Pseudoconvexity in Lorentzian doubly warped products

A Lorentzian manifold M is said to be null (resp. causally) pseudoconvex if, given any compact set K in M, there exists a compact set K' in M such that any null (resp. causal) geodesic segment with both endpoints in K lies in K'. Various implications of causal and null pseudoconvexity on the geodesic structure of a Lorentzian manifold have been studied in several recent papers by Beem and Parker, Beem and Ehrlich, and Low. We provide sufficient conditions for a Lorentzian doubly warped product manifold to be null pseudoconvex. These conditions are not necessary and provide new examples of non-globally hyperbolic spacetimes which are null pseudoconvex.