Gravitational stressenergy tensor is
a symmetric tensor of the second valence (rank), which describes the energy and
momentum density of gravitational field in the Lorentzinvariant theory of gravitation. This tensor in the covariant theory of
gravitation is included in the
equation for determining the metric along with the acceleration stressenergy tensor, pressure stressenergy tensor, dissipation stressenergy tensor and
electromagnetic stressenergy tensor. The covariant derivative of the gravitational
stressenergy tensor determines the density of gravitational force acting on
the matter.
Contents

In LITG the gravitational
stressenergy tensor is determined through the gravitational tensor and the metric tensor in
the Lorentzian metrics: ^{[1]}
where is the gravitational constant, is
the speed of gravitation.
After replacing by the strong gravitational constant the gravitational stressenergy tensor can be
used to describe strong gravitation at
the level of atoms and elementary particles in gravitational model of strong interaction.
Since the gravitational tensor in
LITG consists of the components of vectors of gravitational field strength and gravitational
torsion field , and the tensor in
4coordinates (ct, x,y,z) consists of the numbers 0, 1, 1 and does not
depend on the coordinates and time, so the components of the gravitational
stressenergy tensor can be written explicitly in terms of components of the
mentioned vectors:
The timelike components of the
tensor denote:
1) The volumetric energy density
of gravitational field, negative in value
2) The vector of momentum density
of gravitational field where is the vector of energy flux density
of gravitational field or the Heaviside
vector
The components of the vector are part of the corresponding tensor
components , and the components of the
vector are part of the tensor components , and due to the symmetry of the
tensor indices .
According to the Heaviside
theorem, the relation holds:
where is
the 3vector of mass current density.
3) The spacelike components of
the tensor form a submatrix 3 x 3, which is the 3dimensional gravitational
stress tensor, taken with a minus sign. Gravitational stress tensor can be
written as ^{[1]}
where is the Kronecker delta, if and if
The calculation of the
threedimensional divergence of the gravitational stress tensor gives:
where denote the components of threedimensional
density of gravitational force, – components of Heaviside vector.
The gravitational stressenergy
tensor has such form that it allows us to find the 4vector of the
gravitational force density by
differentiation in fourdimensional space:
As we can see from formula (1),
the 4vector of gravitational force density can be calculated in a different
way, through the gravitational tensor with mixed indices and the 4vector of mass current
density . This is due to the fact that in LITG the
gravitational field equations have the form:
Expressing
from the latter equation in
terms of
and substituting in (1) and also using the definition of the
gravitational stressenergy tensor, we can prove the validity of equation (1).
The components of the 4vector of the gravitational force density are as
follows:
where
is the 3vector of the gravitational force
density, is
the density of the moving matter, is
the 3vector of the mass current density, is
the 3vector of the matter unit velocity.
The
integral of (1) over the threedimensional volume of a small particle or a
matter unit, calculated in the reference frame comoving with the particle,
gives the gravitational fourforce:
In the
integration it was taken into account that ,
where is
the mass density in the comoving reference frame, is
the invariant mass, is
the 4velocity of the particle, is
the 4momentum of the particle, is
the relativistic momentum, is
the relativistic energy of the particle. It is also assumed that the mass
densities
and
include contributions from the massenergy of the proper gravitational field
and the electromagnetic field of the particle. The obtained 4force is acting
on the particle in the presence of the
gravitational field with the tensor , and in
some cases we can neglect the proper gravitational field of the particle and
consider its motion only in the external field.
The
gravitational stressenergy tensor contains timelike components ,
integrating which over the moving volume we can calculate the 4vector of
energymomentum of the free gravitational field, separated from its sources:
where
is
the total energy of gravitational field, is
the total momentum of the field.
If in
this volume there is matter as the source of proper gravitational
(electromagnetic) field, we should consider the total 4vector of
energymomentum, which includes contributions from all the fields in the given
volume, including acceleration field and
pressure field. In particular, for a
uniform spherical body with the radius the
4vector of energymomentum in view of the proper gravitational field of the
body has the form: ^{[2]}
where is
the gravitational mass, which equal to the mass , defined
through mass density and volume,
is the total gravitational energy of the body in the reference frame in
which the body is at rest, is
the invariant inertial mass, which includes the contributions of the
massenergy from all the fields.
It is
assumed that the mass equal
to the mass of
matter particles without their gravitational binding energy, and when the
particles combine in the whole body the bulk of their gravitational energy is
compensated by internal energy of the particles motion and the energy of the
body pressure. Since the energy is
negative, then the condition holds , that is
as long as the scattered matter is undergoing gravitational contraction into
the body of finite size, the gravitational mass is increasing. We can
find also the invariant
energy of the physical system:
A more accurate analysis of the mass of the system shows that the following
relationship holds:^{[3]}
where the gauge mass is related to the cosmological constant and
represents the massenergy of the matter’s particles in the fourpotentials of
the system’s fields; the inertial mass ; the auxiliary mass is equal to the product of the particles’
mass density by the volume of the system.
This can
be compared with the approach of the general relativity, in which such mass is
used that when we add to it the mass from the gravitational field energy, we
obtain the relativistic mass: , and there is a relation: .
CTG is the generalization of LITG
to any reference frames and phenomena that occur in the presence of fields and
accelerations of the acting forces. In CTG all deviations from LITG relations
are described by the metric tensor which becomes the function of coordinates
and time. In addition, in the equations the operator of 4gradient is
replaced by the covariant derivative . After replacing by the gravitational stressenergy tensor
becomes in the following form:
Gravitational field is considered at the same time as a component of general field.
Transforming the contravariant
indices in the gravitational tensor into the covariant indices with the help of
the metric tensor, and interchanging some indices, which are summed up, we
obtain:
Since the gravitational
tensor with covariant indices consists of the
components of vector of gravitational field strength , divided by the speed , and the components of vector of
gravitational torsion field , the formula shows that in the curved
spacetime the gravitational stressenergy tensor is the sum of the products of
the components of these vectors with the corresponding coefficients of the
components of the metric tensor. And it turns out that the energy density of
gravitational field contains mixed products of the form , etc. There are no such products in
the flat Minkowski space, which leads to the fact that in the spacetime of
special relativity the energy associated with the strength of gravitational
field is not mixed with the energy of the torsion field. The same situation
takes place in electromagnetism: in Minkowski space the energy of the electric
field is calculated separately from the energy of the magnetic field, but in
the curved spacetime in the energy density of the electromagnetic field there
is additional energy from the mixed components with the products of the components
of the strengths of the electric and magnetic fields.
Due to the use of covariant
differentiation in fourdimensional space in CTG the gravitational field
equations are changed as well as the expression for the 4vector of
gravitational force density (1), while the expression for the gravitational
4force remains the same: ^{[4]}
In CTG the
metric tensor is determined by solving the equation similar to HilbertEinstein
equation. In covariant indices this equation can be written as follows: ^{[5]}
where is the Ricci tensor, is
the scalar curvature, is
the coefficient to be determined, , , and
are the stressenergy tensors of the acceleration field, pressure field,
gravitational and electromagnetic fields, respectively, and it is assumed that
the speed of gravitation is
equal to the speed of light.
In
contrast to the general theory of relativity, in this equation, there is no
cosmological constant , there
is an additional constant
and the metric is dependent on the gravitational stressenergy tensor. The
latter is the consequence of the fact that in CTG gravitation is an independent
physical force as well as the electromagnetic force, and therefore participates
in determining the metric according to the principles of the metric theory of relativity.
Using the
principle of least action allows us to deduce not only the formula for the
gravitational stressenergy tensor, but also gives the equation of motion
written in tensor form:
The
covariant derivative of the acceleration
stressenergy tensor defines up to the sign the density of the 4force
acting on the field from the matter. At the same time the operator of propertimederivative is
applied to the 4potential of acceleration field in the
Riemannian space:^{ [6]}
where is
the acceleration tensor, is
the proper dynamic time of the particle in the reference frame at rest.
The total
density of the 4force of the gravitational and electromagnetic fields and
pressure field is determined by transfer of the stressenergy tensors of the
fields to the right side of the equation of motion (2) and then applying the
covariant derivative:
where is
the electromagnetic
tensor, is
the pressure field tensor, is
the 4vector of the electromagnetic current density, is
the density of electric charge of the matter unit in the reference frame at
rest.
In the weak
field limit, when the covariant derivative can be replaced by the partial
derivative, for the timelike component in (2), which has the index , the
local conservation law of energymomentum of matter and gravitational and
electromagnetic fields can be written as follows: ^{[7] [}^{8}^{]}
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, which are determined in
the special
relativity.
According
to this law, the work of the field to accelerate the masses and charges is
compensated by the work of the matter to create the field. As a result, the
change in time of the total energy in a certain volume is possible only due to
the inflow of energy fluxes into this volume.
The
analysis of equation (2) for the spacelike components with the index in
a weak field shows that, taking into account the equation of motion of the
matter in the field, all the force densities and the change rates of the
momentums of the matter and fields are cancelled.
However
in general case, when the spacetime is significantly curved by the existing
fields and matter, in (2) we should consider the contributions with the
additional nonzero components of the metric tensor and their derivatives,
which are absent in the special theory of relativity. This follows from the
fact that the covariant derivative is expressed through the partial derivative
and the Christoffel coefficients. Since in CTG the purpose of using the metric
tensor is correction of the motion equations in order to take into account the
dependence on the fields of the time intervals and spatial distances, measured
by the electromagnetic (gravitational) waves, then such correction changes
notation of many physical quantities in the form of 4vectors and tensors. In
particular, if we consider equation (2) as the local conservation law of energy
and momentum of the matter and the gravitational and electromagnetic fields,
then appearing of additional contributions with the components of the metric
tensor leads to specification of the theory for the case of the curved space.
And the physical meaning of the results obtained for the flat spacetime and the
weak field remains unchanged.
As another example we can consider the
integral of the left side of (2) over the entire fourdimensional space. In
flat Minkowski space the covariant divergence of the sum of the tensors becomes
an ordinary 4divergence, to which we can apply the divergence theorem. By this
theorem the integral of the 4divergence of some tensor over the 4space can be
replaced by the integral of the tensor over the hypersurface surrounding the
4volume, over which the integration is done. If we choose a projection of this
hypersurface on the hyperplane of the constant time in the form of a
threedimensional volume the integral of the left side of (2) is transformed
into the integral of the sum of time components of the tensors in (2) over the
volume, which must equal to conserved integral vector of the physical system
under consideration:
For these tensor components the integral vector vanishes. ^{[7]}
Vanishing of the 4vector allows us to explain the 4/3 problem, according to
which the massenergy of the gravitational or electromagnetic 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, ^{[8]} 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 stressenergy tensors do not form fourvectors, and therefore
they cannot define the same mass in the fields’ energy and momentum in
principle. It follows that
the integral vector cannot be considered as a fourmomentum of the
system, determining only the distribution of energy and energy fluxes of the
fields of the system.^{[9]}
In GTR, there is a problem of determining
the gravitational stressenergy tensor. The reason is that due to the applied
principle of geometrization of physics, all manifestations of gravitation are
completely replaced by the geometric effect – the spacetime curvature. Thus,
the gravitational field is reduced to the metric field, set by the metric
tensor and its derivatives with respect to coordinates and time. Since in each
reference frame there is its own metric, so gravitation is not independent
physical interaction but is assumed as the consequence of the metric of the
reference frame. This leads to the fact that instead of the gravitational
stressenergy tensor in GTR there is a pseudotensor which depends on the
metric. The feature of this pseudotensor is that locally it can be made equal
to zero at any point by choosing the corresponding reference frame. As a
result, GTR have to refuse from the possibility to accurately determine the
localization of gravitational energy and the momentum of gravitational field in
a physical system, which significantly impedes understanding the physics of
gravitation on a fundamental level, and describing the phenomena in a classical
form. The purpose of HilbertEinstein equations in GTR is to
find the metric:
These equations do not
contain the gravitational stressenergy tensor, which is present in the right
side of the equations for the metric in CTG. The metric in GTR depends only on
the matter and the electromagnetic field in the considered reference frame.
After finding such metric it is used to find the pseudotensor of the
energymomentum of gravitational field, and this pseudotensor must depend only
on the metric tensor, be symmetrical with respect to the indices, and when added
to the stressenergy tensors of matter and electromagnetic field
it must give zero divergence so that the conservation law of
energymomentum of the matter and field would be satisfied. This pseudotensor
in a weak field, that is, in the special theory of relativity, must vanish in
order to ensure the principle of equivalence of free fall in the gravitational
field and inertial motion.
The LandauLifshitz pseudotensor
corresponds to the mentioned criteria. ^{[}^{10}^{]} There is also an Einstein pseudotensor, but it is not symmetrical with
respect to the indices. ^{[}^{11}^{]} Note that in contrast to GTR, in mathematics under pseudotensor we
understand something different, more precisely a tensor quantity, which changes
its sign when it is transformed into the coordinate reference frame with the
opposite orientation of the basis.
Since in GTR there is no
gravitational stressenergy tensor, the gravitational energy of a body is
determined indirectly. For example, one method is when by the given substance
density and body size the mass of the body is calculated, first in the absence
of the metric’s influence, and then taking into account the metric’s influence
and the corresponding change in the volume differential in the integral of the
mass under influence of the gravitational field. The difference of the
mentioned masses is equated to the massenergy of gravitational field as to the
manifestation of the metric field. ^{[}^{12}^{]} In this case, the components of the metric tensor are used, found in GTR
from the HilbertEinstein equation for the metric.