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 (GC Ts) which leave the line element of a generic Bianchi-type geometry quasifo rm invariant; i.e., preserve manifest spatial homogeneity. We find that the se GCTs, 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, differe nt from those induced by the group SAut(G) advocated in earlier investigati ons as the relevant symmetry group; they are used to simplify the form of t he line element-and thus simplify Einstein's equations as well, without los ing generality. As far as this simplification procedure is concerned, the t ransformations found are proved to be essentially unique. For the case of B ianchi types II and V, where the most general solutions are known, Taub's a nd Joseph's, respectively, it is explicitly verified that our transformatio ns and only those, suffice to reduce the generic line element to the previo usly known forms. It thus becomes possible-for these types-to give in close d form the most general solution, containing all the necessary "gauge" free dom. (C) 2001 American Institute of Physics.