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×≔M∖{point}. In this paper we prove that if X× is a smooth four-manifold homeomorphic but not necessarily diffeomorphic to M× (more precisely, it carries a smooth structure à la Gompf) then X× 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× with this metric is in fact a hyper-Kähler manifold. In particular we find that the largest member of the Gompf–Taubes radial family of large exotic R4’s admits a complete Ricci-flat metric (and in fact it is a hyper-Kähler manifold).These Riemannian solutions are then converted into Ricci-flat Lorentzian ones thereby exhibiting lot of new vacuum solutions which are not accessible by the initial value formulation. A natural physical interpretation of them in the context of the strong cosmic censorship conjecture and topology change is discussed.