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On differentiability of volume time functions. (English) Zbl 1351.83006

Summary: 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.

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
53Z05 Applications of differential geometry to physics
83C10 Equations of motion in general relativity and gravitational theory
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