Causality theory of spacetimes with continuous Lorentzian metrics revisited

We consider the usual causal structure (I ⁺, J ⁺) on a spacetime, and a number of alternatives based on Minguzzi's D ⁺ and Sorkin and Woolgar's K ⁺, in the case where the spacetime metric is continuous, but not necessarily smooth. We compare the different causal structures based on three key properties, namely the validity of the push-up lemma, the openness of chronological futures, and the existence of limit causal curves. Recall that if the spacetime metric is smooth, (I ⁺, J ⁺) satisfies all three properties, but that in the continuous case, the push-up lemma fails. Among the proposed alternative causal structures, there is one that satisfies push-up and open futures, and one that has open futures and limit curves. Furthermore, we show that spacetimes with continuous metrics do not, in general, admit a causal structure satisfying all three properties at once. (paper)