Global hyperbolicity through the eyes of the null distance

Annegret Burtscher, Leonardo García-Heveling

Published 2022-09-30Version 1

No Hopf--Rinow Theorem is possible in Lorentzian Geometry. Nonetheless, we prove that a spacetime is globally hyperbolic if and only if it is metrically complete with respect to the null distance of a suitable time function. Our approach is based on the observation that null distances behave particularly well for temporal functions, leading also to bi-Lipschitz equivalences and the encoding of the causal relation.