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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.15035 |
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| _version_ | 1866929569023393792 |
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| author | Bonnefous, Albert Brenier, Yann Mohayaee, Roya |
| author_facet | Bonnefous, Albert Brenier, Yann Mohayaee, Roya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_15035 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monge-Ampère gravity, optimal transport theory and their link to the Galileons Bonnefous, Albert Brenier, Yann Mohayaee, Roya General Relativity and Quantum Cosmology Cosmology and Nongalactic Astrophysics 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. |
| title | Monge-Ampère gravity, optimal transport theory and their link to the Galileons |
| topic | General Relativity and Quantum Cosmology Cosmology and Nongalactic Astrophysics |
| url | https://arxiv.org/abs/2405.15035 |