Continuum Mechanics and Thermodynamics, Vol. 33, Issue 3, pp. 817834 (2021).
https://rdcu.be/ccV9o.
https://doi.org/10.1007/s00161020009607
The
potentials of the acceleration field and pressure field in rotating
relativistic uniform system
Sergey G. Fedosin
PO box 614088, Sviazeva str. 2279, Perm, Perm Krai, Russia
Email: fedosin@hotmail.com
Abstract: The scalar and vector potentials of
the acceleration field and the pressure field are calculated for the first time
for a rotating relativistic uniform system, and the dependence of the
potentials on the angular velocity is found. These
potentials are compared with the potentials for the nonrotating uniform system
that have been found previously. The rotation leads to the appearance of vector
potentials, which at each point turn out to be directed along the corresponding
linear velocity of rotation. The calculation shows that for rotating stellar
objects the contribution to the fields’ vector potentials from the proper
random motion of particles is small compared to the contribution from rotation and may not be
taken into account. From the expression for the pressure field potential a relativistic
formula follows that relates the pressure, mass density, and mean square
velocity of the particles. This formula in the limit of low speeds corresponds
to the expression for the pressure in molecular kinetic theory. When calculating the
potentials, a new method is used that takes into account the potentials of two
different bodies, a cylinder and a sphere, for solving the wave equation of
rotating system.
Keywords: acceleration field; pressure field; scalar
potential; vector potential; rotation; uniform system.
The derivation of formulas for the pressure field in a
moving matter is not a trivial problem and in the general case requires the solution
of differential equations. One of the methods is to solve the NavierStokes
equation for a given velocity field and known boundary conditions [1]. The
expression for the pressure in [2] was obtained by generalizing the Bernoulli
equation, in [34] the pressure field was found numerically using equations
containing pressure and a velocity field. In this case, the pressure field in
the matter is considered simplified as a scalar field depending on time and
coordinates. In [5], the pressure field, taking into account electromagnetic
phenomena, is considered within the framework of hydrodynamics and
thermodynamics, using the approaches of Euler and Lagrange when describing the
motion of matter. There are also known works on determining the potentials of
the electromagnetic field in rotating bodies [611].
The aim of this article is to study a rotating
relativistic uniform system using the methods of analytical mechanics in the
approximation of continuously distributed matter.
The relativistic uniform system is one of the models
of modern physics and finds its application in various areas [1216]. This
model describes various gravitationally coupled systems based on field theory
and therefore turns out to be much more accurate compared to the model of a uniform system of classical mechanics. The relativistic
model generalizes the classical uniform system model and uses fourdimensional
formalism instead of threedimensional vectors. Unlike the phenomenological thermodynamic approach, the pressure field
is treated not as a scalar, but as a vector field with its own fourpotential,
pressure tensor and energymomentum tensor.
The field equations allow using the principle of least
action to derive the relations arising between the forces acting in a matter.
The smaller the density gradients of the matter inside the system, the closer
in its properties this system becomes to a relativistic uniform system. As a
result, the theoretical results obtained can be applied, for example, to the
dynamics of cosmic gas clouds and to the Metagalaxy as a whole. It may be interesting to study the motion of magnetized matter in the
Earth’s core in connection with the problem of the origin of magnetic field and
change of magnetic poles.
In [1721] the potentials and strengths of various
vector fields in a uniform system at rest were calculated. In such a system,
the global vector field potentials are equal to zero. The next step is
calculating the potentials for a rotating relativistic uniform system. The need
for this step follows from the fact that only taking into account the vector
potentials that appear during rotation allows us to correctly calculate such
physical quantities as angular momentum and relativistic energy of system. It
is well known that in the presence of gravitational and electromagnetic fields,
instead of the usual momentum and angular momentum, generalized momenta are
used, expressed through the vector potentials of the fields. These quantities
can be used to estimate the corresponding quantities for the observed bodies,
such as nucleons and neutron stars, since these objects are quite close in
their properties to the rotating relativistic uniform system.
If we talk about the potentials of the acceleration field
and the pressure field, then using these potentials it becomes possible to
calculate the dependence of mass density on coordinates, the distribution of
velocities of typical particles, pressure and temperature in the system under
consideration. Previously, similar calculations were made without taking into
account the rotation of a relativistic uniform system and were used to estimate
the parameters of stars and planets [13], to calculate the kinetic energy of
the system [14], to determine the dependence of the root mean square velocity
on coordinates [16].
Further, we will assume that the physical system has a
spherical shape and is so densely filled with particles that we can use the
continuous medium approximation and calculate various quantities using
integrals over the system’s volume. The main fields are usually considered the
gravitational and electromagnetic fields, acceleration field and pressure
field, created by the particles, and these fields maintain both the system’s
integrity and the state of its constant rotation.
In this article, in order to reduce its volume, we
will focus only on finding the potentials of the acceleration field and
pressure field. The peculiarity of these fields has one more reason – usually
they act locally, in the sense that they vanish outside the matter. The
locality of action of the acceleration field and pressure field allows us to
significantly simplify the solution of the equations for the potentials of
these fields.
In contrast, the gravitational and electromagnetic
fields can also exist outside the matter. Apparently, this is due to the
fundamental difference between these fields. There is a theoretical approach,
according to which the gravitons and charged particles of the
electrogravitational vacuum act inside and outside the bodies, leading to the
corresponding forces in the matter of these bodies [2223]. In order to
describe this action, the concepts of the gravitational and electromagnetic
fields are introduced. The pressure field takes into account the forces that
appear during mechanical collisions of the particles with each other. In order
to describe the force of deceleration of the motion of the adjacent directed
and interacting particles’ fluxes, the dissipation field is introduced [12].
If we consider all these fields as vector fields, then
the acceleration field that defines the particles’ acceleration must also be
assumed to be a vector field. Indeed, in the equation of the particles’ motion
the acceleration is associated with the acting forces, which are determined
through field tensors [24]. In addition, taking into account the acceleration
field allows one to obtain in covariant form not only acceleration, but also
energy, including the rest energy of the system [15]. This becomes possible due to the fact that the
acceleration field as an independent field is introduced into the Lagrange
function. The subsequent application of the principle of least action makes it
possible to derive the equations of the acceleration field and obtain the
equation of motion [19]. In the simplest case, the acceleration field sets the inertial force in
the form of the product of mass and acceleration, while other fields set the
forces acting on a given mass. Then the connection between the force of inertia
and the sum of forces from other fields is expressed by the wellknown second
Newton's law. In particular, the acceleration field is used to accurately
reproduce the NavierStokes equation [12].
2. The
acceleration field
The equations for the acceleration field resemble in
their form Maxwell equations and are presented in [19]. Within the framework of the
special theory of relativity they look as follows:
,
,
,
,
(1)
where is the acceleration field
coefficient, is the Lorentz factor for the
particles moving inside the sphere, is the particles’ velocity in the
reference frame associated with the center of the
sphere, is the mass density of an arbitrary
particle in the comoving reference frame, is the speed of light, and are the strength and the solenoidal
vector of the acceleration field, respectively.
The fourpotential is defined with the help of the
scalar potential and the vector potential of the acceleration field in such
a way that the following relations hold true:
, . (2)
Substitution of (2) into (1) leads to the wave
equations for the potentials:
According to (12), the
acceleration field is described by two potentials and , and two vectors and . The
vector in
its properties is similar to the magnetic field arising due to the movement of
charged particles and due to the delay in the propagation of electromagnetic
influence. It is known that for a stationary particle both its vector potential
and its magnetic field are equal to zero. The same is true for the solenoidal
vector reflecting
the relativistic nature of the acceleration field.
If we consider the
average quantities, then in the stationary system the particles’ velocity , the vector of the mass current density , as well as the potentials and do
not depend on the time, but are functions of the space coordinates. As a
result, only the Laplacians remain in the wave equations:
, . (3)
In order to simplify, we will assume that the matter
inside the sphere rotates with the same constant angular
velocity relative to the axis that passes
through the center of the sphere. Using the rule of relativistic addition of
velocities, for the absolute velocity and the Lorentz factor of an arbitrary
particle in the reference frame , we can write the following:
, , (4)
where is the velocity of the random motion
of the particle in the reference frame that rotates with the matter
at the angular velocity ; is the linear velocity of motion of the reference frame at the location of the particle,
resulting from rotation in the reference frame ; is
the Lorentz factor
for the velocity , is
the Lorentz factor
for the velocity .
Let us average expressions (4) over the volume in a
small neighborhood around the point under consideration, so that a sufficient
number of particles would be present in this volume. The velocity of random
motion is presented in (4) in such a way
that, after averaging, the contributions from the terms containing become equal to zero. This happens because the velocity of different particles has a
different direction, including the opposite. Then it follows from (4) for the
average quantities: , .
For the spherical system with the particles in the
absence of the matter’s general rotation, the Lorentz factor depends on the
current radius and on the value of the Lorentz
factor at the center of the sphere [17]:
.
This expression was obtained under the assumption that
the matter’s particles do not have their proper rotation and move inertially
and randomly. We will further assume that in the reference frame , rotating together with the matter at the angular
velocity , the Lorentz factor is determined by a similar formula
. (5)
with the exception that the Lorentz factor is used at the center of the rotating sphere,
which may differ in magnitude from for the sphere at rest.
This means that if in the reference frame there is a rotating relativistic
uniform system, then when turning to the comoving rotating reference frame (5) would hold true and the general
rotation would not change the dependence of the Lorentz factor on the current radius.
In its meaning, such a situation corresponds to the
principle of relativity, when in an inertial reference frame a simultaneous
increase of the velocity of all the system’s particles by the same value does
not change the state of the system. The only difference is that the rotation
changes the particles’ velocities differently depending on the current radius,
which makes the reference frame noninertial. Therefore, the assumption of
application of (5) for the rotating reference frame can be considered as the first
approximation to reality.
In spherical coordinates the amplitude of the linear
rotation velocity is expressed by the formula: , where the angle is measured from the axis , and the matter’s particles rotate in the planes parallel to the plane . In cylindrical coordinates the velocity is given by the
expression .
Consequently, the Lorentz factor can be written as
follows:
where is the distance from the axis of
rotation to the point under consideration.
Instead of the real matter’s particles, it is
convenient to assume that the matter consists of typical particles, which
characterize on the average all the system’s properties. After averaging of all the quantities over the volume, it turns out that
the equations for typical particles relate the averaged physical quantities
with each other.
Substitution of instead of and instead of in (3), taking into account (5) and
(6) gives the following:
. (7)
. (8)
In spherical coordinates for the arbitrary function the Laplacian of this function
for the case has the form:
. (9)
For the sphere at rest at , equation (3) for the scalar potential and the solution of this
equation have the form:
The potential in (10) depends only on the radius
. Substitution of instead of in (9) allows us to calculate the Laplacian and compare the result with the
first expression in (10).
Let us solve an auxiliary problem and find the scalar
potential of the acceleration field for the
case of purely rotational motion of uniform system in the form of an infinite long cylinder in the absence of random motion of
the particles. From (3) and (6) it follows:
. (11)
The Laplacian of the arbitrary function in cylindrical coordinates has
the following form:
. (12)
After replacing with in (12), taking into account
(11), we arrive at the equation:
. (13)
In (13) the potential does not depend on the angle due to symmetry with respect to the axis of
rotation. It is also assumed that also does not depend on the coordinate inside the cylinder. This corresponds to the
fact that is some function of the Lorentz factor (6) of
the particles of the rotating matter, which depends only on the coordinate and the angular velocity of rotation . It should be noted that when a
free particle or particles of matter of a relativistic uniform system move, the
acceleration field potential is simply proportional to the corresponding
Lorentz factor. The fact that depends only on is the simplest choice for the dependence of
the potential of the rotating cylinder. We will consider as an auxiliary potential for determining the
total potential of the rotating spherical uniform relativistic system.
The solution of equation (13) is as follows:
The constants and must be chosen in such a way that
must not depend either on nor on , which extend to infinity at . This gives the following:
The question of choosing the constant will be solved in (20), after
solving equation (7) for the rotating spherical uniform relativistic system.
In the case of rectilinear motion of a solid material
point without its proper rotation, its scalar potential is equal to , where is the Lorentz factor of the
particle. As for the vector potential of the acceleration field for such a
solid point, it is equal to , where is the particle’s velocity [25].
In this
connection, we will multiply by and will consider the quantity as the first approximation to the sought value
of the scalar potential . Taking into account that and , we will express in (5) and in (6) in terms of the coordinates , and will calculate the Laplacian in Cartesian coordinates:
(15)
The result of calculating on the righthand side of (15) is expressed in
terms of the radius and the cylindrical coordinate . Using the expansion of functions
of the form , , we will expand the cosine and sine
in the last two terms on the righthand side of (15). In this case, the main
expansion terms will be cancelled out, and there will remain a term containing
the square of the speed of light in the
denominator:
We transform the second term on the righthand side of (15), assuming
that :
Then, as the first approximation, we can write:
Now we will find such a function that its Laplacian would be equal to the sum of the last three terms on the righthand side of (16):
Taking into account the form of the Laplacian (9) in spherical
coordinates, and, accordingly, the form of the Laplacian (12) in cylindrical
coordinates, we find a solution that does not contain infinities at and at :
Subtracting term by term the equality for from equality (16), we obtain:
.
From comparison of this equality with (7), taking into account and equalities (5), (6), (17), we obtain an approximate expression for the
scalar potential of the acceleration field:
We will consider the inner region inside the sphere at
small and , that is, the region near the axis
of rotation and near the center of the sphere. In this case, the sine in (18)
can be replaced by its argument:
. (19)
Using the freedom to calibrate the scalar potential, let us equate the
potential of the rotating cylinder near the axis of the cylinder and the
potential near the center of the rotating sphere. For this purpose, let us send
the cylindrical coordinate in equalities
(14) and in (19) approach zero and equate these equalities. This allows us to estimate
the constant and specify the form of the scalar potential in (14):
.
.
Hence it follows that on the axis of rotation, where , the potential is . If we put the constant , then the potential on the axis of
rotation of the cylinder will not depend on the angular velocity of rotation . In this case, we find:
. (20)
At small , the potential in (20) ceases to
depend on and becomes approximately equal to the value , and there is a Lorentz factor at the center of the sphere and at the center
of the corresponding cylinder. To check this formula, it is sufficient to solve
equation (13) at for the case of a stationary cylinder. In this
case, the solution is:
.
In the solution, it is necessary to set in order to avoid infinity at , and take into account that at the potential must be equal to . Therefore, there is , and we come to that .
,
where is a chemical potential in
the absence of rotation, depending on pressure and temperature ; is the mass of a molecule; in the general case is the
potential energy of a molecule in a certain field, this energy for the case of
a field of rotation was equated to the value of kinetic energy of the molecule taken with a
negative sign.
As the coordinate grows and moves away from the
rotation axis, the chemical potential decreases, and the same follows for
dependence of the potential in (20). In this case, the
potential can be considered an analogue for
, and the first term on the righthand side of (20) an analogue for .
Substitution in (18) gives:
(21)
If the angular velocity of rotation vanishes in (21), the potential becomes equal to the potential in (10).
3. The vector potential of
the acceleration field
In the relativistic uniform system, which is generally
at rest, the vector potentials of the fields, including the acceleration field,
are equal to zero due to the particles’ random motion. This is due to the
superposition principle, when the vector potentials of individual particles are
summed in a vector way.
For the case of rotation of the cylinder without taking
into account the proper motion of the matter’s particles from (3) and (6) in
cylindrical coordinates, we obtain the following equation for the vector
potential :
From the relationship between Cartesian and cylindrical coordinates in
the form , , , it follows that in the absence of
motion of the matter’s particles in the direction of the axis the components of the linear velocity are
determined as follows:
In (23) the angular velocity of rotation is present.
We will project equation (22) on the axis taking into account (23) and
expression (12) for the Laplacian in cylindrical coordinates:
. (24)
We will substitute into (24) the expression , where the function depends only on . This allows us to get rid of :
.
Here , . The differential equation for calculating can be transformed by multiplying
by and then integrating the righthand
side:
.
.
The quantity can also be integrated by parts:
.
Comparing the integrals for in the last two expressions, we
find:
This equation has the following general solution:
Now we can find the first component of the vector
potential:
. (25)
In (25) it is still necessary to determine more
precisely the values of the constant coefficients and . By analogy with gaugefixing of the scalar potential in (14), we will assume that if tends to zero, then the potential must not extend to infinity. For this purpose, it suffices to choose the value of the constant . This gives:
. (26)
At low angular velocities the vector potential must be proportional to the linear velocity and the Lorentz factor , as is the case for the motion of the solid point: . For the component , on condition that , in view of (23), (26) and the expression in cylindrical coordinates , according to (6), this can be
written as follows:
.
Hence it follows that , and for projection of the vector potential on the axis we obtain the following:
Repeating all the steps, starting from (24), and projecting equation
(22) on the axes and , we find the remaining components
of the vector potential, respectively:
, .
(28)
From comparison of (23), (27), and (28) it follows that for the vector
potential the following formula holds true:
According to (29), the vector potential is directed along the linear
velocity of rotation of the matter of the cylinder. Near
the axis of rotation, where , and also at small values of the angular velocity , the potential in its value tends to the
velocity of rotation .
In order to take into account
the contribution to the vector potential from the particles’ proper motion
within the framework of the spheric relativistic uniform system, we will
compare equation (8) for the potential and equation (22) for the
potential . The righthand side of (8) turns out to be times greater than the righthand
side of (22), where is the Lorentz factor (5). In the
first approximation, it should be assumed that the potential will be times greater than the potential . Since is a function of the current
radius , we will determine the average value as the average over the sphere’s
volume , using in this case the volume element :
In most cases , and then , where is the product of the mass density by the sphere’s volume , denotes the Lorentz factor of the
particles at the center of the sphere, and is the radius of the sphere.
Multiplying in (29) by (30), we obtain the estimate of the
vector potential of the acceleration field for the
rotating relativistic uniform system:
. (31)
For example, consider
a neutron star as a relativistic uniform system [21]. Such the star with the
mass of 1.35 solar masses and the radius of km has the
average mass density kg/m^{3},
and the Lorentz factor at the center can reach the value . For the star we need to take into
account the relation , where is the
gravitational constant, and the angle radians. Based on these data, the averaged Lorentz factor for the proper
motion of particles in (30) turns out to be , that is less than , as it could be expected for the
volumeaveraged value. This calculation shows that the contribution of to the vector potential (31) for stars is
rather small.
4. The pressure field
The equations for the strengths and potentials of the
vector pressure field within the framework of the special theory of relativity
have the following form [17], [19]:
, , , . (32)
, . (33)
, , . (34)
Here and are the strength and the
solenoidal vector of the pressure field, respectively, is the pressure field coefficient, is the fourpotential of the
pressure field, and are the scalar and vector
potentials, . Wave equations (33) for the potentials are obtained from equations
(32), taking into account (34).
If the system of particles has a spherical shape and
rotates at the constant angular velocity , the potentials will not depend on the time. Then from (33) it follows:
, . (35)
We suppose that in the reference frame rotating together with the matter
at the angular velocity , the Lorentz factor has the same form as in (5), corresponding to the
system of particles that are moving only randomly in a generally nonrotating
system. Substituting instead of into (35), we find the scalar potential of the pressure
field for a generally nonrotating system [17]:
.
(36)
If we do not take into account the particles’ proper
motion at the velocity and the Lorentz factor , and take into account only the particles’ motion due to rotation, in
(35) we should substitute the Lorentz factor (6) instead of . This gives the equation for the scalar potential of rotating cylinder
in cylindrical coordinates, similar to (11):
. (37)
Consequently, the solution of equation (37) will be
similar to (14):
For comparison, in [25] for the case of rectilinear motion
at the constant velocity of a solid particle without proper rotation, the
fourpotential of the pressure field was determined as follows:
where and denote the invariant pressure and
mass density in the particle’s reference frame , the dimensionless ratio is proportional to the particle’s
pressure energy per unit of the particle’s mass, is the particle’s fourvelocity with
a covariant index. In this case, the scalar potential in (39) equals , and the vector potential equals .
Before determining the constant in (38), we will find the
solution of equation (35) for a rotating spherical system taking into account
the particles’ proper random motion. This means that in (35), we must use the
expression instead of :
An approximate solution of equation (40) is similar to the solution of
equation (7), and it can be written similarly to (18):
The constant is present in (41), since the potential in
(39) is determined with an accuracy up to a constant. For small , we can assume that in (41) . This gives the following:
. (42)
We can equate expressions (38) and (42) to each other, if in these
expressions we direct the cylindrical coordinate to zero. Thus, we will carry out
one of the possible options for calibrating the potential of a rotating
cylinder. This allows expressing and refining the form (38):
.
(43)
On the axis of rotation at in (43) will be: . We choose so that on the axis of
rotation does not depend on the angular velocity of rotation.
This is possible if , and does not
depend on . In this case, it will be .
Substitution of in (41) and
(43) gives:
. (44)
. (45)
In order to find , in (44) we can assume the angular
velocity of rotation equal to and compare the result with (36) for the case
of a nonrotating sphere. This gives the following:
On the other hand, the following expressions were
found in [27]:
, , . (47)
Hence it follows
that the scalar potential of the pressure field at the center of the sphere is
expressed through the root mean square velocity of particles at the center of the sphere:
, (48)
Moreover, in (46) will be:
.
(49)
Taking into account (4649) expressions (4445) will
be written as follows:
. (51)
At small , the potential of the pressure field (51) of a rotating cylinder ceases
to depend on and becomes approximately equal
to .
5. The vector potential of
the pressure field
Substituting in (35) the Lorentz factor
instead of , we obtain the equation for the vector potential of the pressure field
for the case of rotation of a cylinder without taking into account the proper
motion of the matter’s particles:
.
(52)
The velocity of the particles’ motion in (52) is determined in (23). Since (52) coincides in its form with equation
(22), then the solution of (52) for the potential component along the axis is an expression that repeats
(26):
.
(53)
According to (39), the scalar potential of the
pressure field of a rectilinearly moving particle is equal to , and the vector potential must be equal to the value . For the scalar potential and the component of the vector potential in case
of low angular velocities of the cylinder’s rotation, in
view of (23) and this can be written in the
following form:
, .
Comparing and (51) for small and , we find an estimate of the ratio , which allows us to determine more precisely the expression for , in view of (47):
, . (54)
Equating in (53) and (54) for low values and , we find the constant and determine more precisely the
expression for :
.
.
Hence, in view of (23), the vector potential of the
pressure field of rotating cylinder will be equal to:
. (55)
In this case, the vector potential is directed along
the linear velocity , and at low angular velocities it becomes equal to . Near the axis of rotation, where , in view of (47), it will be equal to .
By analogy with (31), we will multiply by in order to take into account the contribution of the
particles’ proper motion to the vector potential of the pressure field of a
rotating sphere:
. (56)
According to (30), the value is close to unity even for a
neutron star.
6. The invariant mass of
all particles
The total invariant mass of particles of a
relativistic uniform system without general rotation is determined by the
formula [18]:
If the system under consideration rotates with angular
velocity , then to calculate the invariant
mass of all particles in the system, the averaged Lorentz factor , which is a consequence of (4), should
be used. Taking into account (5) and the expression for rotating volume element
, we have in spherical coordinates:
Since the total invariant mass does not depend on the
nature of the particle motion, the masses and should be equal to each other. Hence there is
the connection between the Lorentz factors at the center of the sphere: .
This means that in the approximation we have adopted,
the particles at the center of the sphere in the rotating frame of reference move with the same speeds as in the case of a system of particles
without general rotation.
7. Conclusion
Wave equations (3) of the acceleration field contain
the scalar potential and the vector potential , which depend on the Lorentz factor and on the velocity of the matter’s particles. In the
general case, the particles move randomly with the Lorentz factor (5) and at the same time they
participate in collective motion, for example, in general rotation with the
Lorentz factor (6). As a result, the Lorentz factor becomes dependent on and . Besides, it becomes necessary to
average the value for the system’s typical
particles in order to use it subsequently in wave equations for the potentials
(78).
In addition to the scalar potential (10) of the acceleration field
for a generally motionless system of particles of spherical shape, we found
expressions for the scalar potential in two other cases. One of these cases
involves rotation of an infinitely long cylinder, in which the proper motion of
the matter’s particles is completely neglected. In this case, the scalar
potential is expressed by formula (20).
Another case describes rotation of the sphere’s matter with angular velocity , whereas the system of particles is a relativistic uniform system. The
fact that the random proper motion of the particles contributes to the scalar
potential of a rotating system of particles
is reflected in formula (21).
The potential in the absence of rotation, when , turns into the potential of the fixed system. As for the vector potential (31) of the acceleration field,
it exceeds the vector potential (29) of a cylinder by a small
coefficient , according to (30).
The considered approach is fully applicable to the
pressure field with the exception that the scalar potential of the acceleration
field is close in its value to the square of the speed of light, and the scalar
potential of the pressure field at the center of the system, according to (48),
approximately equals 1/3 of the square of the particle’s root mean square
velocity.
If we use the
definition of the scalar
potential of pressure field in (39), then for the potential at the center of
the sphere we can write: , where and denote the
pressure and density of the matter moving at the center [27]. Comparing this
with (48), we arrive at a relativistic formula for the pressure at the center
for the case of a nonrotating sphere, that at low velocities transforms into a
formula from the molecular kinetic theory: . Here is the rootmeansquare velocity
of typical particles and , accordingly, the Lorentz factor at the center of the sphere.
On the other hand, from (10), (36) and (4748) the
relation follows:
,
so that the scalar potential of the pressure field at
some point inside the sphere without rotation is three times less than the
square of the speed of motion of typical particles at the given point.
From (21) and (50), a similar relation is obtained that connects the potentials of
the pressure field and the field of accelerations in the matter of rotating
sphere. Previously, such
connections between the potentials of the acceleration field and the pressure
field were unknown.
Now, after calculating the scalar and vector
potentials of the acceleration field and pressure field, it becomes possible to
determine the strengths and solenoidal vectors of these fields by formulas (2)
and (34). In turn, the strengths and solenoidal vectors of these fields are
required to set the equation of motion of the matter and calculate the
relativistic energy of a rotating body, taking into account the energy of not
only the particles, but also the fields themselves. Thus, it will be possible
to find all the most important parameters of the system. Such work is planned
to be carried out in the next article.
As an example, we can take the scalar potential and the vector potential of the acceleration field, substitute these
potentials in (2) and obtain vectors and . The acceleration of a typical particle in the limit of the
special theory of relativity will be determined by the formula [19]:
, (57)
where is the velocity of a particle.
Let us now compare the acceleration in (57) with the formula for the acceleration of a particle presented in [28, Eq. (5) ]:
.
(58)
The potential coincides in meaning with the potential in (58), however, vector potentials and have different meanings. This happens because
(57) is obtained in a covariant way from the principle of least action for
continuous variables, while the quantities in (58) are
presented in the framework of discrete classical mechanics, using the
HodgeHelmholtz decomposition for acceleration.
, (59)
where is the velocity of liquid
particles; is the acceleration from mass
forces acting in the matter; is the mass density of the
matter; is the pressure.
If we designate and apply to both sides of
equation (59) the rotor operation, we get the Friedmann equation, which is
often used for rotary motion:
. (60)
In the simplest case, when the velocity lies in planes parallel to the
plane of the coordinate system, and the
angular velocity of rotation is directed along the axis and is constant at each point of
the system, there will be . Equations (59) and (60) imply that there are no viscosity and thermal
conductivity in the liquid and the motion occurs adiabatically without a change
in entropy. In addition, pressure is considered as a scalar field and therefore
participates in the formation of the force only through the gradient in the
form of a term . By solving the equations of motion (59) and (60), either the velocity is found, or , if the acceleration ,mass density and pressure are known as functions of
coordinates and time.
In contrast, a much more general field theory approach gives the
following equation of motion taking into account the acceleration field, the
vector pressure field, and the dissipation field [15], [19]:
.
(61)
Here is the fouracceleration,
expressed through the tensor of the acceleration field; is the mass fourcurrent; is the charge fourcurrent; , , and are tensors of the gravitational field, electromagnetic field, pressure
field, and dissipation field, respectively. The indicated tensors are
determined through strengths and solenoidal vectors. Thus, the components of
the tensor are vectors and in (12), found through the
potentials of the acceleration field, and the components of the tensor are vectors and in (3234), found through the potentials of the
pressure field. The equation of motion of matter according to (61) in the limit
of the special theory of relativity is written as follows:
. (62)
where is the Lorentz factor; is the dissipation function
associated with the scalar potential of the dissipation field by the
ratio ; is the acceleration from the mass
gravitational and electromagnetic forces acting in the matter. According to
[25], [27], the following can be written: , where is the scalar potential of the
pressure field.
Comparison of (62) and (59) reveals a difference related to taking
relativistic effects into account. So, in (62), the Lorentz factor is taken into account, as well as
the contribution of the invariant pressure , invariant mass density , and the dissipation function on the lefthand side of the
equality. In addition, on the righthand side of (62) is under the gradient sign , while in (59) the value is outside the gradient.
Equation (61) can be written not in terms of tensors, but directly in
terms of the fourpotentials of the fields [24]:
, (63)
. (64)
Here, , and denote the vector potentials of the
gravitational field, pressure field and electromagnetic field, respectively; is the charge density of an
arbitrary particle in the accompanying reference frame; the index runs through the values 0, 1, 2,
3; the index runs through the values 1, 2, 3;
in Cartesian coordinates, is the fourradius of the
particle; , and denote the fourpotentials of the
gravitational field, pressure field and electromagnetic field, respectively; , and there are scalar potentials of
the gravitational field, pressure field and electromagnetic field,
respectively.
If necessary, other vector fields, for example, the dissipation field,
can be taken into account in a similar way in (6364). Equation (63) describes
the energy balance in the system, and equation (64) is the equation of motion
of matter. Thus, if the wave equations for the field potentials are solved in
the system, similar to equations (33) for the pressure field, then with the
help of these potentials the motion of energy and particles in equations
(6364) can be found. This approach has proven itself well, for example, when
solving typical problems in electrodynamics.
The references to the formulas for the scalar and
vector field potentials obtained in this article are given in Table 1.
Таблица 1. The index of formulas for
field potentials

Fixed relativistic system 
Rotating cylindrical system 
Rotating spherical relativistic system 
The scalar potential of the acceleration field 
(10) 
(20) 
(21) 
The vector potential of the acceleration field 

(29) 
(31) 
The scalar potential of the pressure field 
(36) 
(51) 
(50) 
The vector potential of the pressure field 

(55) 
(56) 
Conflict of interest
The authors declare that they have no conflict of interest.
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