The Cosmic Censor Forbids Naked Topology

TL;DR

This paper proves that under certain causal and focusing conditions, the topology of asymptotically flat spacetimes is constrained, preventing the existence of naked singularities and ensuring the domain of outer communications is simply connected.

Contribution

It establishes active topological censorship without assuming global hyperbolicity, relying instead on causal structure assumptions and focusing conditions.

Findings

Active topological censorship holds under specified conditions.

The domain of outer communications is shown to be simply connected.

Naked singularities would imply violations of the causal assumptions.

Abstract

For any asymptotically flat spacetime with a suitable causal structure obeying (a weak form of) Penrose's cosmic censorship conjecture and satisfying conditions guaranteeing focusing of complete null geodesics, we prove that active topological censorship holds. We do not assume global hyperbolicity, and therefore make no use of Cauchy surfaces and their topology. Instead, we replace this with two underlying assumptions concerning the causal structure: that no compact set can signal to arbitrarily small neighbourhoods of spatial infinity (``$i^0$-avoidance''), and that no future incomplete null geodesic is visible from future null infinity. We show that these and the focusing condition together imply that the domain of outer communications is simply connected. Furthermore, we prove lemmas which have as a consequence that if a future incomplete null geodesic were visible from infinity, then given our $i^0$-avoidance assumption, it would also be visible from points of spacetime that can communicate with infinity, and so would signify a true naked singularity.