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 space-times Hawking’s time function can be uniformly approximated by smooth time functions with timelike gradient.
CITATION STYLE
Chruściel, P. T., Grant, J. D. E., & Minguzzi, E. (2016). On Differentiability of Volume Time Functions. Annales Henri Poincare, 17(10), 2801–2824. https://doi.org/10.1007/s00023-015-0448-3
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