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.

Automated summary

The paper compares different causal structures based on three key properties and shows that spacetimes with continuous metrics do not generally admit a causal structure that satisfies all three properties simultaneously.