Quantum cosmology of a Bianchi III LRS geometry coupled to a source free electromagnetic field

TL;DR

This paper explores the classical and quantum properties of a Bianchi III LRS cosmological model coupled with an electromagnetic field, revealing a connection to black hole solutions and producing singularity-free semi-classical geometries.

Contribution

It introduces a novel analysis of Bianchi III LRS cosmology with electromagnetic fields, linking classical solutions to black hole metrics and deriving quantum wave functions with singularity avoidance.

Findings

Classical solutions include time-dependent electric and constant magnetic charges.

The mini-superspace resembles the Reissner-Nordström black hole metric.

Quantum states lead to semi-classical, mostly singularity-free geometries.

Abstract

We consider a Bianchi type III axisymmetric geometry in the presence of an electromagnetic field. A first result at the classical level is that the symmetry of the geometry need not be applied on the electromagnetic tensor $F_{\mu\nu}$; the algebraic restrictions, implied by the Einstein field equations to the stress energy tensor $T_{\mu\nu}$, suffice to reduce the general $F_{\mu\nu}$ to the appropriate form. The classical solution thus found contains a time dependent electric and a constant magnetic charge. The solution is also reachable from the corresponding mini-superspace action, which is strikingly similar to the Reissner-Nordstr{\"o}m one. This points to a connection between the black hole geometry and the cosmological solution here found, which is the analog of the known correlation between the Schwarzschild and the Kantowski-Sachs metrics. The configuration space is drastically modified by the presence of the magnetic charge from a 3D flat to a 3D pp wave geometry. We map the emerging linear and quadratic classical integrals of motion, to quantum observables. Along with the Wheeler-DeWitt equation these observables provide unique, up to constants, wave functions. The employment of a Bohmian interpretation of these quantum states results in deterministic (semi-classical) geometries most of which are singularity free.