Gravity from uncertainty (almost!)

Heisenberg's uncertainty principle applies for all quantum (tiny) particles: their momentum uncertainty (dp) times their position uncertainty (dx) equals hbar (Planck's constant). This means, that if you measure their position with light, then you lose information about their momentum because the light gives them a random kick, and the more you use short wavelength high energy light to pinpoint their position, the more of a random kick they get, so the less you know their momentum.

I recently published a paper applying this at the Solar system scale, where it is not supposed to apply, by looking at the orbit of two planets of mass M and m. If you couple each Planck mass in mass M with each one in mass m and calculate the total uncertainty for all the mutual interactions between each possible pair, you get dp*dx = a big summation*hbar. The dx is the separation between M and m (the orbital radius, r). If you use the formulae: E=pc and dE=F*dx (again dx=r) you get F = constant*Mm/r^2 which has the form of Newton's gravity law (see eq. 8 in the paper below). This is nice because it has been derived from quantum mechanics which is only supposed to work for tiny quantum objects and is not supposed to be on speaking terms with gravity. One way to think about it is that as an orbital system loses position uncertainty (an orbit becomes tighter) it must gain momentum uncertainty (it orbits faster).

I published the paper last year (see reference below) but the last step falls short. The 'gravitational constant' I derived (the constant G) was  hc/mp^2 (eq. 8). If there was an independent way to measure the Planck mass (mp) then I could claim to have derived Newton's gravity law, but the Planck mass is found by comparing the gravitational potential energy of two Planck masses with separation r with the energy of a photon of the same energy with a wavelength r which makes the final part of my derivation (the step from eq. 8 to eq. 9) circular. Nevertheless, the part leading up to eq. 8 is a valid and interesting approach and I'd like to build on it. The idea has been published in the journal below, and a pdf of the first two pages (the important bit) is freely available to all. So if you want to see an intriguing idea which produces the 'form' of Newton's gravity law from quantum mechanics, here it is. If you can think of a new way to get from eq 8 to eq 9, all the better!

McCulloch, M.E., 2013. Gravity from the uncertainty principle. Astrophysics and Space Science, 349, 2, 957-959. link