An optimal transport formulation of the Einstein equations of general relativity

Abstract

The goal of the paper is to give an optimal transport formulation of the full Einstein equations of general relativity, linking the (Ricci) curvature of a space-time with the cosmological constant and the energy-momentum tensor. Such an optimal transport formulation is in terms of convexity/concavity properties of the Shannon-Bolzmann entropy along curves of probability measures extremizing suitable optimal transport costs. The result gives a new connection between general relativity and optimal transport; moreover it gives a mathematical reinforcement of the strong link between general relativity and thermodynamics/information theory that emerged in the physics literature of the last years.

60 pages. Improved overall exposition, added a non-smooth example in the introduction and an appendix about a synthetic non-smooth framework for Einstein's equations. Final version accepted by the Journal of the European Mathematical Society (JEMS)