Gravitational stress-energy tensor

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

Lorentz-invariant theory of gravitation (LITG)
In LITG the gravitational stress-energy tensor is determined through the gravitational tensor $$ ~\Phi_{ik}$$ and the metric tensor $$~ \eta^{ik}$$ in the Lorentzian metrics:
 * $$~ U^{ik} = - \frac{c^2_{g}} {4 \pi G }\left( -\eta^{im}\Phi_{n m}\Phi^{n k}+ \frac{1} {4} \eta^{ik}\Phi_{mr}\Phi^{mr}\right) ,$$

where $$~ G $$ is the gravitational constant, $$~ c_{g}$$ is the speed of gravitation.

After replacing $$ ~ G $$ by the strong gravitational constant $$ ~ \Gamma $$ the gravitational stress-energy tensor can be used to describe strong gravitation at the level of atoms and elementary particles in gravitational model of strong interaction.

Components of the gravitational stress-energy tensor
Since the gravitational tensor in LITG consists of the components of vectors of gravitational field strength $$ ~ \mathbf{\Gamma }$$ and gravitational torsion field $$ ~ \mathbf{\Omega}$$, and the tensor $$~ \eta^{ik}$$ in 4-coordinates (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 stress-energy tensor can be written explicitly in terms of components of the mentioned vectors:


 * $$~ U^{ik} = \begin{vmatrix} u & \frac {H_x}{c_{g}} & \frac {H_y}{c_{g}} & \frac {H_z}{c_{g}} \\ c_{g} P_{gx} & u+ \frac{\Gamma^2_x+c^2_g \Omega^2_x}{4\pi G} & \frac{\Gamma_x \Gamma_y+c^2_g \Omega_x\Omega_y }{4\pi G} & \frac{\Gamma_x \Gamma_z+c^2_g \Omega_x\Omega_z }{4\pi G} \\ c_{g} P_{gy} & \frac{\Gamma_x \Gamma_y+c^2_g \Omega_x\Omega_y }{4\pi G} & u+\frac{\Gamma^2_y+c^2_g \Omega^2_y }{4\pi G}  & \frac{\Gamma_y \Gamma_z+c^2_g \Omega_y\Omega_z }{4\pi G} \\ c_{g} P_{gz} & \frac{\Gamma_x \Gamma_z+c^2_g \Omega_x\Omega_z }{4\pi G}  & \frac{\Gamma_y \Gamma_z+c^2_g \Omega_y\Omega_z }{4\pi G} & u+\frac{\Gamma^2_z+c^2_g \Omega^2_z }{4\pi G}  \end{vmatrix}. $$

The time-like components of the tensor denote:

1) The volumetric energy density of gravitational field, negative in value
 * $$~ U^{00} = u= -\frac{1}{8 \pi G }\left(\Gamma^2+ c^2_{g} \Omega^2 \right).$$

2) The vector of momentum density of gravitational field $$~\mathbf{P_g} =\frac{ 1}{ c^2_{g}} \mathbf{H}, $$ where is the vector of energy flux density of gravitational field or the Heaviside vector
 * $$~\mathbf{H} =-\frac{ c^2_{g} }{4 \pi G }[\mathbf{\Gamma }\times \mathbf{\Omega }].$$

The components of the vector $$~\mathbf{H} $$ are part of the corresponding tensor components $$~ U^{01}, U^{02}, U^{03}$$, and the components of the vector $$~\mathbf{P_g} $$ are part of the tensor components $$~ U^{10}, U^{20}, U^{30}$$, and due to the symmetry of the tensor indices $$~ U^{01}= U^{10}, U^{02}= U^{20}, U^{03}= U^{30}$$.

According to the Heaviside theorem, the relation holds:
 * $$~ \nabla \cdot \mathbf{H} = - \frac{ \partial U^{00}}{\partial t} - \mathbf{\Gamma }\cdot \mathbf{J},$$

where $$~\mathbf{J} $$ is the 3-vector of mass current density.

3) The space-like components of the tensor form a submatrix 3 x 3, which is the 3-dimensional gravitational stress tensor, taken with a minus sign. Gravitational stress tensor can be written as


 * $$~ \sigma^{p q} = \frac {1}{4 \pi G } \left( - \Gamma^p \Gamma^q - c^2_g \Omega^p \Omega^q + \frac {1}{2} \delta^{pq} (\Gamma^2 + c^2_g \Omega^2 ) \right) ,$$

where $$~p,q =1,2,3, $$ $$~\Gamma^1=\Gamma_x, $$ $$~\Gamma^2=\Gamma_y, $$ $$~\Gamma^3=\Gamma_z, $$ $$~\Omega^1=\Omega_x, $$ $$~\Omega^2=\Omega_y, $$ $$~\Omega^3=\Omega_z, $$ $$~\delta^{pq}$$ is the Kronecker delta, $$~\delta^{pq}=1 $$ if $$~p=q, $$ and $$~\delta^{pq}=0 $$ if $$~p \not=q. $$

The calculation of the three-dimensional divergence of the gravitational stress tensor gives:
 * $$~ \partial_q \sigma^{p q} = f^p +\frac {1}{c^2_g} \frac{ \partial H^p}{\partial t}, $$

where $$~ f^p $$ denote the components of three-dimensional density of gravitational force, $$~ H^p $$ – components of Heaviside vector.

Gravitational force
The gravitational stress-energy tensor has such form that it allows us to find the 4-vector of the gravitational force density $$~ f^\alpha $$ by differentiation in four-dimensional space:
 * $$~ f^\alpha = -\partial_\beta U^{\alpha \beta} = {\Phi^\alpha}_{i} J^i . \qquad (1) $$

As we can see from formula (1), the 4-vector of gravitational force density can be calculated in a different way, through the gravitational tensor with mixed indices $$ {\Phi^\alpha}_{i} $$ and the 4-vector of mass current density $$ ~J^i$$. This is due to the fact that in LITG the gravitational field equations have the form:
 * $$~ \partial_n \Phi_{ik} + \partial_i \Phi_{kn} + \partial_k \Phi_{ni}=0, $$


 * $$~\partial_k \Phi^{ik} = \frac {4 \pi G }{c^2_{g}} J^i .$$

Expressing from the latter equation $$ ~J^i$$ in terms of $$ \Phi^{ik} $$ and substituting in (1) and also using the definition of the gravitational stress-energy tensor, we can prove the validity of equation (1). The components of the 4-vector of the gravitational force density are as follows:
 * $$~ f^\alpha = (\frac {\mathbf{\Gamma } \cdot \mathbf{J} }{c_g}, \mathbf{f} ),$$

where $$~ \mathbf{f}= \rho \mathbf{\Gamma } + [\mathbf{J} \times \mathbf{\Omega} ]$$ is the 3-vector of the gravitational force density, $$~\rho$$ is the density of the moving matter, $$~\mathbf{J} =\rho \mathbf{v} $$ is the 3-vector of the mass current density, $$~\mathbf{v} $$ is the 3-vector of the matter unit velocity.

The integral of (1) over the three-dimensional volume of a small particle or a matter unit, calculated in the reference frame co-moving with the particle, gives the gravitational four-force:
 * $$~ F^\alpha = {\Phi^\alpha}_{i} M u^i= {\Phi^\alpha}_{i} p^i = (\frac {\mathbf{\Gamma } \cdot \mathbf{p} }{c_g}, \frac{E}{ c^2_g } \mathbf{\Gamma }+ [\mathbf{p} \times \mathbf{\Omega}]) . $$

In the integration it was taken into account that $$ ~J^i = \rho_0 u^i$$, where $$ ~ \rho_0$$ is the mass density in the co-moving reference frame, $$ ~ M$$ is the invariant mass, $$ ~ u^i$$ is the 4-velocity of the particle, $$ ~ p^i$$ is the 4-momentum of the particle, $$ ~\mathbf{p} $$ is the relativistic momentum, $$ ~E $$ is the relativistic energy of the particle. It is also assumed that the mass densities $$ ~ \rho_0$$ and $$ ~ \rho$$ include contributions from the mass-energy of the proper gravitational field and the electromagnetic field of the particle. The obtained 4-force is acting on the particle in the presence of the gravitational field with the tensor $$ ~ {\Phi^\alpha}_{i} $$, and in some cases we can neglect the proper gravitational field of the particle and consider its motion only in the external field.

Relation to the 4-vector of energy-momentum
The gravitational stress-energy tensor contains time-like components $$~ U^{0k}$$, integrating which over the moving volume we can calculate the 4-vector of energy-momentum of the free gravitational field, separated from its sources:
 * $$~ Q^k = \int {\frac { U^{0k}}{ c_g } dV} = (\frac {U}{c_g}, \mathbf {Q}), $$

where $$~ U = \int { U^{00} dV}$$ is the total energy of gravitational field, $$~ \mathbf {Q} = \int { \mathbf {P_g}dV}$$ 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 4-vector of energy-momentum, 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 $$~ R $$ the 4-vector of energy-momentum in view of the proper gravitational field of the body has the form:
 * $$~ p^k = (m_g + \frac {U_0}{2c^2_g}) u^k =M u^k ,$$

where $$~ m_g $$ is the gravitational mass, which equal to the mass $$~ m_b $$, defined through mass density and volume, $$~ U_0 = - \frac {3 G m^2_g }{5R}$$ is the total gravitational energy of the body in the reference frame in which the body is at rest, $$~ M$$ is the invariant inertial mass, which includes the contributions of the mass-energy from all the fields.

It is assumed that the mass $$ ~ M $$ equal to the mass $$ ~ m'$$ 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 $$~ U_0$$ is negative, then the condition holds $$~ m_b =m_g > M =m' $$, 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:
 * $$~ E_0 =Mc^2= c \sqrt {g_{ik}p^i p^k } . $$

A more accurate analysis of the mass of the system shows that the following relationship holds:
 * $$~m' < M < m < m_b = m_g ,$$

where the gauge mass $$~m' $$ is related to the cosmological constant and represents the mass-energy of the matter’s particles in the four-potentials of the system’s fields; the inertial mass $$~M $$; the auxiliary mass $$~m $$ 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 $$~ m_b $$ is used that when we add to it the mass from the gravitational field energy, we obtain the relativistic mass: $$~ M = m_b + \frac {U_0}{c^2}$$, and there is a relation: $$~ m_g = M < m_b  <  m' $$.

Covariant theory of gravitation (CTG)
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 $$~ g^{ik}$$ which becomes the function of coordinates and time. In addition, in the equations the operator of 4-gradient $$~ \partial_k$$ is replaced by the covariant derivative $$~ \nabla_k$$. After replacing $$~ \eta^{ik}$$ by $$~ g^{ik}$$ the gravitational stress-energy tensor becomes in the following form:
 * $$~ U^{ik} = - \frac{c^2_{g}} {4 \pi G }\left( -g^{im}\Phi_{n m}\Phi^{n k}+ \frac{1} {4} g^{ik}\Phi_{mr}\Phi^{mr}\right) .$$

Gravitational field is considered at the same time as a component of general field.

Transforming the contravariant indices in the gravitational tensor $$~ \Phi^{mr}$$ into the covariant indices with the help of the metric tensor, and interchanging some indices, which are summed up, we obtain:
 * $$~ U^{ik} = - \frac{c^2_{g}} {4 \pi G }g^{ms} \left( - g^{ir} g^{n k} + \frac{1} {4} g^{ik} g^{n r} \right) \Phi_{m r}\Phi_{s n}.$$

Since the gravitational tensor $$~ \Phi_{mr}$$ with covariant indices consists of the components of vector of gravitational field strength $$ ~ \mathbf{\Gamma }$$, divided by the speed $$ ~ c_g$$, and the components of vector of gravitational torsion field $$ ~ \mathbf{\Omega}$$, the formula shows that in the curved spacetime the gravitational stress-energy 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 $$~ U^{00}$$ contains mixed products of the form $$~ \Gamma_x \Omega_y $$, 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 four-dimensional space in CTG the gravitational field equations are changed as well as the expression for the 4-vector of gravitational force density (1), while the expression for the gravitational 4-force remains the same:


 * $$~ \nabla_n \Phi_{ik} + \nabla_i \Phi_{kn} + \nabla_k \Phi_{ni}=0, $$


 * $$~\nabla_k \Phi^{ik} = \frac {4 \pi G }{c^2_{g}} J^i ,$$


 * $$~ f^\alpha = -\nabla_\beta U^{\alpha \beta} = {\Phi^\alpha}_{i} J^i, $$


 * $$~ F^\alpha = {\Phi^\alpha}_{i} M u^i= {\Phi^\alpha}_{i} p^i. $$

Equation for the metric
In CTG the metric tensor is determined by solving the equation similar to Hilbert-Einstein equation. In covariant indices this equation can be written as follows:
 * $$~ R_{ik} - \frac{1} {4 }g_{ik}R = \frac{8 \pi G \beta }{ c^4} \left( B_{ik}+ P_{ik}+ U_{ik}+ W_{ik} \right), $$

where $$~ R_{ik}$$ is the Ricci tensor, $$~ R=R_{ik}g^{ik}$$ is the scalar curvature, $$~ \beta $$ is the coefficient to be determined, $$~ B_{ik}$$, $$~ P_{ik}$$, $$~ U_{ik}$$ and $$~ W_{ik}$$ are the stress-energy tensors of the acceleration field, pressure field, gravitational and electromagnetic fields, respectively, and it is assumed that the speed of gravitation $$~ c_g $$ is equal to the speed of light.

In contrast to the general theory of relativity, in this equation, there is no cosmological constant $$~ \Lambda $$, there is an additional constant $$~ \beta $$ and the metric is dependent on the gravitational stress-energy 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.

Equation of motion
Using the principle of least action allows us to deduce not only the formula for the gravitational stress-energy tensor, but also gives the equation of motion written in tensor form:
 * $$~ \nabla_\beta \left( {B_\alpha}^\beta + {U_\alpha}^\beta +{W_\alpha}^\beta + {P_\alpha}^\beta \right) = 0. \qquad (2)$$

The covariant derivative of the acceleration stress-energy tensor defines up to the sign the density of the 4-force acting on the field from the matter. At the same time the operator of proper-time-derivative is applied to the 4-potential $$ ~ U_\alpha $$ of acceleration field in the Riemannian space:


 * $$ ~ f_\alpha = \nabla_\beta {B_\alpha}^\beta = - u_{\alpha k} J^k = \rho_0 \frac {DU_\alpha }{D \tau}- J^k \nabla_\alpha U_k = \rho_0 \frac {dU_\alpha }{d \tau} - J^k \partial_\alpha U_k ,$$

where $$ ~ u_{\alpha k}$$ is the acceleration tensor, $$ ~\tau $$ is the proper dynamic time of the particle in the reference frame at rest.

The total density of the 4-force of the gravitational and electromagnetic fields and pressure field is determined by transfer of the stress-energy tensors of the fields to the right side of the equation of motion (2) and then applying the covariant derivative:
 * $$ ~ f_\alpha = - \nabla_\beta \left( {U_\alpha}^\beta + {W_\alpha}^\beta + {P_\alpha}^\beta \right) = \Phi_{\alpha k} J^k + F_{\alpha k} j^k  + f_{\alpha k} J^k , $$

where $$ ~ F_{\alpha k}$$ is the electromagnetic tensor, $$ ~ f_{\alpha k}$$ is the pressure field tensor, $$~j^k = \rho_{0q} u^k $$ is the 4-vector of the electromagnetic current density, $$~\rho_{0q}$$ is the density of electric charge of the matter unit in the reference frame at rest.

Conservation laws
In the weak field limit, when the covariant derivative can be replaced by the partial derivative, for the time-like component in (2), which has the index $$~ i=0$$, the local conservation law of energy-momentum of matter and gravitational and electromagnetic fields can be written as follows:
 * $$~ \nabla \cdot (\mathbf{ K }+ \mathbf{H}+\mathbf{P}+ \mathbf{F} ) = -\frac{\partial (B^{00}+U^{00}+W^{00}+P^{00} )}{\partial t},$$

where $$~ \mathbf{ K }$$ is the vector of the acceleration field energy flux density, $$~ \mathbf{H}$$ is the Heaviside vector, $$~ \mathbf{ P }$$ is the Poynting vector, $$~ \mathbf{F}$$ 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 space-like components with the index $$~ i=1,2,3$$ 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 non-zero 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 4-vectors 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 four-dimensional space. In flat Minkowski space the covariant divergence of the sum of the tensors becomes an ordinary 4-divergence, to which we can apply the divergence theorem. By this theorem the integral of the 4-divergence of some tensor over the 4-space can be replaced by the integral of the tensor over the hypersurface surrounding the 4-volume, 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 three-dimensional 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:
 * $$~ \mathbb{Q}^i= \int{ \left( B^{i0}+ U^{i0} +W^{i0}+P^{i0} \right) dV }. $$

For these tensor components the integral vector $$ ~ \mathbb {Q}^i $$ vanishes. Vanishing of the four-vector allows us to explain the 4/3 problem, according to which the mass-energy 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, 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. It follows that the integral vector $$~ \mathbb{Q}^i$$ cannot be considered as a four-momentum of the system, determining only the distribution of energy and energy fluxes of the fields of the system.

General theory of relativity (GTR)
In GTR, there is a problem of determining the gravitational stress-energy 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 stress-energy 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 Hilbert-Einstein equations in GTR is to find the metric:
 * $$~ R_{ik} - \frac{1} {2 }g_{ik}R + g_{ik} \Lambda = \frac{8 \pi G }{ c^4} \left( t_{ik} + W_{ik} \right). $$

These equations do not contain the gravitational stress-energy 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 energy-momentum 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 stress-energy tensors of matter $$~ t_{ik} $$ and electromagnetic field $$~ W_{ik} $$ it must give zero divergence so that the conservation law of energy-momentum 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 Landau-Lifshitz pseudotensor corresponds to the mentioned criteria. There is also an Einstein pseudotensor, but it is not symmetrical with respect to the indices. 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 stress-energy tensor, the gravitational energy of a body is determined indirectly. For example, one method is when by the given mass 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 mass-energy of gravitational field as to the manifestation of the metric field. In this case, the components of the metric tensor are used, found in GTR from the Hilbert-Einstein equation for the metric.