Cosmological solutions, Hubble's law, and accelerated expansion of the universe from the principle of least action

Abstract: In classic textbooks [1-3], the Hubble constant is defined in terms of the metric. Here, we define it, as expected, in terms of matter, following Milne and McCrea, extending their theory of an 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, or new particles, as an exact consequence of Einstein's classical 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]), 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 previously. A method for transition from kinetic equations to hydrodynamic consequences is proposed [5–6], as was previously done by A.A. Vlasov himself [4]: this can be interpreted as a transition from a kinetic turbulent description using a distribution function to a laminar description of the hydrodynamic type. This yields cosmological solutions of the Milne–McCrea type. In the case of Hamiltonian mechanics, a transition from the hydrodynamic consequences of the Liouville equation to the Hamilton-Jacobi equation is possible, as was already done in quantum mechanics by E. Madelung, and more generally by V.V. Kozlov [7] and V.P. Maslov. This yields Milne–McCrea solutions in the nonrelativistic case, as well as nonrelativistic and relativistic analyses of Friedmann-type solutions to the nonstationary evolution of the Universe. This allows us to obtain the fact of the accelerated expansion of the Universe as a relativistic effect [8-10] without artificial additions such as Einstein's lambda, dark energy, and new fields, from the classical relativistic principle of least action. This places general relativity and cosmology on a solid mathematical foundation and makes it possible to explain the accelerated expansion, a well-tested experiment (with a Nobel Prize in 2011).

References.

1. Dubrovin, B. A., Novikov, S. P., and Fomenko, A. T. “Modern Geometry: Methods and Applications.” Moscow: Nauka, 1986.

2. Landau, L. D., Lifshitz, E. M. “Field Theory.” Moscow: Nauka, 1988.

3. Weinberg, S. “Gravitation and Cosmology.” Moscow: Mir, 1975, 696 p.

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 the Vlasov–Poisson equation system.” European Physical Journal Plus. 2021. Vol. 136. No. 1. P. 71.

6. 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. Kozlov V. V., General Theory of Vortices, Udmurt University Press, Izhevsk, 1998, 239 p.

8. V. V. Vedenyapin, “Mathematical theory of the expanding Universe based on the principle of least action”, Russ. Comput. Math. and Math. Phys., 64:11 (2024), 2114–2131

9. V. V. Vedenyapin, Ya. G. Batishcheva, M. V. Goryunova, and A. A. Russkov, “Mathematical theory of the accelerating expansion of the Universe based on the principle of least action”, CMFD, 71:4 (2025), 562–584.

Russianmathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

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Mathematical models and integration methods

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