The gravitational
tensor or gravitational field tensor, (sometimes called the gravitational
field strength tensor) is an antisymmetric tensor, combining two components of gravitational field – the gravitational
field strength and the gravitational torsion field – into one. It
is used to describe the gravitational field of an arbitrary physical system and
for invariant formulation of gravitational equations in the covariant theory of gravitation. The gravitational field is a
component of general field.
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Tensor of gravitational field is
defined by the gravitational fourpotential as
follows: ^{[1] [2]}
Due to the antisymmetry of this
formula the difference of two covariant derivatives is equal to the difference
between the two partial derivatives with respect to the 4coordinates.
If we consider the definition of
the 4potential of gravitational field:
where is
the scalar potential, is
the vector potential of the gravitational field, is
the propagation speed of gravitational effects (speed of gravity),
and if we introduce for Cartesian
coordinates the gravitational field strengths by the
rules:
where is the gravitational field strength or gravitational acceleration, is
the torsion field,
then the covariant components of
the gravitational field tensor according to (1) will
be:
According to the rules of tensor
algebra, raising (lowering) of the tensors’ indices, that is the transition
from the covariant components to the mixed and contravariant components of
tensors and vice versa, is done by means of the metric tensor . In particular , as well
as
In Minkowski space the metric tensor
turns into the tensor , which does not depend on the coordinates
and time. In this space, which is used in the special relativity, the
contravariant components of the gravitational field tensor are as follows:
Since the vectors of gravitational field strength and torsion field are the components of
the gravitational field tensor, they are transformed not as vectors, but as the
components of the tensor of the type (0,2). The law of transformation of these
vectors in the transition from the fixed reference frame K into the reference
frame K', moving at the velocity V along the axis X, has the following form:
In the more general case where the velocity of the reference frame K’
relative to the frame K is aimed in any direction, and the axis of the
coordinate systems parallel to each other, the gravitational field strength and
the torsion field are converted as follows:
The first expression is the
contraction of the tensor, and the second is defined as the pseudoscalar
invariant. In the latter expression the LeviCivita symbol is used for the fourdimensional space,
which is a completely antisymmetric unit tensor, with its gauge
Let us
consider the following expression:
Equation
(2) is satisfied identically, which is proved by substituting into it the
definition for the gravitational field tensor according to (1). If in (2) we
use nonrecurring combinations 012, 013, 023 and 123 as the indices , and if
we pass from the field potentials to the strengths, this leads to two vector
equations:
Equations
(3) and (4) are two of the four Heaviside's equations for the gravitational
field strengths in the Lorentzinvariant theory of
gravitation. According to (3), the change in time
of the torsion field creates circular gravitational field
strength, which leads to the effect of gravitational induction, and
equation (4) states that the torsion field, as well as the magnetic field, has
no sources. Equations (3) and (4) can also be obtained from equality to zero of
the 4vector, which is found by the formula:
Another
couple of gravitational field equations is also expressed in terms of the
gravitational field tensor:
where
is the 4vector of mass current density, is the
matter density in the comoving reference frame, is
the velocity of the matter unit, is the gravitational constant.
In the
expanded form the equation for the field strengths with field sources are as
follows:
where
is the density of the moving mass, is
the mass current density.
According
to the first of these equations, the gravitational field strength is generated
by the matter density, and according to the second equation the circular
torsion field is always accompanied by the mass current, or emerges due to the
change in time of the gravitational field strength vector.
Gravitational fourforce acting
on the mass of
a body can be expressed in terms of the gravitational field tensor and the 4velocity of
the body:
This
expression can be derived, in particular, as the consequence of the axiomatic
construction of the covariant theory of gravitation in the language of
4vectors and tensors. ^{[3]}
If we
take the covariant divergence of both sides of (5), and
taking into account (1) we obtain: ^{[4]}
The continuity equation for the mass 4current is a gauge condition that is used to derive
the field equation (5) from the principle of least action. Therefore, the
contraction of the gravitational tensor and the Ricci tensor must be zero: . In Minkowski space the Ricci tensor equal to zero, the covariant derivative becomes the
partial derivative, and the continuity equation becomes as follows:
The wave equation for the gravitational tensor is written
as: ^{[5]}
Total
Lagrangian for the matter in gravitational and electromagnetic fields includes
the gravitational field tensor and is contained in the action function: ^{[4]} ^{[6}
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 electromagnetic 4potential, where is
the electric scalar potential, and is
the electromagnetic vector potential, is
the electromagnetic 4current,
is the electric constant, is
the electromagnetic
tensor,
is
the 4potential of acceleration field,
and are
the constants of acceleration field and pressure field, respectively, is
the acceleration tensor, is
the 4potential of pressure field, is
the pressure field tensor, is the invariant 4volume, 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 4coordinates leads to the equation of
motion of the matter unit in gravitational and electromagnetic fields and pressure field:^{
[5]}
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 gravitational fourpotential, we
obtain the equation of gravitational field (5).
With the help of gravitational field
tensor in the covariant theory of gravitation the gravitational stressenergy
tensor is constructed:
.
The covariant derivative of the gravitational
stressenergy tensor determines the 4vector of gravitational force density:
By
definition, the generalized momentum
characterizes the total momentum of the matter unit taking into account the
momenta from the gravitational and electromagnetic fields. In the covariant
theory of gravitation the generalized force, as the
rate of change of the generalized momentum by the coordinate time, depends also
on the gradient of the energy of gravitational field associated with the matter
unit and determined by the gravitational field tensor. ^{[7]}
In the
weakfield approximation Hamiltonian as the relativistic energy of a body with
the mass and the charge with equals:
If we use
the covariant 4vector of generalized velocity
then in
the general case the Hamiltonian has the form: ^{[4]}
where
and
are timelike components of 4vectors and .
If we
move to the reference frame that is fixed relative to the center of mass of
system, Hamiltonian will determine the invariant energy of
the system.