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Pré-Publication, Document De Travail Année : 2024

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

Résumé

Mathematicians have been proposing for sometimes that Monge-Ampère 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-Ampère equation is affine invariant, has rich geometric properties, connects to optimal transport theory, and remains bounded at short distances. Monge-Ampère gravity, that uses a slightly different form of the Monge-Ampère 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-Ampère gravity cannot replace the Newtonian gravity as it does not withstand the solar-system test. We then show that Monge-Ampère gravity can describe a scalar field, often evoked in modified theories of gravity such as Galileons. We show that Monge-Ampère 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-Ampère 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-Ampère 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.

Dates et versions

hal-04600830 , version 1 (04-06-2024)

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Albert Bonnefous, Yann Brenier, Roya Mohayaee. Monge-Ampère gravity, optimal transport theory and their link to the Galileons. 2024. ⟨hal-04600830⟩
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