Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes

Abstract

The folk questions in Lorentzian Geometry, which concerns the smoothness of time functions and slicings by Cauchy hypersurfaces, are solved by giving simple proofs of: (a) any globally hyperbolic spacetime $(M,g)$ admits a smooth time function $\tau$ whose levels are spacelike Cauchy hyperfurfaces and, thus, also a smooth global splitting $M=\mathbb{R}\times \mathcal{S}$, $g=-\beta (\tau ,x)d{\tau }^2+{\bar{g}}_{\tau }$, (b) if a spacetime $M$ admits a (continuous) time function $t$ (i.e., it is stably causal) then it admits a smooth (time) function $\tau$ with timelike gradient $\nabla \tau$ on all $M$.

9 pages, Latex, to appear in Commun. Math. Phys. Some comments on time functions and stably causal spacetimes are incorporated, and referred to gr-qc/0411143 for further details

References in corpus (1)

• On smooth Cauchy hypersurfaces and Geroch's splitting theorem

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