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) 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 = T (0) is constructed -thus, the spacetime splits orthogonally as R × 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 nonacausal). Concretely, we construct a smooth function τ : M → R such that the levels St = τ (t), t ∈ R satisfy: (i) S = S0, (ii) each St is a (smooth) spacelike Cauchy hypersurface for any other t ∈ R\{0}. If S is also acausal then function τ becomes a time function, i.e., it is strictly increasing on any future-directed causal curve. PACS 2003: 04.20.Gz ; 02.40.Ma; 04.62.+v MSC 2000: primary 53C50, secondary 53C80, 81T20