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

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 = T −1 (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 S t = τ −1 (t), t ∈ R satisfy: (i) S = S 0 , (ii) each S t 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.

Effective action for a quantum scalar field in warped spaces

“…The full appreciation of this problem in a background containing a given brane has led to an interesting constraint on the warp factor itself in order to compute the one-loop correction. This criterion is in fully agreement with the more or less recent advances in the characterization of globally hyperbolic spacetimes [13,14]. This paper is organized as follows: in the next section we give an outlook of the basic formalism, indicating the overall procedure to extract the effective lagrangian.…”

“…From a different, but complementary, point of view it was recently shown that a given warped spacetime is hyperbolic (i.e., it is suitable for a well-defined Cauchy problem) if, and only if, the warp factor is of C ∞ class [13,14]. The previous discussion was, then, an illustrative example of the general formalism developed in Refs.…”

“…The previous discussion was, then, an illustrative example of the general formalism developed in Refs. [13,14].…”

Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime

“…We adopt definitions and results about causal structures as in [44,56], but we take recent results [8,9] into account, too. If (M , g) and (M , g ) are spacetimes and S ⊂ M ∩ M , then J ± (S; M ) (I ± (S; M )) and J ± (S; M ) (I ± (S; M )) indicate the causal (resp.…”

The causal hierarchy of spacetimes

“…Even more, we detail the consequences of these folk questions for the structure of a globally hyperbolic spacetime (Theorem 3.78). Although the description of the smoothing procedure lies out of the scope of the present review (see [44], in addition to the original articles [4,6,7]), the main difficulties are stressed, Remark 3.77.…”