Open Universe Models as Analogs of Multidimensional Electric Capacitors

Abstract

A new approach to obtaining open Universes models as exact solutions of gravitational equations is considered. The proposed method is based on an analogy between electrostatics of conductors and open cosmological models which have a conformally-flat 4-metric in the Fock form. These cosmological models are solutions of the Einstein equations with the energy-momentum tensor of a Pascal perfect fluid. As a result, it is shown that the capacity of the multidimensional "ball" electric capacitor in multidimensional Euclidean auxiliary space is connected with harmonic functions by which the potentials of multidimensional capacitors are described. In turn, the conformal factor of the metric describing an open universe model in the Fock representation is a power function of this potential for each dimensionality of the Euclidean space. In particular, for 3D auxiliary Euclidean space in 4D space-time, there is the Friedman solution for an open universe filled with incoherent dust (with flat metric at spatial infinity). Further for 4D Euclidean space, we have an analogue of the open cosmological model with the equation of state of an ultra-relativistic gas. Other cases are listed in a table. A table of the matter states which generalizes this approach to multidimensional space-times is also constructed. Thus, the possibility of replacing the modelling problem in cosmology with the equivalent electrostatic problem for finding the capacity of capacitors is shown under specified boundary conditions. Exceptions from the general approach are also presented in the article.