After
simplification they look as follows:
, 
,
, 
.
This shows that
the cosmological constant is defined by the rest energy density of the
particles with addition of the energy density of the particles in their proper
fields. If we average (5) with respect to the cosmic space, then at 
, where 
is a constant
of the order of unity, the value 
is obtained. Substituting here the standard
estimate of the cosmological constant 
m–2,
we find the corresponding matter density: 
kg/m, which is sufficiently close to the
observed average mass density.
Suppose now
that a certain amount of matter under the action of gravitation is drawn
together into such objects as gas clouds, and then into planets and stars. In
[5] we integrated (4) over the volume of a fixed sphere, filled with the
particles, which were held in that state by the action of gravitation and the
electromagnetic field, taking into account the internal acceleration field and
pressure field. If we denote by 
the gauge mass-energy of the system’s particles, associated with the cosmological constant in (4), the integration results, in view of more precise values of the energies of particles in the potentials of the gravitational and
electromagnetic fields [8, 9], would be as follows:
. (6)
The terms on the
right-hand side of (6) are contained
in (2), so that the energy can be written
as follows:
. (7)
In (2), the
mass 
and the charge 
are used as
auxiliary quantities. However, outside the sphere, the potentials of the
gravitational and electromagnetic fields are determined with the help of the
mass 
and the
electric charge 
. For these masses and charges we obtain [5], [11]:
, 
. (8)
The equation of
motion of the matter inside the sphere [3] and the generalized Poynting theorem
[9] lead to the following relation between the fields’ coefficients:
.
(9)
In view of (8)
and (9), the expression (7) can be
rewritten as follows:
. (10)
This shows that
the relativistic energy of the system is equal to the total energy of the
system’s particles in proper fields,
from which we should subtract the energy of the gravitational and
electromagnetic fields outside the volume, filled with the system’s particles.
3. Comparison of the masses
In [12], we
calculated the total rest energy 
of the sphere’s
particles from the viewpoint of the observer associated with the sphere, the
kinetic energy 
, the scalar potential 
of the pressure
field at the center of the sphere, and found the relations between the fields’
coefficients, as well as the Lorentz factor 
for the
particles’ motion at the center of the sphere :
, 
,
, 
, 
,
. (11)
In (11), the
energy 
is expressed in
terms of mass 
and charge 
. From comparison of (8), (10) and (11) we obtain
relations for the mass-energies of the system under consideration:
,
(12)
, 
.
.
(13)
In (12), the
mass-energy of the gravitational and electromagnetic fields outside the matter
is equal to 
. The difference in masses in (12) and (13) is expressed
in units 
and can also be
expressed in terms of 
.
In (13) the
mass 
, which has an auxiliary character, is equal to the
product of the mass density 
by the body
volume, in this case the density 
is measured in the
reference frames associated with the particles. The gauge mass 
refers to the
mass-energy of the particles under action of fields, and is associated with the
energy gauge and the cosmological constant. The mass 
is the
invariant mass of the system under Lorentz transformations and it can be
considered the inertial mass. The mass 
is greater than
other masses because the particles inside the sphere move faster under the
influence of the fields and have an increased Lorentz factor [5]. In this case,
the mass as the total rest mass of particles
is equal to the gravitational mass 
, since it is the mass that is present in the expressions
for the strength, scalar potential and energy of the gravitational field
outside the sphere [8].
From [12] it follows that
.
(14)
This means that
the binding energy 
divided by the square of the speed
of light, in view of the relation 
, is the difference between the gravitational mass 
and the
invariant mass 
of the system
under consideration.
In (12), taking into account that mass assumed to be constant, the case is
possible when addition of the electric charge to the uniformly charged body
starts to decrease the mass 
. Though transfer of the charge to the body increases
the mass due to the mass
of the charge carriers,
but the electric field energy increases quadratically with respect to the
charge 
, and the contribution of the mass-energy of the field
could exceed the contribution from 
. For this the following condition must be met:
.
If 
, 
, where 
and 
denote the
charge and mass of one electron, 
is the number
of electrons transferred to the body, then this condition can be rewritten as
the relation for the electric potential, which should take place on the surface
of the body:
, 
V.
4. Conclusion
In Section 2 we
describe the gauge mass , and in Section 3 we consider different masses that
characterize the system and express their relation with each other. In (11) we express the relativistic energy in terms of the
mass and the charge
of the system , and in (7) and (10) – in
terms of the mass . The gauge mass is associated
with the energy of particles in the potentials of all the four fields and is
expressed in (6).
The difference
of from the invariant
inertial mass of the system 
in (2) consists
in the fact that the mass according to
(12) contains the contribution of the mass-energy of the
electromagnetic and gravitational fields in the entire volume outside the
system, taken with the opposite sign. The mass-energies of the fields inside
the system do not make any contribution, since the acceleration field and the
pressure field appear inside the sphere, and in view of (9) the sum of the
contributions of all the four fields becomes equal to zero.
In order to
describe the mass of the physical system we introduce in (13) five different masses, the lowest of which is and the largest
is the mass 
. The mass is greater than
the invariant mass of the system , because the contribution of the gravitational field
into the total energy is negative and exceeds the contributions of other fields for cosmic systems. The mass is close in
value to the total mass of the matter’s baryons and electrons, which is used in
the general theory of relativity. However, in our approach the mass coincides with the gravitational mass 
and is not
equal to the inertial mass 
, in contrast to the general theory of relativity,
where 
. This is due to the fact that we calculated the
masses 
and in the framework
of the covariant theory of gravitation.
We can assume
that the masses and are likely
intended for an external observer, estimating the gravitational and inert
properties of the system. The mass is associated
with the gravitational action exerted by the system on other bodies or with the
action exerted by the external bodies on the system under consideration. The
invariant mass of the system
is included in the expression for the energy and momentum of the system, and
therefore it reflects the inert properties of the system from the point of view
of action of non-gravitational forces on this system.
The masses 
and can be
significant for the internal observer, who is trying to determine the sum of
the invariant masses of the system’s particles and the interaction energy of
these particles. The mass 
allows us to
estimate the average density of the particles in their rest reference frame by
dividing by the system’s
volume.
In addition to
masses, the system in question is characterized by two electric charges. One of
these charges is given by the expression 
and is
proportional to the volume 
of the system
and to the invariant charge density 
, measured in the reference frames associated with the
moving particles. The other charge 
, according to (8), determines the electromagnetic
field of the system beyond its limits, and 
. The difference between these charges is due to the
motion of particles and different charge densities of the moving and fixed
particles from the standpoint of the theory of relativity.
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Source: http://sergf.ru/mhen.htm