  We define a Lorentzian distance function on the group of contactomorphisms of
a closed contact manifold. This distance function is continuous with respect to
the Hofer norm on the group of contactomorphisms defined by Shelukhin and
finite if and only if the group of contactomorphisms is orderable. To prove
this we show that intervals defined by the positivity relation are open with
respect to the topology induced by the Hofer norm. For orderable Legendrian
isotopy classes we show that the Chekanov-type metric defined by Rosen and
Zhang is non-degenerate. In this case similar results hold for a Lorentzian
distance function on Legendrian isotopy classes. This leads to a natural class
of metrics associated to a globally hyperbolic Lorentzian manifold such that
its Cauchy hypersurface has a unit co-tangent bundle with orderable isotopy
class of the fibres.
