Some remarks on marginally trapped surfaces and geodesic incompleteness

TL;DR

This paper explores the properties of marginally trapped surfaces in non-globally hyperbolic spacetimes, establishing singularity theorems under weaker causality assumptions and introducing the concept of generic MTS.

Contribution

It extends singularity theorems to chronological spacetimes with weaker causality conditions by analyzing generic MTS and their implications for geodesic incompleteness.

Findings

Proves a Hawking-Penrose-type singularity theorem for generic MTS.

Shows that incomplete geodesics are generally not extendable in non-globally hyperbolic spacetimes.

Introduces the notion of generic MTS and their relevance in dynamical black hole horizons.

Abstract

In a recent paper, Eichmair, Galloway and Pollack have proved a Gannon-Lee-type singularity theorem based on the existence of marginally outer trapped surfaces (MOTS) on noncompact initial data sets for globally hyperbolic spacetimes. However, one might wonder whether the corresponding incomplete geodesics could still be complete in a possible non-globally hyperbolic extension of spacetime. In this note, some variants of that result are given with weaker causality assumptions, thus suggesting that the answer is generically negative, at least if the putative extension has no closed timelike curves. However, unlike in the case of MOTS, on which only the outgoing family of normal geodesics is constrained, we have found it necessary in our proofs to impose also a weak convergence condition on the ingoing family of normal geodesics. In other words, we consider marginally trapped surfaces (MTS) in chronological spacetimes, introducing the natural notion of a generic MTS. In particular, a Hawking-Penrose-type singularity theorem is proven in chronological spacetimes with dimensions greater than 2 containing a generic MTS. Such surfaces naturally arise as cross-sections of quasi-local generalizations of black hole horizons, such as dynamical and trapping horizons. We end with some comments on the existence of MTS in initial data sets.