Symmetries and metrics of homogeneous cosmologies

Journal Article · Tue Jul 01 04:00:00 EDT 1980 · J. Math. Phys. (N.Y.); (United States)

DOI:https://doi.org/10.1063/1.524612· OSTI ID:5338848

Melvin, M A; Michalik, T R

It is shown that an assumption of ''historical homogeneity'': or the persistence of spatial homogeneity in time: leads to the synchronous (time-orthogonal) form ds/sup 2/=-dt/sup 2/+..gamma../sub A/B (t)epsilon/sup A/xepsilon/sup B/, of the metrics for general spatially homogeneous (Bianchi) cosmologies. The epsilon/sup A/ 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 epsilon/sup A/ 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 ..gamma../sub A/B(t). These do correspond to the choice of the canonical basis to describe the symmetry but, in contrast to the epsilon/sup A/, they also depend on the space--time physical theory. Here the ..gamma../sub A/B(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 D 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 ..gamma../sub A/B(t) for all types come solely from the diagonal factor D in C.

• Research Organization: Temple University, Philadelphia, Pennsylvania 19122
• OSTI ID: 5338848
• Journal Information: J. Math. Phys. (N.Y.); (United States), Journal Name: J. Math. Phys. (N.Y.); (United States) Vol. 21:7; ISSN JMAPA
• Country of Publication: United States
• Language: English

Related Subjects

640106* -- Astrophysics & Cosmology-- Cosmology
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CANONICAL TRANSFORMATIONS
COSMOLOGY
METRICS
SPACE-TIME
SYMMETRY GROUPS