Time-dependent automorphism inducing diffeomorphisms in vacuum Bianchi cosmologies and the complete closed form solutions for types II and V

We investigate the set of space–time general coordinate transformations (GCTs) which leave the line element of a generic Bianchi-type geometry quasiform invariant; i.e., preserve manifest spatial homogeneity. We find that these GCTs, induce special time-dependent automorphic changes, on the spatial scale factor matrix γαβ(t)—along with corresponding changes on the lapse function N(t) and the shift vector Nα(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 essentially 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 tra...