# Gauss's Law and the Source for Poisson's Equation in Modified Gravity with Varying G

## TL;DR

This paper explores how a variable gravitational constant G, dependent on acceleration, affects Gauss's law and the source term in Poisson's equation, unifying aspects of MOND and Weyl gravity.

## Contribution

It derives the differential form of Gauss's law and the source for Poisson's equation in G(a) gravity, connecting modified gravity theories with classical laws.

## Key findings

- Gauss's law in integral form remains valid in G(a) gravity.
- The differential form depends on the distribution of G(a)M(r).
- The source term varies with surface density and differs across acceleration regimes.

## Abstract

We have recently shown that the baryonic Tully-Fisher and Faber-Jackson relations imply that the gravitational "constant" $G$ in the force law varies with acceleration $a$ as $G\propto 1/a$ and vice versa. These results prompt us to reconsider every facet of Newtonian dynamics. Here we show that the integral form of Gauss's law in spherical symmetry remains valid in $G(a)$ gravity, but the differential form depends on the precise distribution of $G(a)M(r)$, where $r$ is the distance from the origin and $M(r)$ is the mass distribution. We derive the differential form of Gauss's law in spherical symmetry, thus the source for Poisson's equation as well. Modified Newtonian dynamics (MOND) and weak-field Weyl gravity are asymptotic limits of $G(a)$ gravity at low and high accelerations, respectively. In these limits, we derive telling approximations to the source in spherical symmetry. It turns out that the source has a strong dependence on surface density $M/r^2$ everywhere in $a$-space except in the deep Newton-Weyl regime of very high accelerations.