Simple holographic models of black hole evaporation

TL;DR

This paper analyzes how two holographic models of black hole evaporation can disentangle conflicting Page-curve predictions arising from bulk Hawking-like dynamics versus entanglement-wedge methods. It introduces a unitary evaporation model built on multi-boundary wormholes where the Page curve decreases and the interior remains smooth, and a contrasting information-loss-like model where the radiation entropy grows without a Page-time crossover. The authors show that both computations can agree with their respective bulk dynamics using classical extremal surfaces, clarifying that the quantum extremal surface gamma' is a kinematic object whose relevance depends on the underlying bulk evolution. They conclude that resolving the information paradox requires non-perturbative bulk dynamics beyond Hawking's semiclassical picture and emphasize a precise mapping between bulk models and their holographic duals.

Abstract

Several recent papers have shown a close relationship between entanglement wedge reconstruction and the unitarity of black hole evaporation in AdS/CFT. The analysis of these papers however has a rather puzzling feature: all calculations are done using bulk dynamics which are essentially those Hawking used to predict information loss, but applying ideas from entanglement wedge reconstruction seems to suggest a Page curve which is consistent with information conservation. Why should two different calculations in the same model give different answers for the Page curve? In this note we present a new pair of models which clarify this situation. Our first model gives a holographic illustration of unitary black hole evaporation, in which the analogue of the Hawking radiation purifies itself as expected, and this purification is reproduced by the entanglement wedge analysis. Moreover a smooth black hole interior persists until the last stages the evaporation process. Our second model gives an alternative holographic interpretation of the situation where the bulk evolution leads to information loss: unlike in the models proposed so far, this bulk information loss is correctly reproduced by the entanglement wedge analysis. This serves as an illustration that quantum extremal surfaces are in some sense kinematic: the time-dependence of the entropy they compute depends on the choice of bulk dynamics. In both models no bulk quantum corrections need to be considered: classical extremal surfaces are enough to do the job. We argue that our first model is the one which gives the right analogy for what actually happens to evaporating black holes, but we also emphasize that any complete resolution of the information problem will require an understanding of non-perturbative bulk dynamics.

Figures (6)

• Figure 1: Constructing wormhole spatial geometries by quotients of the Poincaré disk. On the left we remove two semicircles to and identify to produce a Cauchy slice of the BTZ wormhole, with the bifurcate horizon being the dotted line marked $\gamma$. The topology is that of a cylinder. On the right we remove four semicircles to construct a Cauchy slice of a three-exit wormhole with the topology of a pair of pants. The exits are labeled $1,2,3$, the shared interior is shaded grey, and a pair of extremal surfaces $\gamma$ and $\gamma'$ which are both homologous to exit $1$ are marked with dotted lines. In our unitary model below, $\gamma'$ will be the HRT surface at late times and $\gamma$ will be the HRT surface at early times.
• Figure 2: Two steps of the "black hole evaporation" process for our first model. The surfaces $\gamma$ and $\gamma'$ give two possible HRT surfaces homologous either to the "Hawking radiation" exits or the "black hole" exit. As the octopus grows more and more legs, the length of the "ankle bracelets" $\gamma$ increases and the length of the "headband" $\gamma'$ decreases: the crossover point where the headband becomes shorter is the "Page time" of the model.
• Figure 3: The Page curve for our unitary model, computed using the area of the HRT surface (we've taken $L_0/\ell=10$ and $4G=1$, and plotted $\frac{S_R}{\ell}$).
• Figure 4: The simplest example of holographic quantum error correction: splitting a single AdS boundary into three regions $R_1$, $R_2$, and $R_3$Almheiri:2014lwa. The white triangular region in the center is an "island" in the same sense as the bodies of the octopi in figure \ref{['octopi']}: it can be accessed from any two of the boundary regions but not from any one. For example if we compare $R_1$ and $R_2$ to the "Hawking radiation" CFTs and $R_3$ to the remaining "black hole" CFT, we can view this as a model for an evaporating black hole after the Page time.
• Figure 5: The bulk dynamics for information loss: exterior Hawking modes end up in the Hawking radiation (the red dotted line), while interior Hawking modes (and also whatever matter formed the black hole) end up in a baby universe (the blue dotted line).