I am wondering what conditions a Lorentzian manifold $(M,g)$ must satisfy to ensure the existence of a global proper-time foliation (i.e. a decomposition of $M$ into spacelike Cauchy hypersurfaces and a global time function whose tangent vector field $\vec{t}$ satisfies $g(\vec{t},\vec{t})=-1$ and $g(\vec{t},\vec{x})=0$ for all $\vec{x}$ tangent to one of the spacelike hypersurfaces. Is global hyperbolicity enough? If anyone could recommend a good reference, that would be much appreciated!


Thanks!



Conditions on a Lorentzian manifold to ensure existence of global proper-time foliation?
Global hyperbolicity only gives you a Cauchy time function, whose gradient is past directed timelike. See Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes - Antonio N. Bernal, Miguel Sanchez - Commun. Math. Phys. 257, 43–50 (2005)


