Exotica and the status of the strong cosmic censor conjecture in four dimensions

Abstract

An immense class of physical counterexamples to the four dimensional strong cosmic censor conjecture---in its usual broad formulation---is exhibited. More precisely, out of any closed and simply connected 4-manifold an open Ricci-flat Lorentzian 4-manifold is constructed which is not globally hyperbolic and no perturbation of it, in any sense, can be globally hyperbolic. This very stable non-global-hyperbolicity is the consequence of our open spaces having a "creased end" i.e., an end diffeomorphic to an exotic ${\mathbb R}^4$. Open manifolds having an end like this is a typical phenomenon in four dimensions. The construction is based on a collection of results of Gompf and Taubes on exotic and self-dual spaces, respectively, as well as applying Penrose' non-linear graviton construction (i.e., twistor theory) to solve the Riemannian Einstein's equation. These solutions then are converted into stably non-globally-hyperbolic Lorentzian vacuum solutions. It follows that the plethora of vacuum solutions we found cannot be obtained via the initial value formulation of the Einstein's equation because they are "too long" in a certain sense (explained in the text). This different (i.e., not based on the initial value formulation but twistorial) technical background might partially explain why the existence of vacuum solutions of this kind have not been realized so far in spite of the fact that, apparently, their superabundance compared to the well-known globally hyperbolic vacuum solutions is overwhelming.