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)

References in corpus (14)

• Thermodynamics of Spacetime: The Einstein Equation of State
• On the Origin of Gravity and the Laws of Newton
• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure
• Reconstruction of the early Universe as a convex optimization problem
• The causal hierarchy of spacetimes
• Emergent Gravity Paradigm: Recent Progress
• Sharp estimate on the first eigenvalue of the p-Laplacian on compact manifold with nonnegative Ricci curvature
• Examples of backreaction of small scale inhomogeneities in cosmology
• Characterizations of Bounded Ricci Curvature on Smooth and NonSmooth Spaces
• Causality for nonlocal phenomena
• Sectional and intermediate Ricci curvature lower bounds via Optimal Transport
• On Interpolation and Curvature via Wasserstein Geodesics
• Theory of optimal transport for Lorentzian cost functions

Cited by in corpus (20)

• Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications
• A review of Lorentzian synthetic theory of timelike Ricci curvature bounds
• Emergence of the Circle in a Statistical Model of Random Cubic Graphs
• The singularity theorems of General Relativity and their low regularity extensions
• A Lorentzian analog for Hausdorff dimension and measure
• Optimal Transport Reconstruction of Baryon Acoustic Oscillations
• Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds
• Geometry of weighted Lorentz-Finsler manifolds I: Singularity theorems
• Gravity from thermodynamics: optimal transport and negative effective dimensions
• Time functions on Lorentzian length spaces
• Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems
• Convergence of Combinatorial Gravity
• Generally covariant $N$-particle dynamics
• Further observations on the definition of global hyperbolicity under low regularity
• Immiscible Color Flows in Optimal Transport Networks for Image Classification
• Optimal transport on null hypersurfaces and the null energy condition
• The representation of spacetime through time functions
• Volume comparison theorems in Finsler spacetimes
• Causal evolution of probability measures and continuity equation
• Concavity of spacetimes