Pressure stress-energy tensor

Pressure field stress-energy tensor is a symmetric tensor of the second valence (rank), which describes the density and the flux of energy and momentum of the pressure 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 acceleration stress-energy tensor, the dissipation stress-energy tensor and the stress-energy tensor of the electromagnetic field. The covariant derivative of the pressure field stress-energy tensor determines the density of the pressure force acting on the matter.

The pressure field stress-energy tensor is relativistic generalization of the three-dimensional Cauchy stress tensor used in continuum mechanics. In contrast to the stress tensor, which is usually used to describe the relative stress emerging during deformations of bodies, the pressure filed stress-energy tensor describes any internal stresses, including stresses in the absence of deformation of bodies from external influences.

Continuum mechanics
Existing of different variants of the pressure field stress-energy tensor shows the absence of any unambiguous definition of this tensor. Besides the 4-velocity, density and pressure, this tensor often includes a function with specified properties so that the tensor could describe the energy and stress in the substance. The arbitrariness of the choice of such function is related to the fact that when the pressure is considered a simple scalar function, there is a need to compensate the vector properties of the pressure forces with the help of some additional function.

Examples of tensors
For matter in equilibrium with uniform pressure, the simplest pressure stress-energy tensor in the metric (+ – – –) can be written as follows:
 * $$~ P^{ik} = \frac{p} {c^2 }u^i u^k - g^{ik}p  ,$$

where $$~ p$$ is the pressure, $$~ c $$ is the speed of light, $$~ u^i $$ is the 4-velocity, $$~ g^{ik} $$ is the metric tensor.

Due to its simplicity, the tensor in this form is often used not only in mechanics, but also in the general theory of relativity.

Fock introduces the elastic energy density per unit mass $$ ~ \Pi $$ and adds this quantity to the pressure stress-energy tensor:
 * $$~ P^{ik} = \frac{p+ \rho^{*} \Pi } {c^2 }u^i u^k - g^{ik}p  ,$$

here $$ ~ \rho^{*} $$ denotes that the mass density, which is independent of pressure and is related to the total invariant mass density $$ ~ \rho_0 $$ by relation:
 * $$~ \rho^{*}= \frac{ \rho_0 } {1+\Pi /c^2 }.$$

Instead of it, Fedosin used the compression function $$~ L$$ :
 * $$~ P^{ik} = \frac{p} {c^2 }u^i u^k + (L-p) g^{ik}.$$

There are other forms of pressure stress-energy tensor, differing from each other in the way of introducing some scalar function into the tensor, additionally to the pressure.

Description of the motion and metric
The standard approach first involves determining the stress-energy tensor of the system $$~ T^{ik}= \phi^{ik}+ P^{ik}+ W^{ik},$$ where $$~ \phi^{ik}= \rho_0 u^i u^k $$ is the stress-energy tensor of the substance and $$ ~ W^{ik} $$ is the stress-energy tensor of the electromagnetic field. Then, taking in account the pressure and other fields the equation of motion is derived from equality to zero of the covariant derivative of the stress-energy tensor of the system: $$~ - \nabla_k T^{ik}=0.$$ In general theory of relativity (GTR) the gravitational field is taken into account in the equation of motion with the help of dependence of the metric tensor components on the coordinates and time.

Tensor $$~ T^{ik}$$ is used in GTR to find the metric from the Hilbert-Einstein equations:
 * $$~ R_{ik} - \frac{1} {2 }g_{ik}R + g_{ik} \Lambda = \frac{8 \pi G } { c^4} T_{ik}, $$

where $$~ R_{ik}={R^n}_{ink}$$ is the Ricci tensor, $$~ R=R_{ik}g^{ik}$$ is the scalar curvature, $$~ G $$ is the gravitational constant.

Thus, the pressure stress-energy tensor changes the metric inside the bodies.

Definition
In contrast to continuum mechanics, in covariant theory of gravitation (CTG) the pressure field is not considered a scalar field but a 4-vector field consisting of the scalar and 3-vector components. Therefore in CTG the pressure field stress-energy tensor is determined with the help of the pressure field tensor $$ ~ f_{ik} $$ and the metric tensor $$ ~ g^{ik} $$ based on the principle of least action:
 * $$~ P^{ik} = \frac{c^2} {4 \pi \sigma } \left( - g^{im} f_{nm} f^{nk}+ \frac {1} {4} g^{ik}f_{mr}f^{mr}\right) ,$$

where $$ ~ \sigma $$ is a constant having its own value in each task. The constant $$ ~ \sigma $$ is not uniquely defined, which is a consequence of the fact that the pressure inside bodies can be caused by any reasons and both internal and external forces. Pressure field is considered as a component of the general field.

Components of the pressure field stress-energy tensor
In the weak field limit, when the spacetime metric becomes the Minkowski metric of the special theory of relativity, the metric tensor $$ ~ g^{ik} $$ becomes the tensor $$ ~ \eta^{ik} $$, consisting of the numbers 0, 1, –1. In this case the form of the pressure field stress-energy tensor is significantly simplified and it can be expressed in terms of the components of the pressure field tensor, i.e. the pressure field strength $$ ~ \mathbf {C} $$ and the solenoidal pressure vector $$ ~ \mathbf {I} $$:


 * $$~ P^{ik} = \begin{vmatrix} \varepsilon_p & \frac {F_x}{c} & \frac {F_y}{c} & \frac {F_z}{c} \\ c P_{px} & \varepsilon_p - \frac{C^2_x+c^2 I^2_x}{4\pi \sigma } & -\frac{C_x C_y+c^2 I_x I_y }{4\pi\sigma } & -\frac{C_x C_z+c^2 I_x I_z }{4\pi\sigma } \\ c P_{py} & -\frac{C_x C_y+c^2 I_x I_y }{4\pi\sigma } & \varepsilon_p -\frac{C^2_y+c^2 I^2_y }{4\pi\sigma }  & -\frac{C_y C_z+c^2 I_y I_z }{4\pi\sigma } \\ c P_{pz} & -\frac{C_x C_z+c^2 I_x I_z }{4\pi\sigma }  & -\frac{C_y C_z+c^2 I_y I_z }{4\pi\sigma } & \varepsilon_p -\frac{C^2_z+c^2 I^2_z }{4\pi\sigma }  \end{vmatrix}. $$

The time components of the tensor denote:

1) The volumetric energy density of the pressure field
 * $$~ P^{00} = \varepsilon_p = \frac{1}{8 \pi \sigma }\left(C^2+ c^2 I^2 \right).$$

2) The vector of the momentum density of the pressure field $$ ~\mathbf{P_p} =\frac{ 1}{ c^2} \mathbf{F}, $$ where the vector of the energy flux density of the pressure field is
 * $$~\mathbf{F} = \frac{ c^2 }{4 \pi \sigma }[\mathbf{C}\times \mathbf{I}].$$

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

3) The space components of the tensor form a submatrix 3 x 3, which is the 3-dimensional stress tensor or the stress tensor of the pressure field, taken with a minus sign. The stress tensor can be written as
 * $$~ \sigma^{p q} = \frac {1}{4 \pi \sigma } \left( C^p C^q + c^2 I^p I^q - \frac {1}{2} \delta^{pq} (C^2 + c^2 I^2 ) \right) ,$$

where $$~p,q =1,2,3, $$ are the components $$C^1=C_x, $$ $$C^2=C_y, $$ $$C^3=C_z, $$ $$ I^1=I_x, $$ $$I^2=I_y, $$ $$I^3=I_z, $$ the Kronecker delta $$~\delta^{pq}$$ equals 1 if $$~p=q, $$ and equals 0 if $$~p \not=q. $$

This stress tensor is a specific expression of Cauchy stress tensor.

Three-dimensional divergence of the stress tensor of the pressure field relates the pressure force density and rate of change of the momentum density of the pressure field:
 * $$~ \partial_q \sigma^{p q} = f^p +\frac {1}{c^2} \frac{ \partial F^p}{\partial t}, $$

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

Pressure force and pressure field equations
The principle of least action implies that the 4-vector of the pressure force density $$ ~ f^\alpha $$ can be found with the help of either the pressure field stress-energy tensor or the product of the pressure field tensor and the mass 4-current:
 * $$~ f^\alpha = -\nabla_\beta P^{\alpha \beta} = {f^\alpha}_{i} J^i . \qquad (1)  $$

Equation (1) is closely related with the pressure field equations:
 * $$~ \nabla_n f_{ik} + \nabla_i f_{kn} + \nabla_k f_{ni}=0, $$


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

Within the special theory of relativity, according to (1), we can write for the components of the pressure four-force density the following:
 * $$~ f^\alpha = (\frac {\mathbf{C} \cdot \mathbf{J} }{c}, \mathbf{f} ),$$

where $$~ \mathbf{f}= \rho \mathbf{C} + [\mathbf{J} \times \mathbf{I} ]$$ is the 3-vector of the pressure force density, $$~\rho$$ is the density of the moving substance, $$~\mathbf{J} =\rho \mathbf{v} $$ is the 3-vector of the mass current density, $$~\mathbf{v} $$ is the 3-vector of the velocity of the substance unit.

In Minkowski space, the field equations are transformed into 4 equations for the pressure field strength $$ ~ \mathbf {C} $$ and the solenoidal pressure vector $$ ~ \mathbf {I} $$


 * $$~\nabla \cdot \mathbf{ C} = 4 \pi \sigma \rho,$$


 * $$~\nabla \times \mathbf{ I} = \frac {1 }{c^2}\frac{\partial \mathbf{ C}}{\partial t}+\frac {4 \pi \sigma \rho \mathbf{ v}}{c^2},$$


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


 * $$~\nabla \times \mathbf{ C} = - \frac{\partial \mathbf{ I}}{\partial t}.$$

Equation for the metric
In the covariant theory of gravitation the pressure field stress-energy tensor in accordance with the principles of metric theory of relativity is one of the tensors defining the metric inside the bodies by means of 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.

Equation of motion
The equation of motion of a point particle inside or outside the substance can be represented in the tensor form, using the pressure field stress-energy tensor $$ P^{ik} $$ or the pressure field tensor $$ f_{nk} $$ :


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

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

The time component of equation (2) at $$ ~ i = 0 $$ describes the energy change and the space component at $$~ i=1{,}2{,}3$$ relates the acceleration with the force density.

Conservation laws
Time-like component in (2) can be considered as the local law of conservation of energy and momentum. In the limit of special relativity, when the covariant derivative becomes the 4-gradient, and the Christoffel symbols vanish, this conservation law takes the simple form:


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

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.

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 integral form of the law of conservation of energy-momentum is obtained by integrating (2) over the 4-volume to accommodate the energy-momentum of the gravitational and electromagnetic fields, extending far beyond the physical system. By the Gauss's formula 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.