On differentiability of volume time functions

Abstract

We show differentiability of a class of Geroch's volume functions on globally hyperbolic manifolds. Furthermore, we prove that every volume function satisfies a local anti-Lipschitz condition over causal curves, and that locally Lipschitz time functions which are locally anti-Lipschitz can be uniformly approximated by smooth time functions with timelike gradient. Finally, we prove that in stably causal spacetimes Hawking's time function can be uniformly approximated by smooth time functions with timelike gradient.

15 pages, 1 figure; minor corrections, the differentiability proof expanded

References in corpus (4)

• Limit curve theorems in Lorentzian geometry
• Elements of causality theory
• Global hyperbolicity is stable in the interval topology
• Convex Sublattices of a Lattice and a Fixed Point Property

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• Properties of the Null Distance and Spacetime Convergence
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• The Null distance encodes causality
• Time functions on Lorentzian length spaces
• Global Hyperbolicity through the Eyes of the Null Distance
• On the existence of smooth Cauchy steep time functions
• Volume Singularities in General Relativity
• A note on stability and Cauchy time functions
• Lorentzian manifolds properly isometrically embeddable in Minkowski spacetime
• Smoothing causal functions
• On the regularity of Cauchy hypersurfaces and temporal functions in closed cone structures
• Spacetime distances: an exploration
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