Nonexistence of spacelike foliations and the dominant energy condition in Lorentzian geometry

Abstract

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.