1 INTRODUCTION

In classical texts (see [1–4]), equations for fields are proposed without deriving right-hand sides. Below, the right-hand sides of the Maxwell and Einstein equations are derived within the framework of the Vlasov–Maxwell–Einstein equations from a classical, but slightly more general principle of least action and Hamilton–Jacobi equations are used to obtain cosmological solutions. Thus, the energy–momentum tensor and a closed system of gravitation and electrodynamic equations have been derived from the principle of least action for the first time.

2 ACTION IN GENERAL RELATIVITY AND EQUATIONS FOR FIELDS

Let \(f(t,{\mathbf{x}},{\mathbf{v}},m,e)\) be the distribution function of particles over space \({\mathbf{x}} \in {{\mathbb{R}}^{3}}\), velocities \({\mathbf{v}} \in {{\mathbb{R}}^{3}}\), masses \(m \in \mathbb{R}\), and charges \(e \in \mathbb{R}\) at time \(t \in \mathbb{R}\). This means that the number of particles in the volume \(d{\mathbf{x}}d{\mathbf{v}}dmde\) is equal to \(f(t,{\mathbf{x}},{\mathbf{v}},m,e)d{\mathbf{x}}d{\mathbf{v}}dmde\). Consider the action

$$\begin{gathered} S = - \,c\int {\mkern 1mu} m{\mkern 1mu} f\left( {t,{\mathbf{x}},{\mathbf{v}},m,e} \right)\sqrt {{{g}_{{\mu \nu }}}{\mkern 1mu} {{u}^{\mu }}{\mkern 1mu} {{u}^{\nu }}} {\mkern 1mu} {{d}^{3}}x{\mkern 1mu} {{d}^{3}}{v}{\mkern 1mu} dm{\mkern 1mu} de{\mkern 1mu} dt \\ - \,\frac{1}{c}\int {\mkern 1mu} e{\mkern 1mu} f\left( {t,{\mathbf{x}},{\mathbf{v}},m,e} \right){\mkern 1mu} {{A}_{\mu }}{\mkern 1mu} {{u}^{\mu }}{\mkern 1mu} {{d}^{3}}x{\mkern 1mu} {{d}^{3}}{v}{\mkern 1mu} dm{\kern 1pt} de{\mkern 1mu} dt \\ + \,{{k}_{1}}\int {\left( {R + \Lambda } \right)} {\mkern 1mu} \sqrt { - g} {\mkern 1mu} {{d}^{4}}x\, + \,{{k}_{2}}\int {{{F}_{{\mu \nu }}}} {{F}^{{\mu \nu }}}{\mkern 1mu} \sqrt { - g} {\mkern 1mu} {{d}^{4}}x, \\ \end{gathered} $$
(1)

where \(c\) is the speed of light, \({{u}^{0}} = c\), \({{u}^{i}} = {{{v}}^{i}}\) (i = 1–3) is the three-dimensional velocity, \({{x}^{0}} = ct,\) \({{x}^{i}}\) (i = 1–3) is the coordinate, \({{g}_{{\mu \nu }}}({\mathbf{x}},t)\) is the metric \((\mu ,\nu = 0{\text{--}}3)\), \({{A}_{\mu }}({\mathbf{x}},t)\) is the electromagnetic field 4-potential, \({{F}_{{\mu \nu }}}({\mathbf{x}},t)\,\, = \,\,\partial {{A}_{\mu }}({\mathbf{x}},t){\text{/}}\partial {{x}^{\nu }}\, - \,\partial {{A}_{\nu }}({\mathbf{x}},t){\text{/}}\partial {{x}^{\mu }}\) are electromagnetic fields, \(R\) is the total curvature, \(\Lambda \) is Einstein’s lambda term, \({{k}_{1}} = - \frac{{{{c}^{3}}}}{{16\pi \gamma }}\) and k2 = \( - \frac{1}{{16\pi c}}\) are constants [1–4], \(g\) is the determinant of the metric \({{g}_{{\mu \nu }}},\) and γ is the gravitational constant; as usual, summation is implied over repeated indices.

The form of action (1) is convenient for obtaining the Einstein and Maxwell equations by variation over the fields \({{g}_{{\mu \nu }}}\) and \({{A}_{\mu }}\). This method for deriving the Vlasov–Maxwell and Vlasov–Einstein equations was used in [5–11]. The variation of (1) over \({{g}_{{\mu \nu }}}\) yields the Einstein equation

$$\begin{gathered} {{k}_{1}}\left( {{{R}^{{\mu \nu }}} - \frac{1}{2}{{g}^{{\mu \nu }}}(R + \Lambda )} \right)\sqrt { - g} \\ = \int m \frac{{f(t,{\mathbf{x}},{\mathbf{v}},m,e)}}{{2\sqrt {{{g}_{{\alpha \beta }}}{{u}^{\alpha }}{{u}^{\beta }}} }}{{u}^{\mu }}{{u}^{\nu }}{{d}^{3}}{v}dmde\, - \,\frac{1}{2}{{k}_{2}}{{F}_{{\alpha \beta }}}{{F}^{{\alpha \beta }}}{{g}^{{\mu \nu }}}\sqrt { - g} . \\ \end{gathered} $$
(2)

By Hilbert’s definition, the first term on the right-hand side of this equation is the energy–momentum tensor. It was first given in [9–11] in a less general form with no mass or charge distribution. To the best of our knowledge, attempts to write the energy–momentum tensor in terms of the distribution function were made only in relativistic kinetic theory for the Vlasov–Einstein equation [5–15]. The electromagnetic field equation is obtained by varying (1) with respect to \({{A}_{\mu }}\) and is known as Maxwell’s equations:

$${{k}_{2}}\frac{{\partial \sqrt { - g} {{F}^{{\mu \nu }}}}}{{\partial {{x}_{\nu }}}} = \frac{1}{{{{c}^{2}}}}\int e {{u}^{\mu }}f\left( {t,{\mathbf{x}},{\mathbf{v}},m,e} \right){{d}^{3}}{v}dmde.$$
(3)

Let us show that the form of action (1) is more general than the one used in [1–4]. To obtain the action in the standard form, we use a distribution function in the form of the delta function for a single particle:

$$\begin{gathered} f\left( {{\mathbf{x}},{\mathbf{v}},m,e,t} \right) \\ = \delta \left( {{\mathbf{x}} - {\mathbf{x}}'\left( t \right)} \right)\delta \left( {{\mathbf{v}} - {\mathbf{v}}'\left( t \right)} \right)\delta \left( {m - m'} \right)\delta \left( {e - e'} \right). \\ \end{gathered} $$
(4)

Substituting (4) into action (1) and omitting primes, we derive standard expressions [1–4] for all terms:

$$\begin{gathered} S = - cm\int {\sqrt {{{g}_{{\mu \nu }}}({\mathbf{x}},t){{u}^{\mu }}{{u}^{\nu }}} } dt - \frac{e}{c}\int {{{A}_{\mu }}({\mathbf{x}},t)} {{u}^{\mu }}dt \\ + \,{{k}_{1}}\int {(R + \Lambda )} \sqrt { - g} {{d}^{4}}x + {{k}_{2}}\int {{{F}_{{\mu \nu }}}{{F}^{{\mu \nu }}}} \sqrt { - g} {{d}^{4}}x. \\ \end{gathered} $$
(5)

The role of particles can be played by electrons and ions in a plasma, planets in galaxies, galaxies in supergalaxies, and a galaxy cluster in the Universe. In (4) we can use a sum of delta functions to obtain the usual action [1–4] for a finite system of particles: this justifies the uniqueness of the choice of the more general action (1).

3 TRANSITION TO FLUID DYNAMICS AND THE HAMILTON–JACOBI EQUATION IN THE RELATIVISTIC CASE OF THE VLASOV–MAXWELL–EINSTEIN EQUATION

To derive the Vlasov–Maxwell–Einstein equations in the Hamilton–Jacobi form, we need to derive it in terms of momenta [16–19]. Following the general scheme from classical textbooks [1–4], Vlasov-type equations are derived as field equations for given motion of particles and as equations of motion of particles in given fields with the transition to Liouville equations. The part of action (5) for the motion of particles in given fields is

$$S = - cm\int {\sqrt {{{g}_{{\mu \nu }}}{{u}^{\mu }}{{u}^{\nu }}} } dt - \frac{e}{c}\int {{{A}_{\mu }}{{u}^{\mu }}dt.} $$

As a result, we obtain the equation of motion

$$\begin{gathered} - cm\frac{d}{{dt}}\left[ {\frac{{{{g}_{{i\beta }}}{{u}^{\beta }}}}{{\sqrt {{{g}_{{\eta \xi }}}{{u}^{\eta }}{{u}^{\xi }}} }} + \frac{e}{c}{{A}_{i}}} \right] \\ = - cm\frac{1}{{\sqrt {{{g}_{{\eta \xi }}}{{{v}}^{\eta }}{{{v}}^{\xi }}} }}\frac{{\partial {{g}_{{\mu \nu }}}}}{{\partial {{x}^{i}}}}{{u}^{\mu }}{{u}^{\nu }} - \frac{e}{c}\frac{{\partial {{A}_{\mu }}}}{{\partial {{x}^{i}}}}{{u}^{\mu }}. \\ \end{gathered} $$

Here, the Latin indices \(i,\;j,\;k \ldots \) run over \(1{-}3,\) while the Greek indices \(\mu ,\;\nu \ldots \) run over 0–3.

For momenta, we obtain the expression

$${{q}_{\mu }} = \frac{{\partial L}}{{\partial {{u}^{\mu }}}} = - mc\frac{{{{g}_{{\mu \alpha }}}{{u}^{\alpha }}}}{{\sqrt {{{g}_{{\eta \xi }}}{{u}^{\eta }}{{u}^{\xi }}} }} + \frac{e}{c}{{A}_{\mu }}.$$
(6)

Here, an expression for q0 is derived by formal differentiation with respect to \({{u}^{0}} = c.\)

These are expressions for long, or canonical momenta, but we will also need the small momenta \({{p}_{\mu }} = {{q}_{\mu }} - \frac{e}{c}{{A}_{\mu }} = - mc\frac{{{{g}_{{\mu \alpha }}}{{u}^{\alpha }}}}{{\sqrt {{{g}_{{\eta \xi }}}{{u}^{\eta }}{{u}^{\xi }}} }}\). Formulas for the relation to velocities are simpler for small momenta, but, in the transition to the Hamilton–Jacobi equation, we have to use the canonical momenta.

Passing to upper indices via multiplication by the inverse matrix \({{g}^{{\mu \beta }}}\) yields

$${{p}^{\beta }} = - mc\frac{{{{u}^{\beta }}}}{{\sqrt {{{g}_{{\eta \xi }}}{{u}^{\eta }}{{u}^{\xi }}} }}.$$

Now, we need to invert this formula, expressing the velocities in terms of momenta, in order to write the action in terms of momenta. For this purpose, in the last formula, the \(\beta \)th component is divided by the zeroth one:

$$\frac{{{{p}^{\beta }}}}{{{{p}^{0}}}} = \frac{{{{u}^{\beta }}}}{c}.$$

The momentum with zeroth component is eliminated from this formula by using the mass equation \({{p}_{\alpha }}{{p}_{\beta }}{{g}^{{\alpha \beta }}} = {{(mc)}^{2}}\) and its solution for \({{p}_{0}}\):

$${{p}_{0}} = \frac{{ - b \pm \sqrt {{{b}^{2}} - 4ac} }}{{2a}},$$

where \(a = {{g}^{{00}}},\quad b = 2{{p}_{i}}{{g}^{{0i}}},\quad c = {{p}_{i}}{{p}_{j}}{{g}^{{ij}}} - {{(mc)}^{2}}.\) Here, a minus sign is taken for consistency with nonrelativistic dynamics.

The mass equation is obtained by substituting the same relations (for eliminating velocities)

$$\frac{{{{p}^{\beta }}}}{{{{p}^{0}}}} = \frac{{{{u}^{\beta }}}}{c}$$

into formula (6) with μ = 0 and taking into account that u0 = c (cf. [1–4]).

The field equation (2) remains the same after changing to integration with respect to momenta by using the formulas \(f(t,{\mathbf{x}},{\mathbf{v}},m,e)d{\mathbf{v}}dmde\) = f(t, x, q, \(m,e)dqdmde\) = \(f(t,{\mathbf{x}},{\mathbf{p}},m,e)d{\mathbf{p}}dmde\). Each of these three quantities is the number of particles in the volume element, which is invariant under the change of variables:

$$\begin{gathered} {{k}_{1}}\left( {{{R}^{{\mu \nu }}} - \frac{1}{2}{{g}^{{\mu \nu }}}(R + \Lambda )} \right)\sqrt { - g} \\ = c\int {m\frac{{f\left( {t,{\mathbf{x}},q,m,e} \right)}}{{2\sqrt {{{g}_{{\mu \nu }}}{{u}^{\mu }}{{u}^{\nu }}} }}} {{u}^{\mu }}{{u}^{\nu }}{{d}^{3}}qdmde - \frac{1}{2}{{k}_{2}}{{F}_{{\mu \nu }}}{{F}^{{\mu \nu }}}{{g}^{{\mu \nu }}}\sqrt { - g} \\ \end{gathered} $$
$${{k}_{2}}\frac{{\partial \sqrt { - g} {{F}^{{\mu \nu }}}}}{{\partial {{x}_{\nu }}}} = \frac{1}{{{{c}^{2}}}}\int {e{{u}^{\mu }}f\left( {t,{\mathbf{x}},q,m,e} \right){{d}^{3}}qdmde} $$

or

$$\begin{gathered} {{k}_{1}}\left( {{{R}^{{\mu \nu }}} - \frac{1}{2}{{g}^{{\mu \nu }}}\left( {R + \Lambda } \right)} \right)\sqrt { - g} \\ = c\int m \frac{{f\left( {t,{\mathbf{x}},q,m,e} \right)}}{2}\frac{{c{{p}^{\mu }}{{p}^{\nu }}}}{{({{q}^{0}}){{{\left( {mc} \right)}}^{2}}}}{{d}^{3}}qdmde \\ - \,\frac{1}{2}{{k}_{2}}{{F}_{{\mu \nu }}}{{F}^{{\mu \nu }}}{{g}^{{\mu \nu }}}\sqrt { - g,} \\ \end{gathered} $$
(7)
$$\begin{gathered} {{k}_{2}}\frac{{\partial {{F}^{{\mu \nu }}}}}{{\partial {{x}_{\nu }}}}\sqrt { - g} \\ = \frac{1}{{{{c}^{2}}}}\int {e\frac{{c{{p}^{\mu }}}}{{{{p}^{0}}}}f\left( {t,{\mathbf{x}},q,m,e} \right){{d}^{3}}qdmde} . \\ \end{gathered} $$

Here, we mean that the velocities in the first equation and the momenta pμ in the second equation have to be expressed in terms of the canonical momenta \({{q}_{\mu }}\).

The equation of motion for particles is obtained in Hamiltonian form, where the Hamiltonian function is \(H = - c\frac{{\partial L}}{{\partial {{u}^{0}}}} = - c{{q}_{0}}\). This formula is derived taking into account that the Lagrangian for the action \(S = - cm\int \,\sqrt {{{g}_{{\mu \nu }}}{{u}^{\mu }}{{u}^{\nu }}} dt - \frac{e}{c}\int \,{{A}_{\mu }}{{u}^{\mu }}dt\) is a function of the first degree with respect to velocities and using the Euler formula \({{u}^{\mu }}\frac{{\partial L}}{{\partial {{u}^{\mu }}}} - L = 0\). Since, by definition, \(H = {{u}^{i}}\frac{{\partial L}}{{\partial {{u}^{i}}}} - L\), we obtain \(c\frac{{\partial L}}{{\partial {{u}^{0}}}} + H = 0\). Here, summation over i = 1–3 and \(\mu = 0{\text{--}}3\) is implied. From this equation, we find the velocity expressions \({{u}^{i}} = \frac{{\partial H}}{{\partial {{q}_{i}}}} = {{u}^{i}}\left( q \right) = - c\frac{{\partial {{q}_{0}}}}{{\partial {{q}_{i}}}}\).

In terms of this Hamiltonian, we write the Liouville equation

$$\frac{{~\partial f}}{{\partial t}} - c\frac{{\partial {{q}_{0}}}}{{\partial {{q}_{i}}}}\frac{{\partial f}}{{\partial {{x}^{i}}}} - c\frac{{\partial {{q}_{0}}}}{{\partial {{x}^{i}}}}\frac{{\partial f}}{{\partial {{q}_{i}}}} = 0~.$$
(8)

As a result, we obtain the closed system (7), (8) of Vlasov–Maxwell–Einstein equations of gravitation and electrodynamics in terms of momenta. According to the general scheme of [16–20], a hydrodynamic consequence of system (7), (8) is derived by making the hydrodynamic substitution \(f(t,x,q,m,e)\) = \(\rho (x,t,m,e)\delta (q - Q(q,t,m,e)){\text{:}}\)

$$\frac{{\partial \rho }}{{\partial t}} + \frac{\partial }{{\partial {{x}^{i}}}}({{u}^{i}}\left( Q \right)\rho ) = 0,$$
(9)
$$\frac{{\partial ~{{Q}_{k}}}}{{\partial t}} - c\frac{{\partial {{q}_{0}}}}{{\partial {{q}_{i}}}}\left( {x,Q} \right)\frac{{\partial ~{{Q}_{k}}}}{{\partial {{x}^{i}}}} + c\frac{{\partial {{q}_{0}}}}{{\partial {{x}^{k}}}} = 0.$$
(10)

The substitution Qα = \(\frac{{\partial W}}{{\partial {{x}^{\alpha }}}}\) or \({{P}_{\alpha }} = \frac{{\partial W}}{{\partial {{x}^{\alpha }}}} - \frac{e}{c}{{A}_{\alpha }}\) yields the Hamilton–Jacobi equations

$$\left( {\frac{{\partial W}}{{\partial {{x}^{\alpha }}}} - \frac{e}{c}{{A}_{\alpha }}} \right)\left( {\frac{{\partial W}}{{\partial {{x}^{\beta }}}} - \frac{e}{c}{{A}_{\beta }}} \right){{g}^{{\alpha \beta }}} = {{(mc)}^{2}}.$$
(11)

Additionally, we need to rewrite the continuity equation

$$\frac{{\partial \rho }}{{\partial t}} + \frac{\partial }{{\partial {{x}^{i}}}}({{u}^{i}}\left( {\nabla W} \right)\rho ) = 0.$$
(12)

To obtain the Hamilton–Jacobi–Vlasov–Maxwell–Einstein equations in closed form, the same hydrodynamic substitution \(f(t,x,q,m,e)\) = \(\rho (x,t,m,e)\delta (q - Q(x,t,m,e))\) has to be made in the field equations as well:

$$\begin{gathered} {{k}_{1}}\left( {{{R}^{{\mu \nu }}} - \frac{1}{2}{{g}^{{\mu \nu }}}\left( {R + \Lambda } \right)} \right)\sqrt { - g} \\ = c\int m \frac{{\rho \left( {t,x,m,e} \right)}}{2}\frac{{c{{P}^{\mu }}{{P}^{\nu }}}}{{({{P}^{0}}){{{\left( {mc} \right)}}^{2}}}}dmde \\ - \frac{1}{2}{{k}_{2}}{{F}_{{\mu \nu }}}{{F}^{{\mu \nu }}}{{g}^{{\mu \nu }}}\sqrt { - g} , \\ \end{gathered} $$
(13)
$${{k}_{2}}\frac{{\partial {{F}^{{\mu \nu }}}}}{{\partial {{x}_{\nu }}}}\sqrt { - g} = \frac{1}{{{{c}^{2}}}}\int {e\frac{{c{{P}^{\mu }}}}{{{{P}^{0}}}}\rho \left( {t,{\mathbf{x}},m,e} \right)dmde} .$$

Here, the macroscopic momenta \({{P}^{\mu }}\) and \({{P}_{\mu }}\) are related by the usual formulas \({{P}_{\mu }} = {{g}_{{\mu \nu }}}{{P}^{\nu }}\). Moreover, in the Hamilton–Jacobi form, it is necessary to take into account that \({{P}_{\alpha }} = \frac{{\partial W}}{{\partial {{x}^{\alpha }}}} - \frac{e}{c}{{A}_{\alpha }}\). We have obtained the Vlasov–Maxwell–Einstein equation as a reduction to hydrodynamic variables (see Eqs. (9), (10), (13)) and as a reduction to the Hamilton–Jacobi equations (see Eqs. (11)–(13)). In principle, the cosmological problem can be considered in the general case, but the expressions will be cumbersome, so we consider examples of specific relativistic systems.

Example 1. Consider the simplest relativistic action with the Lorentz metric:

$$\begin{gathered} S = - cm\int \left( {\sqrt {{{c}^{2}} - {{{\left( {\frac{{dx}}{{dt}}} \right)}}^{2}}} + \frac{U}{c}} \right)dt \hfill \\ - \frac{1}{{8\pi \gamma }}\int \,{{(\nabla U)}^{2}}dxdt - \frac{{{{c}^{2}}\Lambda }}{{8\pi \gamma }}\int \,Udxdt. \hfill \\ \end{gathered} $$

Varying S with respect to the coordinates \(x(t)\) yields the usual relativistic equations in the Lorentz metric with the Hamiltonian [1–4]

$$H\left( {x,q} \right) = c\sqrt {{{{(mc)}}^{2}} + {{q}^{2}}} + U.$$

Now we pass to an action that can be varied with respect to fields according to our usual scheme:

$$\begin{gathered} S = - c\int {m\left( {\sqrt {{{c}^{2}} - {{{\left( {\frac{{dx}}{{dt}}} \right)}}^{2}}} + U} \right)f(x,p,t,m)d{\mathbf{p}}dmdxdt} \\ - \frac{1}{{8\pi \gamma }}\int {{{{\left( {\nabla U} \right)}}^{2}}d{\mathbf{x}}dt - \frac{{{{c}^{2}}\Lambda }}{{8\pi \gamma }}\int {Udxdt.} } \\ \end{gathered} $$

Varying S with respect to the potential U yields equations for fields:

$$\Delta U = 4\pi \gamma \int {mf\left( {t,x,q,m,e} \right)dqdmde - \frac{1}{2}{{c}^{2}}\Lambda .} $$

Immediately passing to the Hamilton–Jacobi equation, we obtain the system of equations

$$\left\{ \begin{gathered} \frac{{\partial \rho }}{{\partial t}} + \frac{\partial }{{\partial {{x}^{i}}}}({{v}^{i}}\left( {\nabla W} \right)\rho ) = 0 \hfill \\ \frac{{\partial W}}{{\partial t}} + c\sqrt {{{{(mc)}}^{2}} + {{{(\nabla W)}}^{2}}} + U = 0 \hfill \\ \Delta U = 4\pi \gamma \int {m\rho dmde} - \frac{{{{c}^{2}}\Lambda }}{2}, \hfill \\ \end{gathered} \right.$$

where \({{{v}}^{i}}\left( q \right) = \frac{{\partial H}}{{\partial {{q}_{i}}}} = \frac{{c{{q}^{i}}}}{{\sqrt {{{{(mc)}}^{2}} + {{q}^{2}}} }}\).

We have obtained a velocity expression showing the Hubble expansion, a closed system of equations, and the possibility of passing to cosmological solutions in the isotropic case and when the density is independent of space.

Example 2. Consider another relativistic action with the weakly relativistic (rather than Lorentz) metric

$${{g}_{{\alpha \beta }}} = \operatorname{diag} \left( {1 + \frac{{2U}}{{{{c}^{2}}}}, - 1, - 1, - 1} \right).$$

The potential in the action is taken inside the square root:

$$\begin{gathered} S = - cm\int \sqrt {{{c}^{2}} - {{{\left( {\frac{{dx}}{{dt}}} \right)}}^{2}} + U} dt \hfill \\ - \frac{1}{{8\pi \gamma }}\int \,{{(\nabla U)}^{2}}dxdt - \frac{{{{c}^{2}}\Lambda }}{{8\pi \gamma }}\int \,Udxdt. \hfill \\ \end{gathered} $$

Proceeding as above, we obtain the Hamiltonian

$$H = - c{{p}_{0}}(x,q,t) = c\sqrt {\left( {{{{(mc)}}^{2}} + {{q}^{2}}} \right)\left( {1 + \frac{{2U}}{{{{c}^{2}}}}} \right)} $$

and the system of equations

$$\left\{ \begin{gathered} \frac{{\partial \rho }}{{\partial t}} + \frac{\partial }{{\partial {{x}^{i}}}}({{{v}}^{i}}\left( {\nabla W} \right)\rho ) = 0 \hfill \\ \frac{{\partial W}}{{\partial t}} + c\sqrt {({{{(mc)}}^{2}} + {{{(\nabla W)}}^{2}})\left( {1 + \frac{{2U}}{{{{c}^{2}}}}} \right)} = 0 \hfill \\ \Delta U = 4\pi \gamma \int {\frac{{m\rho \left( {m,x,t} \right)}}{{\sqrt {{{c}^{2}} - \left( {{v}(\nabla W} \right){{)}^{2}} + U} }}dm - \frac{{{{c}^{2}}\Lambda }}{2},} \hfill \\ \end{gathered} \right.$$

where \({{{v}}^{i}}\left( q \right) = \frac{{\partial H}}{{\partial {{q}_{i}}}} = \frac{{c{{q}^{i}}\sqrt {1 + \frac{{2U}}{{{{c}^{2}}}}} }}{{\sqrt {{{{(mc)}}^{2}} + {{q}^{2}}} }}\).

Once again, we have obtained a closed system of equations, which shows the origin of the root on the right-hand side of the Einstein equations. Additionally, we have derived a velocity expression showing the Hubble expansion. Moreover, the possibility of passing to cosmological solutions in the isotropic case and when the density is independent of space has been demonstrated.

CONCLUSIONS

Closed equations of electrodynamics and gravitation have been derived from the principle of least action in the form of the Vlasov equation (cf. [5–15]). The meaning of Vlasov-type equations was clarified. Specifically, this is still the only way of deriving both equations of gravitation and electrodynamics from the principle of least action. Moreover, this is still the only method for closing the system of electrodynamic and gravitation equations with the help of the principle of least action by using the velocity and space distribution functions of objects (such as electrons, ions, stars in galaxies, and galaxies in supergalaxies or the Universe). Corresponding hydrodynamic equations (e.g., magnetohydrodynamics equations or gravitating gas dynamics equations) are also naturally obtained from Vlasov-type equations by making the hydrodynamic substitution (still the only way of relation to the classical action for these equations as well). In [20, 21], the system of Vlasov–Maxwell–Einstein equations was obtained for velocities, while, in this paper, we derived it for momenta, which makes it possible to investigate cosmological solutions via the transition to the Hamilton–Jacobi equation. An issue of interest is to study stationary solutions of the resulting equations, as was done for the Vlasov–Poisson equations in [22]. In [20, 21] cosmological solutions in the nonrelativistic case were obtained and the Milne–McCrea model [23, 24] was derived and generalized. On this basis, Gurzadyan’s potential \(U(r) = - \frac{\gamma }{r} + a{{r}^{2}}\) [25] was justified, where the second term is related to Einstein’s lambda term. A task of great interest to do the same work for the above-proposed models in order to estimate Einstein’s lambda term and various relativistic and weakly relativistic approximations.