Genericity of singularities in spacetimes with weakly trapped submanifolds

TL;DR

This paper demonstrates that in certain classes of spacetimes with weakly trapped submanifolds, the presence of causal incomplete geodesics is a generic feature, using Whitney topologies and singularity theorems.

Contribution

It establishes the $C^ abla$-genericity of causal incomplete geodesics in spacetimes with weakly trapped submanifolds under specific curvature and causality conditions.

Findings

Causal incomplete geodesics are generic in spacetimes with weakly trapped submanifolds.

The results extend to submanifolds of any codimension greater than two.

Singularity theorems are applied to prove genericity results.

Abstract

Using the standard Whitney topologies on spaces of Lorentzian metrics, we show that the existence of causal incomplete geodesics is a $C^\infty$-generic feature within the class of spacetimes of a given dimension $n\geq 3$ that are stably causal, satisfy the timelike convergence condition (``strong energy condition'') and contain a codimension-two spacelike weakly trapped closed submanifold such as, e.g., a marginally outer trapped surface (MOTS). By using a singularity theorem of Galloway and Senovilla for spacetimes containing trapped closed submanifolds of codimension higher than two we also prove an analogous $C^\infty$-genericity result for stably causal spacetimes with a suitably modified curvature condition and weakly trapped closed spacelike submanifold of any codimension $k> 2$.