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      Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions

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          Abstract

          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) 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) Given any spacelike Cauchy hypersurface S, a Cauchy temporal function T (i.e., a smooth function with past-directed timelike gradient everywhere, and Cauchy hypersurfaces as levels) with S equal to one of the levels, is constructed -thus, the spacetime splits orthogonally as \(R \times S\) in a canonical way. Even more, accurate versions of this result are obtained if the Cauchy hypersurface S were non-spacelike (including non-smooth, or achronal but non-acausal).

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          Author and article information

          Journal
          15 December 2005
          Article
          10.1007/s11005-006-0091-5
          gr-qc/0512095
          cb05da8a-37ea-4c3a-8aaa-9f3523aabfb9
          History
          Custom metadata
          Lett.Math.Phys. 77 (2006) 183-197
          This paper includes all the results in the two previous ones gr-qc/0507018 and gr-qc/0511016 (which will be withdrawn). 15 pages, 1 figure
          gr-qc math.DG

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