Kiefer C. Quantum gravity

CANONICAL QUANTIZATION OF THE SCHWARZSCHILD BLACK HOLE 215
An interesting analogy with (7.49) is the plane-wave solution for a free nonrelativistic
particle,
exp(ikx − ω(k)t) . (7.52)
As in (7.49), the number of parameters is one less than the number of arguments,
since ω(k) =k 2 /2m. A quantization for extremal holes on their own would correspond
to choosing a particular value for the momentum at the classical level,
say p 0 , and demanding that no dynamical variables (x, p) existforp = p 0 .This
is, however, not the usual way to find classical correspondence—such a correspondence
is not obtained from the plane-wave solution (7.52) but from wave
packets which are constructed by superposing different wave numbers k. This
then yields quantum states which are sufficiently concentrated around classical
trajectories such as x = p 0 t/m.
It seems, therefore, appropriate to proceed similarly for black holes: construct
wave packets for non-extremal holes that are concentrated around the classical
values (7.50) and (7.51) and then extend them by hand to the extremal limit.
This would correspond to ‘extremization after quantization’, in contrast to the
‘quantization after extremization’ made above. Expressing in (7.49) M as a function
of A and q and using Gaussian weight functions, one has
∫
Ψ(α, τ, λ) = dAdq exp
[− (A − A 0) 2
2(∆A) 2 − (q − q 0) 2 ]
2(∆q) 2
A>4πq 2 [ ( )]
i Aα
× exp
8πG − M(A, q)τ − qλ . (7.53)
The result of this calculation is given and discussed in Kiefer and Louko (1999).
As expected, one finds Gaussian packets that are concentrated around the classical
values (7.50) and (7.51). Like for the free particle, the wave packets exhibit
dispersion with respect to Killing time τ. Usingfor∆A the Planck-length
squared, ∆A ∝ G ≈ 2.6 × 10 −66 cm 2 , one finds for the typical dispersion time
in the Schwarzschild case
τ ∗ = 128π2 R 3 S
G
≈ 10 65 ( M
M ⊙
) 3
years . (7.54)
Note that this is just of the order of the black-hole evaporation time (7.14). The
dispersion of the wave packet gives the time scale after which the semiclassical
approximation breaks down.
Coming back to the charged case, and approaching the extremal limit √ GM =
|q|, one finds that the widths of the wave packet (7.53) are independent of τ for
large τ. This is due to the fact that for the extremal black hole, κ =0and
therefore no evaporation takes place. If one takes, for example, ∆A ∝ G and
∆q ∝ √ G, one finds for the α-dependence of (7.53) for τ →∞,thefactor
)
exp
(− α2
128π 2 , (7.55)