Causality theory of spacetimes with continuous Lorentzian metrics revisited

Leonardo García-Heveling (Radboud Universiteit) We revisit the causal structures $J^ $ and $K^ $ on a spacetime, and introduce a new one, called $k^ $. The $k^ $-relation can be used to characterize causal curves, and for smooth Lorentzian metrics, it yields the same result as the standard $J^ $-relation. If, on the other hand, the metric is merely continuous, then the different causal structures become inequivalent. We compare them by investigating three properties, namely the validity of the push-up lemma, the openness of chronological futures, and the existence of limit causal curves. Depending on the definition of causal structure chosen, we show that at most two of these three properties hold for continuous metrics. In particular, by using the new relation $k^ $, the push-up lemma holds even when the metric is continuous, while it generally does not for the standard $J^ $-relation. Finally, we argue that, in general, no rea