Gravity from thermodynamics: optimal transport and negative effective dimensions

TL;DR

This paper establishes a novel link between gravity equations and entropy concavity using optimal transport, revealing bounds on Kaluza-Klein masses and constructing new scale-separated AdS solutions in M-theory.

Contribution

It introduces a new framework connecting gravity, entropy, and optimal transport, and derives bounds on KK spectra and scale separation in string theory.

Findings

Entropy concavity characterizes vacuum gravity equations.

Bounds on Kaluza-Klein masses depend only on the cosmological constant.

Constructed a new explicit scale-separated AdS solution in M-theory.

Abstract

We prove an equivalence between the classical equations of motion governing vacuum gravity compactifications (and more general warped-product spacetimes) and a concavity property of entropy under time evolution. This is obtained by linking the theory of optimal transport to the Raychaudhuri equation in the internal space, where the warp factor introduces effective notions of curvature and (negative) internal dimension. When the Reduced Energy Condition is satisfied, concavity can be characterized in terms of the cosmological constant $\Lambda$; as a consequence, the masses of the spin-two Kaluza-Klein fields obey bounds in terms of $\Lambda$ alone. We show that some Cheeger bounds on the KK spectrum hold even without assuming synthetic Ricci lower bounds, in the large class of infinitesimally Hilbertian metric measure spaces, which includes D-brane and O-plane singularities. As an application, we show how some approximate string theory solutions in the literature achieve scale separation, and we construct a new explicit parametrically scale-separated AdS solution of M-theory supported by Casimir energy.