Thurston's Geometrization Conjecture and cosmological models

Abstract

We investigate a class of spatially compact inhomogeneous spacetimes. Motivated by Thurston's Geometrization Conjecture, we give a formulation for constructing spatially compact composite spacetimes as solutions for the Einstein equations. Such composite spacetimes are built from the spatially compact locally homogeneous vacuum spacetimes which have two commuting Killing vectors by gluing them through a timelike hypersurface admitting a homogeneous spatial slice spanned by the commuting Killing vectors. Topology of the spatial section of the timelike boundary is taken to be the torus. We also assume that the matter which will arise from the gluing is compressed on the boundary, i.e. we take the thin-shell approximation. By solving the junction conditions, we can see dynamical behavior of the connected (composite) spacetime. The Teichm\"uller deformation of the torus also can be obtained. We apply our formalism to a concrete model. The relation to the torus sum of 3-manifolds and the difficulty of this problem are also discussed.