# 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 components of the two three-dimensional vectors – pressure field strength and the solenoidal pressure vector. With the pressure field tensor the pressure stress-energy tensor, the pressure field equations and pressure force in matter are defined. Pressure field is a component of general field.

## Definition

Expression for the pressure field tensor can be found in papers by Sergey Fedosin, [1] where the tensor is defined using 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 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 pressure field, ${\displaystyle ~c}$ – 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 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 convert 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 system K, and the axes of the coordinate systems parallel to each other, the pressure field strength and the solenoidal pressure vector are converted 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 tensor

• ${\displaystyle ~f_{\mu \nu }}$ is the antisymmetric tensor of rank 2, it follows from this condition ${\displaystyle ~f_{\mu \nu }=-f_{\nu \mu }}$. Three of the six independent components of the pressure field tensor 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 the antisymmetry such 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 tensor with itself ${\displaystyle f_{\mu \nu }f^{\mu \nu }}$ is an invariant, and the contraction of tensor product with Levi-Civita symbol as ${\displaystyle {\frac {1}{4}}\varepsilon ^{\mu \nu \sigma \rho }f_{\mu \nu }f_{\sigma \rho }}$ is the pseudoscalar invariant. These invariants in the special 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 Lorentz invariant:
${\displaystyle \det \left(f_{\mu \nu }\right)={\frac {4}{c^{2}}}\left(\mathbf {C} \cdot \mathbf {I} \right)^{2}.}$

## Pressure field

Through the pressure field tensor the equations of pressure field are written:

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

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

Instead of (2) it is possible 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 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 as its divergence vanishes. From (4) follows that the time variation of the solenoidal pressure vector leads to a curl of the pressure field strength.

Equation (3) relates the pressure field to its source in the form of mass 4-current. In Minkowski space of special relativity the form of the equation 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 change in time of the pressure field strength generate the circular field of the solenoidal pressure vector.

From (3) and (1) it can be obtained:[1]

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

The continuity equation for the mass 4-current ${\displaystyle ~\nabla _{\alpha }J^{\alpha }=0}$ is a gauge condition that is used to derive the field equation (3) from the principle of least action. Therefore, the contraction of the acceleration tensor and the Ricci tensor must be zero: ${\displaystyle ~R_{\mu \alpha }f^{\mu \alpha }=0}$. In Minkowski space the Ricci tensor ${\displaystyle ~R_{\mu \alpha }}$ equal to zero, the covariant derivative becomes the partial derivative, and the continuity equation becomes as follows:

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

## Covariant theory of gravitation

### Action and Lagrangian

Total Lagrangian for the matter in gravitational and electromagnetic fields includes the pressure field tensor and is contained in 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 Lagrangian, ${\displaystyle ~dt}$ is differential of 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 4-potential of acceleration field, ${\displaystyle ~\eta }$ and ${\displaystyle ~\sigma }$ are the constants of acceleration field and pressure field, respectively, ${\displaystyle ~u_{\mu \nu }}$ is the acceleration tensor, ${\displaystyle ~\pi _{\mu }}$ is the 4-potential of pressure field, ${\displaystyle ~f_{\mu \nu }}$ is 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 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 matter unit in gravitational and electromagnetic fields and pressure field: [2]

${\displaystyle ~-u_{\beta \sigma }\rho _{0}u^{\sigma }=\rho _{0}{\frac {dU_{\beta }}{d\tau }}-\rho _{0}u^{\sigma }\partial _{\beta }U_{\sigma }=\Phi _{\beta \sigma }\rho _{0}u^{\sigma }+F_{\beta \sigma }\rho _{0q}u^{\sigma }+f_{\beta \sigma }\rho _{0}u^{\sigma },}$

where the first term on the right is the gravitational force density, expressed with the help of the gravitational field tensor, 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 equation of pressure field (3).

### Pressure stress-energy tensor

With the help of 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 stress-energy tensor determines the pressure four-force density:

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

### Generalized velocity and Hamiltonian

Covariant 4-vector of generalized velocity is given by:

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

Given 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 timelike components of 4-vectors ${\displaystyle ~s_{\mu }}$ and ${\displaystyle ~J^{\mu }}$.

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