A proof of the Geroch-Horowitz-Penrose formulation of the strong cosmic censor conjecture motivated by computability theory

TL;DR

This paper proves a mathematical version of the strong cosmic censorship conjecture, linking it to computability theory and Malament-Hogarth space-times, revealing deep connections between general relativity and the Church-Turing thesis.

Contribution

It provides a rigorous proof of a formulation of the strong cosmic censorship conjecture and relates it to Malament-Hogarth space-times and computability theory.

Findings

Future-inextendible causal curves often have infinite length in relevant space-times.

Physically relevant non-globally hyperbolic space-times are related to Malament-Hogarth space-times.

A geometric formulation links the conjecture to computational limits in gravitational systems.

Abstract

In this paper we present a proof of a mathematical version of the strong cosmic censor conjecture attributed to Geroch-Horowitz and Penrose but formulated explicitly by Wald. The proof is based on the existence of future-inextendible causal curves in causal pasts of events on the future Cauchy horizon in a non-globally hyperbolic space-time. By examining explicit non-globally hyperbolic space-times we find that in case of several physically relevant solutions these future-inextendible curves have in fact infinite length. This way we recognize a close relationship between asymptotically flat or anti-de Sitter, physically relevant extendible space-times and the so-called Malament-Hogarth space-times which play a central role in recent investigations in the theory of "gravitational computers". This motivates us to exhibit a more sharp, more geometric formulation of the strong cosmic censor conjecture, namely "all physically relevant, asymptotically flat or anti-de Sitter but non-globally hyperbolic space-times are Malament-Hogarth ones". Our observations may indicate a natural but hidden connection between the strong cosmic censorship scenario and the Church-Turing thesis revealing an unexpected conceptual depth beneath both conjectures.