While Einstein's theory of gravity is formulated in a smooth setting, the celebrated singularity theorems of Hawking and Penrose
describe many physical situations in which this smoothness must eventually breakdown. In positive-definite signature, there is a highly successful theory of metric and metric-measure geometry which includes Riemannian manifolds as a special case, but permits the extraction of nonsmooth limits under dimension and curvature bounds analogous to the energy conditions in relativity:
here sectional curvature is reformulated through triangle comparison, while and Ricci curvature is reformulated using entropic convexity along geodesics of probability measures. This lecture explores recent progress in the development of an analogous theory in Lorentzian signature, whose ultimate goal is to provide a nonsmooth theory of gravity.
We begin with a simplified approach to Kunzinger and Saemann's theory of (globally hyperbolid, regularly localizable) Lorentzian length spaces in which the time-separation function takes center stage. We show compatibility of two different notions of timelike geodesic used in the literature. We then propose a synthetic (i.e. nonsmooth) reformulation of the null energy condition by relating to the timelike curvature-dimension conditions of Cavalletti \& Mondino (and Braun), and discuss its consistency and stability properties.