Lorentzian distance functions on the group of contactomorphisms

The notion of positive (non-negative) contact isotopy, defined by Eliashberg and Polterovich, leads to two relations on the group of contactomorphisms. These relations resemble the causal relations of a Lorentzian manifold. In this talk we will introduce a class of Lorentzian distance functions on the group of contactomorphisms, that are compatible with these relations. The Lorentzian distance functions turn out to be continuous with respect to the Hofer-norm of a contactomorphism defined by Shelukhin.