Global solvability of the vacuum Einstein equation and the strong cosmic censorship in four dimensions

Abstract

Let $M$ be a connected, simply connected, oriented, closed, smooth four-manifold which is spin (or equivalently having even intersection form) and put $M^\times:=M\setminus\{{\rm point}\}$.In this paper we prove that if $X^\times$ is a smooth four-manifold homeomorphic but not necessarily diffeomorphic to $M^\times$ (more precisely, it carries a smooth structure \`a la Gompf) then $X^\times$ can be equipped with a complete Ricci-flat Riemannian metric. As a byproduct of the construction it follows that this metric is self-dual as well consequently $X^\times$ with this metric is in fact a hyper-K\"ahler manifold. In particular we find that the largest member of the Gompf--Taubes radial family of large exotic ${\mathbb R}^4$'s admits a complete Ricci-flat metric (and in fact it is a hyper-K\"ahler manifold). These Riemannian solutions are then converted into Ricci-flat Lorentzian ones thereby exhibiting lot of new vacuum solutions which are not accessable by the initial vaule formulation. A natural physical interpretation of them in the context of the strong cosmic censor conjecture and topology change is discussed.