Pressure stress-energy tensor

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

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

1 Continuum mechanics

1.1 Examples of tensors
1.2 Description of motion and metrics

2 Covariant theory of gravitation

2.1 Definition
2.2 Components of pressure stress-energy tensor
2.3 Pressure force and pressure field equations
2.4 Equation for metric
2.5 Equation of motion
2.6 Conservation laws

3 See also
4 References
5 External links

Continuum mechanics

The existence of different variants of the pressure stress-energy tensor shows absence of any unambiguous definition of this tensor. Besides the 4-velocity, density and pressure in the tensor a function is often added with desired properties such that the tensor can describe the energy and stress in the matter. The arbitrariness of the choice of such function is related to the fact that when the pressure is believed a simple scalar function, there is a need to add 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:

$~ P^{ik} = \frac{p} {c^2 }u^i u^k - g^{ik}p ,$

where is the pressure, is the speed of light, is the four-velosity, $~ g^{ik}$ is the metric tensor.

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

Fock introduces the pressure energy density per unit mass $~ \Pi$ and adds this quantity to the pressure stress-energy tensor: ^[1]

$~ P^{ik} = \frac{p+ \rho^{*} \Pi } {c^2 }u^i u^k - g^{ik}p ,$

here $~ \rho^{*}$ denotes that the mass density, which is independent of pressure, and related to the total invariant mass density $~ \rho_0$ by relation:

$~ \rho^{*}= \frac{ \rho_0 } {1+\Pi /c^2 }.$

Instead of it, Fedosin used a compression function : ^[2]

$~ P^{ik} = \frac{p} {c^2 }u^i u^k + (L-p) g^{ik}.$

There are other forms of pressure stress-energy tensor, differing from each other in the way of introducing some additional scalar function in the tensor. ^[3] ^[4] ^[5]

Description of motion and metrics

The standard approach involves first determining 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 matter and $~ W^{ik}$ is the stress-energy tensor of electromagnetic field. Thereafter, taking in account the pressure and other fields the equation of motion follows as a result of vanishing of covariant derivative of the stress-energy tensor of the system: $~ - \nabla_k T^{ik}=0.$ In general relativity (GR) account of the gravitational field in equation of motion is carried out through the dependence of the metric tensor components on position and time .

Tensor $~ T^{ik}$ is used in GR for finding metrics of 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, is the gravitational constant.

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

Covariant theory of gravitation

Definition

In contrast to the continuum mechanics in covariant theory of gravitation (CTG) pressure field is not a scalar field and considered as 4-vector field consisting of scalar and 3-vector components. Therefore in CTG the pressure stress-energy tensor is defined by the pressure field tensor $~ f_{ik}$ and the metric tensor $~ g^{ik}$ by the principle of least action: ^[6]

$~ P^{ik} = \frac{c^2} {4 \pi \sigma } \left( - g^{im} f_{nm} f^{nk}+ \frac {1} {4} g^{ik}f_{mr}f^{mr}\right) ,$

where $~ \sigma$ is a constant having its own value in each task. The constant $~ \sigma$ is not uniquely defined, and it is a consequence of the fact that the pressure inside the bodies may have been caused by any reasons and both internal and external forces. Pressure field is considered as a component of the general field.^[⁷^]

Components of pressure stress-energy tensor

In the weak field limit, when the space-time metric becomes the Minkowski metric of special 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 stress-energy tensor is greatly simplified and can be expressed in terms of the components of the pressure field tensor, i.e. the pressure field strength $~ \mathbf {C}$ and solenoidal pressure vector $~ \mathbf {I}$ :

$~P^{{ik}}={\begin{vmatrix}\varepsilon _{p}&{\frac {F_{x}}{c}}&{\frac {F_{y}}{c}}&{\frac {F_{z}}{c}}\\{\frac {F_{x}}{c}}&\varepsilon _{p}-{\frac {C_{x}^{2}+c^{2}I_{x}^{2}}{4\pi \sigma }}&-{\frac {C_{x}C_{y}+c^{2}I_{x}I_{y}}{4\pi \sigma }}&-{\frac {C_{x}C_{z}+c^{2}I_{x}I_{z}}{4\pi \sigma }}\\{\frac {F_{y}}{c}}&-{\frac {C_{x}C_{y}+c^{2}I_{x}I_{y}}{4\pi \sigma }}&\varepsilon _{p}-{\frac {C_{y}^{2}+c^{2}I_{y}^{2}}{4\pi \sigma }}&-{\frac {C_{y}C_{z}+c^{2}I_{y}I_{z}}{4\pi \sigma }}\\{\frac {F_{z}}{c}}&-{\frac {C_{x}C_{z}+c^{2}I_{x}I_{z}}{4\pi \sigma }}&-{\frac {C_{y}C_{z}+c^{2}I_{y}I_{z}}{4\pi \sigma }}&\varepsilon _{p}-{\frac {C_{z}^{2}+c^{2}I_{z}^{2}}{4\pi \sigma }}\end{vmatrix}}.$

The time-like components of the tensor denote:

1) The volumetric energy density of pressure field

$~ P^{00} = \varepsilon_p = \frac{1}{8 \pi \sigma }\left(C^2+ c^2 I^2 \right).$

2) The vector of energy flux density of 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 due to the symmetry of the tensor indices $P^{01}= P^{10}, P^{02}= P^{20}, P^{03}= P^{30}$ .

3) The space-like components of the tensor form a submatrix 3 x 3, which is the 3-dimensional stress tensor, 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 the components the Kronecker delta $~\delta^{pq}$ equals 1 if and equals 0 if $~p \not=q.$

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

Three-dimensional divergence of stress tensor of pressure field relates the pressure force density and rate of change of energy flux density of pressure field:

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

where denote the components of the three-dimensional pressure force density, – 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 pressure force density $~ f^\alpha$ can be found through the pressure stress-energy tensor, either through the product of pressure field tensor and 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:

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

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

In the special theory of relativity, according to (1) for the components of the pressure four-force density can be written:

$~ f^\alpha = (\frac {\mathbf{C} \cdot \mathbf{J} }{c}, \mathbf{f} ),$

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 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 4 equations for the pressure field strength $~ \mathbf {C}$ and 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 metric

In the covariant theory of gravitation the pressure 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:^[8]

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

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 matter can be represented in tensor form, with pressure stress-energy tensor $P^{ik}$ or 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 tensor , $~F_{nk}$ is the electromagnetic 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.

Time-like component of the equation (2) at describes the change in the energy and spatial component at connects 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 energy flux. 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: ^[9] ^[10]

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

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 and energy flux is obtained by integrating (2) over the 4-volume to accommodate the energy and energy flux of 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 flux of field of the moving system in 4/3 more than in the field energy of fixed system. On the other hand, according to, ^[10] 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 the energy flux in principle. ^[11]

References

V. A. Fock, The Theory of Space, Time and Gravitation (Pergamon Press, London, 1959).
Fedosin S.G. Fizika i filosofiia podobiia: ot preonov do metagalaktik, Perm: Style-MG, 1999, ISBN 5-8131-0012-1. 544 pages, Tabl.66, Ill.93, Bibl. 377 refs
Herglotz G. // Ann. d. Phys. 1911, Bd 36, S. 493.
Ignatowsky W.V. // Phys. Ztschr. 1911, Bd 12, S. 441.
Lamla E. // Berl. Diss., 1911; Ann. d. Phys. 1912, Bd 37, S. 772.
Fedosin S.G. About the cosmological constant, acceleration field, pressure field and energy. Jordan Journal of Physics. Vol. 9 (No. 1), pp. 1-30 (2016). http://dx.doi.org/10.5281/zenodo.889304.
Fedosin S.G. Two components of the macroscopic general field. Reports in Advances of Physical Sciences, Vol. 1, No. 2, 1750002, 9 pages (2017). http://dx.doi.org/10.1142/S2424942417500025.
Fedosin S.G. Lagrangian formalism in the theory of relativistic vector fields. International Journal of Modern Physics A, Vol. 40, No. 02, 2450163 (2025). https://doi.org/10.1142/S0217751X2450163X.
Fedosin S.G. The Integral Energy-Momentum 4-Vector and Analysis of 4/3 Problem Based on the Pressure Field and Acceleration Field. American Journal of Modern Physics. Vol. 3, No. 4, 2014, pp. 152-167. http://dx.doi.org/10.11648/j.ajmp.20140304.12.
^8.0 ^8.1 Fedosin S.G. The generalized Poynting theorem for the general field and solution of the 4/3 problem. International Frontier Science Letters, Vol. 14, pp. 19-40 (2019). https://doi.org/10.18052/www.scipress.com/IFSL.14.19.
Fedosin S.G. What should we understand by the four-momentum of physical system? Physica Scripta, Vol. 99, No. 5, 055034 (2024). https://doi.org/10.1088/1402-4896/ad3b45.

External links

Pressure stress-energy tensor in Russian

Source: http://sergf.ru/psen.htm

На русском языке