On the regularity of Cauchy hypersurfaces and temporal functions in closed cone structures

Abstract

We complement our work on the causality of upper semi-continuous distributions of cones with some results on Cauchy hypersurfaces. We prove that every locally stably acausal Cauchy hypersurface is stable. Then we prove that the signed distance dS from a spacelike hypersurface S is, in a neighborhood of it, as regular as the hypersurface, and by using this fact we give a proof that every Cauchy hypersurface is the level set of a Cauchy temporal (and steep) function of the same regularity of the hypersurface. We also show that in a globally hyperbolic closed cone structure compact spacelike hypersurfaces with boundary can be extended to Cauchy spacelike hypersurfaces of the same regularity. We end the work with a separation result and a density result. Lemma 2.3. Let (M, C) be a closed cone structure. Let S be a locally stably acausal Cauchy hypersurface then J + (S) and J − (S) are closed, J + (S) ∩ J − (S) = S, J + (S) ∪ J − (S) = M . The set M \S is disconnected, for it is the union of the disjoint non-empty open sets J + (S)\S = M \J − (S) and J − (S)\S = M \J + (S). Proof. Through every point passes an inextendible continuous causal curve intersecting S, thus J + (S) ∪ J − (S) = M . The acausality of S implies J + (S) ∩ J − (S) = S.Suppose that there is a point p ∈ J + (S)\J + (S) then necessarily p ∈ J − (S)\S. Since p ∈ J + (S)\J + (S) by a standard limit curve argument [20, Thm. 2.14]

Low regularity extensions beyond Cauchy horizons

“…The interior IntX of a set X ⊂ M might be denoted X for shortness. The reader is referred to [22,24] for the C 0 causality theory and to [23] for all the other conventions adopted without mention in this work.…”

“…Galloway, Ling and Sbierski [11] proved that in globally hyperbolic spacetimes timelike completeness guarantees C 0 inextendibility, a result improved by Minguzzi and Suhr [25] who showed that global hyperbolicity could in fact be dropped. Bernard and Suhr [2,3] and Minguzzi [22,24] obtained several results on regularity of time functions and Cauchy hypersurfaces under even weaker conditions. Grant, Kunzinger, Sämann and Steinbauer clarified properties and pathologies of the chronological future under low regularity [13].…”

Green operators in low regularity spacetimes and quantum field theory

“…These are Cauchy hypersurfaces which are also Cauchy hypersurfaces for some metric g g with strictly wider lightcones than g. Bernard and Suhr [7,Corollary 2.4] show that a smooth spacelike Cauchy hypersurface is a stable Cauchy hypersurface and that furthermore one can construct a smooth temporal function such that Σ = t −1 (0) [7, Theorem 1]. A full discussion of this issue is given in the paper by Minguzzi [40]. The approach in [40] is complementary to that in [7] and consists of using topological arguments to show that the causal cones can be widened while preserving the Cauchy property of the hypersurface.…”

“…A full discussion of this issue is given in the paper by Minguzzi [40]. The approach in [40] is complementary to that in [7] and consists of using topological arguments to show that the causal cones can be widened while preserving the Cauchy property of the hypersurface. One may then use the methods of Bernal Sanchez [5] to construct a smooth time function with Σ = t −1 (0) which as shown in [43] is a smooth temporal function for the original spacetime (M, g).…”