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 $M^{\times }:=M\setminus \{\mathrm{point}\}$.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 à 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-Kähler 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-Kähler 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.

LaTeX, 31 pp., 7 figures; this is the final published version based on our earlier works on arXiv:1503.04945, arXiv:1707.09180 and arXiv:1905.03952

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Cited by in corpus (1)

• Particulate exotica