**Dissipation field** is a two-component vector force field,
which describes in a covariant way the friction force and energy dissipation
emerging in systems with a number of closely interacting particles. The
dissipation 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 dissipation field and the term with the field energy. ^{[1]} ^{[2]} The dissipation field is included in the
equation of motion by means of the dissipation
field tensor and in the equation for the metric – by means the dissipation stress-energy tensor.

By energy dissipation is meant
conversion of the energy of directed motion of particles into the energy of
random motion of these particles and the particles of the surrounding medium,
as well as conversion into the energy of intramolecular and atomic motion,
while the energy of motion of fast moving fluxes of particles decreases due to
the friction with slower fluxes. The typical examples of dissipation of
mechanical energy are damping of motion of a jet in a liquid and heating of
falling meteorites during their motion in the Earth's atmosphere.

The dissipation field is generally
considered as a macroscopic field with its energy and momentum, describing the
averaged interaction of particles in an arbitrary small volume of a system. The
cause of the dissipation field emerging at the micro level is different
interactions leading to the effects of friction and deceleration of individual
particles or their fluxes. At the atomic level, the electromagnetic forces and strong gravitation are prevailing, by
which the particles interact with each other and exchange their energy. The
friction forces in a system of particles appear as a collective effect and are
proportional not only to the velocity, but also to its derivatives with respect
to time and coordinates. Since the friction force is described by the
dissipation field tensor and the corresponding stress-energy tensor, the
dissipation field in each small volume obtains its energy density and energy
flux density. In dissipative processes, some change occurs in the internal
energy of the system, mainly due to the change in the quantity of heat or in
the energy of phase transitions, which can be considered as a change in the
dissipation field energy. The internal energy also changes when the pressure field energy and the energy of the
particles’ acceleration field change, as
well as due to the change in the energy of the electromagnetic, gravitational
and other fields. In turn, the dissipation field energy flux makes its
contribution into the flux of internal energy and the flux of the relativistic
energy of the system.

- 1 Classical mechanics
- 2 Relativistic hydrodynamics
- 3 The dissipation field as a
vector field

- 3.1 Mathematical
description
- 3.2 Action,
Lagrangian and energy
- 3.3 Equations
- 3.4 Stress–energy
tensor
- 3.5 Application

- 4 See also
- 5 References
- 6 External links

The main processes that lead to
energy dissipation are the viscous friction of the layers of liquid or gas
against each other, the friction during the motion of solid bodies due to
interaction with the surrounding medium, thermal conductivity and diffusion in
gases and liquids. All of these processes belong to transport phenomena:
friction occurs during transfer of momentum, heat conductivity occurs during
the internal energy transfer, and diffusion is associated with the transfer of
mass (charge, electric and magnetic moment, etc.). Friction is the main source
of energy dissipation. Thermal conductivity and diffusion also make some
contribution, since in real processes all kinds of transfer are intertwined
with each other. In order to describe the friction forces in Euler–Lagrange
equation the friction forces are introduced: ^{[3]}

where is the Lagrangian, are the generalized coordinates. In addition, the dissipation function is introduced into consideration, so that the following relation holds:

The Rayleigh dissipation function
is given by the expression:

If the tensor does not depend on the velocities, the corresponding friction force is equal
to:

The dissipation function has the
dimension of power and the tensor must have the dimension kg/s. If the tensor is symmetrical, the friction
force appears to be directed oppositely to the velocity of the particle’s
motion relative to the surrounding medium.

This shows that the dissipation
function represents some form of a scalar potential, which depends on the
products of projections of relative velocities, and the dissipation field is
considered as the corresponding scalar field. But unlike the standard scalar
potentials of fundamental fields, the friction force is not in the form of a
gradient of dissipative functions but in the form of a derivative with respect
to the motion velocity. This approach cannot be considered as a fully covariant
description of friction processes, it can only serve as a first approximation,
because it does not take into consideration the friction in accelerated motion.

The Newton's law of fluid
friction describes the internal friction force in the liquid layer, moving
relative to other parallel layers:

here is the coefficient of internal friction or the coefficient of dynamic
viscosity, i s the gradient of the velocity of the layers’
motion in the direction perpendicular to the layer surface, is the surface area of the layer.

A more accurate description of
motion in a viscous medium is given by the Navier–Stokes equations:

where is the mass density of the matter, is the "second viscosity", or the volume viscosity, is the Kronecker delta, is the Cauchy stress tensor, is the acceleration due to the mass forces (including the gravitational and
electromagnetic forces and the force of inertia), while the second equation is
the continuity equation.

Instead of the tensor in the relativistic case the four-dimensional viscous stress tensor is used
to describe the equation of motion of the viscous and heat-conducting media: ^{[4]}

where is the four-velocity with a contravariant index, is the four-velocity with a covariant index, is the coefficient of the second (volume) viscosity, is the metric tensor, is the speed of light.

The density of the four-force, arising from viscosity, is
calculated using the covariant derivative of the tensor and is present in the right side of the Navier-Stokes equations. By its
meaning the tensor is the stress-energy tensor, but it cannot be derived from the principle of
least action.

The dissipation field 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.
The equations of this field were developed as a consequence of the principle of
least action, ^{[5]} and a special procedure was
used. ^{[6]}

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

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

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

This yields the following:

In the covariant theory of
gravitation, the four-potential of the dissipation 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. 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 dissipation field is a
component of the general field, then the corresponding dissipation 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: ^{[7]}

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 dissipation 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 dissipation field:

In Minkowski space of the special
theory of relativity, the form of the dissipation 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 dissipation
field potentials from the field equations:

In curved space 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 :^{ [8]}

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

The dissipation stress-energy tensor is
calculated with the help of the dissipation field 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 dissipation field determines the density of the
dissipation four-force:

The stress-energy tensor of the
dissipation 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 pressure 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
dissipation 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: ^{[6]}

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

where is
the dissipation function. 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 dissipation field in the matter
changes and is different from the dissipation field of a single particle at the
observation point. The dissipation 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. As a rule, in a
gravitationally-bound system the radial gradients of different field strengths
appear, including the dissipation field strength and if some part of the particles is moving
synchronously or rotating, then the vector appears. From (2) and (3) we derive a general expression for the four-force
density with a covariant index, which arises from the dissipation field:

where denotes the four-dimensional space-time
interval.

The relativistic equation of
motion of viscous compressible matter, taking into account the four-potential
of the dissipation field, the dissipation field tensor and the stress-energy
tensor of the dissipation field, within the limits of low curvature of
spacetime can be represented as follows: ^{[5]}

where is the acceleration of gravitational and electromagnetic mass forces, is
the gravitational field strength, is
the gravitational torsion field, is
the electric field strength, is
the magnetic field.

At low velocities, for the
Lorentz factor we can assume . Under ordinary conditions, we can also
neglect the contribution of the pressure and the dissipation function on
the left side of the equation. Determining the dynamic viscosity by the
expression ,
where is
the acceleration field coefficient, and
denoting , we obtain the following:

The main difference of this equation
from the Navier–Stokes equation is a small additional term, which contains the
second time derivative of the flux velocity and the square of the speed of light in
the denominator.

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

^{1.0}^{1.1}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.
- Torby, Bruce (1984). "Energy Methods". Advanced
Dynamics for Engineers. HRW Series in Mechanical Engineering. United
States of America: CBS College Publishing. ISBN
0-03-063366-4. (p. 271).
- Ландау Л. Д., Лифшиц Е. М. Гидродинамика. – Издание
7-е, исправленное. – М.: Наука, 1988. – 731 с. – (Теоретическая физика,
том VI).
^{ }^{5.0}^{5.1}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.^{6.0}^{6.1}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.- 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.