Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions
The authors prove the following two theorems: Theorem 1: Let \((M,g)\) be a globally hyperbolic space-time, and let \(H\subset M\) be a space-like and acausal compact \(m\)-submanifold with boundary. Then, there exists a space-like Cauchy hypersurface \(S\) such that \(H\subset S\); and Theorem 2. Let \((M,g)\) be globally hyperbolic, and \(S\) a Cauchy hypersurface (possibly non-space-like, nonsmooth, or achronal but non-acausal). Then there exists a smooth function \(\tau:{\mathcal M}\to \mathbb{R}\) such that the levels \(S_t=\tau^{-1}(t)\), \(t\in\mathbb{R}\) satisfy: (i) \(S=S_0\), (ii) each \(S_t\) is a (smooth) space-like Cauchy hypersurface for any other \(t\in\mathbb{R}\setminus\{0\}\). Even more: (A) If \(S\) is also accausal then the function \(\tau\) becomes a time function, i.e., it is strictly increasing on any future-directed causal curve. (B) If \(S\) is space-like (and thus smooth and acausal), then \(\tau\) can be modified to obtain a function with, additionally, time-like gradient everywhere. That is, there exists a Cauchy temporal function \({\mathcal T}:M\to\mathbb{R}\) such that \({\mathcal T}^{-1}(0)\).
scientific article