The dissipation field tensor
is an antisymmetric tensor describing the energy dissipation due to viscosity
and consisting of six components. Tensor components are at the same time
components of the two three-dimensional vectors – dissipation field strength
and the solenoidal dissipation vector. With the dissipation field tensor the
dissipation stress-energy tensor, the dissipation field equations and
dissipation force in matter are defined. Dissipation
field is a
component of general field.
Expression for the dissipation
field tensor can be found in papers by Sergey Fedosin, ^{[1]}
where the tensor is defined using 4-curl:
Here dissipation 4-potential is
given by:
where is
the scalar potential, is
the vector potential of dissipation field, – speed of light.
The dissipation field strength
and the solenoidal dissipation vector are found with the help of (1):
and the same in vector notation:
The dissipation field tensor
consists of the components of these vectors:
The transition to the dissipation
field tensor with contravariant indices is carried out by multiplying by double
metric tensor:
In the special relativity, this
tensor has the form:
To convert the components of the
dissipation field tensor from one inertial system to another we must take into
account the transformation rule for tensors. If the reference frame K' moves
with an arbitrary constant velocity with respect to the fixed reference system K,
and the axes of the coordinate systems are parallel to each other, the
dissipation field strength and the solenoidal dissipation vector are converted
as follows:
Through the dissipation field
tensor the equations of dissipation field
are written:
where is the mass 4-current, is
the mass density in comoving reference frame, is
the 4-velocity, is a constant.
Instead of (2) it is possible to
use the expression:
Equation (2) is satisfied
identically, which is proved by substituting into it the definition for the dissipation
field tensor according to (1). If in (2) we insert tensor components , this leads to two vector equations:
According to (5), the solenoidal
dissipation vector has no sources as its divergence vanishes. From (4) it
follows that the time variation of the solenoidal dissipation vector leads to a
curl of the dissipation field strength.
Equation (3) relates the
dissipation field to its source in the form of mass 4-current. In Minkowski
space of special relativity the form of the equation is simplified and becomes:
где – плотность движущейся массы, – плотность тока массы.
According to the first of these
equations, the dissipation field strength is generated by the mass density, and
according to the second equation the mass current or change in time of the
dissipation field strength generate the circular field of the solenoidal
dissipation vector.
From (3) and (1) we can obtain continuity equation:
This equation means that due to
the curvature of space-time when the Ricci tensor is
non-zero, the dissipation field tensor is
a possible source of divergence of mass 4-current. If space-time is flat, as in
Minkowski space, the left side of the equation is set to zero, the covariant
derivative becomes the 4-gradient and remains the following:
Total Lagrangian for the matter
in gravitational and electromagnetic fields includes the dissipation field
tensor and is contained in the action function: ^{[1]}
where is
Lagrangian, is
differential of coordinate time, is
a certain coefficient, is
the scalar curvature, is
the cosmological constant, which is a function of the system, is
the speed of light as a measure of the propagation speed of electromagnetic and
gravitational interactions, is
the gravitational four-potential, is the gravitational constant, is
the gravitational tensor, is
the electromagnetic 4-potential, is
the electromagnetic (charge) 4-current, is the vacuum permittivity, is
the electromagnetic tensor, is
the covariant 4-velocity, , and are some constants, is
the acceleration tensor, is
the 4-potential of pressure field, is
the pressure field tensor, is
the 4-potential of dissipation field, is
the dissipation field tensor, is the
invariant 4-volume, is
the square root of the determinant of
metric tensor, taken with a negative sign, is
the product of differentials of the spatial coordinates.
The variation of the action
function by 4-coordinates leads to the equation of motion of the matter unit in
gravitational and electromagnetic fields, pressure
field and dissipation field:
where is
the 4-acceleration with the covariant index, the operator of proper-time-derivative with
respect to proper time is
used, the first term on the right is the gravitational force density, expressed
with the help of the gravitational field tensor, second term is the Lorentz
electromagnetic force density for the charge density measured in the comoving reference frame, and
the two last term set the pressure force density and the dissipation force
density, respectively.
If we vary the action function by
the dissipation 4-potential, we obtain the equation of dissipation field (3).
With the help of dissipation
field tensor in the covariant theory of
gravitation the dissipation
stress-energy tensor is constructed:
.
The covariant derivative of the
dissipation stress-energy tensor determines the dissipation four-force density:
Covariant 4-vector of generalized
velocity is given by:
With regard to the generalized
4-velocity, the Hamiltonian contains the dissipation field tensor and has the
form:
where and are timelike components of 4-vectors and .
In the reference frame that is
fixed relative to the center of mass of system, the Hamiltonian will determine
the invariant energy of the system.