Quantum Gravity

228 QUANTIZATION OF BLACK HOLESψ κλ (t, p) =ψ κλ (p)e −ipt . (7.95)More interesting is the evolution of the wave packet in the r-representation.This is obtained by the integral transform (7.86) of ψ κλ (t, p) with respect to theeigenfunctions (7.90). It leads to the exact resultΨ κλ (t, r) = √ 1 κ!(2λ) κ+1/2 []i√2π (2κ)! (λ +it +ir) κ+1 − i(λ +it − ir) κ+1 . (7.96)One interesting consequence can be immediately drawn:lim Ψ κλ(t, r) =0. (7.97)r→0This means that the probability of finding the shell at vanishing radius is zero!In this sense, the singularity is avoided in the quantum theory. It must be emphasizedthat this is not a consequence of a certain boundary condition—it isa consequence of the unitary evolution. If the wave function vanishes at r =0for t →−∞(asymptotic condition of ingoing shell), it will continue to vanish atr = 0 for all times. It follows from (7.96) that the quantum shell bounces andre-expands. Hence, no absolute event horizon can form, in contrast to the classicaltheory. The resulting object might still be indistinguishable from a blackhole due to the huge time delay from the gravitational redshift—the re-expansionwould be visible from afar only in the far future. Similar features follow froma model by Frolov and Vilkovisky (1981) who consider a null shell for the casewhere ‘loop effects’ in the form of Weyl curvature terms are taken into account.Of interest also is the expectation value of the shell radius; see Hájíček (2001,2003) for details. Again one recognizes that the quantum shell always bouncesand re-expands. An intriguing feature is that an essential part of the wave packetcan even be squeezed below the expectation value of its Schwarzschild radius.The latter is found from (7.92) to read (re-inserting G),〈R 0 〉 κλ ≡ 2G〈E〉 κλ =(2κ +1) l2 Pλ , (7.98)while its variation follows from (7.94),∆(R 0 ) κλ =2G∆E κλ = √ 2κ +1 l2 Pλ . (7.99)The main part of the wave packet is squeezed below the Schwarzschild radius if〈r〉 κλ +(∆r) κλ < 〈R 0 〉 κλ − ∆(R 0 ) κλ .It turns out that this can be achieved if either λ ≈ l P (and κ>2) or, for biggerλ, ifκ is larger by a factor of (λ/l P ) 4/3 . The wave packet can thus be squeezedbelow its Schwarzschild radius if its energy is bigger than the Planck energy—agenuine quantum effect.