We generalize some geometric features (in particular, timelike incompleteness) of warped product manifolds with non-negative timelike Ricci curvature to spacetimes that admit irrotational, but not necessarily geodesic, observer fields. It turns out that the strong energy condition still implies properties such as slowing expansion rates on these more general asymmetric spacetimes, provided the restspace is compact. Rigid observer fields are also studied in the context of curvature.