Weak-field approximation of effective gravitational theory with local Galilean invariance

TL;DR

This paper explores a Galilean-invariant gravitational theory in a 5D framework, deriving key equations and demonstrating compatibility with MOND's AQUAL model, offering new insights into weak-field gravity.

Contribution

It introduces a 5D Galilean-invariant approach that derives fundamental equations and aligns with MOND's AQUAL, expanding the theoretical landscape of weak-field gravity.

Findings

Derived Poisson, diffusion, and Schrödinger equations for the fluctuation field.

Demonstrated compatibility with AQUAL, a MOND-based modified gravity theory.

Presented a wave solution for the Schrödinger equation.

Abstract

We examine the weak-field approximation of locally Galilean invariant gravitational theories with general covariance in a $(4+1)$-dimensional Galilean framework. The additional degrees of freedom allow us to obtain Poisson, diffusion, and Schr\"odinger equations for the fluctuation field. An advantage of this approach over the usual $(3+1)$-dimensional General Relativity is that it allows us to choose an ansatz for the fluctuation field that can accommodate the field equations of the Lagrangian approach to MOdified Newtonian Dynamics (MOND) known as AQUAdratic Lagrangian (AQUAL). We investigate a wave solution for the Schr\"odinger equations.