The pressure field
tensor is an antisymmetric tensor describing the pressure field and consisting of six
components. Tensor components are at the same time components of the two
three-dimensional vectors – pressure field strength and the solenoidal pressure
vector. With the pressure field tensor the pressure stress-energy tensor, the
pressure field equations and pressure force in matter are defined. Pressure field is a component of general field.
Expression for the pressure field tensor can be found in
papers by Sergey Fedosin, ^{[1]} where the tensor
is defined using 4-curl:
Here pressure 4-potential is
given by:
where is
the scalar potential, is
the vector potential of pressure field, –
speed of light.
The pressure field strength and the solenoidal pressure
vector are found with the help of (1):
and the same in vector notation:
The pressure field tensor consists of the components of
these vectors:
The transition to the pressure 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 pressure 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 parallel to each other, the pressure
field strength and the solenoidal pressure vector are converted as follows:
Through the pressure field tensor the equations of
pressure 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 use the expression:
Equation (2) is satisfied identically, which is proved by
substituting into it the definition for the pressure field tensor according to
(1). If in (2) we insert tensor components , this leads to two vector equations:
According to (5), the solenoidal pressure vector has no
sources as its divergence vanishes. From (4) follows that the time variation of
the solenoidal pressure vector leads to a curl of the pressure field strength.
Equation (3) relates the pressure 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:
where is
the density of moving mass, is
the density of mass current.
According to the first of these equations, the pressure
field strength is generated by the mass density, and according to the second
equation the mass current or change in time of the pressure field strength
generate the circular field of the solenoidal pressure vector.
From (3) and (1) can be obtained continuity equation:
This
equation means that thanks to the curvature of space-time when the Ricci
tensor is non-zero, the pressure 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 pressure 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 4-current, is
the electric constant, is
the electromagnetic tensor, is the 4-potential of
acceleration field, and are the constants of acceleration field and
pressure field, respectively,
is the acceleration
tensor, is
the 4-potential of pressure field, is
pressure 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 and pressure field:^{ }^{[2]}
where 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
last term sets the pressure force density.
If we vary the action function by the pressure
4-potential, we obtain the equation of pressure field (3).
With the help of pressure field tensor in the covariant theory of gravitation the pressure stress-energy tensor is
constructed:
.
The covariant derivative of the pressure stress-energy tensor
determines the pressure four-force density:
Covariant 4-vector of generalized velocity is given by:
Given the generalized 4-velocity the Hamiltonian contains
the pressure 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, Hamiltonian will determine the invariant energy of the system.
See also
References
1.
^{1,0} ^{1,1} 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). http://dx.doi.org/10.5281/zenodo.889304.
2.
Федосин
С.Г. Уравнения движения в теории релятивистских векторных полей. Препринт.
Январь, 2018.
External links
· Pressure
field tensor in Russian
Source:
http://sergf.ru/tpden.htm