Characterization of the null energy condition via displacement convexity of entropy

We characterize the null energy condition for an $(n+1)$-dimensional Lorentzian manifold in terms of convexity of the relative $(n-1)$-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 $N$-null energy condition that we characterize in terms of null displacement convexity of the relative $N$-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.