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

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 $Σ_{t}$. 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.