Exotica or the failure of the strong cosmic censorship in four dimensions

TL;DR

This paper constructs a smooth Ricci-flat Lorentzian metric on an exotic 4-dimensional space, providing a plausible counterexample to the strong cosmic censorship conjecture in four dimensions.

Contribution

It introduces the concept of a robust counterexample and constructs a specific Lorentzian metric on an exotic 4-manifold as a potential counterexample to the conjecture.

Findings

Existence of a smooth Ricci-flat Lorentzian metric on an exotic R^4

This metric serves as a plausible generic counterexample in four dimensions

Counterexample relies on twistor theory and exotic smooth structures

Abstract

In this letter a generic counterexample to the strong cosmic censor conjecture is exhibited. More precisely---taking into account that the conjecture lacks any precise formulation yet---first we make sense of what one would mean by a "generic counterexample" by introducing the mathematically unambigous and logically stronger concept of a "robust counterexample". Then making use of Penrose' nonlinear graviton construction (i.e., twistor theory) and a Wick rotation trick we construct a smooth Ricci-flat but not flat Lorentzian metric on the largest member of the Gompf--Taubes uncountable radial family of large exotic ${\mathbb R}^4$'s. We observe that this solution of the Lorentzian vacuum Einstein's equations with vanishing cosmological constant provides us with a sort of counterexample which is weaker than a "robust counterexample" but still reasonable to consider as a "generic counterexample". It is interesting that this kind of counterexample exists only in four dimensions.