On the regularity of Cauchy hypersurfaces and temporal functions in closed cone structures

Abstract

We complement our work on the causality of upper semi-continuous distributions of cones with some results on Cauchy hypersurfaces. We prove that every locally stably acausal Cauchy hypersurface is stable. Then we prove that the signed distance $d_S$ from a spacelike hypersurface $S$ is, in a neighborhood of it, as regular as the hypersurface, and by using this fact we give a proof that every Cauchy hypersurface is the level set of a Cauchy temporal (and steep) function of the same regularity as the hypersurface. We also show that in a globally hyperbolic closed cone structure compact spacelike hypersurfaces with boundary can be extended to Cauchy spacelike hypersurfaces of the same regularity. We end the work with a separation result and a density result.

arXiv comment17 pages. v2: updated some references, fixed some typos

References in corpus (11)

• On smooth Cauchy hypersurfaces and Geroch's splitting theorem
• Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions
• Causality theory for closed cone structures with applications
• On the definition and examples of cones and Finsler spacetimes
• Convex neighborhoods for Lipschitz connections and sprays
• Lyapounov Functions of closed Cone Fields: from Conley Theory to Time Functions
• Normally preordered spaces and utilities
• On differentiability of volume time functions
• Globally hyperbolic spacetimes can be defined without the 'causal' condition
• An equivalence of Finslerian relativistic theories
• Affine sphere relativity

Cited by in corpus (3)

• Globally hyperbolic spacetimes: slicings, boundaries and counterexamples
• Lorentzian manifolds properly isometrically embeddable in Minkowski spacetime
• Low regularity extensions beyond Cauchy horizons