Volume Comparison for Hypersurfaces in Lorentzian Manifolds and Singularity Theorems

Abstract

We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along timelike curves, and an upper bound on the mean curvature of the hypersurface. Using these results, we give a new proof of Hawking's singularity theorem.

15 pages, LaTeX

References in corpus (1)

• Globally hyperbolic spacetimes can be defined as "causal" instead of "strongly causal"

Cited by in corpus (7)

• The 1965 Penrose singularity theorem
• The Hawking-Penrose singularity theorem for $C^{1,1}$-Lorentzian metrics
• A review of Lorentzian synthetic theory of timelike Ricci curvature bounds
• Volume comparison for $C^{1,1}$ metrics
• Hawking-type singularity theorems for worldvolume energy inequalities
• Volume Singularities in General Relativity
• Mean Curvature, Singularities and Time Functions in Cosmology