Lee Smolin, Surely You Were Wrong
There have been a heated discussion by Marcus et al:
Which is about these paper:
And an earlier one by Lee Smolin:
What interested those people are the claim that they obtained a dimensionless coupling constant, which equals G*Lambda, i.e., gravity constant times the cosmological constant, the number is claimed to be very small, 10^-120, so that "it becomes amenable for perturbation theory which doesn't break general covariance".
These folks have been completely wrong in this and I am going to talk about why. To start, let me re-write their dimensionless coupling constant in a more explicit form, so omitted parameters like hbar and C can be properly recognized.
In their foot note they said:
"We work in units where c and 16*PI*hbar = 1, so G means 16*PI*hbar*G = lp^2 which is the Planck length"
So, they are claiming a coupling constant:
alpha_g = lp^2/L^2, with lp the Planck length and L the radius of the universe.
Or in another note:
alpha_g = G * hbar /(C^3*L^2)
The units indeed work out to cancel each other to yield a dimensionless number, and that number can be convenient calculated using my natural unit sets:
G = 1/(2N), L = PI*N, hbar = C = 1, N ~= 1.5x10^40
alpha_g = (1/(2N)) /(PI*N)^2 = 1/(2*PI^2*N^3) = 1.5x10^-122
That agrees with their estimate of 10^-120, though they are off by two oders of magnitude. But that small discrepancy is NOT what I am talking about why they were wrong.
The form of
alpha_g = G * hbar /(C^3*L^2) (1)
is wrong, and the result is off by 80 orders of magnitude!!!
Comparing with the EM fine structure constant
alpha = e^2/(hbar*C)
We would expect the gravity coupling constant to take the form
alpha_g = G*M^2/(hbar*C) (2)
Where M is an appropriate characteristic mass.
So far, both the super string camp and LQG camp took the Planck Scale for granted. So they would naturally use M = planck mass as the characteristic mass. You immediately see that that is wrong and is not going any where, since once you plug it in, alpha_g=== 1, which trivializes it to a meanless parameter!!!! The Planck Scale is wrong and leads both camp to no where!
Now, let's see what is the characteristic mass, in the new claim of alpha_g = 10^-120:
alpha_g = G*hbar/(C^3*L^2) = G*M^2/(hbar*C)
M^2 = hbar^2/(C^2*L^2)
M*C^2 = hbar * (C/L) = hbar * frequency
We immediately recognize what it is: the characteristic mass energy equals to the mass-energy of a photon, whose wavelength is about equal to the radius of the universe, or in another word, whose period is not 10^-14second, but about equal to the age of the universe!!!! The EM wave corresponding to such a photon are of such low frequency that it has NOT finished oscillation of a single cycle yet since the "birth" of the universe!!!!
What a ridiculously small characteristic mass. No wonder their alpha_g is wrong.
Why do theoretical physicists have to go extreme? They either use the Planck mass, which is ridiculously too high. Or they use the mass-energy of a photon which oscillate at the cycle of age of the universe, which is ridiculously too small!!!
Can't they use a characteristic mass which MAKES SENSE? Such a characteric mass should be the mass scales of elementary particles. My theory says it should be the unit of natural mass unit, which is about 137.036 times the mass of an electron.
Using that correct characteristic mass, the correct alpha_g coupling constant should be
alpha_g = G = 1/(2N) = 3.3x10^-41