Quantum Gravity

218 QUANTIZATION OF BLACK HOLEScluded, for example, from the ‘Euclidean’ wave function (7.56): if one imposedan ad hoc Bohr–Sommerfeld quantization rule, one would find, recalling (7.48),∮2πn = π αE dα E =∫ 2π0Adα E8πG = A4G . (7.57)(Recall from above that α E ranges from 0 to 2π.) A similar result follows ifthe range of the time parameter τ in the Lorentzian version is assumed to becompact, similar to momentum quantization on finite spaces (Kastrup 1996).A different argument to fix the factor in the area spectrum goes as follows(Mukhanov 1986; Bekenstein and Mukhanov 1995). One assumes the quantizationconditionA n = αlP 2 n,n∈ N , (7.58)with some undetermined constant α. The energy level n will be degenerate withmultiplicity g(n), so one would expect the identificationS ≡ A4lP2 + constant = ln g(n) . (7.59)Demanding g(1) = 1 (i.e. assuming that the entropy of the ground state vanishes),this leads with (7.58) tog(n) =e α(n−1)/4 .Since this must be an integer, one has the optionsα =4lnk, k=2, 3,... (7.60)and thus g(n) =k n−1 . Note that the spectrum would then slightly differ from(7.57). From information-theoretic reasons (‘it from bit’; cf. e.g. Wheeler (1990))one would prefer the value k = 2, leading to A n =(4ln2)l 2 P n.In the Schwarzschild case, the energy spacing between consecutive levels isobtained from∆A =32πG 2 M∆M =(4lnk)l 2 Pto readwith the fundamental frequency∆M =∆E ≡ ˜ω k =˜ω k = ln k8πGM , (7.61)ln k8πGM =(lnk)T BH . (7.62)The black-hole emission spectrum would then be concentrated at multiples ofthis fundamental frequency—unlike the continuous thermal spectrum of Hawkingradiation. In fact, one would have a deviation from the Hawking spectrumeven for large black holes, that is, black holes with masses M ≫ m P .Another