Characterization of the null energy condition via displacement convexity of entropy

We characterize the null energy condition for an (n+1)$(n+1)$‐dimensional, time‐oriented Lorentzian manifold in terms of convexity of the relative (n−1)$(n-1)$‐Renyi entropy along displacement interpolations on null hypersurfaces. More generally, we also consider Lorentzian manifolds with a smooth weight function and introduce the Bakry–Emery N$N$‐null energy condition that we characterize in terms of null displacement convexity of the relative N$N$‐Renyi entropy. As application we then revisit Hawking's area monotonicity theorem for a black hole horizon and the Penrose Singularity Theorem from the viewpoint of this characterization and in the context of weighted Lorentzian manifolds.