  It is shown that the warped product spacetime P=M *_f H, where H is a
complete Riemannian manifold, 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. For
instance, it is shown that if diamH=+\infty, the direct product spacetime P=M*H
is causally simple if and only if (M,g) is causally simple, the Lorentzian
distance on M is continuous and any two causally related events at finite
distance are connected by a maximizing geodesic. Similar conditions are found
for the causal continuity property. Some new results concerning the behavior of
the Lorentzian distance on distinguishing, causally continuous, and causally
simple spacetimes are obtained. Finally, a formula which gives the Lorentzian
distance on the direct product in terms of the distances on the two factors
(M,g) and (H,h) is obtained.
