Rigidity Aspects of Penrose's Singularity Theorem.

Area of Science:

• General Relativity and Gravitational Physics
• Differential Geometry and Topology in Physics

Background:

• Penrose's singularity theorem establishes conditions for the existence of singularities in spacetimes.
• The theorem relies on the presence of trapped surfaces, which pose challenges for analysis.
• Weakly trapped surfaces offer a potential relaxation of these conditions.

Purpose of the Study:

• To investigate the global structure of spacetimes satisfying Penrose's singularity theorem hypotheses with weakly trapped surfaces instead of trapped surfaces.
• To determine the implications of null geodesic completeness for these modified spacetimes.
• To explore rigidity results under these relaxed conditions.

Main Methods:

• Analysis of spacetimes with weakly trapped surfaces and null geodesic completeness.
• Construction of foliations using marginally outer trapped surfaces (MOTS).
• Derivation of properties of generated null hypersurfaces.

Main Results:

• Demonstration that weakly trapped surfaces, when combined with null geodesic completeness, lead to a foliation of MOTS.
• These MOTS generate totally geodesic null hypersurfaces.
• Establishment of either local or global rigidity results depending on specific assumptions.

Conclusions:

• The study provides rigidity results for spacetimes under relaxed singularity theorem conditions.
• Applications to cosmological spacetimes and topological censorship scenarios are discussed.
• This work extends understanding of spacetime structure and singularity theorems.