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The Principle of Least Action in Covariant Theory of Gravitation

The stress-energy tensor of gravitational field in covariant theory of gravitation:

$U^{\alpha \beta }={\frac {c^{2}}{4\pi G}}\left(g^{\alpha \nu }\Phi _{\kappa \nu }\Phi ^{\kappa \beta }-{\frac {1}{4}}g^{\alpha \beta }\Phi _{\mu \nu }\Phi ^{\mu \nu }\right)=-{\frac {c^{2}}{4\pi G}}\left(\Phi _{\,\,\,\kappa }^{\alpha }\Phi ^{\kappa \beta }+{\frac {1}{4}}g^{\alpha \beta }\Phi _{\mu \nu }\Phi ^{\mu \nu }\right).$

Another form of the stress-energy tensor:

$U_{\mu \nu }=-{\frac {c^{2}}{4\pi G}}g^{\alpha \kappa }\left(-\delta _{\mu }^{\beta }\delta _{\nu }^{\sigma }+{\frac {1}{4}}g_{\mu \nu }g^{\sigma \beta }\right)\Phi _{\alpha \beta }\Phi _{\kappa \sigma }.$

Here $c$ is the speed of light, $G$ is the gravitational constant, $g^{\alpha \beta }$ is the metric tensor, $\Phi _{\mu \nu }$ is the tensor of gravitational field.

About the cosmological constant, acceleration field, pressure field and energy

The action function for continuously distributed matter in an arbitrary frame of reference is:

$~S=\int {Ldt}=\int (kR-2k\Lambda -{\frac {1}{c}}D_{\mu }J^{\mu }+{\frac {c}{16\pi G}}\Phi _{\mu \nu }\Phi ^{\mu \nu }-{\frac {1}{c}}A_{\mu }j^{\mu }-{\frac {c\varepsilon _{0}}{4}}F_{\mu \nu }F^{\mu \nu }-$

$~-{\frac {1}{c}}U_{\mu }J^{\mu }-{\frac {c}{16\pi \eta }}u_{\mu \nu }u^{\mu \nu }-{\frac {1}{c}}\pi _{\mu }J^{\mu }-{\frac {c}{16\pi \sigma }}f_{\mu \nu }f^{\mu \nu }){\sqrt {-g}}d\Sigma ,$

where $L$ is the Lagrange function or Lagrangian; $dt$ is the time differential of the used reference frame; $k$ is some coefficient; $R$ is the scalar curvature; $\Lambda$ is the cosmological constant, which characterizes the energy density of the considered system as a whole, and therefore is the function of the system; $c$ is the speed of light as the measure of the propagation speed of the electromagnetic and gravitational interactions; $D_{\mu }$ is the four-potential of the gravitational field; $J^{\mu }$ is the mass four-current; $G$ is the gravitational constant; $\Phi _{\mu \nu }$ is the tensor of gravitational field; $A_{\mu }=\left({\frac {\varphi }{c}},-\mathbf {A} \right)$ is the electromagnetic 4-potential, where $\varphi$ is the scalar potential and $\mathbf {A}$ is the vector potential; $j^{\mu }$ is the electric four-current; $\varepsilon _{0}$ is the electric constant; $F_{\mu \nu }$ is the electromagnetic tensor; $U_{\mu }$ is the four-potential of the acceleration field; $\eta$ and $\sigma$ are the constants of acceleration field and pressure field, respectively; $u_{\mu \nu }$ is the tensor of acceleration field; $\pi _{\mu }$ is the four-potential of pressure field; $f_{\mu \nu }$ is the tensor of pressure field; ${\sqrt {-g}}d\Sigma ={\sqrt {-g}}cdtdx^{1}dx^{2}dx^{3}$ is the invariant four-volume, expressed through the differential of the time coordinate $dx^{0}=cdt$, through the product $dx^{1}dx^{2}dx^{3}$ of the differentials of the spatial coordinates, and through the square root ${\sqrt {-g}}$ of the determinant $g$ of the metric tensor, taken with the negative sign.

The covariant field equations have the form:

$\nabla _{k}\Phi ^{ik}={\frac {4\pi G}{c^{2}}}J^{i},\qquad \nabla _{n}\Phi _{ik}+\nabla _{i}\Phi _{kn}+\nabla _{k}\Phi _{ni}=0,$

$\nabla _{k}u^{ik}=-{\frac {4\pi \eta }{c^{2}}}J^{i},\qquad \nabla _{n}u_{ik}+\nabla _{i}u_{kn}+\nabla _{k}u_{ni}=0,$

$\nabla _{k}f^{ik}=-{\frac {4\pi \sigma }{c^{2}}}J^{i},\qquad \nabla _{n}f_{ik}+\nabla _{i}f_{kn}+\nabla _{k}f_{ni}=0.$

The application of the principle of least action, taking into account the energy gauge by means of the cosmological constant within the framework of the covariant theory of gravitation and the four vector fields, leads to the equation for finding the metric tensor components:

$R^{\alpha \beta }-{\frac {1}{4}}Rg^{\alpha \beta }=-{\frac {1}{2ck}}\left(U^{\alpha \beta }+W^{\alpha \beta }+B^{\alpha \beta }+P^{\alpha \beta }\right),$

where $R^{\alpha \beta }$ is the Ricci tensor; $R$ is the scalar curvature; $g^{\alpha \beta }$ is the metric tensor; $c$ is the speed of light, $k=-{\frac {c^{3}}{16\pi G\beta }}$, $G$ is the gravitational constant; $\beta$ is a certain coefficient of the order of unity to be determined; $U^{\alpha \beta }$, $W^{\alpha \beta }$, $B^{\alpha \beta }$ and $P^{\alpha \beta }$ are the stress-energy tensors of the gravitational and electromagnetic fields, the acceleration field and the pressure field, respectively.

$B^{\alpha \beta }={\frac {c^{2}}{4\pi \eta }}\left(-g^{\alpha \nu }u_{\kappa \nu }u^{\kappa \beta }+{\frac {1}{4}}g^{\alpha \beta }u_{\mu \nu }u^{\mu \nu }\right).$

$P^{\alpha \beta }={\frac {c^{2}}{4\pi \sigma }}\left(-g^{\alpha \nu }f_{\kappa \nu }f^{\kappa \beta }+{\frac {1}{4}}g^{\alpha \beta }f_{\mu \nu }f^{\mu \nu }\right).$

Four-Dimensional Equation of Motion for Viscous Compressible and Charged Fluid with Regard to the Acceleration Field, Pressure Field and Dissipation Field

Covariant equations of motion of matter particles with tensors of fields:

$-u_{\alpha k}J^{k}=\rho _{0}{\frac {dU_{\alpha }}{d\tau }}-J^{k}\partial _{\alpha }U_{k}=\Phi _{\alpha k}J^{k}+F_{\alpha k}j^{k}+f_{\alpha k}J^{k}+h_{\alpha k}J^{k},$

where $\rho _{0}$ is the invariant mass density; $u_{\mu \nu }=\nabla _{\mu }U_{\nu }-\nabla _{\nu }U_{\mu }={\frac {\partial U_{\nu }}{\partial x^{\mu }}}-{\frac {\partial U_{\mu }}{\partial x^{\nu }}}$ is the tensor of acceleration field; $U_{\mu }=\left({\frac {\vartheta }{c}},-\mathbf {U} \right)$ is the four-potential of acceleration field; $\vartheta$ is the scalar potential, $\mathbf {U}$ is the vector potential of acceleration field; $h_{\alpha k}$ is the tensor of dissipation field, and the operator of proper-time-derivative is used.

Another form of covariant equations of motion:

$-u_{\alpha k}J^{k}=\nabla _{k}{B_{\alpha }}^{k}=-\nabla _{k}\left({U_{\alpha }}^{k}+{W_{\alpha }}^{k}+{P_{\alpha }}^{k}+{Q_{\alpha }}^{k}\right),$

where

$Q^{\alpha \beta }={\frac {c^{2}}{4\pi \tau }}\left(-g^{\alpha \nu }h_{\kappa \nu }h^{\kappa \beta }+{\frac {1}{4}}g^{\alpha \beta }h_{\mu \nu }h^{\mu \nu }\right)$

is the stress-energy tensor of the dissipation field, $\tau$ is the constant of the dissipation field.

The relativistic energy of system of particles and fields:

$E={\frac {1}{c}}\int {(\rho _{0}\psi +\rho _{0q}\varphi +\rho _{0}\vartheta +\rho _{0}\wp ++\rho _{0}\varepsilon )u^{0}{\sqrt {-g}}dx^{1}dx^{2}dx^{3}-}$

$-\int {\left({\frac {c^{2}}{16\pi G}}\Phi _{\mu \nu }\Phi ^{\mu \nu }-{\frac {c^{2}\varepsilon _{0}}{4}}F_{\mu \nu }F^{\mu \nu }-{\frac {c^{2}}{16\pi \eta }}u_{\mu \nu }u^{\mu \nu }-{\frac {c^{2}}{16\pi \sigma }}f_{\mu \nu }f^{\mu \nu }-{\frac {c^{2}}{16\pi \tau }}h_{\mu \nu }h^{\mu \nu }\right){\sqrt {-g}}dx^{1}dx^{2}dx^{3}},$

where $\psi$, $\varphi$, $\vartheta$, $\wp$ and $\varepsilon$ are the scalar potentials of gravitational field, electromagnetic field, acceleration field, pressure field and dissipation field, respectively.

Equations of Motion in the Theory of Relativistic Vector Fields

The wave equation for electromagnetic tensor $F_{\mu \nu }$ in curved space-time:

$\nabla ^{\sigma }\nabla _{\sigma }F_{\mu \nu }=\mu _{0}\nabla _{\mu }j_{\nu }-\mu _{0}\nabla _{\nu }j_{\mu }+F_{\nu \rho }R_{\,\,\mu }^{\rho }-F_{\mu \rho }R_{\,\,\nu }^{\rho }+R_{\mu \nu \lambda \eta }F^{\eta \lambda },$

where $\mu _{0}$ is the magnetic constant, $j_{\mu }$ is the charge four-current, $R_{\,\,\mu }^{\rho }$ is the Ricci tensor, $R_{\mu \nu \lambda \eta }$ is the Riemann curvature tensor.

Covariant equations of motion of matter particles with strengths of fields in the matter:

$\mathbf {S} \cdot \mathbf {v} +\mathbf {\Gamma } \cdot \mathbf {v} +{\frac {\rho _{0q}}{\rho _{0}}}\mathbf {E} \cdot \mathbf {v} +\mathbf {C} \cdot \mathbf {v} =0,$

$\mathbf {S} +[\mathbf {v} \times \mathbf {N} ]+\mathbf {\Gamma } +[\mathbf {v} \times \mathbf {\Omega } ]+{\frac {\rho _{0q}}{\rho _{0}}}\left(\mathbf {E} +[\mathbf {v} \times \mathbf {B} ]\right)+\mathbf {C} +[\mathbf {v} \times \mathbf {I} ]=0,$

where $\mathbf {S}$ is the strength of the acceleration field, $\mathbf {v}$ is the velocity of particles, $\mathbf {\Gamma }$ is the strength of the gravitational field, $\rho _{0q}$ is invariant charge density, $\rho _{0}$ is invariant mass density, $\mathbf {E}$ is the strength of the electromagnetic field, $\mathbf {C}$ is the strength of the pressure field, $\mathbf {N}$ is the solenoidal vector of the acceleration field, $\mathbf {\Omega }$ is the solenoidal vector of the gravitational field or torsion field, $\mathbf {B}$ is the magnetic field, $\mathbf {I}$ is the solenoidal vector of the pressure field.

Covariant equations of motion of matter particles with potentials of fields in the matter:

${\frac {d(\vartheta +\psi +\wp )}{dt}}+{\frac {\rho _{0q}}{\rho _{0}}}{\frac {d\varphi }{dt}}={\frac {dx^{\nu }}{dt}}{\frac {\partial U_{\nu }}{\partial t}}+{\frac {dx^{\nu }}{dt}}{\frac {\partial D_{\nu }}{\partial t}}+{\frac {\rho _{0q}}{\rho _{0}}}{\frac {dx^{\nu }}{dt}}{\frac {\partial A_{\nu }}{\partial t}}+{\frac {dx^{\nu }}{dt}}{\frac {\partial \pi _{\nu }}{\partial t}},$

${\frac {d(\mathbf {U} +\mathbf {D} +\mathbf {\Pi } )}{dt}}+{\frac {\rho _{0q}}{\rho _{0}}}{\frac {d\mathbf {A} }{dt}}=-{\frac {dx^{\nu }}{dt}}\partial _{i}U_{\nu }-{\frac {dx^{\nu }}{dt}}\partial _{i}D_{\nu }-{\frac {\rho _{0q}}{\rho _{0}}}{\frac {dx^{\nu }}{dt}}\partial _{i}A_{\nu }-{\frac {dx^{\nu }}{dt}}\partial _{i}\pi _{\nu },$

where $\vartheta$, $\psi$, $\wp$ and $\varphi$ are the scalar potentials of the acceleration field, of the gravitational field, of the pressure field and of the electromagnetic field, respectively; $dx^{\nu }$ is the displacement four-vector; $U_{\nu }$, $D_{\nu }$, $A_{\nu }$ and $\pi _{\nu }$ are the four-potentials of the acceleration field, of the gravitational field, of the electromagnetic field and of the pressure field, respectively; $\mathbf {U}$, $\mathbf {D}$, $\mathbf {\Pi }$ and $\mathbf {A}$ are the vector potentials of the acceleration field, of the gravitational field, of the pressure field and of the electromagnetic field, respectively; index $i=1,2,3$.

The relativistic uniform model: the metric of the covariant theory of gravitation inside a body

The standard expression for the square of the interval between two close points in all metric theories is the following:

$ds^{2}\ =\ g_{\mu \nu }(x)\ dx^{\mu }\ dx^{\nu }.$

For the static metric with the spherical coordinates $x^{0}=ct,$ $x^{1}=r,$ $x^{2}=\theta ,$ $x^{3}=\phi ,$ there are four nonzero components of the metric tensor: $g_{00},$ $g_{11},$ $g_{22},$ and $g_{33}=g_{22}\sin ^{2}\theta .$ As a result, there is

$ds^{2}\ =g_{00}c^{2}dt^{2}+g_{11}dr^{2}+g_{22}d\theta ^{2}+g_{22}\sin ^{2}\theta d\phi ^{2}.$

As it was found for the components of the metric inside a spherical body within the framework of the relativistic uniform model, $g_{22}=-r^{2},$ and

$(g_{00})_{i}=-{\frac {1}{(g_{11})_{i}}}=1+{\frac {8\pi G\beta r^{2}}{3c^{4}}}\left(\rho _{0}c^{2}\gamma _{c}+\rho _{0}\psi _{a}-{\frac {Gm\rho _{0}\gamma _{c}}{2a}}+\rho _{0q}\varphi _{a}+{\frac {q\rho _{0q}\gamma _{c}}{8\pi \varepsilon _{0}a}}+\rho _{0}\wp _{c}\right),$

where $G$ is the gravitational constant; $\beta$ is the coefficient to be determined; $r$ is the radial coordinate; $c$ is the speed of light; $\rho _{0}$ is the invariant mass density of matter particles, moving inside the body; $\gamma _{c}$ is the Lorentz factor of particles moving at the center of body; $\psi _{a}=-{\frac {Gm_{g}}{a}}$ is the gravitational potential at the surface of sphere with radius $a$ and gravitational mass $m_{g}$; quantities $m={\frac {4\pi a^{3}\rho _{0}}{3}}$ and $q={\frac {4\pi a^{3}\rho _{0q}}{3}}$ are auxiliary values; $\rho _{0q}$ is the invariant charge density of matter particles, moving inside the body; $\varphi _{a}={\frac {q_{b}}{4\pi \varepsilon _{0}a}}$ is the electric scalar potential at the surface of sphere with total charge $q_{b}$; $\wp _{c}$ is the potential of pressure field at the center of body.

On the surface of the body, with $r=a$, the component $(g_{00})_{i}$ of the metric tensor inside the body must be equal to the component $(g_{00})_{o}$ of the metric tensor outside the body. This allows us to refine the expression for the metric tensor components outside the body:

$(g_{00})_{o}=-{\frac {1}{(g_{11})_{o}}}=1+{\frac {2Gm\gamma _{c}\beta }{c^{2}r}}+{\frac {2G\beta }{c^{4}r}}\left(m\psi _{a}+{\frac {1}{2}}m_{g}(\psi -\psi _{a})-{\frac {Gm^{2}\gamma _{c}}{2a}}+q\varphi _{a}+{\frac {1}{2}}q_{b}(\varphi -\varphi _{a})+{\frac {q^{2}\gamma _{c}}{8\pi \varepsilon _{0}a}}+m\wp _{c}\right),$

where $\psi =-{\frac {Gm_{g}}{r}}$ is the gravitational potential outside the body; $\varphi ={\frac {q_{b}}{4\pi \varepsilon _{0}r}}$ is the electric potential outside the body.