Cambridge Journal of Mathematics

Volume 8 (2020)

Number 3

Displacement convexity of Boltzmann’s entropy characterizes the strong energy condition from general relativity

Pages: 609 – 681



Robert J. McCann (Department of Mathematics, University of Toronto, Ontario, Canada)


On a Riemannian manifold, lower Ricci curvature bounds are known to be characterized by geodesic convexity properties of various entropies with respect to the Kantorovich–Rubinstein–Wasserstein square distance from optimal transportation. These notions also make sense in a (nonsmooth) metric measure setting, where they have found powerful applications. This article initiates the development of an analogous theory for lower Ricci curvature bounds in timelike directions on a (globally hyperbolic) Lorentzian manifold. In particular, we lift fractional powers of the Lorentz distance (a.k.a. time separation function) to probability measures on spacetime, and show the strong energy condition of Hawking and Penrose is equivalent to geodesic convexity of the Boltzmann–Shannon entropy there. This represents a significant first step towards a formulation of the strong energy condition and exploration of its consequences in nonsmooth spacetimes, and hints at new connections linking the theory of gravity to the second law of thermodynamics.

2010 Mathematics Subject Classification

Primary 53C50. Secondary 49J52, 58Z05, 82C35, 83C99.

The author acknowledges partial support of his research by Natural Sciences and Engineering Research Council of Canada Grants 217006-08, 2015-04383 and 2020-04162, by a Simons Foundation Fellowship, and by U.S. National Science Foundation Grant No. DMS-144041140 while in residence at MSRI during thematic programs in 2013 and 2016.

Received 13 May 2019

Published 2 October 2020