Symmetries and metrics of homogeneous cosmologies

Abstract

It is shown that an assumption of ’’historical homogeneity’’—or the persistence of spatial homogeneity in time—leads to the synchronous (time-orthogonal) form ds2=−dt2+γAB (t)εA⊗εB, of the metrics for general spatially homogeneous (Bianchi) cosmologies. The εA are group-representation 1-forms associated with the spatial symmetry group of the given cosmology, and are independent of any particular space–time theory; in particular they do not depend on Einstein’s equations and the local physics. Expressions for the εA in the canonical basis used by Estabrook, Wahlquist, and Behr (EWB) have been tabulated in the companion paper on ’’Spatially homogeneous neutrino cosmologies’’ (SHNC). In the present paper explicit expressions are given for the scale factors γAB(t). These do correspond to the choice of the canonical basis to describe the symmetry but, in contrast to the εA, they also depend on the space–time physical theory. Here the γAB(t) come out as observable geometric–kinematic quantities (generalized Hubble constants) which also appear in the solutions of Einstein’s equations for a given cosmology. Central to the discussion is the matrix C(t) which relates the time-independent canonical basis describing the symmetry to a time-dependent orthonormalizing basis. With the imposed requirement that the Riemannian geometry of the evolving spatial hypersurfaces retain its quasi-canonical form in time (’’quasi-canonical gauge’’) it is shown that C(t) is a product of a diagonal matrix D̀ and a rotation matrix R; and a table of these forms of C(t) for the various Bianchi types is given. The simple Hubble-constant expressions for the metric scale factors γAB(t) for all types come solely from the diagonal factor D̀ in C. The metric, then, in all cases takes the form ds2=−dt2+γABεA ⊗εB=−dt2 +(εI/α)2 +(εII/β)2 +(εIII/γ)2. Thus, the specializing assumption of Misner and of Ryan and Shepley, requiring C to be symmetric, does not limit the form of the metric but does restrict the relation between the invariant symmetry basis and the quasi-canonical orthonormal basis to the cases where the orthogonal factor R reduces to the identity, i.e., to those cases where the two bases remain parallel at all times. Thus, the gauge assumed by them is not compatible with the preservation of the quasi-canonical form of the spatial geometry in almost all cases where there is degeneracy involved in the spatial curvatures.