Kiefer C. Quantum gravity

QUANTUM THEORY OF COLLAPSING DUST SHELLS 221
modes are characteristic oscillations of the black hole before it settles to its
stationary state; see for example, Kokkotas and Schmidt (1999) for a review. As
was conjectured by Hod (1998) on the basis of numerical evidence and shown by
Motl (2003), the frequency of the quasi-normal modes is for n →∞given by
ω n = − i(n + 1 2 )
4GM + ln 3 (
8πGM + O n −1/2)
(
= −iκ n + 1 )
+ κ (
2 2π ln 3 + O n −1/2) ; (7.69)
see also Neitzke (2003). The imaginary part indicates that one is dealing with
damped oscillations. It might be that the black-hole entropy arises from the
quantum entanglement between the black hole and the quasi-normal modes
(Kiefer 2004a). The quasi-normal modes would then serve as an environment
leading to decoherence (Section 10.1). This, however, would still have to be
shown. Interestingly, at least for the Schwarzschild black hole, a quantum measurement
of the quasi-normal modes would introduce a minimal noise temperature
that is exactly equal to the Hawking temperature (Kiefer 2004b). If one had
performed this analysis before the advent of Hawking’s work, one would have
concluded that there is a temperature associated with the real part in (7.69),
which is proportional to and which is equal to (1.33).
The imaginary part of the frequency (7.69) in this limit is equidistant in
n. This could indicate an intricate relation with Euclidean quantum gravity and
provide an explanation of why the Euclidean version readily provides expressions
for the black-hole temperature and entropy: if one considered in the Euclidean
theory a wave function of the form
ψ E ∼ e inκtE ,
one would have to demand that the Euclidean time t E be periodic with period
8πGM. This, however, is just the inverse of the Hawking temperature, in accordance
with the result that Euclidean time must have this periodicity if the line
element is to be regular (see e.g. Hawking and Penrose 1996).
7.4 Quantum theory of collapsing dust shells
In this section, a particular model will be described in some detail, but without
too many technicalities. This concerns the collapse of a null dust shell. In the
classical theory, the collapse leads to the formation of a black hole. We shall see
that it is possible to construct an exact quantum theory of this model in which
the dynamical evolution is unitary with respect to asymptotic observers (since
one has an asymptotically flat space, a semiclassical time exists, which is just the
Killing time at asymptotic infinity). As a consequence of the unitary evolution,
the classical singularity is fully avoided in the quantum theory: if the collapsing
shell is described by a wave packet, the evolution leads to a superposition of
black-hole and white-hole horizon yielding a vanishing wave function for zero