Abstract
When absorbing boundary conditions are used to evaporate a black hole in AdS/CFT, we show that there is a phase transition in the location of the quantum Ryu-Takayanagi surface, at precisely the Page time. The new RT surface lies slightly inside the event horizon, at an infalling time approximately the scrambling time β/2πlogSBH into the past. We can immediately derive the Page curve, using the Ryu-Takayanagi formula, and the Hayden-Preskill decoding criterion, using entanglement wedge reconstruction. Because part of the interior is now encoded in the early Hawking radiation, the decreasing entanglement entropy of the black hole is exactly consistent with the semiclassical bulk entanglement of the late-time Hawking modes, despite the absence of a firewall.By studying the entanglement wedge of highly mixed states, we can understand the state dependence of the interior reconstructions. A crucial role is played by the existence of tiny, non-perturbative errors in entanglement wedge reconstruction. Directly after the Page time, interior operators can only be reconstructed from the Hawking radiation if the initial state of the black hole is known. As the black hole continues to evaporate, reconstructions become possible that simultaneously work for a large class of initial states. Using similar techniques, we generalise Hayden-Preskill to show how the amount of Hawking radiation required to reconstruct a large diary, thrown into the black hole, depends on both the energy and the entropy of the diary. Finally we argue that, before the evaporation begins, a single, state-independent interior reconstruction exists for any code space of microstates with entropy strictly less than the Bekenstein-Hawking entropy, and show that this is sufficient state dependence to avoid the AMPSS typical-state firewall paradox.
Highlights
By studying the entanglement wedge of highly mixed states, we can understand the state dependence of the interior reconstructions
We have argued that the key expected features of unitary black hole evaporation in AdS/CFT can be derived from the bulk semiclassical description of an evaporating black hole, so long as we assume entanglement wedge reconstruction
We studied entanglement wedge reconstruction in an evaporating black hole, formed from collapse, where the Hawking radiation was extracted out of the AdS space containing the black hole, with boundary Hilbert space HCFT, and into an auxiliary Markovian reservoir Hrad
Summary
We study an evaporating black hole formed from collapse. For simplicity, we assume throughout that the collapsing matter, and the entire spacetime, is rotationally symmetric. The Ryu-Takayanagi surface for Hrad will be the same as the RT surface for HCFT, and the entanglement wedge of Hrad will contain the outgoing modes that were extracted into the reservoir, along with the bulk domain of dependence of any spacelike surface in the black hole interior that is bounded only by the RT surface. It was argued there that the RT surface should be the minimal generalised entropy quantum extremal surface, homologous to the boundary region, defined with the nonholographic system automatically included as part of the fields in Sbulk (see for example eq 4.14 of [38]). The surface of minimal generalised entropy within this Cauchy slice is the empty surface, with generalised entropy Srad.16 It follows that the RT surface is empty and the interior of the black hole is in the entanglement wedge of the boundary CFT, and not the entanglement wedge of the Hawking radiation reservoir. The main focus of this section will be on identifying the location of, and the consequences of the location of, this non-empty quantum extremal surface
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