  In these notes we discuss Lorentz-Finsler metrics, a notion originated in
relativity theory, on certain groups of symplectic and contact transformations.
Some basic geometric questions arising in this context concerning distance,
geodesics and their conjugate points, and existence of a time function, turn
out to be related to a variety of subjects including the contact systolic
problem, group quasi-morphisms, the Monge-Amp\`ere equation, and a subtle
interplay between symplectic rigidity and flexibility. We discuss these
interrelations, providing necessary preliminaries, and formulate a number of
open questions.
