**Pressure stress-energy tensor** is a symmetric tensor of the second
valence (rank), which describes the density and flux of energy and momentum of pressure field in matter. This tensor in
the covariant theory of gravitation is
included in the equation for determining the metric along with the gravitational stress-energy tensor, the acceleration stress-energy tensor , the dissipation stress-energy tensor and the stress-energy tensor of
electromagnetic field. The covariant derivative of the pressure stress-energy
tensor determines the density of pressure force acting on the matter.

The pressure stress-energy tensor
is relativistic generalization of the three-dimensional Cauchy stress tensor
used in continuum mechanics. In contrast to the stress tensor, which is usually
used to describe the relative stress appearing at deformations of bodies, the
pressure stress-energy tensor describes any internal stresses, including
stresses in the absence of deformation of bodies from external influences.

- 1 Continuum mechanics

- 1.1 Examples of
tensors
- 1.2 Description
of the motion and metrics

- 2 Covariant theory of gravitation

- 2.1 Definition
- 2.2 Components
of the pressure stress-energy tensor
- 2.3 Pressure
force and pressure field equations
- 2.4 Equation for
the metric
- 2.5 Equation of
motion
- 2.6 Conservation
laws

- 3 See also
- 4 References
- 5 External links

The existence of different
variants of the pressure stress-energy tensor shows absence of any unambiguous
definition of this tensor. Besides the 4-velocity, density and pressure in the
tensor a function is often added with desired properties such that the tensor
can describe the energy and stress in the matter. The arbitrariness of the
choice of such function is related to the fact that when the pressure is
believed a simple scalar function, there is a need to add vector properties of
the pressure forces with the help of some additional function.

For matter in equilibrium with
uniform pressure, the simplest pressure stress-energy tensor in the metric (+ –
– –) can be written as:

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

Because of its simplicity, the
tensor in this form is often used not only in mechanics, but also in the
general relativity.

Fock introduces the pressure energy density
per unit mass and adds this quantity to the pressure stress-energy tensor: ^{[1]}

here denotes that the mass density, which is
independent of pressure, and related to the total invariant mass density by relation:

Instead of it, Fedosin used a
compression function : ^{[2]}

There are other forms of pressure
stress-energy tensor, differing from each other in the way of introducing some
additional scalar function in the tensor. ^{[3]} ^{[4]} ^{[5]}

The standard approach involves
first determining stress-energy tensor of the system where is the stress-energy tensor of matter and is the stress-energy tensor of electromagnetic field. Thereafter, taking in
account the pressure and other fields the equation of motion follows as a
result of vanishing of covariant derivative of the stress-energy tensor of the
system: In general relativity (GR) account of the
gravitational field in equation of motion is carried out through the dependence
of the metric tensor components on position and time .

Tensor is used in GR for finding metrics of
Hilbert-Einstein equations:

where is the Ricci tensor, is the scalar curvature, is the gravitational constant.

Thus, the pressure stress-energy
tensor changes metric inside the bodies.

In contrast to the continuum
mechanics in covariant theory of gravitation
(CTG) pressure field is not a scalar field and considered as 4-vector field
consisting of scalar and 3-vector components. Therefore in CTG the pressure
stress-energy tensor is defined by the pressure
field tensor and the metric tensor by the principle of least action: ^{[6]}

where is a constant having its own value in each task. The constant is not uniquely defined, and it is a consequence of the fact that the
pressure inside the bodies may have been caused by any reasons and both
internal and external forces. Pressure field is considered as a component of the general field.

In the weak field limit, when the
space-time metric becomes the Minkowski metric of special relativity, the
metric tensor becomes the tensor , consisting of the numbers 0, 1, –1. In
this case the form of the pressure stress-energy tensor is greatly simplified
and can be expressed in terms of the components of the pressure field tensor, i.e.
the pressure field strength and solenoidal pressure vector :

The time-like components of the tensor
denote:

1) The volumetric energy density
of pressure field

2) The vector of momentum density
of pressure field where the vector of energy flux density of
pressure field is

The components of the vector are part of the corresponding tensor components , and the components of the vector are part of the tensor components , and due to the symmetry of the tensor indices .

3) The space-like components of
the tensor form a submatrix 3 x 3, which is the 3-dimensional stress tensor,
taken with a minus sign. The stress tensor can be written as

where the components the Kronecker delta equals 1 if and equals 0 if

This stress tensor is a concrete
expression of Cauchy stress tensor.

Three-dimensional divergence of
the stress tensor of pressure field connects the pressure force density and
rate of change of momentum density of the pressure field:

where denote the components of the three-dimensional pressure force density, – the components of the energy flux density of the pressure field.

The principle of least action
implies that the 4-vector of pressure force density can be found through the pressure
stress-energy tensor, either through the product of pressure field tensor and
mass 4-current:

Equation (1) is closely related
with the pressure field equations:

In the special theory of
relativity, according to (1) for the components of the pressure four-force density can be written:

where is
the 3-vector of the pressure force density, is
the density of the moving matter, is the 3-vector of the mass current
density, is
the 3-vector of velocity of the matter unit.

In Minkowski space, the field
equations are transformed into 4 equations for the pressure field strength and
solenoidal pressure vector

In the covariant theory of
gravitation the pressure stress-energy tensor in accordance with the principles
of metric theory of relativity is one
of the tensors defining metrics inside the bodies by the equation for the
metric:

where is
the coefficient to be determined, , , and are the stress-energy tensors of the
acceleration field, pressure field, gravitational and electromagnetic fields,
respectively.

The equation of motion of a point
particle inside or outside matter can be represented in tensor form, with
pressure stress-energy tensor or
pressure field tensor :

where is
the acceleration tensor, is
the gravitational tensor , is
the electromagnetic tensor, is the charge 4-current, is
the density of electric charge of the matter unit in the reference frame at
rest.

Time-like component of the
equation (2) at describes the change in the energy and
spatial component at connects the acceleration with the force
density.

Time-like component in (2) can be considered as the local law of conservation
of energy and momentum. In the limit of special relativity, when the covariant
derivative becomes the 4-gradient, and the Christoffel symbols vanish, this
conservation law takes the simple form: ^{[7]} ^{[8]}

where is the vector of the
acceleration field energy flux density, is the Heaviside
vector, is the Poynting vector, is the vector of the pressure field energy
flux density.

According to this law, the work of the field to accelerate the masses
and charges is compensated by the work of the matter to create the field. As a
result, the change in time of the total energy in a certain volume is possible
only due to the inflow of energy fluxes into this volume.

The integral form of the law of conservation of energy-momentum is
obtained by integrating (2) over the 4-volume to accommodate the
energy-momentum of the gravitational and electromagnetic fields, extending far
beyond the physical system. By the Gauss's formula the integral of the
4-divergence of some tensor over the 4-space can be replaced by the integral of
time-like tensor components over 3-volume. As a result, in Lorentz coordinates
the integral vector equal to zero may be obtained:

Vanishing of the integral vector allows us to explain the 4/3 problem,
according to which the mass-energy of field in the momentum of field of the
moving system in 4/3 more than in the field energy of fixed system. On the
other hand, according to, ^{[8]} the generalized Poynting theorem and
the integral vector should be considered differently inside the matter and beyond
its limits. As a result, the occurrence of the 4/3 problem is associated with
the fact that the time components of the stress-energy tensors do not form
four-vectors, and therefore they cannot define the same mass in the fields’
energy and momentum in principle.

- Pressure field
- Gravitational stress-energy tensor
- Electromagnetic
stress-energy tensor
- Acceleration stress-energy tensor
- Dissipation
stress-energy tensor
- Pressure field tensor
- Cauchy
stress tensor
- General field
- Dissipation
field
- Acceleration field

- V. A. Fock, The Theory of
Space, Time and Gravitation (Pergamon Press,
London, 1959).
- Fedosin S.G. Fizika i filosofiia
podobiia: ot preonov do metagalaktik,
Perm: Style-MG, 1999, ISBN 5-8131-0012-1. 544 pages, Tabl.66, Ill.93,
Bibl. 377 refs
- Herglotz G. // Ann. d. Phys. 1911, Bd
36, S. 493.
- Ignatowsky W.V. // Phys. Ztschr.
1911, Bd 12, S. 441.
- Lamla E. // Berl. Diss., 1911;
Ann. d. Phys. 1912, Bd 37, S. 772.
- 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. 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.
^{8.0}^{8.1}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.

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