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

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.

💡 Research Summary

The paper tackles a mathematically precise version of the Strong Cosmic Censorship (SCC) conjecture, namely the formulation attributed to Geroch‑Horowitz‑Penrose (GHP) and later spelled out by Wald. The authors’ central claim is that in any non‑globally‑hyperbolic spacetime possessing a future Cauchy horizon, one can always find a future‑inextendible causal curve lying entirely within the causal past of a point on that horizon. By constructing such curves explicitly, they demonstrate that, for a wide class of physically relevant solutions—charged Reissner‑Nordström black holes, rapidly rotating Kerr black holes, and asymptotically anti‑de Sitter (AdS) spacetimes—the inextendible curves have infinite proper length.

The proof proceeds in several steps. First, the paper reviews the hierarchy of SCC statements, emphasizing that the GHP version replaces the vague “singularities are hidden” slogan with a concrete geometric condition: every point on a future Cauchy horizon must admit a causal curve in its past that cannot be extended any further into the future. Using the causal hierarchy developed by Hawking and Ellis, the authors identify the future Cauchy horizon ( \mathcal{H}^{+} ) and pick an arbitrary point ( q \in \mathcal{H}^{+} ). They then analyze the set ( J^{-}(q) ) and show that, because the spacetime fails global hyperbolicity, the boundary of ( J^{-}(q) ) contains a null or timelike generator that terminates on the horizon without encountering any curvature singularity. This generator is the desired future‑inextendible curve ( \gamma ).

The novelty lies in the second part of the argument, where the authors examine the proper length of ( \gamma ). In the Reissner‑Nordström case with charge exceeding the mass, the inner Cauchy horizon is a regular null surface; the affine parameter along ( \gamma ) diverges as it approaches the horizon, yielding an infinite proper time. An analogous calculation for the Kerr geometry with angular momentum close to the extremal limit shows the same divergence, thanks to the frame‑dragging effect that stretches the generator indefinitely. For asymptotically AdS spacetimes, reflective boundary conditions cause null rays to bounce back and forth, effectively creating an infinite sequence of causal segments that accumulate on the horizon, again giving ( \gamma ) infinite length.

These infinite‑length, future‑inextendible curves are precisely the defining feature of Malament‑Hogarth (MH) spacetimes. An MH spacetime allows an observer with finite proper time to receive signals from a region where an infinite amount of proper time has elapsed, a property that underlies the notion of “gravitational computers” capable of performing non‑Turing‑computable tasks. By establishing that all the examined physically relevant non‑globally‑hyperbolic solutions are MH, the authors propose a sharper geometric formulation of SCC: every physically relevant, asymptotically flat or AdS spacetime that is not globally hyperbolic must be a Malament‑Hogarth spacetime.

The paper then explores the conceptual bridge between SCC and the Church‑Turing thesis. If SCC excludes MH spacetimes, then the laws of classical general relativity would enforce the Church‑Turing limit on what can be computed in the physical universe. Conversely, if MH spacetimes are admissible, general relativity would permit “hypercomputational” processes, thereby challenging the universality of the Church‑Turing thesis. This duality suggests that the two seemingly unrelated conjectures may be manifestations of a deeper principle governing the computational structure of spacetime.

The authors acknowledge several limitations. Their proof relies on specific families of solutions where the inner horizon is regular and the energy conditions hold; exotic matter or quantum effects could alter the causal structure and invalidate the MH property. Moreover, the analysis is classical; incorporating semiclassical back‑reaction or a full quantum gravity treatment might remove the Cauchy horizon altogether, thereby restoring global hyperbolicity.

In conclusion, the paper offers a rigorous demonstration that many non‑globally‑hyperbolic, physically interesting spacetimes contain infinite‑length, future‑inextendible causal curves, thereby qualifying as Malament‑Hogarth. This result refines the strong cosmic censorship conjecture into a concrete geometric statement and opens a dialogue between relativistic causality and computational theory, hinting that the ultimate fate of singularities and the limits of computation may be governed by a common underlying principle.