On the genericity of singularities in spacetimes with weakly trapped submanifolds

TL;DR

This paper studies conditions under which singularities are common in spacetimes with weakly trapped submanifolds, showing that certain types of geodesic incompleteness are prevalent or generic in specific mathematical senses.

Contribution

It introduces new results on the genericity of singularities in spacetimes with weakly trapped submanifolds, using advanced topological and functional-analytic methods.

Findings

Singular Lorentzian metrics are prevalent near metrics with weakly trapped submanifolds.

Null geodesic incompleteness is shown to be generic around initial data sets containing MOTS.

Results apply to both codimension 2 and higher codimension cases.

Abstract

We investigate suitable, physically motivated conditions on spacetimes containing certain submanifolds - the so-called {weakly trapped submanifolds} - that ensure, in a set of neighboring metrics with respect to a convenient topology, that the phenomenon of nonspacelike geodesic incompleteness (i.e., the existence of singularities) is generic in a precise technical sense. We obtain two sets of results. First, we use strong Whitney topologies on spaces of Lorentzian metrics on a manifold $M$, in the spirit of Lerner and obtain that while the set of singular Lorentzian metrics around a fiducial one possessing a weakly trapped submanifold $\Sigma$ is not really generic, it is nevertheless prevalent in a sense we define, and thus still quite ``large'' in this sense. We prove versions of that result both for the case when $\Sigma$ has codimension 2, and for the case of higher codimension. The second set of results explore a similar question, but now for initial data sets containing MOTS. For this case, we use certain well-known infinite dimensional, Hilbert manifold structures on the space of initial data and use abstract functional-analytic methods based on the work of Biliotti, Javaloyes, and Piccione to obtain a true genericity of null geodesic incompleteness around suitable initial data sets containing MOTS.