Initial data sets, conformal geometry and the topology of physical space in general relativity

Abstract

We discuss the interplay between the energetic content of physical space and the topology of the underlying three-manifold. Within the context of the conformal approach to the initial value problem we examine both the case of axymptotically euclidean data given on a complete, non-compact three-manifold, and the case of data assigned on a closed three-manifold. In the former case we provide a description, in the full theory, of the topological changes induced by large concentration of gravitational radiation, and of the formation of apparent horizons for time-symmetric data. In the closed case, we show that the presence of enough matter and radiation necessarily implies that the topology of the underlying three-manifold is (up to identifications) the three-sphere topology, or the (S 1 x S 2)-wormhole topology, or that of a connected sum of a denumerable number of such manifolds. We also show that such topologies leave, as for as the field equations are concerned, more room to possible gravitational initial data sets.