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

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.