Time-Dependent Automorphism Inducing Diffeomorphisms In Vacuum Bianchi Cosmologies And The Complete Closed Form Solutions For Type II & V

TL;DR

This paper identifies a specific set of time-dependent coordinate transformations that preserve spatial homogeneity in Bianchi cosmologies, enabling the derivation of complete closed-form solutions for Types II and V.

Contribution

It introduces a unique group of automorphic transformations that simplify Einstein's equations for Bianchi Types II and V, leading to the explicit derivation of their most general solutions.

Findings

Transformations form a group distinct from SAut(G)

Transformations simplify line element without loss of generality

Closed-form solutions for Types II and V verified with known solutions

Abstract

We investigate the set of spacetime general coordinate transformations (G.C.T.) which leave the line element of a generic Bianchi Type Geometry, quasi-form invariant; i.e. preserve manifest spatial Homogeneity. We find that these G.C.T.'s, induce special time-dependent automorphic changes, on the spatial scale factor matrix $\gamma_{\alpha\beta}(t)$ -along with corresponding changes on the lapse function $N(t)$ and the shift vector $N^{\alpha}(t)$. These changes, which are Bianchi Type dependent, form a group and are, in general, different from those induced by the group SAut(G) -advocated in earlier investigations as the relevant symmetry group-, they are used to simplify the form of the line element -and thus simplify Einstein's equations as well-, without losing generality. As far as this simplification procedure is concerned, the transformations found, are proved to be essentialy unique. For the case of Bianchi Types II and V, where the most general solutions are known -Taub's and Joseph's, respectively-, it is explicitly verified that our transformations and only those, suffice to reduce the generic line element, to the previously known forms. It becomes thus possible, -for these Types- to give in closed form, the most general solution, containing all the necessary ``gauge'' freedom.