**Relativistic uniform system** is an ideal
physical system, in which the mass density (or any other physical quantity)
depends on the Lorentz factor of the system’s particles, but is constant in the
reference frames associated with the moving particles.

**Содержание**

- 1 Difference
from the classical uniform system
- 2 Field
functions for the bodies of spherical shape
- 2.1 Acceleration field
- 2.2 Pressure field
- 2.3 Gravitational field
- 2.4 Electromagnetic field
- 3 Tensor field
invariants
- 4 Energies of
particles in the field potentials
- 5 Relation
between the field coefficients
- 6 Relation
between the energies of the internal and external fields
- 7 Relativistic energy
- 8 Relation
between the energy and the cosmological constant
- 9 Lagrange function and motion integrals
- 10 Integral vector
- 11 Virial
theorem and the kinetic energy of particles
- 12 Extreme objects
- 13 Cosmological
constant and scalar curvature
- 14 References
- 15 See also
- 16 External links

**Difference from the classical uniform system**

In classical physics, the ideal uniform body
model is widely used, in which the mass density is constant throughout the
volume of the body or is given as the volume-averaged quantity. This model
simplifies solution of physical problems and allows us to quickly estimate
different physical quantities. For example, the body mass is calculated by
simply multiplying the mass density by the body volume, which is easier than
integrating the density over the volume in case of dependence of the density on
the coordinates. The disadvantage of the classical model is that the majority
of real physical systems are far from this ideal uniformity.

The use of the concept of the relativistic
uniform system is based on the special theory of relativity (STR) and is the
next step towards a more precise description of physical systems. In STR
particular importance is given to invariant physical quantities, which can be
calculated in each inertial reference frame and are equal to the values that
these quantities have in the proper reference frame of the body. For example,
multiplication of the invariant mass by the four-velocity gives the
four-momentum of the body containing the invariant energy, and the
multiplication of the corresponding invariant quantities by the four-velocity
allows us in the case of motion of solid point
particles to find the four-potentials of any vector fields
and to develop their complete theory. ^{[1]} Another example is that
for determination of the four-velocity or four-acceleration
as a rule the operator
of proper-time-derivative is used instead of the time derivative.
Therefore, the use of the invariant mass density and charge density of the
moving particles that make up the system does not only conform to the
principles of STR but also significantly simplifies solution of the
relativistic equations of motion.

**Field functions for the bodies of spherical shape**

Field equations are most easily solved in case
of spherical symmetry in the absence of general rotation of particles. In this
case all the physical quantities depend only on the current radius, which
starts at the center of the sphere. Below the solutions are provided of the
equations for different fields in the framework of STR, including the scalar
potentials, field strengths and solenoidal vectors. Due to the random motion of
the particles in the system, the vector field potentials become equal to zero.
This leads to zeroing of the solenoidal vectors of fields, including the
magnetic field and the __gravitational torsion field__.

**Acceleration field**

The four-potential of the __acceleration field__ includes the
scalar potential and the vector potential . Applying
the four-curl to the four-potential gives the
__acceleration tensor__ . In
the curved spacetime the acceleration field equation with the field sources is
derived from the principle of least action: ^{[1]}

This equation after expressing the acceleration
tensor in terms of the four-potential turns into the
wave equation for finding the four-potential of the acceleration field:

which, taking into account the calibration
condition of the four-potential , can
be transformed as follows:

where is the speed of light, is the acceleration field coefficient, is the mass four-current with the covariant
index, is the metric tensor, is the Ricci tensor, is the four-velocity, is the invariant mass density of the
particles in the comoving reference frames, which is the same for all the
particles.

In Minkowski spacetime within the framework of
STR, the covariant derivatives of the form turn into the partial derivatives of the form
,
while the result of the action of the partial derivatives does not depend on
the order of their action. As a consequence of the calibration of the 4-potential,
the equality holds: . As a result, the four-potential of the
acceleration field can be found from the wave equation:

This equation can be divided into two equations
– one for the scalar potential and the other for the vector potential of the
acceleration field. In this system under consideration the vector potential is
equal to zero, and the scalar potential of the acceleration field is given by:

where are the time components of the metric tensor,
is the Lorentz factor of the particles in the
reference frame K' associated with the
center of the sphere.

Since the scalar potential of the stationary
system does not depend on time, the wave equation for the scalar potential
turns into the Poisson equation: ^{[2]}

and the following formula is obtained for the
Lorentz factor of the particles: ^{[3]}

where is the Lorentz factor of the particles at the
center of the sphere, is the current radius.

The acceleration field strength and the
corresponding solenoidal vector are expressed by the formulas:

**Pressure field**

The four-potential of the __pressure field__ includes the
scalar potential and the vector potential , and obeys the calibration condition:
.

The pressure field equation with the field
sources, the __pressure field tensor__ and the equation for finding the
four-potential of the pressure field have the form: ^{[1]}

where is the pressure field coefficient.

In STR the latter equation turns into the wave
equation:

In the stationary case the potentials do not
depend on time and the time component of the wave equation turns into the
Poisson equation for the scalar potential of the pressure field:

The solution of this equation inside the sphere
with particles is as follows: ^{[3]}

where is the scalar potential at the center of the
sphere. This potential is approximately equal to: ^{[}^{4]}

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

The strength of the pressure field and the
corresponding solenoidal vector are found as follows:

**Gravitational field**

The
__gravitational
four-potential __ of the gravitational field is made up with
the use of the scalar and vector potentials. The calibration condition of the four-potential
is: .

The gravitational field equation with the field
sources, the __gravitational
tensor__ and the equation for finding the
four-potential of the gravitational field in the __covariant theory of gravitation__ have the
form: ^{[5]} ^{[6]}

where is the gravitational
constant.

In STR the latter equation is simplified and
becomes the wave equation:

From the wave equation in the stationary case,
the Poisson equation follows for the scalar potential inside the sphere with
randomly moving particles in the framework of the __Lorentz-invariant theory of
gravitation__ (LITG):

The right-hand side of this equation contains
the Lorentz factor ,
which depends on the radius according to (1). In addition, the internal scalar
potential near the surface of the sphere must coincide with the scalar
potential of the external field of the system, in view of the standard
potential gauge, that is with equality of the potential to zero at infinity.

As a result, the dependence of the scalar
potential on the current radius differs from the dependence in the classical
case of the uniform sphere with the radius and is equal to it only
approximately: ^{[3]}

For
the __gravitational field strength__
and the __gravitational torsion field__ inside the
sphere we obtain the following: ^{[7]}

The solutions for the external gravitational
field potential and for the field strength according to LITG are as follows:

Here, the auxiliary mass is equal to the product of the mass density by the volume of the sphere: .
From the expressions for the potential and strength of the external gravitational
field we can see that the role of the gravitational mass is played by the
mass Since then the relation is satisfied.

To
understand the difference between these masses we should calculate the total
relativistic mass of the particles moving inside the sphere.
For the motion of particles there should be some voids between them. Both the
average accelerations and the average velocities of the particles inside the
sphere are functions of the current radius. Dividing the particles’ velocities
by their acceleration, we can find the dependence of the average period of the
oscillatory motion of particles on the radius. Finally, multiplying the
velocity by the average period of motion, we can obtain an estimate of the size
of the voids between the particles.

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 mass density in the
reference frames associated with the particles, is the Lorentz factor of the moving
particles, the product gives the mass density of the particles from
the viewpoint of an observer, who is stationary with respect to the sphere, and
the volume element inside the sphere corresponds to the volume
of the particle from the viewpoint of this observer. This leads to the fact
that the total volume of the particles moving inside the sphere becomes
approximately equal to the volume of the sphere. For the mass, in view of the
Lorentz factor (1), the following relation is obtained:

This implies the equality of the gravitational
mass and the total relativistic mass of the particles moving inside the sphere.
The both masses are greater than the mass
.
By the method of its calculation, the mass is equal to the sum of the invariant masses
of the particles that make up the system.

The external torsion field is equal to zero:

**Electromagnetic field**

The electromagnetic four-potential of the electromagnetic field includes the
scalar potential and the vector potential .
The
covariant Lorentz calibration for four-potential is: . For a fixed
uniformly charged spherical body with random motion of charges the total
electromagnetic field on the average is purely electric and the vector
potential is equal to zero.

The electromagnetic field equation with the
field sources, the electromagnetic tensor and the equation for finding the
four-potential are expressed as follows:

where is the electric
constant, is the electromagnetic four-current.

The latter equation in STR turns into the wave
equation:

Due to the absence of time-dependence in the
case under consideration, the wave equation becomes the Poisson equation for
the scalar potential inside the sphere:

where is the charge density in the reference frames
associated with the charges.

The dependence of the scalar potential on the
current radius in the general case differs from the dependence in the classical
case of the potential of a uniformly charged sphere with the radius ,
coinciding with it only in the first approximation: ^{[8]}

The field strength of the electric field and the
magnetic field inside the sphere have the form:

Outside the system under consideration the
charge density is equal to zero and the Poisson equation for the scalar
potential turns into the Laplace equation:

The solution for the external electric field
potential, corresponding to the potential gauge and the Maxwell's equations for
the electric field strength is given by:

The external magnetic field is equal to zero:

In these expressions, the charge is an auxiliary quantity equal to the product
of the charge density by the volume of the sphere: .

In this case, the following quantity serves as
the total charge of the system:

while The charge
is calculated in the same way as the mass and has the meaning of the sum of the charges
of all the system’s particles.

**Tensor field invariants**

The knowledge of the field strengths and the
solenoidal components of fields allows us to find the tensor components of the
corresponding fields with the covariant indices. To pass on to the field
tensors with the contravariant indices we need to know the metric tensor. In
STR the metric tensor does not depend on the coordinates and time, is uniquely
defined, and in Cartesian coordinates consists of zeros and unities. As a
result, it is easy to find the tensor field invariants , , and ,
where , , and are the __acceleration tensor__,
the __pressure field tensor__,
the __gravitational tensor__ and the electromagnetic
tensor, respectively.

The tensor field invariants are included in the
Lagrangian, the Hamiltonian. the action function and the relativistic energy of
the system, and they are located there inside the integrals over the space
volume. In addition, they are included in the corresponding stress-energy
tensors of the fields. ^{[2]}
Since in the system under consideration the solenoidal vectors are zero,
the tensor invariants depend only on the field strengths:

The volume integrals of the tensor invariants
multiplied by the corresponding factors were calculated in the article. ^{[7]} For the acceleration field and the pressure
field the integrals are taken only over the volume of the sphere:

The gravitational and electromagnetic fields of
the system are present not only inside but also outside the sphere, where they
extend to infinity, while the field strengths of the internal and external
fields behave differently. The field strengths
and are substituted respectively into the
integrals of the tensor invariants of these fields taken over the volume of the
sphere, which gives the following:

Into the volume integrals of the tensor
invariants of the gravitational and electromagnetic fields of the system
outside the sphere the field strengths and are substituted, respectively:

**Energies of particles in the field potentials**

All the four fields act on the particles inside
the sphere, and therefore each particle of the system acquires the
corresponding energy in a particular field. The energy of the particle in the
field is calculated as the volume integral of the product of the effective mass
density by the corresponding scalar potential, and
for the electric field the energy is determined as the volume integral of the
product of the effective charge density by the scalar potential , where the
Lorentz factor from (1) is used. In STR the energies of
particles in the acceleration field, pressure field, gravitational and electric
fields in the uniform relativistic spherical system, in view of the expressions
for the field potentials ^{[7]} and the corrections to calculations, ^{[8]
[9]} ^{[10] [11]} are, respectively:

It
should be noted that all the fields, which contain particles, are not the fields
arising from the external sources, but are generated by the particles
themselves. As a result, the particles’ energies calculated above in the scalar
potentials of the fields are twice as large as the potential energy of one or
another interaction. For example, in order to calculate the electrostatic
energy of the system of two charges, it is sufficient to take the potential of
the first charge at the location of the second charge and to multiply it by the
value of the second charge. But if we use the formula for the energy in the form of an
integral, then the electrostatic energy will be taken into account twice,
because the term is added, which contains the potential of the second charge at
the location of the first charge multiplied by the value of the first charge. On the
other hand, the electrostatic energy must consist of two components that take
into account both the energy of particles in each other’s fields and the energy
of the electric field itself.

Instead, in electrostatics, the
electrostatic energy is calculated either through the scalar potential or
through the field strength by integrating the time component of the
stress-energy tensor over the volume. Both methods provide the same result, but
the connection between the field energy and the energy of particles in the
potential is lost in this case, and it is not clear why these energies should
coincide.

**Relation between the field coefficients**

For the four fields under consideration the
equation of motion of matter in the concept of the general field is as follows:
^{[12]}

where is the mass four-current, is the electromagnetic four-current.

The components of the field tensors are the
field strengths and the corresponding solenoidal vectors, but in the physical
system under consideration the latter are equal to zero. As a result, the space
component of the equation of motion is reduced to the relation:

If we substitute here the expression for the
field strengths inside the sphere, we obtain the relation between the field
coefficients: ^{[13]}

The same is obtained for the time component of
the equation of motion, which leads to the generalized Poynting theorem. ^{[9]}

**Relation between the energies of the internal and external
fields**

In article ^{[14]} it was found that the
energy of particles in the gravitational field inside the stationary sphere is
up to a sign two times greater than the total energy associated with the tensor
invariants of the gravitational field inside and outside the body. A similar
situation takes place in the system under consideration with the random motion
of particles and zero solenoidal vectors both for the gravitational ^{[10]} and electromagnetic fields. ^{[8]} In particular, we can write the following:

These expressions relate the energy of particles
in the scalar field potentials with the energy found with the help of the field
strengths.

**Relativistic energy**

In the curved spacetime the system’s energy for
the continuously distributed matter is given by the formula: ^{[2]} ^{[}^{4}^{]}

This formula is valid in the case when we can assume that the potentials
and strengths of the fields at each point of space do not depend on the
velocities of the motion of the individual particles of the system.

In the STR the metric tensor determinant is ,
the time component of the four-velocity is
,
and in order to calculate the energy of the spherical system with particles, taking
into account the fields’ energies, we can use the above-mentioned energies of
particles in the field potentials and the energies in the form of the tensor
invariants of the fields:

The expression for the energy is simplified if we
use the relation between the field coefficients (2):

Taking into account the relations between the
energies of the internal and external fields also simplifies the expression for
the system’s energy:

**Relation between the energy and the cosmological constant**

In the approach under consideration, the
relativistic energy of the system is not an absolute value and requires
gauging. For this purpose the cosmological constant is used. The gauge condition for the four
main fields is related to the sum of the products of the fields’
four-potentials by the corresponding four-currents and has the following form: ^{[2]}
^{[4}^{]}

where for large cosmic systems , and is the constant of the order of unity.

Within the framework of the STR the gauge
condition has the following form:

If we divide the system’s particles and remove
them to infinity and leave there at rest, the terms with the products of the
vector field potentials by the velocity of particles would vanish, and the Lorentz factor of an
arbitrary particle would be .
On the right-hand side we will have only the sum of the terms specifying the
energy density of the particles located in the potentials of their proper
fields. Since , we see that the cosmological constant for
each system’s particle is up to the multiplier equal to the rest energy density of this
particle with a certain addition from its proper fields. Then the integral over
the volume of all the particles gives a certain energy:

where the gauge mass is related to the gauge condition of the
energy.

In the process of gravitational clustering the particles
that were initially far from each other are united into closely bound systems,
in which the field potentials increase manyfold. In the system under
consideration , the solenoidal vectors of the fields are
considered equal to zero due to the random motion of the particles, which gives
the following:

The expression on the right-hand side is part of
the relativistic energy of the system, so that the energy can be written as
follows:

The mass is related to the relativistic energy of the
generally stationary system and is the inertial mass of the system. In view of
(2), the energy will be equal to:

This shows that the relativistic energy of this
system is equal to the gauge mass-energy
,
from which the gravitational and electromagnetic energy of the fields outside
the system should be subtracted.

**Lagrange function and motion integrals**

The Lagrange function for a system of particles and
four main vector fields has the following form:^{[1]} ^{[2]}

Here is the scalar curvature. With the help of
such Lagrange function, one can calculate the generalized momentum of the
system:^{[15]}

This vector depends on the vector potentials of
all four fields and is preserved in a closed physical system, that is, it is an
integral of motion. Another integral of motion is the relativistic energy of
the system , which is found by formula (5). Further, it
is assumed that one can neglect the contributions from the gravitational and
electromagnetic fields outside the matter and take into account only the
generalized momentum. Then we can assume that these values form a four-momentum
of the system, written with a covariant index:

The angular momentum of the system is also an
integral of motion:

The antisymmetric angular momentum pseudotensor
is determined through the four-radius , taken with a covariant index, and through
the four-momentum :

The spatial components of the angular momentum
pseudotensor are the components of the angular momentum of the system:

The radius-vector of the center of momentum of a
physical system is determined by the formula:

The time components of the pseudotensor are the components of the three-dimensional
vector , which is often called the time-varying
dynamic mass moment:

If we take into account definition of the
radius-vector of the center of momentum and the relationship between the
momentum and the velocity of the center of the momentum in the form , we get the relationship:

In a closed system the pseudotensor must be conserved, and its components must be
some constants. For the space components of the pseudotensor this results in
conservation of the angular momentum: . From the equality of the pseudotensor’s
time components and the components of the vector it follows that it should be .
Given the expression for , it can be written as , where the constant vector specifies the position of the system’s center
of momentum at .
Thus, in this reference frame we obtain the equation of motion of the center of
momentum at the constant velocity ,
as a property of the motion of a closed system.

**Integral vector**

The equation used to find the metric tensor
components in the __covariant
theory of gravitation__ for the tensors with mixed indices has the
following form:^{[2]}

here is the Ricci tensor with mixed indices; is the unit tensor or the Kronecker delta; , , and are the stress-energy tensors of the acceleration
field and pressure field, gravitational and electromagnetic fields,
respectively.

With the help of the covariant derivative we can find the four-divergence of both sides
of the above equation for the metric. The divergence of the left-hand side is
zero due to equality to zero of the divergence of the Einstein tensor, , and also as a consequence of the fact that
outside the body the scalar curvature vanishes, ,
and inside the body it is constant. The latter follows from the gauge condition
of the energy of the closed system. The divergence of the right-hand side of
equation for the metric is also zero:

where the tensor
with mixed indices represents the sum of the
stress-energy tensors of all the fields acting in the system.

The resulting expression for the tensors’ space
components is nothing but the differential equation of the matter’s motion
under the action of forces generated by the fields, which is written in a
covariant form. As for the tensors’ time components, for them the expression is
expression of the generalized Poynting theorem for all the fields. ^{[9]}

In a weak field and at low velocities of motion
of the particles, the equation can be integrated over the four-volume,
taking into account the divergence theorem. As a result, at the initial moment
of time for the system under consideration, the following relation will be
valid:

In a closed system, the four-dimensional
integral vector must be constant. ^{[15]} For a
stationary sphere with randomly moving particles in the continuous medium approximation,
the energy fluxes of the fields defining the components , where
, are missing , so that the spatial components
are zero, . As for the time component of the integral vector, then for the volume
occupied by the matter inside the sphere, it also vanishes due to relation (4)
for the field coefficients. However, outside the sphere, where there are only
gravitational and electromagnetic fields, the time component of the integral
vector is not equal to zero. As a result, the contribution to this component is
made by the energies of the external fields:

It follows from the above that the integral
vector shows the distribution of energy and energy fluxes in the system under
consideration. For the nonzero space components
of the integral vector to appear some stationary
motion of the matter and fields is required, for example, general rotation,
volume pulsations or mixing of matter. In this case, solenoidal vectors and the
fields’ energy fluxes appear in the system.

Since the integral vector is associated with the energies and energy
fluxes of the fields in the energy-momentum tensors, it differs from the
four-momentum , which includes the invariant mass and
proportional to its rest energy. It turns out that the difference between and is due to the fundamental difference between
particles and fields, they cannot be reduced to each other, although they are
interrelated with each other.

**Virial theorem and the kinetic energy of particles**

In article ^{[16]} the kinetic energy of
the particles of the system under consideration is estimated by three methods: from
the virial theorem, from the relativistic definition of energy and using the
generalized momenta and the proper fields of the particles. In the limit of low
velocities, all these methods give for the kinetic energy the following:

The possibility to use the generalized momenta
to calculate the energy of the particles’ motion is associated with the fact
that despite zeroing of the vector potentials and the solenoidal vectors on the
large scale, in the volume of each randomly moving particle these potentials
and vectors are not equal to zero. As a result, the energy of motion of the
system’s particles can be found as the half-sum of the scalar products of the
vector field potentials by the particles’ momentum, while for the
electromagnetic field we should take not the momentum, but the product of the
charge by the velocity and the Lorentz factor.

If we square the equation for in (1), we can obtain the dependence of the
squared velocity of the particles’ random motion on the current radius:

On the other hand, we can assume that where denotes the averaged velocity component
directed along the radius, and is the averaged velocity component perpendicular
to the current radius. In addition, from statistical considerations, it follows
that

This implies the dependence of the radial velocity
on the radius:

Next, from the virial theorem we find the
squared velocity of the particles at the center of the sphere:

This makes it possible to estimate the Lorentz
factor in the center:

In the ordinary interpretation of the virial
theorem the time-averaged kinetic energy of the system of particles must be two
times less than the averaged energy associated with the forces holding the particles at the radius-vectors :

However, in the relativistic uniform system this
equation is changed:

while the quantity exceeds the kinetic energy of particles, , and it becomes equal to it only in the limit
of low velocities.

In contrast to the classical case, the total
time derivative of the virial in the stationary system is other than zero due
to the virial’s dependence on the radius:

An analysis of the integral theorem of
generalized virial makes it possible to find, on the basis of field theory, a
formula for the root-mean-square speed of typical particles of a system without
using the notion of temperature: ^{[17]}

**Extreme objects**

In formula (2) for the gravitational field strength
outside
the body there is a quantity , where . As was shown in article, ^{[11]} at the
value radians
the gravitational field strength vanishes and
the gravitational acceleration disappears. Therefore, in real physical objects
the following condition must hold: . If the angle is
increased, then the quantity would
first increase, and then would begin to decrease and even change its sign. So,
at we will
have , at we will have , at we will
have .

Let us now consider the observable Universe, which on
a scale 100 Mpc or more can be considered as a
relativistic uniform system. The total mass-energy density of the Universe is
close to the critical value kg/m^{3} and
the size of the Universe can be estimated as the Hubble length m, where
is the
Hubble parameter.

Using the approximate equality according
to, ^{[13]} we find the value radians. Since the
angle is sufficiently large,
then for modeling of the gravitational field of the Universe it is necessary to
use refined formulas with sines and cosines. For example, if we take the size
of the observable Universe equal to , then we will have , and the
gravitational field at the boundaries of the Universe will tend to zero. This
is what we observe in the form of a large-scale cellular structure consisting
of clusters of galaxies. The reason for the gravitation action weakening is
assumed to be graviton scattering by the particles of the space medium. ^{[18]}