On differentiability of volume time functions

TL;DR

This paper proves the differentiability of certain volume functions in globally hyperbolic spacetimes and shows they can be approximated by smooth time functions with timelike gradients, impacting causality theory.

Contribution

It establishes the differentiability of Geroch's volume functions and their approximation by smooth functions with timelike gradients in various causality conditions.

Findings

Volume functions are differentiable on globally hyperbolic manifolds.

Volume functions satisfy a local anti-Lipschitz condition.

Hawking's time function can be approximated by smooth functions with timelike gradients in stably causal spacetimes.

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.