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 formulationfor constructing spatially compact composite spacetimes as solutions forthe Einstein equations. Such composite spacetimes are built from thespatially compact locally homogeneous vacuum spacetimes which havetwo commuting local Killing vector fields and are homeomorphic to torusbundles over the circle by gluing them through a timelike hypersurfaceadmitting a homogeneous spatial torus spanned by the commuting localKilling vector fields. We also assume that the matter which will arisefrom the gluing is compressed on the boundary, i.e. we take the thin-shellapproximation. By solving the junction conditions, we can see dynamicalbehaviour of the connected (composite) spacetime. The Teichmüllerdeformation of the torus canalso be obtained. We apply our formalism to aconcrete model. The relation to the torus sum of 3-manifolds and thedifficulty of this problem are also discussed.