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
vacuum permittivity, is
the electromagnetic tensor, is
covariant 4-velocity, and are some constants, 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:
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 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.
·
Electromagnetic tensor
·
Pressure
stress-energy tensor
·
Lorentz-invariant theory
of gravitation
·
Covariant theory of
gravitation
Source:
http://sergf.ru/tpden.htm