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

Antonio N. Bernal·Miguel Sánchez
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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 $τ$ whose levels are spacelike Cauchy hyperfurfaces and, thus, also a smooth global splitting $M= \R \times {\cal S}$, $g= - β(τ,x) dτ^2 + \bar g_τ$, (b) if a spacetime $M$ admits a (continuous) time function $t$ (i.e., it is stably causal) then it admits a smooth (time) function $τ$ with timelike gradient $\nabla τ$ on all $M$.

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Antonio N. Bernal, Miguel Sánchez. 2005-02-15. Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. https://doi.org/10.1007/s00220-005-1346-1

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