Annales Henri Poincaré

, Volume 17, Issue 10, pp 2801–2824 | Cite as

On Differentiability of Volume Time Functions

  • Piotr T. ChruścielEmail author
  • James D. E. Grant
  • Ettore Minguzzi


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.


Time Function Conjugate Point Cauchy Hypersurface Causal Curve Global Hyperbolicity 
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Copyright information

© Springer International Publishing 2015

Authors and Affiliations

  • Piotr T. Chruściel
    • 1
    Email author
  • James D. E. Grant
    • 2
  • Ettore Minguzzi
    • 3
  1. 1.Fakultät für Physik and Erwin Schrödinger InstituteUniversität WienWienAustria
  2. 2.Department of Mathematics, Faculty of Engineering and Physical SciencesUniversity of SurreyGuildfordUK
  3. 3.Dipartimento di Matematica e Informatica “U. Dini”Università degli Studi di FirenzeFlorenceItaly

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