Another limit on the fine structure constant - based on a formula by Lubos Motl This paper by Hod entitled "Gravitation, thermodynamics and the fine-structure constant" (PDF) starts from a formula on black hole resonance frequencies.  If you follow the references through, Lubos Motl has also derived this formula in L. Motl, Adv. Theor. Math. Phys. 6, 1135 (2003). Now Hod claims that the formula and the third principle of thermodynamics, taken together, imply that
$$\alpha > \frac {{\rm ln} \,3}  {48 \, \pi}$$
This is a very good limit, because it means that $\alpha > 1/137.26$ - which is very close to the real value $1/137.036$. Is Hod's argument correct?
 A: The fine-structure constant cannot be derived from considerations about gravity and thermodynamics only because it's a constant determining the strength of a particular non-gravitational force, electromagnetism, which is mediated by a spin-one boson, the photon.
Quite generally, almost all expert physicists will surely agree with me that no such simple derivation of the fine-structure constant – which ignores the detailed spectrum of quantum fields we know in Nature – may exist. Also, the fine-structure constant with the value $\alpha \sim 1/137.036$ is an "accidental" outcome of some running of constants from the high scales to the low ones. You may check that 7 years later, Hod's paper has 0 citations.
He makes some assumptions that make be "morally right" but they cannot be taken literally. In particular, he assumes that the black holes have to be strictly "heavier than the Planck mass". While the lightest black holes are of order the Planck mass, the precise numerical coefficient in front of the Planck mass may be any number of order one, so no specific numerical constant on the right hand side may be correctly derived.
Moreover, I don't really believe that the statement that Hod derives is correct. He says that the dimensionless coupling can never be smaller than a number of order one. Well, I think that the string vacua with $\alpha\sim 1/150$ almost certainly exist and are logically consistent and consistent with quantum gravity, thermodynamics, and its third law. Inequalities of the form "the coupling can't be smaller than a universal constant of order one" just aren't true, so they cannot be proven.
Maybe I should mention some question about the credit. Shahar Hod's interpretations may be invalid but he is an extremely playful and prolific discoverer of various numerological formulae. So the insight that the number in the quasinormal frequencies seems to be a simple multiple of $\ln 3$ was his insight and he gave some numerical evidence it was true. 
I only proved that the number was indeed precisely $\ln 3$, first in my paper based on continued fraction and then using the monodromy method in the complex plane in a paper we wrote with Andy Neitzke. The proof was mine or ours, the conjecture was his. In our paper – and in papers by many – it was also argued that the far-reaching interpretations of Shahar Hod and others, such as Olaf Dreyer who introduced the jargon of loop quantum gravity, don't work, e.g. because the predictions for other kinds of black holes are ruled out by the calculations of quasinormal modes of these other black holes.
While I said that his derivation of the inequality for the fine-structure constant isn't right, I must also say that efforts to derive similar inequalities or estimates are thriving these days and some of the similar attempts look intriguing and justified. Similar inequality "electromagnetism is stronger than something" follow from the weak gravity conjecture of ours. There are also heuristic efforts to derive the value of the fine-structure constant from the multiple critical point conjecture. Like Hod's paper, the approach based on this conjecture may say something about the low-energy values of constants, regardless of their apparent high-energy origin. It's because the multiple critical point conjecture is a new principle that says something about the phases how they should be observable at long distances. 
