Acceleration stress-energy tensor

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.

Definition
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 $$ ~ u_{ik} $$ and the metric tensor $$ ~ g^{ik} $$ by the principle of least action:
 * $$~ B^{ik} = \frac{c^2} {4 \pi \eta } \left( - g^{im} u_{nm} u^{nk}+ \frac {1} {4} g^{ik}u_{mr}u^{mr}\right) ,$$

where $$ ~ \eta $$ 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.

Components of the acceleration stress-energy tensor
Since acceleration tensor consists of the components of the acceleration field strength $$ ~ \mathbf {S} $$ and the solenoidal acceleration vector $$ ~ \mathbf {N} $$, 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:
 * $$~ B^{ik} = \begin{vmatrix} \varepsilon_a & \frac {K_x}{c} & \frac {K_y}{c} & \frac {K_z}{c} \\ c P_{ax} & \varepsilon_a - \frac{S^2_x+c^2 N^2_x}{4\pi \eta } & -\frac{S_x S_y+c^2 N_x N_y }{4\pi\eta } & -\frac{S_x S_z+c^2 N_x N_z }{4\pi\eta } \\ c P_{ay} & -\frac{S_x S_y+c^2 N_x N_y }{4\pi\eta } & \varepsilon_a -\frac{S^2_y+c^2 N^2_y }{4\pi\eta }  & -\frac{S_y S_z+c^2 N_y N_z }{4\pi\eta } \\ c P_{az} & -\frac{S_x S_z+c^2 N_x N_z }{4\pi\eta }  & -\frac{S_y S_z+c^2 N_y N_z }{4\pi\eta } & \varepsilon_a -\frac{S^2_z+c^2 N^2_z }{4\pi\eta }  \end{vmatrix}. $$

The time-like components of the tensor denote:

1) The volumetric energy density of acceleration field
 * $$~ B^{00} = \varepsilon_a = \frac{1}{8 \pi \eta }\left(S^2+ c^2 N^2 \right).$$

2) The vector of momentum density of acceleration field $$ ~\mathbf{P_a} =\frac{ 1}{ c^2} \mathbf{K}, $$ where the vector of energy flux density of acceleration field is
 * $$~\mathbf{K} = \frac{ c^2 }{4 \pi \eta }[\mathbf{S}\times \mathbf{N}].$$

Due to the symmetry of the tensor indices, $$ P^{01}= P^{10}, P^{02}= P^{20}, P^{03}= P^{30}$$, so that $$  \frac{ 1}{ c} \mathbf{K}=  c \mathbf{P_a} .$$

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
 * $$~ \sigma^{p q} = \frac {1}{4 \pi \eta } \left( S^p S^q + c^2 N^p N^q - \frac {1}{2} \delta^{pq} (S^2 + c^2 N^2 ) \right) ,$$

where $$~p,q =1,2,3, $$ the components $$S^1=S_x, $$ $$S^2=S_y, $$ $$S^3=S_z, $$ $$ N^1=N_x, $$ $$N^2=N_y, $$ $$N^3=N_z, $$ the Kronecker delta $$~\delta^{pq}$$ equals 1 if $$~p=q, $$ and equals 0 if $$~p \not=q. $$

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:
 * $$~ \partial_q \sigma^{p q} = - f^p +\frac {1}{c^2} \frac{ \partial K^p}{\partial t}, $$

where $$~ f^p $$ denote the components of the three-dimensional acceleration force density, $$~ K^p $$ – the components of the energy flux density of the acceleration field.

4-force density and field equation
The principle of least action implies that the 4-vector of force density $$ ~ f_\alpha $$ can be found through the acceleration stress-energy tensor, either through the product of acceleration tensor and mass 4-current:
 * $$~ f_\alpha = \nabla_\beta {B_\alpha}^\beta = - u_{\alpha k} J^k. \qquad (1)  $$

The field equations of acceleration field are as follows:
 * $$~ \nabla_n u_{ik} + \nabla_i u_{kn} + \nabla_k u_{ni}=0, $$


 * $$~\nabla_k u^{ik} = -\frac {4 \pi \eta }{c^2} J^i .$$

In the special theory of relativity, according to (1) for the components of the four-force density can be written:
 * $$~ f_\alpha = (- \frac {\mathbf{S} \cdot \mathbf{J} }{c}, - \mathbf{f} ),$$

where $$~ \mathbf{f}= - \rho \mathbf{S} - [\mathbf{J} \times \mathbf{N} ]$$ is the 3-vector of the 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 velocity of the matter unit.

In Minkowski space, the field equations are transformed into four equations for the acceleration field strength $$ ~ \mathbf {S} $$ and solenoidal acceleration vector $$ ~ \mathbf {N} $$


 * $$~\nabla \cdot \mathbf{ S} = 4 \pi \eta \rho,$$


 * $$~\nabla \times \mathbf{ N} = \frac {1 }{c^2}\frac{\partial \mathbf{ S}}{\partial t}+\frac {4 \pi \eta \rho \mathbf{ v}}{c^2},$$


 * $$~\nabla \cdot \mathbf{ N} = 0,$$


 * $$~\nabla \times \mathbf{ S} = - \frac{\partial \mathbf{ N}}{\partial t}.$$

Equation for the metric
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:
 * $$~ R_{ik} - \frac{1} {4 }g_{ik}R = \frac{8 \pi G \beta }{ c^4} \left( B_{ik}+ P_{ik}+ U_{ik}+ W_{ik} \right), $$

where $$~ \beta $$ 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, $$~ G $$ is the gravitational constant.

Equation of motion
The equation of motion of a point particle inside or outside matter can be represented in tensor form, with acceleration stress-energy tensor $$ B^{ik}$$ or acceleration tensor $$ u_{nk}$$ :


 * $$~ - \nabla_k \left( B^{ik}+ U^{ik} +W^{ik}+ P^{ik}  \right) = g^{in}\left(u_{nk} J^k + \Phi_{nk} J^k + F_{nk} j^k  + f_{nk} J^k  \right)  =0. \qquad (2)$$

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

We now recognize that $$ ~ J^k = \rho_{0} u^k $$ is the mass 4-current and the acceleration tensor is defined through the covariant 4-potential as $$~ u _{nk}= \nabla_n U_k - \nabla_k U_n. $$ This gives the following:


 * $$~ \nabla_\beta {B_n}^\beta = - u_{n k} J^k = - \rho_{0} u^k (\nabla_n U_k - \nabla_k U_n)= \rho_{0} \frac {DU_n}{D \tau } - \rho_{0} u^k \nabla_n U_k . \qquad (3)$$

Here operator of proper-time-derivative $$~ u^k \nabla_k = \frac {D}{D \tau }$$ is used, where $$ ~ D $$ is the symbol of 4-differential in curved spacetime, $$ ~ \tau $$ is the proper time, $$ ~ \rho_0 $$ is the mass density in the comoving frame.

Accordingly, the equation of motion (2) becomes:
 * $$~ \rho_{0} \frac {DU_n}{D \tau }- \rho_{0} u^k \nabla_n U_k = - \nabla^k \left(U_{nk} +W_{nk}+ P_{nk}  \right) =  \Phi_{nk} J^k + F_{nk} j^k  + f_{nk} J^k. $$

Time-like component of the equation at $$~ n=0$$ describes the rate of change of the scalar potential of the acceleration field, and spatial component at $$~ n=1{,}2{,}3$$ connects the rate of change of the vector potential of the acceleration field with the force density.

Conservation laws
When the index $$ ~ i = 0 $$ 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:


 * $$~ \nabla \cdot (\mathbf{ K }+ \mathbf{H}+\mathbf{P}+ \mathbf{F} ) = -\frac{\partial (B^{00}+U^{00}+W^{00}+P^{00} )}{\partial t},$$

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.

This equation can be regarded as a local conservation law of energy-momentum of the four fields.

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:
 * $$~ \mathbb{Q}^i= \int{ \left( B^{i0}+ U^{i0} +W^{i0}+P^{i0} \right) dV }. $$

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, 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.

Relativistic mechanics
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: $$~ \phi_{ n \beta }= \rho_0 u_n u_\beta $$. In GR, the tensor $$~ \phi_{ n \beta }$$ is substituted into the equation for the metric and its covariant derivative gives the following:


 * $$~ \nabla^\beta \phi _{n \beta} = \nabla^\beta (\rho_0 u_n u_\beta) = u_n \nabla^\beta J_\beta + \rho_0 u_\beta \nabla^\beta u_n . $$

In GR it is assumed that there is the continuity equation in the form $$~ \nabla^\beta J_\beta =0 .$$ Then, using the operator of proper-time-derivative the covariant derivative of the tensor $$~ \phi_{ n \beta }$$ gives the product of the mass density and four-acceleration, i.e. the density of 4-force:


 * $$~ \nabla^\beta \phi _{n \beta} = \rho_0 u_\beta \nabla^\beta u_n = \rho_0 \frac {Du_n}{D \tau }. \qquad (4)$$

However, the continuity equation is valid only in the special theory of relativity as $$~ \partial^\beta J_\beta = \partial_\beta J^\beta =0 .$$ In curved space-time instead would have to be the equation $$~ \nabla^\beta J_\beta =0 $$, but instead of zero on the right side of this equation there appears an additional non-zero term with Riemann curvature tensor. Consequently, (4 ) is not an exact expression, and tensor $$~ \phi_{ n \beta }$$ 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 $$~ B_{ n \beta }$$ describes well the acceleration field of matter particles in curved Riemannian space-time.