Letters in Mathematical Physics

, Volume 77, Issue 2, pp 183–197 | Cite as

Further Results on the Smoothability of Cauchy Hypersurfaces and Cauchy Time Functions



Recently, folk questions on the smoothability of Cauchy hypersurfaces and time functions of a globally hyperbolic spacetime M, have been solved. Here we give further results, applicable to several problems:
  1. (1)

    Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy hypersurface S. If H were only achronal, counterexamples to the smooth extension exist, but a continuous extension (in fact, valid for any compact achronal subset K) is still possible.

  2. (2)

    Given any spacelike Cauchy hypersurface S, a Cauchy temporal function \(\mathcal{T}\) (i.e., a smooth function with past-directed timelike gradient everywhere, and Cauchy hypersurfaces as levels) with \(S= \mathcal{T}^{-1}(0)\) is constructed – thus, the spacetime splits orthogonally as \(\mathbb{R} \times S\) in a canonical way.

Even more, accurate versions of this last result are obtained if the Cauchy hypersurface S were non-spacelike (including non-smooth, or achronal but non-acausal).


Causality global hyperbolicity Cauchy hypersurface smoothability time and temporal functions Geroch’s theorem submanifolds quantum fields on curved spacetimes 

Mathematics Subject Classification (2000)

primary 53C50 secondary 53C80 81T20 


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Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Departamento de Geometría y TopologíaUniversidad de Granada. Facultad de CienciasGranadaSpain

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