We show that many Lorentzian manifolds of dimension >2 do not admit a
spacelike codimension-one foliation, and that almost every manifold of
dimension >2 which admits a Lorentzian metric at all admits one which satisfies
the dominant energy condition and the timelike convergence condition. These two
seemingly unrelated statements have in fact the same origin.
We also discuss the problem of topology change in General Relativity. A
theorem of Tipler says that topology change is impossible via a spacetime
cobordism whose Ricci curvature satisfies the strict lightlike convergence
condition. In his theorem, the boundary of the cobordism is required to be
spacelike. We show that topology change with the strict lightlike convergence
condition and also the dominant energy condition is possible in many cases when
one requires instead only that there exists a timelike vector field which is
transverse to the boundary.Comment: 31 page
