Mathematical Physics
[Submitted on 4 Aug 2018 (v1), last revised 23 Sep 2023 (this version, v3)]
Title:Displacement convexity of Boltzmann's entropy characterizes the strong energy condition from general relativity
View PDFAbstract: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.
Submission history
From: Robert McCann [view email][v1] Sat, 4 Aug 2018 21:30:11 UTC (103 KB)
[v2] Sat, 6 Oct 2018 20:43:35 UTC (105 KB)
[v3] Sat, 23 Sep 2023 01:52:39 UTC (84 KB)
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