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 the four-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, 4-potential of which consists of the
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
the acceleration field
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 four 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-potential as This gives the following: ^{[2]}
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 rate of change of the scalar potential of
the acceleration field, and
spatial component at connects the rate
of change of the vector potential of the acceleration field 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. ^{[}^{3}^{]}
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 integral vector equal to zero
may be obtained: ^{[4]}
Vanishing of the integral 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. On the
other hand, according to, ^{[3]} the generalized Poynting theorem and
the integral vector should be considered differently inside the matter and
beyond its limits. As a result, the occurrence of the 4/3 problem is associated
with the fact that the time components of the stress-energy tensors do not form
four-vectors, and therefore they cannot define the same mass in the fields’
energy and momentum in principle.
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