Blackhole Entropy, LQG and Super String
There is recently a heated debate between the super string camp, Lubos et al, the loop quantum gravity (LQG) camp, Smolin et al, and the Other, Peter Woit et al, triggered by this review paper by Hermann Nicolai, Kasper Peeters, Marija Zamaklar:
Note that both the super string camp and the LQG camp claimed their derivations of the Bekenstein-Hawking black hole entropy as their biggest success of their theories. In my judgement, claiming the derivation of Bekenstein Hawking entropy, such a trivial feat, as their biggest success, is completely "childish" and only shows the lack of "innate" ability on the part of each camp to comprehend what is the REAL physics behind the blackhole entropy!
I am going to show one very trivial derivation of the black hole entropy and how it is proportional to the event horizen surface area divided by Planck area. One that is different from Hawking's but much simpler.
But first, one has to realize two things:
1.Hawking entropy is not an empirical experimental evidence, but merely the result of a gedanken "experiment", e.g., mind exercise.
2. The entropy is a DIMENTIONLESS physical quantity.
Since Hawking entropy is just a mind exercise instead of empirical experimental result. Any claim of deriving the same result as Hawking merely shows that your theory does not have any logical inconsistency or conflict against the line of logic that Hawking's gedanken mind exercise. That's all. You still have not made any connection with the physics reality, unless, of course, that the Hawking formula is confirmed by a REAL experiment, not a gedanken one.
So, if a theory is able to derive Hawking entropy, it's good but really not a big deal. But if it can not, then there is a huge trouble in that it is logically inconsistent with Hawking's reasoning and what Hawing based his reasonings on.
Actually, any consistent theory at all will always leads to the Hawking formula, give or take a trivial numerical factor which is of the order of one, a numerical factor that both LQG and super string had struggled a bit to get right.
Now back to the dimentionless-ness of entropy. Given some basic physics quantities and known physics constants for a black hole, how would you construct an entropy formula that gives a dimentionless quantity? To construct such a formula, all the units should cancel out. That gives you pretty good clue to almost certainly arrive at the only correct answer.
We are given:
1.G, the gravity constant that is involved in any thing related to gravity
2.hbar, anything that envolved entropy needs to count quantum micro states.
3.C, light speed is certainly involved in anything related to spacetime.
4.M, the mass of black hole. We certainly need that.
There is nothing more we need. How would you construct a dimentionless number out of these 4 quantities? An immediate possibility is similar to the electromagnetic coupling constant, the fine structure constant, we can construct a gravity coupling constant here using:
S = G*M^2/(hbar*C) (1)
And that is the Hawking entropy formula, give or take a numerical factor!!! Actual it is only bigger than the Hawking entropy by a factor of 4*PI. Actually that is the only simple way to get a dimentionless number out of the 4 quantities!!!
How come? We know the radius of a black hole is proportional to its mass:
R = 2*G*M/C^2 (2)
M = R*C^2/(2*G) (3)
So the (1) becomes:
S = (1/4)*R^2/(G*hbar/C^3) = (1/4) * R^2/lp^2 = (1/(4*PI)) * (1/4) * A/lp^2 (4)
So it differ from Hawking formula by 1/(4*PI).
Now, let me try to use a totally different but much simpler gedanken experiment to derive an entropy formular similar to Hawkings. It's trivial. Any one can think out a hundred different gedanken experiments, all arrive at the same result, differing by only a numerical factor.
Let's start with a black hole of almost zero mass, and gradually increase its size by throwing in photons in appropriate wavelength.
We do not want to throw in photons whose wavelength is much smaller than the size of the blackhole, since they will lose a great portion of their energy by gravity red-shift, and we do not know how much the mass of blackhole is increase. We do not want to throw in photons of wavelenth much larger than the blackhole size either, since then the photo will diffract around the blackhole completely, without absorbtion.
Let's choose photo wavelength
Lambda = diameter of blackhole = 2*R (5)
Such photons will be absorbed with its energy largely unchanged, increasing the blackhole mass by the equivalent mass of the photon energy. The increase of mass is:
delta M = delta E * C^-2 = 2*PI*hbar * C^-1/lambda = 2*PI*hbar*C^-1/(2*R) (6)
We know that each photon carries an entropy of exactly one, regardless of the photon's energy, so the increase of black hole entropy for each photon is one:
delta S = 1 (7)
dM = PI*hbar*C^-1/R * dS (8)
dS = 1/(PI*hbar) * C * R * dM (9)
R = 2*G*M/C^2, which is M = C^2 * R/(2G) (10)
dM = C^2/(2G) * dR (11)
Put it into equ. (9):
dS = 1/(2*PI*G*hbar) * C^3 * R * dR (12)
Integrate equ. (12) from zero:
S = (1/(4*PI)) * R^2/ (hbar*G/C^3) (13)
S = (1/4*PI^2)^2 * (1/4)* A / lp^2 (14)
Again we ontained virtually the same Hawking entropy, except for a small numerical factor. I could easily get the factor correct if I am willing to try a little bit numerology like the LQG and super string camp did in their crackpot theories.
So you see it is really not a big deal at all to have derived the black hole entropy proportional to horizen area divided by Planck area.