In the case of a globally hyperbolic spacetime, what you want is a smooth *Cauchy temporal function* (the gradient is everywhere timelike, not just causal, and each level set is a Cauchy surface that is necessarily spacelike). That global hyperbolicity is also sufficient the the existence of a smooth temporal function was also shown by Bernal and Sanchez, in the followup [arXiv:gr-qc/0512095] to their original paper that you cite.

I'm not sure if you are also interested in foliations of *non globally hyperbolic* spacetimes. I'm not completely sure what the right conditions would be then. See comment about *stable casuality* below.

This post imported from StackExchange MathOverflow at 2015-03-11 13:20 (UTC), posted by SE-user Igor Khavkine