After recalling the general setting of Lorentzian synthetic spaces (including important examples fitting the framework), I will discuss some basics of optimal transport theory thereof in order to define "timelike Ricci curvature and dimension bounds" for a possibly non-smooth Lorentzian space.
Some cases of such bounds have remarkable physical interpretations (like the attractive nature of gravity) and can be used to give a characterisation of the Einstein's equations for a non-smooth space.
Based partly on joint work with S. Suhr and partly on joint work with F. Cavalletti.