The (2+1)-dim Axial Universes -- Solutions to the Einstein Equations, Dimensional Reduction Points, and Klein-Fock-Gordon Waves

Abstract

The paper presents a generalization and further development of our recent publications where solutions of the Klein-Fock-Gordon equation defined on a few particular $D=(2+1)$-dim static space-time manifolds were considered. The latter involve toy models of 2-dim spaces with axial symmetry, including dimension reduction to the 1-dim space as a singular limiting case. Here the non-static models of space geometry with axial symmetry are under consideration. To make these models closer to physical reality, we define a set of "admissible" shape functions $\rho(t,z)$ as the $(2+1)$-dim Einstein equations solutions in the vacuum space-time, in the presence of the $\Lambda$-term, and for the space-time filled with the standard "dust". It is curious that in the last case the Einstein equations reduce to the well-known Monge-Amp\`{e}re equation, thus enabling one to obtain the general solution of the Cauchy problem, as well as a set of other specific solutions involving one arbitrary function. A few explicit solutions of the Klein-Fock-Gordon equation in this set are given. An interesting qualitative feature of these solutions relates to the dimension reduction points, their classification, and time behavior. In particular, these new entities could provide us with novel insight into the nature of P- and T-violation, and of Big Bang. A short comparison with other attempts to utilize dimensional reduction of the space-time is given.