Comments on Closed Bianchi Models

TL;DR

This paper investigates the intrinsic properties of Bianchi cosmological models with compact spatial sections, revealing restrictions on anisotropic expansion and the non-existence of certain Bianchi types under these conditions.

Contribution

It provides a geometric analysis of spatial homogeneity in Bianchi models with compact sections and clarifies the relation between symmetry and spatial compactness, including conditions for their existence.

Findings

No anisotropic expansion in Bianchi type V and VII with closed hypersurfaces.

Types IV and VI with certain parameters do not exist with compact spatial sections.

Clarifies the relation between Bianchi symmetry and spatial compactness.

Abstract

We show several kinematical properties that are intrinsic to the Bianchi models with compact spatial sections. Especially, with spacelike hypersurfaces being closed, (A) no anisotropic expansion is allowed for Bianchi type V and VII(A\not=0), and (B) type IV and VI(A\not=0,1) does not exist. In order to show them, we put into geometric terms what is meant by spatial homogeneity and employ a mathematical result on 3-manifolds. We make clear the relation between the Bianchi type symmetry of space-time and spatial compactness, some part of which seem to be unnoticed in the literature. Especially, it is shown under what conditions class B Bianchi models do not possess compact spatial sections. Finally we briefly describe how this study is useful in investigating global dynamics in (3+1)-dimensional gravity.