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

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

2004-01-28

arxiv.org/abs/gr-qc/0401112v3

Commun.Math.Phys. 257 (2005) 43-50

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

9 pages, Latex, to appear in Commun. Math. Phys. Some comments on time functions and stably causal spacetimes are incorporated

Scientific paper

10.1007/s00220-005-1346-1

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 $M$.

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