Interposing a Varying Gravitational Constant Between Modified Newtonian
  Dynamics and Weak Weyl Gravity
  The Newtonian gravitational constant $G$ obeys the dimensional relation $[G]
[M] [a] = [v]^4$, where $M$, $a$, and $v$ denote mass, acceleration, and speed,
respectively. Since the baryonic Tully-Fisher (BTF) and Faber-Jackson (BFJ)
relations are observed facts, this relation implies that $G\, a = {\rm
constant}$. This result cannot be obtained in Newtonian dynamics which cannot
explain the origin of the BTF and BFJ relations. An alternative, modified
Newtonian dynamics (MOND) assumes that $G=G_0$ is constant in space and derives
naturally a characteristic constant acceleration $a=a_0$, as well as the BTF
and BFJ relations. This is overkill and it comes with a penalty: MOND cannot
explain the origin of $a_0$. A solid physical resolution of this issue is that
$G \propto a^{-1}$, which implies that in lower-acceleration environments the
gravitational force is boosted relative to its Newtonian value because $G$
increases. This eliminates all problems related to MOND's empirical cutoff
$a_0$ and yields a quantitative method for mapping the detailed variations of
$G(a)$ across each individual galaxy as well as on larger and smaller scales.
On the opposite end, the large accelerations produced by $G(a)$ appear to be
linked to the weak-field limit of the fourth-order theory of conformal Weyl
gravity.
