# Pressure field tensor

The pressure field tensor is an antisymmetric tensor describing the pressure field and consisting of six components. Tensor components are at the same time the components of two three-dimensional vectors – the pressure field strength and the solenoidal pressure vector. The pressure field stress-energy tensor, the pressure field equations and the pressure force in the substance are determined with the help of the pressure field tensor. Pressure field is a component of general field.

## Definition

The expression for the pressure field tensor can be found in papers by Sergey Fedosin, [1] where the tensor is defined using the 4-curl:

${\displaystyle f_{\mu \nu }=\nabla _{\mu }\pi _{\nu }-\nabla _{\nu }\pi _{\mu }={\frac {\partial \pi _{\nu }}{\partial x^{\mu }}}-{\frac {\partial \pi _{\mu }}{\partial x^{\nu }}}.\qquad \qquad (1)}$

Here the pressure 4-potential ${\displaystyle ~\pi _{\mu }}$ is given by:

${\displaystyle ~\pi _{\mu }=\left({\frac {\wp }{c}},-\mathbf {\Pi } \right),}$

where ${\displaystyle ~\wp }$ is the scalar potential, ${\displaystyle ~\mathbf {\Pi } }$ is the vector potential of the pressure field, ${\displaystyle ~c}$ is the speed of light.

## Expression for the components

The pressure field strength and the solenoidal pressure vector are found with the help of (1):

${\displaystyle ~C_{i}=c(\partial _{0}\pi _{i}-\partial _{i}\pi _{0}),}$
${\displaystyle ~I_{k}=\partial _{i}\pi _{j}-\partial _{j}\pi _{i},}$

and the same in vector notation:

${\displaystyle ~\mathbf {C} =-\nabla \wp -{\frac {\partial \mathbf {\Pi } }{\partial t}},}$
${\displaystyle ~\mathbf {I} =\nabla \times \mathbf {\Pi } .}$

The pressure field tensor consists of the components of these vectors:

${\displaystyle ~f_{\mu \nu }={\begin{vmatrix}0&{\frac {C_{x}}{c}}&{\frac {C_{y}}{c}}&{\frac {C_{z}}{c}}\\-{\frac {C_{x}}{c}}&0&-I_{z}&I_{y}\\-{\frac {C_{y}}{c}}&I_{z}&0&-I_{x}\\-{\frac {C_{z}}{c}}&-I_{y}&I_{x}&0\end{vmatrix}}.}$

The transition to the pressure field tensor with contravariant indices is carried out by multiplying by double metric tensor:

${\displaystyle ~f^{\alpha \beta }=g^{\alpha \nu }g^{\mu \beta }f_{\mu \nu }.}$

In the special theory of relativity, this tensor has the form:

${\displaystyle ~f^{\alpha \beta }={\begin{vmatrix}0&-{\frac {C_{x}}{c}}&-{\frac {C_{y}}{c}}&-{\frac {C_{z}}{c}}\\{\frac {C_{x}}{c}}&0&-I_{z}&I_{y}\\{\frac {C_{y}}{c}}&I_{z}&0&-I_{x}\\{\frac {C_{z}}{c}}&-I_{y}&I_{x}&0\end{vmatrix}}.}$

To transform the components of the pressure field tensor from one inertial system to another we must take into account the transformation rule for tensors. If the reference frame K' moves with an arbitrary constant velocity ${\displaystyle ~\mathbf {V} }$ with respect to the fixed reference frame K, and the axes of the coordinate systems are parallel to each other, the pressure field strength and the solenoidal pressure vector are transformed as follows:

${\displaystyle \mathbf {C} ^{\prime }={\frac {\mathbf {V} }{V^{2}}}(\mathbf {V} \cdot \mathbf {C} )+{\frac {1}{\sqrt {1-{V^{2} \over c^{2}}}}}\left(\mathbf {C} -{\frac {\mathbf {V} }{V^{2}}}(\mathbf {V} \cdot \mathbf {C} )+[\mathbf {V} \times \mathbf {I} ]\right),}$
${\displaystyle \mathbf {I} ^{\prime }={\frac {\mathbf {V} }{V^{2}}}(\mathbf {V} \cdot \mathbf {I} )+{\frac {1}{\sqrt {1-{V^{2} \over c^{2}}}}}\left(\mathbf {I} -{\frac {\mathbf {V} }{V^{2}}}(\mathbf {V} \cdot \mathbf {I} )-{\frac {1}{c^{2}}}[\mathbf {V} \times \mathbf {C} ]\right).}$

## Properties of the tensor

• ${\displaystyle ~f_{\mu \nu }}$ is the antisymmetric tensor of rank 2, from this the condition follows: ${\displaystyle ~f_{\mu \nu }=-f_{\nu \mu }}$. Three of the six independent components of the pressure field tensor are associated with the components of the pressure field strength ${\displaystyle ~\mathbf {C} }$, and the other three – with the components of the solenoidal pressure vector ${\displaystyle ~\mathbf {I} }$. Due to antisymmetry such an invariant as the contraction of the tensor with the metric tensor vanishes: ${\displaystyle ~g^{\mu \nu }f_{\mu \nu }=f_{\mu }^{\mu }=0}$.
• Contraction of the tensor with itself ${\displaystyle f_{\mu \nu }f^{\mu \nu }}$ is an invariant, and contraction of the tensor product with Levi-Civita symbol as ${\displaystyle {\frac {1}{4}}\varepsilon ^{\mu \nu \sigma \rho }f_{\mu \nu }f_{\sigma \rho }}$ is a pseudoscalar invariant. These invariants in the special theory of relativity can be expressed as follows:
${\displaystyle f_{\mu \nu }f^{\mu \nu }=-{\frac {2}{c^{2}}}(C^{2}-c^{2}I^{2})=inv,}$
${\displaystyle {\frac {1}{4}}\varepsilon ^{\mu \nu \sigma \rho }f_{\mu \nu }f_{\sigma \rho }=-{\frac {2}{c}}\left(\mathbf {C} \cdot \mathbf {I} \right)=inv.}$
• Determinant of the tensor is also a Lorentz invariant:
${\displaystyle \det \left(f_{\mu \nu }\right)={\frac {4}{c^{2}}}\left(\mathbf {C} \cdot \mathbf {I} \right)^{2}.}$

## Pressure field

The equations of pressure field are written with the pressure field tensor:

${\displaystyle \nabla _{\sigma }f_{\mu \nu }+\nabla _{\mu }f_{\nu \sigma }+\nabla _{\nu }f_{\sigma \mu }={\frac {\partial f_{\mu \nu }}{\partial x^{\sigma }}}+{\frac {\partial f_{\nu \sigma }}{\partial x^{\mu }}}+{\frac {\partial f_{\sigma \mu }}{\partial x^{\nu }}}=0.\qquad \qquad (2)}$
${\displaystyle ~\nabla _{\nu }f^{\mu \nu }=-{\frac {4\pi \sigma }{c^{2}}}J^{\mu },\qquad \qquad (3)}$

where ${\displaystyle J^{\mu }=\rho _{0}u^{\mu }}$ is the mass 4-current, ${\displaystyle \rho _{0}}$ is the mass density in the comoving reference frame, ${\displaystyle u^{\mu }}$ is the 4-velocity, ${\displaystyle ~\sigma }$ is a constant.

Instead of (2) it is possible to use the expression:

${\displaystyle ~\varepsilon ^{\mu \nu \sigma \rho }{\frac {\partial f_{\mu \nu }}{\partial x^{\sigma }}}=0.}$

Equation (2) is satisfied identically, which is proved by substituting into it the definition for the pressure field tensor according to (1). If in (2) we insert the tensor components ${\displaystyle f_{\mu \nu }}$, this leads to two vector equations:

${\displaystyle ~\nabla \times \mathbf {C} =-{\frac {\partial \mathbf {I} }{\partial t}},\qquad \qquad (4)}$
${\displaystyle ~\nabla \cdot \mathbf {I} =0.\qquad \qquad (5)}$

According to (5), the solenoidal pressure vector has no sources since its divergence vanishes. From (4) it follows that the time variation of the solenoidal pressure vector leads emerging of the curl of the pressure field strength.

Equation (3) relates the pressure field to its source in the form of the mass 4-current. In Minkowski space of the special theory of relativity the equation form is simplified and becomes:

${\displaystyle ~\nabla \cdot \mathbf {C} =4\pi \sigma \rho ,}$
${\displaystyle ~\nabla \times \mathbf {I} ={\frac {1}{c^{2}}}\left(4\pi \sigma \mathbf {J} +{\frac {\partial \mathbf {C} }{\partial t}}\right),}$

where ${\displaystyle ~\rho }$ is the density of moving mass, ${\displaystyle ~\mathbf {J} }$ is the density of mass current.

According to the first of these equations, the pressure field strength is generated by the mass density, and according to the second equation the mass current or the change in time of the pressure field strength generate the circular field of the solenoidal pressure vector.

From (3) and (1) can be obtained continuity equation:

${\displaystyle ~R_{\mu \alpha }f^{\mu \alpha }={\frac {4\pi \sigma }{c^{2}}}\nabla _{\alpha }J^{\alpha }.}$

This equation means that thanks to the curvature of space-time when the Ricci tensor ${\displaystyle ~R_{\mu \alpha }}$ is non-zero, the pressure field tensor ${\displaystyle ~f^{\mu \alpha }}$ is a possible source of divergence of mass 4-current. If space-time is flat, as in Minkowski space, the left side of the equation is set to zero, the covariant derivative becomes the 4-gradient and remains the following:

${\displaystyle ~\partial _{\alpha }J^{\alpha }={\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0.}$

## Covariant theory of gravitation

### Action and Lagrangian

The total Lagrangian for the substance in the gravitational and electromagnetic fields includes the pressure field tensor and is part of the action function: [1]

${\displaystyle ~S=\int {Ldt}=\int (kR-2k\Lambda -{\frac {1}{c}}D_{\mu }J^{\mu }+{\frac {c}{16\pi G}}\Phi _{\mu \nu }\Phi ^{\mu \nu }-{\frac {1}{c}}A_{\mu }j^{\mu }-{\frac {c\varepsilon _{0}}{4}}F_{\mu \nu }F^{\mu \nu }-}$
${\displaystyle ~-{\frac {1}{c}}u_{\mu }J^{\mu }-{\frac {c}{16\pi \eta }}u_{\mu \nu }u^{\mu \nu }-{\frac {1}{c}}\pi _{\mu }J^{\mu }-{\frac {c}{16\pi \sigma }}f_{\mu \nu }f^{\mu \nu }){\sqrt {-g}}d\Sigma ,}$

where ${\displaystyle ~L}$ is the Lagrangian, ${\displaystyle ~dt}$ is the differential of the coordinate time, ${\displaystyle ~k}$ is a certain coefficient, ${\displaystyle ~R}$ is the scalar curvature, ${\displaystyle ~\Lambda }$ is the cosmological constant, which is a function of the system, ${\displaystyle ~c}$ is the speed of light as a measure of the propagation speed of electromagnetic and gravitational interactions, ${\displaystyle ~D_{\mu }}$ is the gravitational four-potential, ${\displaystyle ~G}$ is the gravitational constant, ${\displaystyle ~\Phi _{\mu \nu }}$ is the gravitational tensor, ${\displaystyle ~A_{\mu }}$ is the electromagnetic 4-potential, ${\displaystyle ~j^{\mu }}$ is the electromagnetic 4-current, ${\displaystyle ~\varepsilon _{0}}$ is the electric constant, ${\displaystyle ~F_{\mu \nu }}$ is the electromagnetic tensor, ${\displaystyle ~u_{\mu }}$ is the covariant 4-velocity, ${\displaystyle ~\eta }$ and ${\displaystyle ~\sigma }$ are some constants, ${\displaystyle ~u_{\mu \nu }}$ is the acceleration tensor, ${\displaystyle ~\pi _{\mu }}$ is the 4-potential of pressure field, ${\displaystyle ~f_{\mu \nu }}$ is the pressure field tensor, ${\displaystyle ~{\sqrt {-g}}d\Sigma ={\sqrt {-g}}cdtdx^{1}dx^{2}dx^{3}}$ is the invariant 4-volume, ${\displaystyle ~{\sqrt {-g}}}$ is the square root of the determinant ${\displaystyle ~g}$ of the metric tensor, taken with a negative sign, ${\displaystyle ~dx^{1}dx^{2}dx^{3}}$ is the product of differentials of the spatial coordinates.

The variation of the action function by 4-coordinates leads to the equation of motion of the substance unit in the gravitational and electromagnetic fields and the pressure field:

${\displaystyle ~\rho _{0}a_{\beta }=\rho _{0}{\frac {Du_{\beta }}{D\tau }}=\rho _{0}u^{k}\nabla _{k}u_{\beta }=\rho _{0}{\frac {du_{\beta }}{d\tau }}-\rho _{0}\Gamma _{k\beta }^{s}u^{k}u_{s}=\Phi _{\beta \sigma }\rho _{0}u^{\sigma }+F_{\beta \sigma }\rho _{0q}u^{\sigma }+f_{\beta \sigma }\rho _{0}u^{\sigma },}$

where ${\displaystyle ~a_{\beta }}$ is the 4-acceleration with the covariant index, the operator of proper-time-derivative with respect to the proper time ${\displaystyle ~\tau }$ is used, the first term on the right is the gravitational force density, expressed with the help of the gravitational field tensor, the second term is the Lorentz electromagnetic force density for the charge density ${\displaystyle ~\rho _{0q}}$ measured in the comoving reference frame, and the last term sets the pressure force density.

If we vary the action function by the pressure 4-potential, we obtain the pressure field equation (3).

### Pressure field stress-energy tensor

With the help of the pressure field tensor in the covariant theory of gravitation the pressure stress-energy tensor is constructed:

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

The covariant derivative of the pressure field stress-energy tensor determines the pressure four-force density:

${\displaystyle ~f^{\alpha }=-\nabla _{\beta }P^{\alpha \beta }=f_{k}^{\alpha }J^{k}.}$

### Generalized velocity and Hamiltonian

The covariant 4-vector of generalized velocity is given by:

${\displaystyle ~s_{\mu }=u_{\mu }+D_{\mu }+{\frac {\rho _{0q}}{\rho _{0}}}A_{\mu }+\pi _{\mu }.}$

With regard to the generalized 4-velocity, the Hamiltonian contains the pressure field tensor and has the form:

${\displaystyle ~H=\int {(s_{0}J^{0}-{\frac {c^{2}}{16\pi G}}\Phi _{\mu \nu }\Phi ^{\mu \nu }+{\frac {c^{2}\varepsilon _{0}}{4}}F_{\mu \nu }F^{\mu \nu }+{\frac {c^{2}}{16\pi \eta }}u_{\mu \nu }u^{\mu \nu }+{\frac {c^{2}}{16\pi \sigma }}f_{\mu \nu }f^{\mu \nu }){\sqrt {-g}}dx^{1}dx^{2}dx^{3}},}$

where ${\displaystyle ~s_{0}}$ and ${\displaystyle ~J^{0}}$ are the time components of the 4-vectors ${\displaystyle ~s_{\mu }}$ and ${\displaystyle ~J^{\mu }}$.

In the reference frame that is fixed relative to the system's center of mass, the Hamiltonian will determine the invariant energy of the system.