Causality theory of spacetimes with continuous Lorentzian metrics revisited

TL;DR

This paper revisits the causal structure of spacetimes with continuous Lorentzian metrics, comparing various causal relations and their properties, revealing limitations and possibilities in the non-smooth setting.

Contribution

It analyzes and compares different causal structures for continuous metrics, identifying which properties hold or fail, and clarifies the limitations of causal theories in non-smooth spacetimes.

Findings

Push-up lemma fails for continuous metrics

Some causal structures satisfy push-up and open futures

No causal structure satisfies all three key properties simultaneously

Abstract

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.