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

TL;DR

This paper proves that globally hyperbolic spacetimes always have smooth time functions with spacelike Cauchy hypersurfaces, and stably causal spacetimes admit smooth time functions with timelike gradients, ensuring smooth metric splittings.

Contribution

It provides simple proofs that globally hyperbolic spacetimes admit smooth time functions and splittings, and that stably causal spacetimes admit smooth time functions with timelike gradients.

Findings

Existence of smooth time functions with spacelike Cauchy hypersurfaces

Smooth global splitting of globally hyperbolic spacetimes

Existence of smooth time functions with timelike gradients in stably causal 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 $M= \R \times {\cal S}$, $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 $M$.