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

Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology

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This paper includes all the results in the two previous ones gr-qc/0507018 and gr-qc/0511016 (which will be withdrawn). 15 pag

Scientific paper

10.1007/s11005-006-0091-5

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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