The acceleration
tensor is an antisymmetric tensor describing the four-acceleration of particles and
consisting of six components. Tensor components are at the same time components
of the two three-dimensional vectors – acceleration field strength and the
solenoidal acceleration vector. With the acceleration tensor the acceleration
stress-energy tensor, the acceleration field equations and the four-force density in matter are defined. Acceleration
field in matter is
a component of general field.
Expression for the acceleration
tensor can be found in papers by Sergey Fedosin, ^{[1]}
where the tensor is defined using 4-curl:
Here the acceleration
4-potential is given by:
where is
the scalar potential, is
the vector potential of acceleration field, –
speed of light.
The acceleration field strength
and the solenoidal acceleration vector are found with the help of (1):
and in the second expression three numbers are composed of non-recurring sets
1,2,3; or 2,1,3;
or 3,2,1 etc.
In vector notation can be
written:
The acceleration tensor
consists of the components of these vectors:
The transition to the
acceleration tensor with contravariant indices is carried out by multiplying by
double metric tensor:
In the special relativity, this
tensor has the form:
For the vectors, related to the specific point
particle, considered as
a solid body, we can write:
where , is the velocity of the particle.
To transform the components of the
acceleration 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 acceleration
field strength and the solenoidal acceleration vector are transformed as
follows:
Through the acceleration tensor
the equations of acceleration field are written:
where is the mass 4-current, is
the mass density in comoving reference frame, is
the 4-velocity, is a constant of
acceleration field.
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
acceleration tensor according to (1). If in (2) we insert tensor components , this leads to two vector equations:
According to (5), the
solenoidal acceleration vector has no sources as its divergence vanishes. From
(4) follows that the time variation of the solenoidal acceleration vector leads
to a curl of the acceleration field strength.
Equation (3) relates the
acceleration 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 acceleration field strength is generated by the mass density,
and according to the second equation the mass current or change in time of the
acceleration field strength generate the circular field of the solenoidal
acceleration 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 acceleration 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:
Acceleration stress-energy tensor
With the help of acceleration tensor in the covariant
theory of gravitation the acceleration
stress-energy tensor is constructed:
.
The covariant derivative of the acceleration
stress-energy tensor with mixed indices determines the four-force density: ^{[2]}
here the operator of proper-time-derivative with respect to proper time is used.
The density of the 4-force can be written for the time
and space component in the form of two expressions:
where denotes
a four-dimensional space-time interval,
Total Lagrangian for the matter
in gravitational and electromagnetic fields includes the acceleration 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:
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 acceleration 4-potential, we obtain the equation of acceleration field
(3).
The covariant 4-vector of generalized velocity,
considered as 4-potential of [[general field]], is given by the expression:
With regard to the generalized 4-velocity, the Hamiltonian contains the
acceleration tensor and has the form:
Where and are the time components of the 4-vectors and .
In the reference frame that is fixed relative
to the system's center-of-momentum frame, Hamiltonian will determine the invariant energy of the system.
Studying Lorentz covariance of
4-force, Friedman and Scarr found not full covariance
expressions for 4-force in the form ^{[3]}
This led them to conclude that
the four-acceleration must be expressed with the
help of some antisymmetric tensor :
Based on the analysis of
different types of motion, they rated their required values of the components
of the acceleration tensor, thereby giving this tensor indirect definition.
From comparison with (6) it follows that the tensor up to a sign and a constant factor coincides
with the acceleration tensor in case when rectilinear motion of a solid
body without rotation is considered. Then indeed the four-potential of the
acceleration field coincides with the four-velocity, . As a result, the quantity on the right-hand side of (6) vanishes, since
the following relations hold true: ,
. With this in mind, in (6) we can raise the
index and
cancel the mass density, which gives the following:
Mashhoon and Muench
considered transformation of inertial reference systems, related to accelerated
frame of reference, and came to the relation: ^{[4]}
Tensor has the same properties as the acceleration
tensor
Other theories
In the articles ^{[5]} ^{[6]} ^{[7]} devoted to the modified Newtonian dynamics
(MOND), in the tensor-vector-scalar gravity appear scalar function or , that defines a scalar field, and 4-vector or , and 4-tensor or
The analysis of these values in the corresponding Lagrangian
demonstrates that scalar function or correspond to scalar potential of the acceleration field; 4-vector or
correspond to 4-potential of the
acceleration field; 4-tensor or correspond to acceleration tensor .
As it is known, the acceleration field is not intended to explain the accelerated
motion, but for its accurate description. In this case, it can be assumed that
the tensor-vector-scalar theories cannot pretend to explain the rotation curves
of galaxies. At best, they can only serve to describe the motion, for example
to describe the rotation of stars in galaxies and the rotation of galaxies in
clusters of galaxies.
Source: http://sergf.ru/aten.htm