  We identify certain general geometric conditions on a foliation of a
spacetime (M,g) by timelike curves that will impede the existence of null
geodesic lines, especially if (M,g) possesses a compact Cauchy hypersurface.
The absence of such lines, in turn, yields well-known restrictions on the
geometry of cosmological spacetimes, in the context of Bartnik's splitting
conjecture. Since the (non)existence of null lines is actually a conformally
invariant property, such conditions only need to apply for some suitable
conformal rescaling of g.
