**Pressure field** is a two-component vector force field, which
describes in a covariant way the dynamic pressure of individual particles and
the pressure emerging in systems with a number of closely interacting
particles. The pressure field is a general
field component, which is represented in the Lagrangian and Hamiltonian of
an arbitrary physical system including the term with the energy of particles in
the pressure field and the term with the field energy. ^{[1]}
^{[2]}

The pressure field is included in the equation of motion
by means of the pressure field tensor
and in the equation for the metric – by means the pressure stress-energy tensor. Any forces
acting on the matter particles and causing a change in their interaction with
each other make a contribution to the pressure field, its energy and momentum.
The pressure field is generally considered as a macroscopic field, describing
the averaged interaction of particles in an arbitrary small volume of a system.
The cause of the pressure field emerging at the microlevel is different
interactions. For example, electromagnetic forces and strong gravitation hold electrons and
nucleons in atoms together. The action of the external forces causes the matter
compression and change in the volume occupied by atoms and electrons in the
matter atoms. This leads to a change in the system’s energy, which can be represented
as a change in the pressure field energy.

- 1 The scalar pressure
field
- 2 The vector pressure
field

- 2.1 Mathematical description
- 2.2 Action, Lagrangian and energy
- 2.3 Equations
- 2.4 Stress–energy tensor
- 2.5 Application in certain
problems

- 3 See also
- 4 References
- 5 External links

In equilibrium states of matter and in the absence of
mass forces, atoms and molecules usually move chaotically and their total
directed motion can be neglected. Under these conditions, the characteristic of
internal motion is the average velocity of particles . In the molecular kinetic theory there is
a formula for the pressure: where is
the average mass of one particle of thermodynamic system, is
the particle concentration.

As a macroscopic thermodynamic variable, the pressure is
part of the equation of state, which relates various thermodynamic variables.
In particular, the pressure as a physical variable is included in the ideal gas
law:

where is
the gas volume, is
the gas mass, is
the molar mass, is
the universal gas constant, is
the temperature, is the Boltzmann constant, is
the Avogadro constant.

Pressure is part of the Bernoulli's principle for a stationary flux
of ideal (i.e., without internal friction) incompressible liquid, which is the
consequence of the conservation of energy:

where is
the liquid mass density, is
the flux velocity, is
the height at which this liquid unit is located, is
the pressure at the point in space, where the center of mass of the liquid unit
under consideration is located, is
the free fall acceleration. The first term of the equation is the dynamic
pressure, the second term gives the pressure from the mass forces (in this case
from gravitation), the third term is the static pressure and the constant in
the right side is called the total pressure.

The scalar pressure characterizes the continuous medium
state, and in case of the equilibrium state in the liquid the pressure becomes
hydrostatic. In this case the pressure is the diagonal component of the
symmetric three-dimensional Cauchy stress tensor:

where is
the Kronecker symbol.

In the general relativity, the pressure stress-energy
tensor is used for the ideal liquid, which is a generalization of the formulas
of classical mechanics: ^{[3]}

where is
the metric tensor, is
the four-velocity, is
the speed of light.

In the concept of a scalar field, under the pressure
field energy the work is meant, which is done by the pressure to change the
system’s volume from the initial state with zero pressure to the current state,
taking into account the contribution of the particles’ kinetic energy from the
mass-energy change due to the pressure field.

The drawback of the scalar pressure field concept is the
inaccurate method of taking into account the energy and momentum of the
pressure field in accelerated reference frames with a number of the field
sources, where the effects of field self-action and addition of individual
pressure waves at a limited propagation velocity of the field are manifested.
In the vector fields an additional degree of freedom appears in the form of a
vector potential. As a result, the energy of one field component can go into
the energy of another component, the field strength becomes a function of the
scalar and vector potentials, and the force is determined by the field
strength, motion velocity and solenoidal vector. The examples of the field
self-action are the electromagnetic induction and gravitational induction.

The pressure as a two-component vector field was
presented by Sergey Fedosin within the framework of the metric theory of relativity and the covariant theory of gravitation, and the
equations of this field were developed as a consequence of the principle of
least action. ^{[4]} ^{[5]}

The four-potential of the pressure field is expressed in
terms of the scalar and vector
potentials:

The antisymmetric pressure
field tensor is calculated with the four-curl of the four-potential:

The pressure tensor components are the vector components
of the pressure field strength and the solenoidal pressure vector :

From here we obtain the following:

In the covariant theory of gravitation, the
four-potential of
the pressure field is part of the four-potential of the general field , which is the sum of the four-potentials
of particular fields, such as electromagnetic and gravitational fields,
acceleration field, pressure field, dissipation field, strong interaction
field, weak interaction field and other vector fields, acting on the matter and
its particles. All these fields in one way or another are represented in the
matter, so that the four-potential cannot consist solely of the four-potential . The energy density of interaction of the
general field with the matter is given by the product of the four-potential of
the general field and the mass four-current: . From the four-potential of the general
field we obtain the general field tensor by applying the four-curl:

The tensor invariant in the form of is up to a constant factor proportional to the
energy density of the general field. As a result, the action function that
contains the scalar curvature and the cosmological constant is
given by the expression: ^{[1]}

where is
the Lagrange function or Lagrangian, is
the time differential of the coordinate reference frame, and are the constants to be determined, is
the speed of light, as a measure of the propagation velocity of electromagnetic
and gravitational interactions, is the
invariant four-volume, expressed in terms of the differential of the time
coordinate , the product of
differentials of the space coordinates and the square root of the determinant of the metric tensor, taken with a negative sign.

Variation of the action function gives the general field
equations, the four-dimensional equation of motion and the equation for determining
the metric. Since the pressure field is a component of the general field, then
the corresponding pressure field equations can be derived from the general
field equations.

Given the gauge conditions of the cosmological constant
are met in the following form:

the system’s energy does not depend on the term with the scalar
curvature and it becomes uniquely determined: ^{[5]}

where and denote the time components of the
four-vectors and .

The four-momentum of the system is given by the formula:

where and denote the system’s momentum and the velocity of the system’s center of
mass.

Main article: Equation
of vector field

The four-dimensional equations of the pressure field are
similar by their form to Maxwell equations and have the following form:

where is the mass four-current, is the mass density in the co-moving reference frame, is the four-velocity of the matter unit, is
a constant determined in each problem, and it is assumed that there is a
balance between all the fields in the physical system under consideration.

The gauge condition for the four-potential of the field
of pressure:

In Minkowski space of the special theory of relativity,
the form of the pressure field equations is simplified and they can be
expressed in terms of the field strength and the solenoidal vector :

where is
the Lorentz factor, is the mass current density , is the matter unit velocity.

If we also use the gauge condition in the form of and relation (1), we can obtain the wave
equations for the pressure field potentials from the field equations:

The equation of motion of the matter unit in the general
field is given by the formula:

.

Since , and the general field tensor is expressed in terms of tensors of
particular fields, then the equation of motion can be represented using these
tensors:^{ [6]}

Here is
the acceleration tensor, is the electromagnetic tensor, is
the charge four-current, is
the gravitational tensor, is the dissipation field tensor, is the strong interaction field tensor, is
the weak interaction field tensor.

The pressure
stress-energy tensor is calculated with the help of the pressure tensor:

.

The tensor includes the three-vector of energy-momentum
flux , which
is similar in its meaning to the Poynting vector and the Heaviside vector. The vector can be represented through the vector product
of the field strength and the solenoidal vector :

here the index is

The covariant derivative of the stress-energy tensor of the
pressure field determines the density of the pressure four-force:

The stress-energy tensor of the pressure field is part of
the stress-energy tensor of the general field
, but
in the general case the tensor also contains the cross-terms with the products of strengths and solenoidal
vectors of particular fields:

where are some coefficients, is the electromagnetic stress-energy tensor, is
the gravitational stress-energy tensor,
is the acceleration stress-energy tensor,
is the dissipation
stress-energy tensor, is
the strong interaction stress-energy tensor, is
the weak interaction stress-energy tensor.

By means of the tensor
, the
stress-energy tensor of the pressure field becomes part of the equation for the
metric:

where is
the Ricci tensor, is the gravitational constant, is a certain constant, and the gauge condition for the cosmological
constant is used.

In the case when a certain
vector potential of a particle is equal to zero in the rest frame of the particle,
the four-potential of this vector field in an arbitrary frame of reference can
be represented as follows: ^{[4]}

where for electromagnetic field and for other fields, and are
the mass density and accordingly charge density in comoving reference frame, is the energy density of the particle in the given
field, is
the covariant four-velocity.

For the pressure field , ,
and according to the definition, for the four-potential of the pressure field
of one particle we have the following:

where is
the scalar pressure. For an arbitrary particle, the components of the
four-potential in the framework of the special relativity (STR) take the
form:

and hence, the vector potential is directed along the
particle’s velocity. If the vector potential components are the functions of time
and do not directly depend on the space coordinates, then for such motion
according to (1) the solenoidal vector vanishes.

Due to the interaction of a set of particles with each
other by means of various fields, including interaction at a distance without
direct contact, the pressure field in the matter changes and is different from
the pressure field of a single particle at the observation point. The pressure
field in the system of particles is specified by the field strength and
solenoid vector, which represent the typical averaged characteristics of the
matter’s motion. For example, in a gravitationally-bound system a radial
gradient of the vector appears, and if some part of the particles
is moving synchronously or rotating, then the vector appears. From (3)
and (4) we derive a general expression for the four-force density with a
covariant index, which arises from the pressure field:

where denotes the four-dimensional space-time
interval.

For a stationary case, when the pressure field potentials
do not depend on time, the wave equation (2) for the scalar potential in STR is
transformed into the equation:

The solution of this equation for a fixed sphere with
randomly moving particles in it has the following form: ^{[7]}

Here is
a coefficient of the acceleration field,
represents the scalar potential of the
pressure field in the center of the sphere, is
the Lorentz factor for the velocities of
the particles in the center of the sphere, and in view of the argument’s
smallness the sine is expanded to the second-order terms. It follows from the
formula that the pressure potential and the scalar pressure reach the maximum
at the center and decrease, when approaching the surface of the sphere with the
radius and the total mass .

The obtained dependence for the pressure at the center holds true for a variety of space objects,
including gas clouds, Bok globules, Earth, and neutron stars. In the center of
the main sequence stars, including the Sun, the main contribution to the total
pressure is made by thermonuclear reactions instead of gravitation. This
contribution was taken into account in the article, ^{[1]}
where the following relation was obtained for the pressure at the center of the
Solar core:

where and denote the radius and mass of the Solar core,
is the constant in the stress-energy tensor
of the strong interaction field, and .

In the system under consideration, the scalar potential becomes the function of the radius, and the
vector potential and solenoidal vector are equal to zero. The pressure field
strength is
found from (1). Next, we can calculate all the functions of the pressure field,
including the four-acceleration of the pressure field, the energy of the
particles in this field and the energy of the pressure field itself. ^{[8]} For cosmic bodies without additional sources of
energy, the main contribution to the four-acceleration in the matter is made by
the gravitational force and the pressure field. In this case we can
automatically derive the relativistic rest energy of the system, taking into
account the motion of particles inside the sphere. For a system of particles
with the acceleration field, pressure field, gravitational and electromagnetic
fields, this approach allowed us to solve the 4/3 problem and showed, where and
in what form the system’s energy is contained. The following relation was found
in this problem for the constant of the pressure field:

where is
the electric constant, and are the total charge and mass of the system.

In articles ^{[9]} ^{[10]} the ratio of the field’s
coefficients for the fields was specified as follows:

If we introduce the parameter as the number of nucleons per
ionized gas particle, then the pressure field constant is expressed as follows:

For the pressure inside the cosmic bodies in the gravitational
equilibrium model we find the dependence on the current radius:

where the coefficients and are included into the dependence
of the mass density on the radius in the relation

Under the
assumption that the system’s typical particles have the mass , where is the mass of one gas particle,
for which the unified atomic mass unit is taken, and that it is typical
particles that define the temperature and pressure, for the pressure field
constant we obtain the following: ^{[11]}

The scalar
potential at the center of the sphere is approximately equal to: ^{[12]}

The relativistic equation of motion of the viscous
compressible fluid, with regard to the four-potential of the pressure field,
pressure field tensor and stress-energy tensor of the pressure field, was
presented within the limits of low curvature of spacetime in the form of the
Navier-Stokes equations in hydrodynamics in the framework of STR. ^{[13]}

Taking into account the vector
pressure field, within the framework of the relativistic
uniform system, it is possible to refine the virial theorem, which in the
relativistic form is written as follows: ^{[14]}

where the value exceeds the kinetic energy of the particles by a factor equal to the Lorentz factor of the particles at the center of the system.
Under normal conditions we can assume that , then we can see that in the virial theorem
the kinetic energy is related to the potential energy not by the coefficient
0.5, but rather by the coefficient close to 0.6. The difference from the
classical case arises due to considering the pressure field and the
acceleration field of particles inside the system, while the derivative of the
virial scalar is not equal to zero and should be considered
as the material derivative.

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: ^{[15]}

The integral
__field
energy theorem__ for pressure field in a curved space-time
is as follows: ^{[16]}

In the
relativistic uniform system, the scalar potential of the pressure field is related
to the scalar potential of the acceleration field: ^{[17]}

The
relativistic expression for pressure is as follows:

where is the mass density of moving
matter, is the speed of light, is the Lorentz factor. In the
limit of low velocities, this relationship turns into the standard formula of
the kinetic theory of gases.

- General field
- Acceleration field
- Dissipation
field
- Covariant theory of gravitation
- Metric theory of relativity
- Pressure field tensor
- Pressure stress-energy tensor
- Four-force
- Equation of vector field

^{1.0}^{1.1}^{1.2}Fedosin S.G. The Concept of the General Force Vector Field. OALib Journal, Vol. 3, pp. 1-15 (2016), e2459. http://dx.doi.org/10.4236/oalib.1102459.- Fedosin S.G. Two components of the
macroscopic general field. Reports in Advances of Physical Sciences, Vol.
1, No. 2, 1750002, 9 pages (2017). http://dx.doi.org/10.1142/S2424942417500025.
- C. W. Misner, K. S. Thorne, and J.
A. Wheeler, Gravitation (W. H. Freeman, San Francisco, CA, 1973).
^{4.0}^{4.1}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, no. 18, pp. 771-779 (2014). http://dx.doi.org/10.12988/astp.2014.47101.^{5.0}^{5.1}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). http://dx.doi.org/10.5281/zenodo.889304.- Fedosin S.G. Equations of Motion in the Theory of
Relativistic Vector Fields. International Letters of Chemistry, Physics
and Astronomy, Vol. 83, pp. 12-30 (2019). https://doi.org/10.18052/www.scipress.com/ILCPA.83.12.
- 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,
pp. 152-167 (2014). https://dx.doi.org/10.11648/j.ajmp.20140304.12.
- Fedosin S.G. Relativistic
Energy and Mass in the Weak Field Limit. Jordan Journal of Physics.
Vol. 8, No. 1, pp. 1-16 (2015). http://dx.doi.org/10.5281/zenodo.889210.
- 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, pp.
370-379 (2016). http://dx.doi.org/10.1139/cjp-2015-0593.
- Fedosin S.G. The generalized Poynting theorem for
the general field and solution of the 4/3 problem. International Frontier
Science Letters, Vol. 14, pp. 19-40 (2019). https://doi.org/10.18052/www.scipress.com/IFSL.14.19.
- Fedosin S.G. The binding energy and the total
energy of a macroscopic body in the relativistic uniform model. Middle
East Journal of Science, Vol. 5, Issue 1, pp. 46-62 (2019). http://dx.doi.org/10.23884/mejs.2019.5.1.06.
- Fedosin S.G. Energy and metric gauging in the
covariant theory of gravitation. Aksaray University
Journal of Science and Engineering, Vol. 2, Issue 2, pp. 127-143 (2018). http://dx.doi.org/10.29002/asujse.433947.
- Fedosin S.G. Four-Dimensional
Equation of Motion for Viscous Compressible and Charged Fluid with Regard
to the Acceleration Field, Pressure Field and Dissipation Field.
International Journal of Thermodynamics. Vol. 18, No. 1, pp. 13-24 (2015).
http://dx.doi.org/10.5541/ijot.5000034003.
- 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. (2017). https://dx.doi.org/10.1007/s00161-016-0536-8.
- Fedosin S.G. The
integral theorem of generalized virial in the relativistic uniform model.
Continuum Mechanics and Thermodynamics, Vol. 31, Issue 3, pp. 627-638
(2019). https://dx.doi.org/10.1007/s00161-018-0715-x.
- Fedosin S.G. The Integral
Theorem of the Field Energy. Gazi University Journal of Science. Vol.
32, No. 2, pp. 686-703 (2019). http://dx.doi.org/10.5281/zenodo.3252783.
- Fedosin S.G. The potentials of the acceleration field and pressure field in
rotating relativistic uniform system Continuum Mechanics and
Thermodynamics, Vol. 33, Issue 3, pp. 817-834 (2021). https://doi.org/10.1007/s00161-020-00960-7.

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