arXiv:2405.15035v1 Announce Type: new
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.
Future Roadmap for Readers: Challenges and Opportunities
Introduction
In this article, we examine the conclusions of a study on Monge-Amp`ere gravity, a non-relativistic description of gravity based on the Monge-Amp`ere equation. We outline a future roadmap for readers, highlighting potential challenges and opportunities on the horizon.
Overview of Monge-Amp`ere Equation
The Monge-Amp`ere equation is a nonlinear generalization of the Poisson equation, where the trace of the Hessian is replaced by its determinant. This equation is affine invariant, has rich geometric properties, and connects to optimal transport theory. It also remains bounded at short distances.
Emergence of Monge-Amp`ere Gravity
The study explores how Monge-Amp`ere gravity naturally emerges through the application of a large-deviation principle to a Brownian system of indistinguishable and independent particles. A physical formulation of this mathematical model is provided, and its theoretical viability is examined and compared to observations.
Limitations of Monge-Amp`ere Gravity
The study finds that Monge-Amp`ere gravity cannot replace Newtonian gravity, as it does not withstand the solar-system test. However, it is shown that Monge-Amp`ere gravity can still describe a scalar field, which is often considered in modified theories of gravity such as Galileons.
Behavior of Monge-Amp`ere Gravity
It is demonstrated 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. However, it still approaches Newtonian gravity asymptotically.
Relativistic Formulation and Connection to Galileons
A relativistic Lagrangian for Monge-Amp`ere gravity in flat spacetime is derived, which represents the field equation of a sum of the Lagrangians of all Galileons. Furthermore, it is shown how the Monge-Amp`ere equation can be obtained from the fully covariant Lagrangian of quartic Galileon in the static limit.
Promising Domain for Further Exploration
The connection between optimal transport theory and modified theories of gravity with second-order field equations, as unraveled in this study, remains a promising domain for further exploration. There are potential challenges and opportunities to investigate the implications and applications of Monge-Amp`ere gravity in various contexts.