Time-Dependent Automorphism Inducing Diffeomorphisms, Open Algebras and the Generality of the Kantowski-Sachs Vacuum Geometry

TL;DR

This paper explores a specific class of coordinate transformations that preserve spatial homogeneity in Kantowski-Sachs vacuum models, revealing their gauge freedoms and providing a more general, explicit form of the metric.

Contribution

It introduces a new analysis of time-dependent automorphism inducing diffeomorphisms with open algebras, extending the understanding of gauge freedoms in Kantowski-Sachs models.

Findings

Derived integrability conditions for GCTs with open Lie algebras.

Provided an explicit, most general form of the Kantowski-Sachs vacuum metric.

Confirmed the maximal gauge freedom in the spatially homogeneous setting.

Abstract

Following the spirit of a previous work of ours, we investigate the group of those General Coordinate Transformations (GCTs) which preserve manifest spatial homogeneity. In contrast to the case of Bianchi Type Models we, here, permit an isometry group of motions $G_{4}=SO(3)\otimes T_{r}$, where $T_{r}$ is the translations group, along the radial direction, while SO(3) acts multiply transitively on each hypersurface of simultaneity $\Sigma_{t}$. The basis 1-forms, can not be invariant under the action of the entire isometry group and hence produce an Open Lie Algebra. In order for these GCTs to exist and have a non trivial, well defined action, certain integrability conditions have to be satisfied; their solutions, exhibiting the maximum expected ``gauge'' freedom, can be used to simplify the generic, spatially homogeneous, line element. In this way an alternative proof of the generality of the Kantowski-Sachs (KS) vacuum is given, while its most general, manifestly homogeneous, form is explicitly presented.