Causality conditions and the lengths of nonspacelike curves

In globally hyperbolic space-times there is a finite upper bound on the proper time lengths of nonspacelike curves between two chronologically related events. It is shown that if global hyperbolicity is replaced by a weaker causality condition, then there can exist chronologically related events in the space-time such that the proper time lengths of nonspacelike curves joining them grow without bound. It is also shown that the existence of a finite upper bound on the lengths of nonspacelike curves between two chronologically related events implies only chronology of the space-time but not causality. (author)