Interior design of a two-dimensional semiclassical black hole: Quantum transition across the singularity

We study the internal structure of a two-dimensional dilatonic evaporating black hole based on the Callan, Giddings, Harvey, and Strominger model. At the semiclassical level, a (weak) spacelike singularity was previously found to develop inside the black hole. We employ here a simplified quantum formulation of spacetime dynamics in the neighborhood of this singularity, using a minisuperspace-like approach. Quantum evolution is found to be regular and well defined at the semiclassical singularity. A well-localized initial wave packet propagating towards the singularity bounces off the latter and retains its well-localized form. Our simplified quantum treatment thus suggests that spacetime may extend semiclassically beyond the singularity, and also signifies the specific extension.

Figure 1

The initial wave function $|{\Psi }_0|$ as a function of $R$, constructed from Eq. (33) (solid black line), and compared with the approximation (32) (dashed gray line). Note the nonexistence of the “mirror” Gaussian on the negative side of $R$. Note also that $r_0=4$ corresponds to initial location $R=3^{2/3}2\approx 4.16$.

Figure 2

A three-dimensional plot describing the time evolution of the wave function $|\Psi (R,t)|$. The bounce at $R=0$ is evident.

Figure 3

Plots of $|\Psi |$ as a function of $R$ at several different times, as it propagates towards the singularity (a), hits the singularity (b), and then recedes from it (c). The solid gray lines and dashed black lines represent the numerical calculations and analytic approximations, respectively. Figure 3a displays three snapshots which, going from right to left, correspond to times $t=0$, 0.04, 0.08. Figure 3b Displays $|\Psi |$ at $t=t_{\mathrm{hit}}=0.1$. Figure 3c again displays three snapshots which, going from left to right, correspond to times $t=0.12$, 0.16, 0.2. The analytic approximation (dashed black line) in the two right-most graphs of both Figs. 3a, 3c corresponds to Eq. (41). In Fig. 3b the analytic approximation corresponds to Eq. (42).

Figure 4

Spacetime diagram of a CGHS black hole which forms by gravitational collapse and subsequently evaporates. The thick null line at the bottom left represents the collapsing massive shell. The event horizon is marked by a dotted line. $D$ represents the future Cauchy development of the singularity line $R=0$. The domain marked by $J$ is discussed in the text.