Monge-Ampère gravity, optimal transport theory, and their link to the Galileons

Mathematicians have been proposing for some time that the Monge-Amp\`ere equation, a nonlinear generalization of the Poisson equation, where a trace of the Hessian is replaced by its determinant, provides an alternative nonrelativistic description of gravity. The Monge-Amp\`ere equation is an affine invariant, has rich geometric properties, connects to optimal transport theory, and remains bounded at short distances. Monge-Amp\`ere gravity, which 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 spacetime, 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.