Information radiation in BCFT models of black holes

TL;DR

The paper develops and analyzes holographic BCFT models where a boundary sector with many degrees of freedom radiates into a lighter bulk CFT, yielding evaporating black-hole dynamics controlled by the ratio $c_{bdy}/c_{bulk}$. Using both holographic calculations of entanglement entropy via HRT surfaces and direct BCFT (twist operator) computations, it demonstrates a first-order transition in the entanglement wedge that after the Page time includes a portion of the black hole interior, enabling interior reconstruction from the radiation. The authors verify a consistent Page-time scaling across BCFT and gravity pictures and extend the analysis to static 2D models as well as dynamical, evaporating scenarios, including single-sided configurations. They also discuss connections to behind-the-horizon microstate physics and outline avenues for higher-dimensional BCFT duals and Vaidya-like evolutions, highlighting the robustness and versatility of the BCFT/ETW-brane framework in studying information flow in evaporating black holes.

Abstract

In this note, following [arXiv:1905.08255, arXiv:1905.08762, arXiv:1908.10996], we introduce and study various holographic systems which can describe evaporating black holes. The systems we consider are boundary conformal field theories for which the number of local degrees of freedom on the boundary ($c_{bdy}$) is large compared to the number of local degrees of freedom in the bulk CFT ($c_{bulk}$). We consider states where the boundary degrees of freedom on their own would describe an equilibrium black hole, but the coupling to the bulk CFT degrees of freedom allows this black hole to evaporate. The Page time for the black hole is controlled by the ratio $c_{bdy}/c_{bulk}$. Using both holographic calculations and direct CFT calculations, we study the evolution of the entanglement entropy for the subset of the radiation system (i.e. the bulk CFT) at a distance $d > a$ from the boundary. We find that the entanglement entropy for this subsystem increases until time $a + t_{Page}$ and then undergoes a phase transition after which the entanglement wedge of the radiation system includes the black hole interior. Remarkably, this occurs even if the radiation system is initially at the same temperature as the black hole so that the two are in thermal equilibrium. In this case, even though the black hole does not lose energy, it "radiates" information through interaction with the radiation system until the radiation system contains enough information to reconstruct the black hole interior.

Figures (16)

• Figure 1: Basic setup. A) Our thermal system, dual to a bulk black hole, is the red boundary. It interacts with a bulk CFT which can serve as an auxiliary system to which the black hole can radiate. B) Higher-dimensional bulk picture: the red surface is a dynamical ETW brane whose tension is monotonically related to the number of local degrees of freedom in the boundary system. For large tension, this ETW brane moves close to the boundary and behaves like a Randall-Sundrum Planck brane. C) The Planck-brane picture suggests an effective lower-dimensional description where a part of the CFT in the central region is replaced with a cutoff CFT coupled to gravity, similar to the setup in AM.
• Figure 2: Time at which the subsystem of the radiation system greater than some distance from the BCFT boundary exhibits a transition in its entanglement entropy, for the case $c_{bnd} / c_{bulk} \sim 50$. After the transition, the entanglement wedge of this subset of the radiation system includes a portion of the black hole interior. After a time equal to the Page time plus the light travel time from the boundary to our subsystem, there is enough information in the subsystem to reconstruct part of the black hole.
• Figure 3: An ETW brane with tension parameter $T$ enters the bulk at coordinate angle $\Theta$ in Fefferman-Graham coordinates. Larger $T$ gives a larger angle $\Theta$. Shown in blue is the RT surface computing the entanglement entropy of the subsystem $A$ which includes the boundary. The area to the right of the dashed line proportional to the boundary entropy.
• Figure 4: a) BCFT path integral defining the thermofield double state of two 1+1 dimensional BCFTs. b) Euclidean geometry dual to the BCFT thermofield double. The red surface is an ETW brane. c) The same geometry represented as part of Euclidean Poincaré-AdS. d) Lorentzian geometry of the original state, looking perpendicular to the boundary. Dashed lines represent horizons on the ETW brane, corresponding to the horizons of the two-sided black hole represented by the boundary system.
• Figure 5: Geometry of the ETW brane and half of the disconnected RT surface in the plane of the RT surface. We have $OQ=1$ and $OA=\tan \Theta$. Thus, $AQ=AH=\sec \Theta$. Also $HB \perp AH$ so $AH^2 + HB^2 = OA^2 + OB^2$. This gives $\pmb{r_H = (r^2 -1)/(2r)}$. Now $OM = OA \tan \alpha = \tan \Theta \tan \alpha$ and $AM = OA \sec \alpha = \tan \Theta \sec \alpha$. So $HM = HA - MA = \sec \Theta - \tan \Theta \sec \alpha$. Finally, $HM/HB = \tan \alpha$ gives $\pmb{ r_H = sec \Theta \cot \alpha - \tan \Theta \csc \alpha}$, while $HP = HB \sin \alpha$ gives $\pmb{z = r_H \sin \alpha}$. The boldface equations allow us to express $z$ and $r_H$ in terms of $r$.