**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]}
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: ^{[2]}

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.
^{[3]} ^{[4]}

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: ^{[4]}

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.

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 and four-acceleration :

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.

The four-potential of any vector field for a single
particle can be represented as: ^{[3]}

where for electromagnetic field and for other fields, and are the mass density and accordingly charge density in comoving reference
frame, is the field energy density of the particle, 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-acceleration with a covariant index, which arises from the pressure field
and is part of the general four-acceleration:

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: ^{[5]}

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. ^{[6]} 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 vacuum permittivity, and are the total charge and mass of the system.

In articles ^{[7]} ^{[8]} 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

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. ^{[7]}

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

^{1.0}^{1.1}^{1.2}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.- C. W. Misner, K. S.
Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman, San Francisco, CA,
1973).
^{3.0}^{3.1}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, 2014, no. 18, 771 - 779. http://dx.doi.org/10.12988/astp.2014.47101.^{4.0}^{4.1}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).- 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. 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).
- 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 generalized Poynting theorem for the general field and solution
of the 4/3 problem. Preprint, February 2016.
- 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.

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