Newton's gravitational constant G tells us about the strength of the static gravitational interaction within our spacetime. The speed of light c can be derived from the properties of spacetime itself, and in some sense describes the "stiffness" of the medium which is spacetime. From here we can derive, together with GR, a complete description of the propagation of gravitational waves. There are, therefore, multiple conceptual ways in which G and c interrelate when describing events unfolding upon the stage of curved spacetime. It seems therefore curious that G and c are not somehow related via some wider and deeper conceptual framework.
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The only relation I know of between G and c comes from Mach's Principle. Before it was known the expansion of the Universe is actually accelerating, General Relativity predicted a time of maximum expansion. At this time, rather crudely
R = GM/c^2 where R is the radius of the Universe, and M is the total mass.
The GR version of this formula is on page 705 of "Gravitation" by Misner, Thorne and Wheeler.
(6/a^2)(da/dt)^2 + 6/a^2 = 16 pi.rho
At the time of maximum expansion (da/dt)^2 = 0 so there is a relation between the radius a and the density rho (in these units G = 1, and c = 1).
For a 3-sphere rho.2(pi)^2(a)^3 = M
This gives a(max) = 4M/3pi or 4GM/3pi(c)^2 in conventional units.
