Acceleration stress-energy
tensor is a symmetric
four-dimensional tensor of the second valence (rank), which describes the
density and flux of energy and momentum of acceleration 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 pressure
stress-energy tensor, the dissipation stress-energy tensor
and the stress-energy tensor of
electromagnetic field. The covariant derivative of the acceleration
stress-energy tensor determines the density of acceleration force acting on the
matter.
In covariant theory of gravitation (CTG) the
acceleration field is not a scalar field and considered as 4-vector field
consisting of scalar and 3-vector components. In CTG the acceleration
stress-energy tensor was defined by Fedosin through the acceleration tensor and the metric tensor by the principle of least action: ^{[1]}
where is
a constant defined in terms of the fundamental constants and physical parameters
of the system. Acceleration field is considered as a component of the general field.
Since acceleration tensor
consists of the components of the acceleration field strength and the solenoidal acceleration vector , then the acceleration stress-energy
tensor can be expressed through these components. In the limit of special
relativity the metric tensor ceases to depend on the coordinates and time, and
in this case the acceleration stress-energy tensor gains the simplest form:
The time-like components of the
tensor denote:
1) The volumetric energy density
of acceleration field
2) The vector of momentum density
of acceleration field where the
vector of energy flux density of acceleration field is
Due to the symmetry of the tensor
indices, , so that
3) The space-like components of
the tensor form a submatrix 3 x 3, which is the 3-dimensional acceleration
stress tensor, taken with a minus sign. The acceleration 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 acceleration field connects the force density and rate of
change of momentum density of the acceleration field:
where denote the components of the three-dimensional
acceleration force density, –
the components of the energy flux density of the acceleration field.
The principle of least action
implies that the 4-vector of force density can
be found through the acceleration stress-energy tensor, either through the
product of acceleration tensor and mass 4-current:
The field equations of
acceleration field are as follows:
In the special theory of
relativity, according to (1) for the components of the four-force density can be written:
where is
the 3-vector of the 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 acceleration field strength and solenoidal acceleration vector
In the covariant theory of
gravitation the acceleration 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, , , and are the stress-energy tensors of the
acceleration field, pressure field,
gravitational and electromagnetic fields, respectively, is the gravitational constant.
The equation of motion of a point
particle inside or outside matter can be represented in tensor form, with
acceleration stress-energy tensor or
acceleration tensor :
where is
the gravitational tensor , is
the electromagnetic tensor, is
the pressure field tensor, is the charge 4-current, is the density of electric charge of the
matter unit in the reference frame at rest, is
the 4-velocity.
We now recognize that is the mass 4-current and the
acceleration tensor is defined through the covariant 4-velocity as This gives the following:
Here operator of proper-time-derivative is used,
where is the symbol of 4-differential in curved
spacetime, is the proper time, is
the mass density in the comoving frame.
Accordingly, the equation of
motion (2) becomes:
Time-like component of the
equation at describes the change in the energy and spatial
component at connects the acceleration with the force density.
When the index in
(2), i.e. for the time-like component of the equation, in the limit of special
relativity from the vanishing of the left side of (2) follows:
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.
This equation can be regarded as
a local conservation law of energy-momentum of the four fields.
The integral form of the law of
conservation of energy-momentum is obtained by integrating (2) over the
4-volume. By the divergence theorem 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 conserved
4-vector equal to zero may be obtained: ^{[2]}
Vanishing of the 4-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.
As in relativistic mechanics, and
in general relativity (GR), the acceleration stress-energy tensor is not used.
Instead it uses the so-called stress-energy tensor of matter, which in the
simplest case has the following form: . In GR,
the tensor is substituted into the equation for the metric and its covariant
derivative gives the following:
In GR it is assumed that there is
the continuity equation in the form Then, using the operator of
proper-time-derivative the covariant derivative of the tensor gives the product of the mass density and four-acceleration,
i.e. the density of 4-force:
However, the continuity equation
is valid only in the special theory of relativity as In
curved space-time instead would have to be the equation , but instead of zero on the right
side of this equation there appears an additional non-zero term with Riemann
curvature tensor. ^{[1]} Consequently, (4 ) is not
an exact expression, and tensor determines the properties of the matter only
in the special theory of relativity. In contrast, in the covariant theory of
gravitation equation (3) is written in covariant form, so that the acceleration
stress-energy tensor describes well the acceleration field of
matter particles in curved Riemannian space-time.
Source:
http://sergf.ru/asen.htm