MATHEMATICS OF RELATIVITY THEORY AND COSMOLOGY: VLASOV EQUATIONS AND THE HUBBLE CONSTANT

The derivation and properties of the Vlasov-Einstein and Vlasov-Poisson equations and cosmological solutions are considered. In classical works, the equations for 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. The resulting derivation of Vlasov-type equations gives Vlasov-Einstein equations that are different from those proposed earlier. A method is proposed for the transition from kinetic equations to hydrodynamic consequences, as was previously done by A. A. Vlasov himself. 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 already done in quantum mechanics by E. Madelung and, in a more general form, by V. V. Kozlov. Thus, in the non-relativistic case, Milne-McCrea solutions are obtained, as well as non-relativistic and relativistic analysis of Friedmann-type solutions of the non-stationary evolution of the Universe. This makes it possible to determine the Hubble constant not on the basis of the metric, as was done previously, but, as expected, on the basis of observable matter, to write equations for it on the basis of matter motion in a given metric, to analyze Einstein's Lambda and the cause of the accelerated expansion of the Universe as a relativistic effect, and not as a consequence of mythical dark energy, hypothetical particles, or lambda terms. This is a triumph and the best confirmation of the General Theory of Relativity. The fact of accelerated expansion also allows us to determine the sign of the curvature: it is negative, and we live in Lobachevsky space.