A spacetime is said to be time-orientable if a continuous designation of wich timelike vectors are to be future/past-directed at each of its points and from point to point over the entire manifold. [Ref. Hawking and Israel (1979) page 225] I want to make sure what conditions must hold in order for a given Lorentzian manifold to be time-orientable,i.e.for the existence of a nonvanishing timelike vector field at each point in the manifold, to express the above in different words. Such a timelike vector field can easily be seen to exist for those Lorentzian manifolds like Minkowski spacetime for instance, with a timelike Killing vector field, but absent such symmetry, for singular spacetimes(say FRW) there is a lack of continuity at singularities where the vector field vanishes that should prevent the existence of a nowhere vanishing timelike vector field. Or under what circumstances is this avoided?