**Dissipation stress-energy
tensor** is a symmetric tensor
of the second valence (rank), which describes the density and flux of energy
and momentum of dissipation 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 pressure stress-energy tensor and
the stress-energy tensor of electromagnetic field. The covariant derivative of
the dissipation stress-energy tensor determines the density of dissipation
force acting on the matter and retarding the movement of flows of matter
relative to each other.

The dissipation stress-energy
tensor is relativistic generalization of the three-dimensional viscous stress tensor used in fluid
mechanics.

- 1 Fluid mechanics
- 2 Covariant theory of gravitation

- 2.1 Definition
- 2.2 Components
of the dissipation stress-energy tensor
- 2.3 Dissipation
force and dissipation 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

For relativistic description of
the equations of motion of viscous and heat-conducting medium in the book ^{[1]} is used the four-dimensional viscous stress tensor:

where is
the coefficient of common (shear) viscosity, is
the 4-velocity with contravariant index, is the 4-velocity with covariant index, is the coefficient of bulk viscosity (or "second viscosity"), is
the metric tensor, is the speed of light.

The form of the tensor is of the
requirements imposed by the law of entropy. This tensor is defined in such a
way that in the reference frame in which the moving element of the matter
rests, tensor and reset. This means that the energy of the
element of matter in the comoving frame must be calculated by other physical
variables that are not related to the viscosity as in the absence of
dissipative processes. As a result, the condition is superimposed at the
tensor:

The tensor is
a part of energy-momentum tensor of matter with pressure and it takes into account the viscosity:

here , is
the energy density of matter without pressure.

The equation of motion of matter
with pressure and viscosity is obtained from the vanishing of the covariant
derivative of the energy-momentum tensor of matter:

A significant drawback of the tensor
is that it is not derived from the principle
of least action, and therefore can not be used, for example,
to calculate the metrics in the system. In addition, in the general case the
tensor components and can not zeroed in the comoving frame, because the
environment is moving relative to the element of matter and energy dissipation
process is not terminated.

In covariant theory of gravitation (CTG)
dissipation field is considered as 4-vector field consisting of scalar and
3-vector components,
and is a component of general field. Therefore in CTG the dissipation stress-energy
tensor is defined by the dissipation field
tensor and the metric tensor by the principle of least action: ^{[2]}

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 dissipation
in liquid medium may have been caused by any reasons and both internal and
external forces.

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 dissipation
stress-energy tensor is greatly simplified and can be expressed in terms of the
components of the dissipation field tensor, i.e. the dissipation field strength
and solenoidal dissipation vector :

The time-like components of the
tensor denote:

1) The volumetric energy density
of dissipation field

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

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

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

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

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

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

where is the 3-vector of the dissipation 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 dissipation field strength and solenoidal dissipation vector :

In the covariant theory of
gravitation the dissipation 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, , , , , are the stress-energy tensors of the acceleration
field, pressure field, gravitational
and electromagnetic fields, dissipation
field, respectively, is
the gravitational constant.

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

where is
the acceleration tensor, is the gravitational tensor , is the electromagnetic tensor, is the pressure field tensor, is
the dissipation field tensor, is the charge 4-current, is the density of electric charge of the matter unit in the comoving
reference frame, is
the 4-velociry.

Time-like component of the
equation (2) at describes the change in the energy and
spatial component at connects the acceleration with the total
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: ^{[3]}^{ [4]}

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, is
the vector of the dissipation 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 sum of
tensor components with energy density 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, ^{[4]} 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.

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

- L.D. Landau, E.M. Lifshitz (1987). Fluid Mechanics.
Vol. 6 (2nd ed.). Butterworth-Heinemann. ISBN 978-0-08-033933-7.
- 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 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). http://dx.doi.org/10.11648/j.ajmp.20140304.12.
^{4.0}^{4.1}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/dsen.htm