Middle East Journal of Science, Vol. 5, Issue 1, pp.
46-62 (2019). http://dx.doi.org/10.23884/mejs.2019.5.1.06
The binding energy and the total
energy of a macroscopic body in the relativistic uniform model
Sergey G. Fedosin
PO box 614088, Sviazeva
str. 22-79, Perm, Perm
Krai, Russia
E-mail: fedosin@hotmail.com , Tel: +7 912 987 0408
The total energy, binding energy, energy of fields,
pressure energy and the potential energy of the system consisting of particles
and four fields is precisely calculated in the relativistic uniform model.
These energies are compared with the kinetic energy of particles. The relations
between the coefficients of the acceleration field and the pressure field
independent of the system’s properties are found, which can be expressed in
terms of each other and in terms of the gravitational constant and the vacuum
permittivity. A noticeable difference is shown between the obtained results and
the relations for simple systems in classical mechanics, in which the
acceleration field and pressure field are not taken into account or the
pressure is considered to be a simple scalar quantity. The conclusion is
substantiated that as increasingly massive relativistic uniform systems are
formed, the average density of these systems decreases as compared to the
average density of the particles or bodies making up these systems. In this
case the inertial mass of the massive system is less than the total inertial
mass of the system’s parts.
Keywords: relativistic uniform system; binding
energy; total energy; pressure energy; potential
energy; kinetic energy.
PACS: 03.30.+p, 03.50.-x, 95.30.Sf
1. Introduction
By definition, the relativistic energy of the system includes all forms
of energy and should be written in a covariant form. In some cases, a constant
term in the form of the rest energy of the system’s particles must be
subtracted from the relativistic energy. Thus we
obtain the total energy of the system, which is usually used in classical
mechanics when simple problems are solved that do not require the relativistic
approach. As a rule, the total energy is divided into two main parts – the
kinetic energy and the potential energy.
In large macroscopic systems several fields can be acting
simultaneously, each of them changes the energy of the particles and can also
have its own potential energy. This significantly complicates the expressions
for the relativistic and total energies and often leads to the fact that
different field theories give the expressions for the energy that are different
in form. For example, in the general theory of relativity, the gravitational
field energy is calculated not directly, but indirectly, with the help of the
stress-energy pseudotensor and the spacetime metric obtained previously [1].
This means that the knowledge of the system metric is necessary to determine
the energy even in case of an extremely weak field. But even with the known metric,
there is some ambiguity in determination of the relativistic energy and
inertial mass of the system and its individual parts [2], [3].
In contrast to this, in the covariant theory of gravitation there is a
stress-energy tensor of the gravitational field, and the metric of any system
in the weak field limit turns gradually into the constant metric of Minkowski
space, where the gravitational energy no longer depends on the type of the
system metric [4]. Similarly, under earthly
conditions the spacetime metric is almost never used to calculate the energy of
the bodies’ electromagnetic field.
With this in mind, in the framework of the covariant theory of
gravitation and the relativistic uniform model we will further determine the
total energy and the binding energy of a macroscopic body, which is in
equilibrium without rotation, and compare the obtained expressions with the
results for simple systems in classical mechanics. We will also calculate the
individual energy components, including the fields’ energy, the pressure energy
and the potential energy of the system. Our approach uses the field theory in
the covariant notation, so that the results obtained for the flat spacetime can
be easily adjusted for the curved spacetime with the corresponding metric, if
necessary.
The thermodynamic properties of the considered physical system of
particles and fields have previously been studied by Chernikov
using the methods of the relativistic kinetic theory of gases and statistical
physics [5]. For the case of a self-gravitating system with charged particles
there is a special name – the Vlasov plasma. Vlasov used the additional idea of a self-consistent field
[6] and pointed to the constraints of the Boltzmann’s model with pair
collisions, which did not take into account the action of the electromagnetic
and gravitational fields at a distance. However, our approach is based not on
the kinetic theory, but on the vector field theory, and instead of the general
theory of relativity we use the covariant theory of gravitation. In addition,
we describe the interaction of particles by means of the fields with the proper
four-potentials, and this applies both to the pressure field and to the
particles’ acceleration field. Thus, obtaining the average values of physical
quantities is not associated either with the distribution functions, the phase
space or the Liouville’s theorem, but with averaging the physical quantities
directly in the equations, arising from the principle of least action.
Despite the fact that the approximation of the constant invariant mass
density in the presence of sufficiently strong gravitational fields is a
certain constraint, this approach is justified, as it gives an accurate
description and can be applied to a number of macroscopic systems, such as the
observable Universe, cosmic gas clouds and even neutron stars. In addition, the
suggested approach can also be adjusted for the case of non-uniform mass
density, as it was done in [7] for white dwarfs and the Sun, which is the
main-sequence star.
2. Relativistic and kinetic energy
In [8] within the framework of the covariant theory of gravitation the
formula was found to calculate the relativistic energy of a physical system of
particles and fields associated with these particles. At the same time, the
electromagnetic and gravitational fields, the acceleration field and the pressure
field were taken into account, and the role of the stress-energy tensor of the
matter was played by the stress-energy tensor of the acceleration field. A
similar formula for the energy was presented in the general field concept for a
macroscopic system in [9, 10].
With the help of this formula in [4] the
energy of the equilibrium system with continuously distributed randomly moving
matter was calculated, taking into account the energy of the fields, in the
explicit form with an accuracy up to the terms that do not contain the square
of the speed of light in the denominators. Taking into account the corrections,
made in [11], the energy is equal to:
. (1)
Expression for the
energy (1) differs noticeably from the energy accepted in the classical
mechanics. The main difference arises from the acceleration field and the
pressure field, which are considered as unique and independent vector fields
that have stress-energy tensors determined in a covariant way.
In (1) it is assumed that the system has a spherical form with the
radius 
; the mass 
and
the charge 
of the system are obtained by multiplying the mass density 
and the
charge density 
by the
volume of the fixed sphere, while and are the
invariant densities of the particles that make up the sphere, measured in the
reference frames associated with the particles. Within the relativistic uniform
model these densities are assumed to be constants. Due to the random motion of
matter, the global vector potentials and the corresponding solenoidal vectors
of the system’s fields are considered equal to zero, so that the contribution
of the solenoidal vectors into the energy (1) is neglected.
At the center of the sphere the velocity of the particles is 
and the
Lorentz factor of the particles equals 
,
where 
is the
speed of light. Each field is
characterized by its own coefficient: 
is the
acceleration field coefficient, 
is the
gravitational constant, 
is the
vacuum permittivity, 
is the
pressure field constant. The quantity 
in (1)
represents the scalar potential of the pressure field at the center of the
sphere, and the mass 
is the relativistic invariant inertial mass of
the system.
The kinetic energy of motion of this system’s particles was found in [12]
by three methods – by the virial theorem, by subtracting the rest energy of the
particles from the relativistic energy of their motion, and using the
generalized three-momenta of the system’s particles. All these methods give the same result:
. (2)
Expressions (1) and (2) are the initial point for determining the
components of the total energy and the binding energy of the system under
consideration. The numerical coefficient 
in (2)
is the consequence of the solution of the quadratic equation for the velocity at the
center of the sphere and is a property of the relativistic uniform system. As a
result, similar numerical coefficients can be found in some other energy
components.
3.
The total energy components
. (3)
The Lorentz factor is maximal at the center of the sphere, where the
particles are moving at the highest velocity. We use the reference frame 
that is associated with the center of the
sphere, so that all the results will refer to the sphere, which is stationary
relative to the observer. If necessary, the energies and momenta of individual
fields and the entire system can be converted into the laboratory reference
frame 
by means of the corresponding Lorentz
transformations at a known velocity of the sphere’s motion in .
Formula (3) was obtained by solving the wave equation for the scalar
potential 
of the acceleration field in the same way as
the potentials of the electromagnetic or gravitational fields are calculated
inside the sphere. In this case in the Minkowski space the acceleration field
potential becomes proportional to the Lorentz factor: 
.
The physical reason for the radial dependence in (3) is the need to
maintain a balance of the acting forces. Thus, the gravitational force tends to
compress the matter, and the pressure of the moving particles resists such
compression. Indeed, the gravitational field strength inside the uniform system
of a spherical form is proportional to the current radius and is directed
towards the center of the sphere. The volumetric density of the gravitational
force is proportional to the field strength and the mass density of the matter.
As the observation point moves from the surface to the center along the radius,
the total thickness and the mass of the observed spherical layer increase, as
well as the total gravitational force from this layer towards the center. The
gravitational pressure is balanced by the dynamic pressure of the particles,
which according to the kinetic theory is two-thirds of the kinetic energy of
the particles per unit volume. Consequently, the closer to the center, the
greater is the pressure of particles in the matter.
Gravitation also forms some spherical boundary of the system with the
radius so that the particles do not on average go
beyond this radius. If 
charged particles were closed inside a sphere
with a rigid boundary, then in the equilibrium state and in the absence of mass
forces we could expect the uniform mass density, the same velocity and the
constant Lorentz factor of the particles at each point of the system, which
would depend only on the temperature. However, here is no rigid boundary in our
model, and the proper gravitational and electromagnetic forces penetrating into
any point of the system act as the mass forces. Thus, it turns out that the
velocity of particles and their Lorentz factor are maximal at the center of the
system under consideration and decrease with increasing current radius.
We will take into account now that as a consequence of the relativistic
effect of length contraction the moving particles in the special theory of
relativity must be considered as if they have a reduced volume and increased
density. Indeed, is the mass density in the reference frames
associated with the particles, 
is the Lorentz factor of the moving particles,
and 
gives the mass density of these particles from
the viewpoint of an observer, who is stationary relative to the sphere.
In order to
calculate the volume of the sphere, it is necessary to sum up the volumes of
all the typical particles moving inside the sphere, as well as the volumes of
the voids between them. Suppose now that the sizes of the typical particles are
much larger than the voids between the particles, and the volume of the voids is
substantially less than the total volume of the particles. In this case, we can
use the approximation of continuous medium, so that the unit of the mass of
matter inside the sphere will be given by the approximate expression 
, where 
is the volume element of the fixed sphere.
The question
whether it is acceptable to increase the sizes of typical particles up to the
limit necessary for using the approximation of continuous medium can be
answered as follows. In the gravitational field the acceleration of particles
depends neither on the mass nor on the density of particles, which follows from
the equivalence principle in the general theory of relativity and from the
equation of motion in the covariant theory of gravitation. For the electric
forces the acceleration is proportional to the ratio of densities 
and does not depend on the mass of particles.
The same applies to the motion velocity, Lorentz factor, kinetic energy and other
quantities, which are determined not by the mass of particles, but by their
density . Thus, with a given density,
we can choose the mass and, consequently, the sizes of typical particles within
the limits we need.
The standard formula to calculate the kinetic energy all of the 
particles of the
system has
the form:
.
The mass 
and the Lorentz factor 
of the particles were substituted in [12] with
and 
,
respectively, and the sum for the particles was substituted with the integral
over the sphere volume. This led to relation (2) for
the kinetic energy of the particles, which also contains the expression for the
total rest energy of the sphere’s particles from the viewpoint of the observer
associated with the sphere:
. (4)
The average rest mass of one
particle is obtained by dividing the total rest mass 
by the number of particles in the system.
In order to obtain the total energy of the system, we need to subtract
the energy 
from the relativistic energy (1):
.
On the other hand, according to (2):
.
After eliminating from these equations, taking into account (1)
and collecting the similar terms, we obtain the expression for the total
energy:
. (5)
The total energy should consist of the kinetic and potential energies, 
, therefore
the potential energy of the system will be as follows:
. (6)
As we can see, the potential energy (6) contains the energy of the
particles in the gravitational and electromagnetic fields and in the pressure
field, as well as the energy of these fields themselves, with addition of the
acceleration field energy.
To simplify expression (6) we will use the definition of the scalar
potential of the pressure field in [14], according to which the
potential at the center of the sphere is 
,
which means that it is defined by the pressure 
in the reference frame 
of the moving particle. On
the other hand, the scalar potential of the pressure field from the viewpoint
of the observer in equals 
,
where 
is the pressure, 
is the mass density at the center of the
sphere for this observer. Consequently, 
, and
the pressure in the system due to the particles’ motion increases more rapidly,
in proportion to the square of the Lorentz factor, as compared to the mass
density, which increases in proportion to the Lorentz factor.
If the radiation
pressure is not taken into account, for the pressure in the reference frame 
of the particle and for the pressure at the
center of the sphere in the reference frame 
we can write the following:
, 
,
where 
is the parameter, which represents the number
of nucleons per unit of relativistic ionized gas, 
is the atomic mass unit, 
is the Boltzmann constant, and the temperature
at the center of the sphere 
in the reference frame 
during transition from 
into is transformed to the temperature 
.
Taking this into consideration, the scalar potential can be expressed in terms of the temperature 
at the center of the sphere:
.
(7)
In [12] the square of the particles’ velocity at the center of the
sphere was estimated:
. (8)
In derivation of (8) the value of the acceleration field coefficient 
was not recorded, as well as the mass of the
system’s particles. Real bodies can contain several types of particles with
different masses at the same time, such as atoms, ions, electrons, and
individual nucleons. It is convenient to assume that the coefficient refers to the particles with the effective
mass 
. In
this case we can write for the mass density 
at an arbitrary point of the sphere the
following:
,
where 
is the concentration of nucleons, 
is the concentration of particles with the
effective mass 
,
contributing to the pressure, so that 
.
We will multiply relation (8) for 
by 
and equate it to 
,
which gives the equality between the kinetic energy of one particle with the
mass 
and the kinetic temperature at the center of
the sphere:
.
Let us substitute the left-hand side of this equation in (7) instead of 
and take into account the definition 
:
. (9)
According to [13], for the scalar potential of the pressure field inside
the sphere the following relation holds:
(10)
If we pass from the Lorentz factors to the squares of velocities in (3),
we will obtain:
.
By multiplying this equation by 
we pass on to the relation for the
temperatures inside the sphere:
.
Let us go ahead and multiply the last equation by 
.
Taking into account the definition and (7) we have:
.
From comparison of this relation with (10) we arrive at the fact that
.
(11)
We will now use the relation between the field coefficients, which was obtained in [7] using the equation of motion:
. (12)
Combination of (11) and (12) gives the following for the coefficients of
the pressure field and the acceleration field:
, 
. (13)
Let us now substitute (11) into (9):
.
(14)
In view of (14), we will sum up all the terms with the pressure in (6):
. (15)
In view of (15), for the potential energy (6) we obtain:
. (16)
Relation (12) can be used to eliminate the coefficient 
in (16) and to express the potential energy of
the system in terms of the corresponding energies of the gravitational and
electromagnetic fields and the pressure field:
. (17)
We will now express all the terms in (16) in terms of the acceleration
field coefficient using (12) and (11):
. (18)
Hence we see that the potential
energy of a relativistic uniform system that is in equilibrium condition is
always negative.
Substituting (2) into (18) gives the following:
. (19)
Accordingly, the total energy is equal to:
. (20)
Despite the fact that in (19) the absolute value of the potential energy
is much greater than the kinetic energy, the virial theorem is satisfied in the
system under consideration. It was shown in [12], where the nonzero virial of
the system and the forces acting on the particles were explicitly calculated.
In particular, the average energy, associated with these forces, in view of (2)
is equal to:
.
We can also transform the pressure energy (15), using (11), and compare
it with the kinetic energy (2):
. (21)
The entire energy, associated with the pressure, appears almost 2 times
less than the kinetic energy of the particles’ motion.
Most cosmic bodies are neutral, we can neglect their electromagnetic
fields and can assume that 
. In
this case, according to (13) 
, and
the potential energy (18) is equal to:
.
In the classical uniform model the body’s
matter is compressed by the gravitational forces, which are opposed by the
internal pressure force, while the pressure is considered to be a scalar
quantity. It is assumed that the main contribution into the potential energy is
made by the gravitational energy, which equals the value
.
We see that the potential energy 
in the relativistic uniform model is very
close in value to the gravitational energy 
. This
explains why in the classical model for estimating the potential energy of the
system it is sufficient to calculate only the total gravitational energy of the
system and there is no need to take into account neither the pressure field
energy nor the acceleration field energy.
4. The binding energy
By definition, the
binding energy of the physical system is
obtained by subtracting the relativistic energy 
(1) from the total rest energy of the
particles 
(4):
. (22)
Taking into account
the definition of the total energy 
, in
view of (20), we find for the binding energy the following:
. (23)
For comparison, in
simple systems, where there are only potential forces in the absence of
pressure, due to the virial theorem 
.
In our physical
system of closely interacting particles and fields in addition to the
electromagnetic field, we also take into account the contributions from the
gravitational field, the acceleration field and the pressure field. In such a
system, according to (19), the potential energy of the fields 
is negative and it is much greater in its
absolute value than the kinetic energy 
in comparison to simple systems. This leads to
the increased binding energy, which is required to separate the system’s
particles from each other and to scatter them to infinity. Thus it is expected
that during the formation of a bound relativistic uniform system with charged
particles under the action of the gravitational field, taking into account the
contributions from the acceleration field and the pressure field, the system
must emit the energy, which is equal to the binding energy 
.
Among all the
energies it's most convenient to calculate the energies of the gravitational
and electromagnetic fields, which go beyond the limits of the system to
infinity. The sum of the energies of these external fields, taking into account
(12) and (11), is equal to:
. (24)
From (4) the relation follows:
.
Similarly,
in [4] the charge 
was associated
with the charge 
of the sphere,
which was found by the observer in 
:
.
Applying this to
(24), we find:
. (25)
In (25) the energy
of the gravitational and electromagnetic fields outside the sphere is expressed
in terms of the total rest mass 
of the particles inside the sphere and the
total charge of the particles 
. From
(25) we see that the mass is actually equal to the gravitational mass 
,
which is responsible for the gravitation outside the body.
Comparing (24) with
the kinetic energy (2) and the binding energy (23) gives the following:
. (26)
With the help of
(26) we can easily estimate the binding energy 
, if
we know the mass, charge and radius of the system, using which in (24) and (25)
we can calculate the sum of the energies 
outside the system. Although cosmic bodies
with the same masses and sizes can differ in their state of matter, the binding
energy of these bodies in (23), according to (26), will be the same. Indeed,
the phase transformations of matter, arising from the energy transfer inside
the system with the constant radius and mass, should not influence the energy
of the external fields and the total binding energy of the system.
5. Estimation of the energy of fields
In this section we
will consider the question about what contribution into the relativistic energy
and the total energy is made by the energy 
,
associated with the system’s fields. The energy is calculated with the help of the volume
integrals of the fields’ tensor invariants, for which it is necessary to know
the strengths and solenoidal vectors of the fields. As part of the relativistic
energy of the system (1), the energy according to [4] equals:
.
In this expression
we will take into account (12), (11), (24), (2), (26) and (23):
. (27)
According to (27),
the energy of all the four fields is approximately 
times greater than the
kinetic energy of the particles 
and 
times greater than the binding energy 
,
which is equal in its absolute value to the total energy of the system 
. In
addition, in (27) 
, that
is, the energy 
is up to a sign equal to the sum of the
energies of the gravitational and electromagnetic fields outside the sphere.
Hence it follows that the sum of the energies of all the fields inside the
sphere is equal to zero.
We will now
calculate the sum of the particles’ energies 
in the gravitational and electromagnetic
fields and in the pressure field. According to (1) we have the following:
.
Using (12), (14),
(11), (2) and (24) we find:
.
Comparison with
(27) gives us the following:
, (28)
that is, the energy
,
associated with the fields, in its absolute value is over 2 times less than the
sum of the particles’ energies in the gravitational and electromagnetic
fields and in the pressure field. Note that all the conclusions are made in the
weak field approximation, while the Lorentz factor 
of the particles at the center of the sphere
does not differ significantly from unity.
As it was shown in
[12], although the global vector potentials of the fields inside the sphere
with the particles are equal to zero, there are also proper vector potentials
of the fields inside the particles due to their motion. These vector potentials
are part of the generalized momentum, with the help of which we can estimate
the kinetic energy of the system’s particles. Because of the proper vector
potentials, during the particles’ motion the corresponding solenoidal fields
emerge, which make an additional contribution into the energy of the particles’
fields .
What can be the
value of this contribution? Due to the motion, the energy of the fields becomes
dependent on the velocity, however the total energy of the fields inside the
body vanishes. If we assume that the same is true for each particle filling the
sphere in the model under consideration, then the proper scalar and vector
potentials of the fields inside the particles do not result in the total energy
of the fields. At the same time the global scalar potentials of the fields,
that are the scalar superposition of the external scalar potentials of
individual particles, give the corresponding field strengths, with the help of
which we can calculate the energy of the fields inside and outside the sphere.
In contrast to
this, the global vector potentials of the fields are the vector superposition
of the external vector potentials of individual particles, and they are equal
to zero due to the randomness of motion of the set of system’s particles. This
implies the equality of all the global solenoidal vectors to zero, and thus
they do not contribute to the energy of the fields.
6. Conclusion
Our
analysis of the energy for the system of a spherical form shows that in real
massive bodies there are noticeable deviations of the total, kinetic and
potential energies from the expressions for the energies of simple systems,
interacting only by means of gravitational and electromagnetic forces at a
distance. This is due to the additional contributions from the acceleration
field and the pressure field.
Taking
into account (23) and (19-20), the binding energy and the total energy can be
expressed in terms of the kinetic energy of the particles or the potential
energy of the system:
, 
,
(29)
Relations (29) that
we obtained can be compared to the standard expression for simple systems and
the corresponding virial theorem in classical mechanics:
, 
, (30)
where the classical
potential energy is expressed as the sum over all the particles in terms of the
radius vectors of the particles and the forces acting on the particles from the
potential fields:
.
In
contrast to (30), in [12] we obtained the following expression for the virial
theorem:
. (31)
From (29) and (31)
it follows that taking into account the acceleration field and the pressure
field in the system under consideration leads to a change from the classical
value 
to approximately 
of the binding energy’s share relative to the
potential energy, and to a corresponding change in the value from 
to approximately 
of the kinetic energy’s share relative to the
energy associated with the action of the potential forces. In addition, (29)
and (31) imply the inequality of the potential energy and the energy associated
with the forces acting on the particles, so that 
.
The
binding energy of the system, according to the (26), can be expressed in terms
of the sum of energies of the gravitational and electromagnetic fields outside
the body:
. (32)
The
unique relation between the total, kinetic and potential energies and the
binding energy in the system under consideration can be obtained due to the
fact that we use the effective mass of the particles 
, which is associated with the system’s state of matter by means of the parameter 
(the number of
nucleons per one particle of relativistic gas). This leads to relations between
the coefficients of the acceleration field and the pressure field in (11) and
to relations between these coefficients and the gravitational constant and the
vacuum permittivity in (12-13), which allows us to compare the values of the
energies.
In the last section
we estimated the contribution of the energy ,
which is made by the fields into the relativistic energy 
, and
we compared it with the energy in the gravitational and electromagnetic
fields, as well as in the pressure field in relation (28). Using (2) and (4) we
obtain the relation:
.
Taking this into
account, from (1) and (26-28) for the relativistic energy of the system we find
the following:
(33)
Expression (33) for
the relativistic energy , in
view of (32), corresponds to expression (22) for the binding energy of the
system.
Since 
is the rest energy of the systems’ particles
according to (4), and the fields’ energy 
outside the sphere is typically negative due
to the prevailing contribution of the gravitational energy over the
electromagnetic energy, we can see that the inertial mass 
of the
system turns out to be less than the total invariant mass of the particles 
,
which is part of the equation 
. We
should also note that the mass 
is exactly equal to the gravitational mass ,
according to [4].
After substituting
(25) into (33) we arrive at the relation, which is only slightly different from
relation (31) in [4]:
. (34)
Hence it follows
that the introduction of the charge 
into the system typically increases the
inertial mass 
, at
least this holds exactly at the constant mass . And
conversely, since 
,
decrease in the gravitational mass 
of the system is possible with increasing of
the charge . To
see this, it suffices to solve (34) as a quadratic equation for , and
to fix the inertial mass while is changing:
.
In the classical uniform
system of a spherical shape with stationary particles the total gravitational
energy summed up with the total electric energy is equal to the following: 
. Consequently, (34) can be
written as follows:
.
As we can see, the
inertial mass of the relativistic uniform system differs
from the rest mass of the particles by approximately half of the total mass-energy
of the gravitational and electric fields of the classical uniform system,
whereas the presence of the electric field increases the mass in contrast to the action of
the gravitational field.
Now let us imagine
that for an external observer the sphere with the particles has the invariant
inertial mass , the
volume 
and the corresponding mass density 
: 
. Let
us substitute (4) into (33) and take into account (24) and the relation 
. This
gives the following:
. (35)
In [12]
we found expression (8) for the squared velocities of the particles at the
center of the sphere, with the help of which we can estimate the Lorentz
factor:
.
Substituting
the expression for 
into (35),
after reduction by 
we find:
. (36)
Hence it follows
that 
,
i.e., as more increasingly massive relativistic uniform systems are formed, the
average density of these systems decreases as compared to the average density
of the particles and bodies that make up these systems.
Let us assume that
this conclusion holds true for neutron stars, mostly consisting of nucleons
only with a small admixture of atomic nuclei and a certain number of electrons,
which give a small contribution into the total mass. We will assume that the
mass density 
in (36) represents the density of the
nucleons’ matter, and 
is the average density of a neutron star, and
we will use (13) for the case of an uncharged star with zero charge density
. This
gives the following:
.
(37)
Substituting here
instead of 
the mass of a typical star of 1.35 solar
masses, taking as the nucleon density 
the proton density of 
kg/m3 with the proton radius of 
m according to [15], and
using the estimate of the star density in the form 
, we
find the corresponding radius of the neutron star: 
km. This radius supposes tight packing of
neutrons in the matter of the star and by the order of magnitude is in
reasonable agreement with the observational data. However, it should be noted
that in the star there are gaps between the nucleons. Therefore, in (37) for we should substitute not the mass density of
the proton, but a smaller quantity. This leads to decreasing of 
, so
that the radius 
km places the lower limit on the radius of the neutron star.
In the theory of
infinite nesting of matter nucleons are similar in their properties to neutron
stars, and for these objects the ratio of the central density to the average
density is approximately 1.5 according to [7, 15]. Thus, in the first
approximation nucleons and neutron stars are close enough in their properties
to relativistic uniform systems. In addition it should
be noted that these objects consist of particles, for which it is necessary at
least to take into account the energy of the proper spin rotation and the
energy of strong interaction. Consequently, our analysis with respect to such
relativistic objects needs clarification, starting with introduction of
additional terms into the Lagrangian and finishing with taking into account the
metric in the equations, arising from the principle of least action. These
calculations have not yet been made, but we can rely on the equation derived by
Tolman, Oppenheimer, and Volkoff in the framework of
the general theory of relativity [16].
From (37) and (33)
it follows that as a certain large relativistic uniform system is formed from a
number of small relativistic uniform systems, the average mass density of the
system decreases as compared to the average density of its parts, while the
inertial (invariant) mass of the large system is less than the sum of inertial
(invariant) masses of the system’s parts. At the same time, the relativistic
energy density and the binding energy density decrease in the transition to
increasingly more massive objects. From a qualitative point of view the density
decrease can be explained by the presence of gaps between the individual parts
of the system with reduced mass density. From a quantitative point of view the
decrease in the average density of the system can be derived from the
contributions of the particles’ energy in the potentials of the system’s proper
fields with addition from the fields’ energy, found through the strengths and
solenoidal vectors of the fields.
In this regard, we
note that the strengths and solenoidal vectors of the fields are the temporal
and spatial rates of change of the field potentials, as they are calculated
using the partial derivatives of the scalar and vector potentials. Therefore,
in case of the known dependencies of the field potentials on time and
coordinates, the full description of the system can be easily achieved and the
main dependencies, including the equation of motion and the stress-energy
tensor [14], can be easily found. In contrast to this, if only the strengths
and solenoidal vector of the fields are given, in order to determine the potentials we need to perform integration and to take into
account the initial conditions. And such integration, as is known, is much more
difficult than differentiation. An additional advantage of the use of
potentials in the field physics is the fact that they are calculated in the
standard way using the corresponding wave equations [4], [13].
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Source: http://sergf.ru/been.htm