Generalizations of the Abstract Boundary singularity theorem

Abstract

The Abstract Boundary singularity theorem was first proven by Ashley and Scott. It links the existence of incomplete causal geodesics in strongly causal, maximally extended spacetimes to the existence of Abstract Boundary essential singularities, i.e., non-removable singular boundary points. We give two generalizations of this theorem: the first to continuous causal curves and the distinguishing condition, the second to locally Lipschitz curves in manifolds such that no inextendible locally Lipschitz curve is totally imprisoned. To do this we extend generalized affine parameters from $C^1$ curves to locally Lipschitz curves.

arXiv comment24 pages

References in corpus (12)

• Singularity Theorems and Their Consequences
• The 1965 Penrose singularity theorem
• The abstract boundary---a new approach to singularities of manifolds
• Further properties of causal relationship: causal structure stability, new criteria for isocausality and counterexamples
• Gromov, Cauchy and causal boundaries for Riemannian, Finslerian and Lorentzian manifolds
• Causality violation and singularities
• Topology of the Misner Space and its g-boundary
• The Attached Point Topology of the Abstract Boundary For Space-Time
• A rigidity result on the ideal boundary structure of smooth space-times
• The Strongly Attached Point Topology of the Abstract Boundary For Space-Time
• The chart based approach to studying the global structure of a spacetime induces a coordinate invariant boundary
• A Correspondence Between Distances and Embeddings for Manifolds: New Techniques for Applications of the Abstract Boundary

Cited by in corpus (4)

• The Endpoint Theorem
• What actually happens when you approach a gravitational singularity?
• The global properties of the finiteness and continuity of the Lorentzian distance
• Examples of covering properties of boundary points of space-times