Comment on "Black hole constraints on varying fundamental constants".

In the Letter [1] (also [2]) there is a claim that the generalised second law of thermodynamics (entropy increase) for black holes provides some limits on the rate of variation of the fundamental constants of nature (electric charge e, speed of light c, etc.). We have come to a different conclusion. The results in [1, 2] are based on assumption that mass of a black hole does not change without radiation and accreation. We present arguments showing that this assumption is incorrect and give an estimate of the black hole mass variation due to α = e 2 /¯ hc variation using entropy (and quantum energy level) conservation in an adiabatic process. No model-independent limits on the variation of the fundamental constants are derived from the second law of thermodynamics. It is convenient to present the dimensionless entropy of a charged black hole [3] in terms of dimensionless parameters: S = π[µ + µ 2 − Z 2 α] 2 (1) Here µ = M/M P , M is the black hole mass,M P = (¯ hc/G) 1/2 is the Plank mass, Ze is the black hole charge, the Boltzmann constant k = 1 (for a rotating black hole S = π([µ + µ 2 − Z 2 α − J(J + 1)/µ 2 ] 2 + J(J + 1)/µ 2)). This expression does not contain explicitly any parameters which have dimension (speed of light, proton electric charge, etc.). Therefore, one may only discuss variation of two dimensionless parameters: mass of the black hole in units of the Plank mass (µ) and α. In all known adia-batic processes the entropy S is conserved. It is natural to assume that this is also valid for a slow variation of the fundamental constants. Then eq. (1) gives µ(t) in terms of α(t) and constant S: µ = M M P = (S/π) + Z 2 α 2 (S/π) (2) The event horizon area A of the black hole is quantized [4]. Because of the relation between the entropy S and the horizon area A we obtain the entropy quantization S = (c 3 /4G¯ h)A = πγ ·n, where γ is a numerical constant, n is an integer. This gives us µ as a function of α: µ = M M P = γ · n + Z 2 α 2 √ γ · n (3) One may compare this result with …