New discussion with Quantoken on Motl's blog
Dear Quantoken, The e^3 result comes as follows:Mass continuity equation for the galaxies in the space-time of the receding universe: dρ/dt + div.(ρv) = 0. Hence: dρ/dt = -div.(ρv). dx = dy = dz = dr, where r is radius. Hence divergence (div) term -div.(ρv) = -3d(ρv)/dx. For spherical symmetry Hubble equation v = Hr. Hence dρ/dt = -div.(ρv) = -div.(ρHr) = -3d(ρHr)/dr= -3ρHdr/dr= -3ρHSo dρ/dt = -3ρH. Rearranging:-3Hdt = (1/ρ) dρ. Integrating:-3Ht = (ln ρ1) – (ln ρ). Using the base of natural logarithms (e) to get rid of the ln’s: e^(-3Ht) = density ratioBecause H = v/r = c/(radius of universe) = 1/(age of universe, t) = 1/t:e-3Ht = density ratio= e^[-3(1/t)t] = e^-3 = 1/20.
All this says is that the mass of the receding universe is the volume times the higher effective volume at earlier times, which is partly offset by divergence. Another way to analyse it is to calculate numerically the gauge boson pressure from increasing shells around the observer, which cause gravity and electromagnetism. If you look at the actual mechanism by which QFT gauge bosons cause forces, you can get the whole thing from pushing. Similar charges recoil apart since the exchange between them is stronger than the red-shifted gauge bosons from the surrounding universe that are pushing them together, whereas opposite charges shield one another and are thus pushed together. Gravity, as D. R. Lunsford concluded his paper on the subject, is a residual of electromagnetism. Electromagnetism is 10^40 times gravity because the gauge boson exchange between similar charges addes up in a statistical drunkard’s walk, whereby the vector sum is equal to one step times the square root of the total number of steps, and there are 10^80 similar charges in the universe. It’s obvious that there is no way string theory is going to deal with gravity strength or mechanism, because these numbers can’t come out of any string theory. You have to look at the heuristic explanation of GR and QFT, plus the empirical facts of the big bang (not ‘facts’ derived from naive assumptions that gravity has no mechanism in the universe). The 377-ohm vacuum impedance discussion has been removed from my page. Everytime someone says there's something on my page which isn't helpful, I remove it. Your GUITAR may have useful elements in it, but it is likely to be incomplete. Best wishes, Nigel
As for 137: Heisenberg’s uncertainty (based on impossible gamma ray microscope thought experiment): pd = h/(2.Pi), where p is uncertainty in momentum and d is uncertainty in distance. The product pd is physically equivalent to Et, where E is uncertainty in energy and t is uncertainty in time. Since, for light speed, d = ct, we obtain: d = hc/(2.Pi.E). This is the formula the experts generally use to relate the range of the force, d, to the energy of the gauge boson, E.Notice that both d and E are really uncertainties in distance and energy, rather than real distance and energy, but the formula works for real distance and energy, because we are dealing with a definite ratio between the two. Hence for 80 GeV mass-energy W and Z intermediate vector bosons, the force range is on the order of 10^-17 m.Since the formula d = hc/(2.Pi.E) therefore works for d and E as realities, we can introduce work energy as E = Fd, which gives us the strong nuclear force law: F = hc/(2.Pi.d^2). This inverse-square law is 137 times Coulomb's law of electromagnetism.