Vlasov type equations, geometry, gravity, electrodynamics and cosmology

The connection between geometry and the general theory of relativity (GTR) [1] has interested Sergei Petrovich Novikov since the 1970s, when in the 1970–1971 season he analyzed the 2nd volume of Landau and Livshits at a seminar together with Veniamin Petrovich Myasnikov and Petr Petrovich Mosolov. The author of these theses, being a 5th-year student, gave a report on cosmology at this seminar on the last chapter of the textbook [2]. Sergei Petrovich's student Oleg Igorevich Bogoyavlensky, a 1970 graduate, defended his doctoral dissertation in cosmology and published a monograph of the same name (O.I.Bogoyavlensky, Methods of Qualitative Theory of Dynamic Systems in Astrophysics and Gas Dynamics, Nauka, Moscow, 1980, 320 p.). These results are also reflected in the famous textbook [1]. However, the latest developments related to the accelerated expansion of the Universe (Nobel Prize 2011) required a dramatic improvement of the entire theory of general relativity.
In classical textbooks [1–3] the Hubble constant is defined through the metric. Here we define it, as expected, through matter, according to Milne and McCrea, extending their theory of the expanding Universe to the relativistic case. This allows us to explain the accelerated expansion as a simple relativistic effect without Einstein's lambda, dark energy and new particles as an exact consequence of the classical Einstein action. The well-verified fact of accelerated expansion allows us to determine the sign of the curvature in the Friedmann model: it turns out to be negative, and we live in Lobachevsky space.
Also in classical works (see [1–4]), the equations for the fields are proposed without deriving the right-hand sides. Here we give a derivation of the right-hand sides of the Maxwell and Einstein equations within the framework of the Vlasov–Maxwell–Einstein equations from the classical, but slightly more general principle of least action [5,6]. The resulting derivation of Vlasov-type equations yields Vlasov–Einstein equations that differ from those proposed earlier. A method is pro-posed for the transition from kinetic equations to hydrodynamic consequences [5,6], as was done earlier by A. A. Vlasov himself [4]. In the case of Hamiltonian mechanics, a transition is possible from the hydrodynamic consequences of the Liouville equation to the Hamilton–Jacobi equation, as was done in quantum mechanics by E. Madelung, and in a more general form by V. V. Kozlov [7]. In this way, the Milne–McCrea solutions are obtained in the nonrelativistic case, as well as the nonrelativistic and relativistic analysis of Friedmann-type solutions of the non-stationary evolution of the Universe. This allows us to obtain the fact of the accelerated expansion of the Universe as a relativistic effect [8,9] without artificial additives such as Einstein's lambda, dark energy and new fields, from the classical relativistic principle of least action. This puts the general theory of relativity and cosmology on a solid mathematical basis.
Список литературы
1. Dubrovin B.A., Novikov S.P., Fomenko A.T. Modern geometry. Methods and applications. M.: Science. 1986.
2. Landau L.D., Lifshits E.M. Field theory. M.: Nauka, 1988.
3. Weinberg S. Gravity and Cosmology. Moscow: Mir, 1975, 696 pp.
4. Vlasov A.A. Statistical distribution functions. Moscow: Nauka, 1966. 356 p.
5. Vedenyapin, V., Fimin, N., Chechetkin, V. The generalized Friedmann model as a self-similar solution of Vlasov–Poisson equa-tion system// European Physical Journal Plus. 2021. Т. 136. № 1. С. 71.
6 В. В. Веденяпин, В. И. Парёнкина, С. Р. Свирщевский, “О выводе уравнений электродинамики и гравитации из принципа наименьшего действия”, Ж. вычисл. матем. и матем. физ., 62:6 (2022), 1016–1029 V. V. Vedenyapin, V. I. Parenkina, S. R. Svirshchevskii, “Derivation of the equations of electrodynamics and gravity from the principle of least action”, Comput. Math. Math. Phys., 62:6 (2022), 983–995
7. Козлов В. В., Общая теория вихрей, Изд-воУдмуртскогого ун-та, Ижевск,1998, 239с. Kozlov V.V., General theory of vortices, Udmurt University Publishing House, Izhevsk, 1998, 239 p.
8. В. В. Веденяпин, Я. Г. Батищева, Ю. А. Сафронов, Д. И. Богданов, “Расширение Вселенной в случае обобщенной метрики Фридмана–Леметра–Робертсона–Уокера”, Препринты ИПМ им. М. В. Келдыша, 2025, 014, 26 стр. V. V. Vedenyapin, Ya. G. Batishcheva, Yu. A. Safronov, D. I. Bogdanov, “Expansion of the Universe in the case of the generalized Friedmann–Lemaitre–Robertson–Walker metric”, Keldysh Institute of Applied Mathematics Preprints, 2025, 014, 26 pp.
9. Веденяпин В.В., “Математическая теория расширения Вселенной на основе принципа наименьшего действия”, Ж. вы-числ. матем. и матем. физ., 64:11 (2024), 2114–2131 V. V. Vedenyapin, “Mathematical theory of the expanding Universe based on the principle of least action”, Comput. Math. Math. Phys., 64:11 (2024), 2624–2642