Kiefer C. Quantum gravity

220 QUANTIZATION OF BLACK HOLES
Taking instead SO(3) as the underlying group, one has j min =1andthus
β =ln3/π √ 2. If a link with spin j min is absorbed or created at the black-hole
horizon, the area changes by
∆A = A 0 =8πβl 2 P√
jmin (j min +1), (7.65)
which is equal to 4(ln 2)lP 2 in the SU(2)-case and to 4(ln 3)l2 P in the SO(3)-case. In
the SU(2)-case, the result for ∆A corresponds to the one advocated by Bekenstein
and Mukhanov (1995), although the spectrum here is not equidistant (which is
why here there is no conflict with the spectrum of the Hawking radiation for
large mass).
The situation is, however, not so simple. As was demonstrated by Domagala
and Lewandowski (2004), spins bigger than the minimal spin are not negligible
and therefore have to be taken into account when calculating the entropy.
Therefore the entropy has been underestimated in the earlier papers leading to
the result (7.64). If one again demands that the result be in the highest order
equal to the Bekenstein–Hawking entropy, the equation which fixes the Barbero–
Immirzi parameter reads
( √ )
∞∑
n(n +2)
2 exp −2πβ
=1,
4
n=1
which can only be solved numerically, with the result (Meissner 2004),
β =0.23753295796592 ... .
Taking also into account the next order in the calculation, one arrives at the
following value for the entropy:
S = A
4lP 2 − 1 ( ) A
2 ln lP
2 + O(1) . (7.66)
The logarithmic correction term is in fact independent of β. Correction terms of
this form (mostly with coefficients −1/2 or−3/2) have also been found in other
approaches; see Page (2005) for a review.
If one nevertheless took the value k = 3 in (7.61), one would find a change
in the mass given by
∆M = ln 3 m2 P
8πM , (7.67)
corresponding to the fundamental frequency
˜ω 3 = ln 3
8πGM ≡ ln 3 T BH
≈ 8.85 M ⊙
kHz . (7.68)
M
It was emphasized by Dreyer (2003) that ˜ω 3 coincides with the real part of
the asymptotic frequency for the quasi-normal modes of the black hole. These