Quantum Gravity

216 QUANTIZATION OF BLACK HOLESwhich is independent of both τ and . It is clear that this packet, althoughconcentrated at the value α = 0 for extremal holes, has support also for α ≠0and is qualitatively not different from a wave packet that is concentrated at avalue α ≠ 0 close to extremality.An interesting question is the possible occurrence of a naked singularity forwhich √ GM < |q|. Certainly, the above boundary conditions do not comprisethe case of a singular three-geometry. However, the wave packets discussed abovealso contain parameter values that would correspond to the ‘naked’ case. Suchgeometries could be avoided if one imposed the boundary condition that thewave function vanishes for such values. But then continuity would enforce thewave function also to vanish on the boundary, that is, at √ GM = |q|. Thiswouldmean that extremal black holes could not exist at all in quantum gravity—aninteresting speculation.A possible thermodynamical interpretation of (7.49) can only be obtained ifan appropriate transition into the Euclidean regime is performed. This transitionis achieved by the ‘Wick rotations’ τ →−iβ, α →−iα E (from (7.46) it is clearthat α is connected to the lapse function and must be treated similar to τ), andλ →−iβΦ. Demanding regularity of the Euclidean line element, one arrivesat the conclusion that α E =2π. But this means that the Euclidean version of(7.50) just reads 2π = κβ, whichwithβ =(k B T BH ) −1 is just the expression forthe Hawking temperature (1.32). Alternatively, one could use (1.32) to deriveα E =2π.The Euclidean version of the state (7.49) then reads( )AΨ E (α, τ, λ) =χ(M,q)exp4G − βM − βΦq . (7.56)One recognizes in the exponent of (7.56) the occurrence of the Bekenstein–Hawkingentropy. Of course, (7.56) is still a pure state and should not be confusedwith a partition sum. But the factor exp[A/(4G)] in (7.56) directly gives theenhancement factor for the rate of black-hole pair creation relative to ordinarypair creation (Hawking and Penrose 1996). It must be emphasized that S BH fullyarises from a boundary term at the horizon (r → 0).It is now clear that a quantization scheme that treats extremal black holes asa limiting case gives S BH = A/(4G) also for the extremal case. 4 This coincideswith the result found from string theory; see Section 9.2.5. On the other hand,quantizing extremal holes on their own would yield S BH = 0. From this point ofview, it is also clear why the extremal (Kerr) black hole that occurs in the transitionfrom the disk-of-dust solution to the Kerr-solution has entropy A/(4G);see Neugebauer (1998). If S BH ≠ 0 for the extremal hole (which has temperaturezero), the stronger version of the Third Law of Thermodynamics (that wouldrequire S → 0forT → 0) apparently does not hold. This is not particularlydisturbing, since many systems in ordinary thermodynamics (such as glasses)4 We set k B = 1 here and in the following.