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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.

auden
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Andrew Palfreyman
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    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. I am open to correction, but I thought $c$ was just an arbitrary figure. http://physics.stackexchange.com/q/56973/ –  Aug 03 '16 at 16:32
  • In that c is derived directly from permittivity and permeability, both themselves properties of spacetime, then c and spacetime are joined inextricably. – Andrew Palfreyman Aug 03 '16 at 18:21
  • That's a good point, but I will have to bow out, and let someone with more expertise deal with it. Best of luck with an answer, and I will mark this question. –  Aug 03 '16 at 18:32
  • The speed of light is simply a choice of spatial and temporal scale. You can make it 1, if you like, which is done often enough in theoretical physics. – CuriousOne Aug 03 '16 at 19:11
  • These are arbitrary scaling constants that we've used to define our units. What you can consider the evolution of intelligent creatures in the universe and then considering their typical length scale, the typical shortest time scale they can perceive, etc.. A system of units where these scales are set to 1 will then yield a typical speed of light expressed in "typical ET's SI units" that is likely going to be a large number (because ET will typically be much smaller than the distance light travels in what for ET is an instant). – Count Iblis Aug 03 '16 at 20:20
  • There is a relation between gravitational potential and c. With higher G the speed of light decreases. – HolgerFiedler Aug 03 '16 at 20:38

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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.

Peter McNeall
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