**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 Virial
theorem and the kinetic energy of particles
- 10 References
- 11 See also
- 12 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 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
equation for finding the four-potential of the acceleration field:

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 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 result, the equality holds: , if we also take into account the continuity equation of the mass
four-current in the form of , this equality is valid in
STR when . 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 .

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:

where is the scalar potential at the center of the
sphere.

The field
strength of the pressure field and the corresponding solenoid 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
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: ^{[4]} ^{[5]}

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:

For the __gravitational field
strength__ and the __gravitational torsion field__ inside the sphere we obtain
the following: ^{[6]}

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.
The mass element inside the sphere is given by the 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. The total volume of the
particles at rest is greater than the volume of the sphere, but as a result of
motion the volume of each particle decreases due to the effect of the length
contraction in STR. This leads to the fact that the total volume of the
particles moving inside the sphere becomes 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 . 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 vacuum permittivity, 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:

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. ^{[6]}
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 ^{[6]} and the corrections to calculations, ^{[7]}
^{[8]} ^{[9]} are, respectively:

**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: ^{[10]}

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: ^{[11]}

The same is
obtained for the time component of the equation of motion, which leads to the
generalized Poynting theorem. ^{[7]}

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

In article ^{[12]}
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 ^{[8]}
and electromagnetic fields. ^{[9]} 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]}

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]}

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. Despite this, the mass as a consequence of the energy gauging
remains unchanged. 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.

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

In article ^{[13]}
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:

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:

**References**

^{1.0}^{1.1}^{1.2}Fedosin S.G. The procedure of finding the stress-energy tensor and vector field equations of any form. Advanced Studies in Theoretical Physics, Vol. 8, 2014, no. 18, 771-779. http://dx.doi.org/10.12988/astp.2014.47101.^{2.0}^{2.1}^{2.2}^{2.3}Fedosin S.G. About the cosmological constant, acceleration field, pressure field and energy. Jordan Journal of Physics. Vol. 9 (No. 1), pp. 1-30, (2016).- Fedosin S.G. The
Integral Energy-Momentum 4-Vector and Analysis of 4/3 Problem Based on the
Pressure Field and Acceleration Field. American Journal of
Modern Physics. Vol. 3, No. 4, 2014, pp. 152‒167. http://dx.doi.org/10.11648/j.ajmp.20140304.12
.
- 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).
- Fedosin S.G. The
Principle of Least Action in Covariant Theory of Gravitation.
Hadronic Journal, February 2012, Vol. 35, No. 1, P. 35-70.
^{ }^{6.0}^{6.1}Fedosin S.G. Relativistic Energy and Mass in the Weak Field Limit. Jordan Journal of Physics. Vol. 8 (No. 1), pp. 1‒16, (2015).^{6.2}^{7.0}^{7.1}Fedosin S.G. The generalized Poynting theorem for the general field and solution of the 4/3 problem. Preprint, February 2016.^{ }^{8.0}^{8.1}Fedosin S.G. The gravitational field in the relativistic uniform model within the framework of the covariant theory of gravitation. 5th Ulyanovsk International School-Seminar “Problems of Theoretical and Observational Cosmology” (UISS 2016), Ulyanovsk, Russia, September 19-30, 2016, Abstracts, p. 23, ISBN 978-5-86045-872-7.^{ }^{9.0}^{9.1}Fedosin S.G. The electromagnetic field in the relativistic uniform model. Preprint, May 2016.- Fedosin S.G. The
Concept of the General Force Vector Field. OALib
Journal, Vol. 3, P. 1‒15 (2016), e2459. http://dx.doi.org/10.4236/oalib.1102459.
- Fedosin S.G. Estimation
of the physical parameters of planets and stars in the gravitational
equilibrium model. Canadian Journal of Physics, Vol. 94, No. 4,
P. 370‒379 (2016). http://dx.doi.org/10.1139/cjp-2015-0593.
- Fedosin S.G. The
Hamiltonian in Covariant Theory of Gravitation. Advances in
Natural Science, 2012, Vol. 5, No. 4, P. 55-75. http://dx.doi.org/10.3968%2Fj.ans.1715787020120504.2023.
- Fedosin S.G. The virial theorem and the kinetic energy of particles of
a macroscopic system in the general field concept. Continuum Mechanics and Thermodynamics, Vol. 29, Issue 2, pp. 361-371 (2016). https://dx.doi.org/10.1007/s00161-016-0536-8.

**See****
also**

- Invariant energy
- General field
__Acceleration field____Pressure field__- Gravitational field
- Electromagnetic field
__Covariant theory of gravitation__- Energy

**External**** links**

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