On the genericity of singularities in spacetimes with weakly trapped submanifolds
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| Format: | Preprint |
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2024
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| author | Espinoza, Victor Luis Silva, Ivan Pontual Costa e |
| author_facet | Espinoza, Victor Luis Silva, Ivan Pontual Costa e |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_09651 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the genericity of singularities in spacetimes with weakly trapped submanifolds Espinoza, Victor Luis Silva, Ivan Pontual Costa e Differential Geometry General Relativity and Quantum Cosmology 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. |
| title | On the genericity of singularities in spacetimes with weakly trapped submanifolds |
| topic | Differential Geometry General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2406.09651 |