Kiefer C. Quantum gravity

216 QUANTIZATION OF BLACK HOLES
which is independent of both τ and . It is clear that this packet, although
concentrated at the value α = 0 for extremal holes, has support also for α ≠0
and is qualitatively not different from a wave packet that is concentrated at a
value α ≠ 0 close to extremality.
An interesting question is the possible occurrence of a naked singularity for
which √ GM < |q|. Certainly, the above boundary conditions do not comprise
the case of a singular three-geometry. However, the wave packets discussed above
also contain parameter values that would correspond to the ‘naked’ case. Such
geometries could be avoided if one imposed the boundary condition that the
wave function vanishes for such values. But then continuity would enforce the
wave function also to vanish on the boundary, that is, at √ GM = |q|. Thiswould
mean that extremal black holes could not exist at all in quantum gravity—an
interesting speculation.
A possible thermodynamical interpretation of (7.49) can only be obtained if
an appropriate transition into the Euclidean regime is performed. This transition
is achieved by the ‘Wick rotations’ τ →−iβ, α →−iα E (from (7.46) it is clear
that α is connected to the lapse function and must be treated similar to τ), and
λ →−iβΦ. Demanding regularity of the Euclidean line element, one arrives
at 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 for
the Hawking temperature (1.32). Alternatively, one could use (1.32) to derive
α E =2π.
The Euclidean version of the state (7.49) then reads
( )
A
Ψ E (α, τ, λ) =χ(M,q)exp
4G − βM − βΦq . (7.56)
One recognizes in the exponent of (7.56) the occurrence of the Bekenstein–Hawking
entropy. Of course, (7.56) is still a pure state and should not be confused
with a partition sum. But the factor exp[A/(4G)] in (7.56) directly gives the
enhancement factor for the rate of black-hole pair creation relative to ordinary
pair creation (Hawking and Penrose 1996). It must be emphasized that S BH fully
arises from a boundary term at the horizon (r → 0).
It is now clear that a quantization scheme that treats extremal black holes as
a limiting case gives S BH = A/(4G) also for the extremal case. 4 This coincides
with 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 of
view, it is also clear why the extremal (Kerr) black hole that occurs in the transition
from 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 temperature
zero), the stronger version of the Third Law of Thermodynamics (that would
require S → 0forT → 0) apparently does not hold. This is not particularly
disturbing, since many systems in ordinary thermodynamics (such as glasses)
4 We set k B = 1 here and in the following.