On the genericity of singularities in spacetimes with weakly trapped submanifolds

TL;DR

The paper addresses whether singularities (null or nonspacelike geodesic incompleteness) are generic in spacetimes containing weakly trapped submanifolds. It develops two parallel routes: (i) a Whitney-topology analysis on the space of Lorentzian metrics around a spacetime with a weakly trapped submanifold, showing prevalence of singular metrics under codimension-two and higher-codimension generalizations; (ii) an initial-data/MOTS framework that uses Banach/Hilbert manifold structures and the Sard–Smale theorem to prove true genericity of incompleteness in the Cauchy developments of initial data containing MOTS. The results rely on energy/convergence conditions, conformal perturbations, and curvature-tidal-force considerations to establish when nearby metrics must yield trapped submanifolds and incomplete null geodesics. Together, these approaches illuminate how singular spacetimes arise generically near weakly trapped structures, with implications for the stability of singularity theorems and the semi-local modeling of gravitational collapse.

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 $Σ$ 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 $Σ$ 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.

Theorems & Definitions (52)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Example 2.5
• Proposition 2.6: lerner_space_1973, prop. 4.7
• Proposition 2.7: lerner_space_1973, prop. 4.2
• Theorem 2.8