On the causal properties of warped product spacetimes

It is shown that the warped product spacetime P = M×f R and the original spacetime M share necessarily the same causality properties, the only exceptions being the properties of causal continuity and causal simplicity which present some subtleties. In this respect it is shown that the direct product spacetime P = M × R is causally simple if and only if (M, g) is causally simple, has a continuous Lorentzian distance and any two causally related events are connected by a maximizing geodesic. Similar conditions are found for the causally continuous property. Some results concerning the behavior of the Lorentzian distance are obtained.