Monge-Amp\`ere gravity, optimal transport theory and their link to the Galileons

TL;DR

This paper explores Monge-Ampère gravity, linking it to optimal transport and Galileons, analyzing its physical viability, and proposing a relativistic formulation connecting it to modified gravity theories.

Contribution

It provides a physical formulation of Monge-Ampère gravity, studies its theoretical viability, and establishes connections with Galileon theories and optimal transport.

Findings

Monge-Ampère gravity cannot replace Newtonian gravity in the solar system.

It can describe a scalar field similar to those in modified gravity theories.

The model is screened at short distances and approaches Newtonian gravity at large scales.

Abstract

Mathematicians have been proposing for sometimes that Monge-Amp\`ere equation, a nonlinear generalization of the Poisson equation, where trace of the Hessian is replaced by its determinant, provides an alternative non-relativistic description of gravity. Monge-Amp\`ere equation is affine invariant, has rich geometric properties, connects to optimal transport theory, and remains bounded at short distances. Monge-Amp\`ere gravity, that uses a slightly different form of the Monge-Amp\`ere equation, naturally emerges through the application of large-deviation principle to a Brownian system of indistinguishable and independent particles. In this work we provide a physical formulation of this mathematical model, study its theoretical viability and confront it with observations. We show that Monge-Amp\`ere gravity cannot replace the Newtonian gravity as it does not withstand the solar-system test. We then show that Monge-Amp\`ere gravity can describe a scalar field, often evoked in modified theories of gravity such as Galileons. We show that Monge-Amp\`ere gravity, as a nonlinear model of a new scalar field, is screened at short distances, and behaves differently from Newtonian gravity above galactic scales but approaches it asymptotically. Finally, we write a relativistic Lagrangian for Monge-Amp\`ere gravity in flat space time, which is the field equation of a sum of the Lagrangians of all Galileons. We also show how the Monge-Amp\`ere equation can be obtained from the fully covariant Lagrangian of quartic Galileon in the static limit. The connection between optimal transport theory and modified theories of gravity with second-order field equations, unravelled here, remains a promising domain to further explore.