Quantum Gravity

QUANTUM THEORY OF COLLAPSING DUST SHELLS 221modes are characteristic oscillations of the black hole before it settles to itsstationary state; see for example, Kokkotas and Schmidt (1999) for a review. Aswas conjectured by Hod (1998) on the basis of numerical evidence and shown byMotl (2003), the frequency of the quasi-normal modes is for n →∞given byω 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 withdamped oscillations. It might be that the black-hole entropy arises from thequantum entanglement between the black hole and the quasi-normal modes(Kiefer 2004a). The quasi-normal modes would then serve as an environmentleading to decoherence (Section 10.1). This, however, would still have to beshown. Interestingly, at least for the Schwarzschild black hole, a quantum measurementof the quasi-normal modes would introduce a minimal noise temperaturethat is exactly equal to the Hawking temperature (Kiefer 2004b). If one hadperformed this analysis before the advent of Hawking’s work, one would haveconcluded 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 inn. This could indicate an intricate relation with Euclidean quantum gravity andprovide an explanation of why the Euclidean version readily provides expressionsfor the black-hole temperature and entropy: if one considered in the Euclideantheory a wave function of the formψ E ∼ e inκtE ,one would have to demand that the Euclidean time t E be periodic with period8πGM. This, however, is just the inverse of the Hawking temperature, in accordancewith the result that Euclidean time must have this periodicity if the lineelement is to be regular (see e.g. Hawking and Penrose 1996).7.4 Quantum theory of collapsing dust shellsIn this section, a particular model will be described in some detail, but withouttoo many technicalities. This concerns the collapse of a null dust shell. In theclassical theory, the collapse leads to the formation of a black hole. We shall seethat it is possible to construct an exact quantum theory of this model in whichthe dynamical evolution is unitary with respect to asymptotic observers (sinceone has an asymptotically flat space, a semiclassical time exists, which is just theKilling time at asymptotic infinity). As a consequence of the unitary evolution,the classical singularity is fully avoided in the quantum theory: if the collapsingshell is described by a wave packet, the evolution leads to a superposition ofblack-hole and white-hole horizon yielding a vanishing wave function for zero