Singularity theorems based on trapped submanifolds of arbitrary co-dimension

Abstract

Standard singularity theorems are proven in Lorentzian manifolds of arbitrary dimension n if they contain closed trapped submanifolds of arbitrary co-dimension. By using the mean curvature vector to characterize trapped submanifolds, a unification of the several possibilities for the boundary conditions in the traditional theorems and their generalization to arbitrary co-dimension is achieved. The classical convergence conditions must be replaced by a condition on sectional curvatures, or tidal forces, which reduces to the former in the cases of co-dimension 1, 2 or n.

11 pages, no figures, some corrections.

References in corpus (4)

• Singularity Theorems and Their Consequences
• Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions
• Trapped surfaces and symmetries
• On the existence of horizons in spacetimes with vanishing curvature invariants

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• The Hawking-Penrose singularity theorem for $C^1$-Lorentzian metrics
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• A critical appraisal of the singularity theorems
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• The codimension 2 null cut locus with applications to spacetime topology
• Ultra-massive spacetimes in 2+1 dimensions with positive Λ
• New effective theories of gravitation and their phenomenological consequences
• On complete trapped submanifolds in globally hyperbolic spacetimes
• On the genericity of singularities in spacetimes with weakly trapped submanifolds
• Gauge-covariant Raychaudhuri dynamics for spin-nondegenerate Lorentz-violating congruences