The effect of a CFT shockwave on the entanglement structure of an eternal black hole in Jackiw-Teitelboim gravity, that is in thermal equilibrium with a thermal bath, is considered. The shockwave carries energy and entropy into the black hole and heats the black hole up leading to evaporation and the eventual recovery of equilibrium. We find an analytical description of the entire relaxational process within the semiclassical high temperature regime. If the shockwave is inserted around the Page time then several scenarios are possible depending on the parameters. The Page time can be delayed or hastened and there can be more than one transition. The final entropy saddle has a quantum extremal surface that generically starts inside the horizon but at some later time moves outside. In general, increased shockwave energy and slow evaporation rate favour the extremal surface to be inside the horizon. The shockwave also disrupts the scrambling properties of the black hole. The same analysis is then applied to a shockwave inserted into the extremal black hole with similar conclusions.
A Page curve for an evaporating black hole in asymptotically flat spacetime is computed by adapting the Quantum Ryu-Takayanagi (QRT) proposal to an analytically solvable semi-classical two-dimensional dilaton gravity theory. The Page time is found to be one third of the black hole lifetime, at leading order in semi-classical corrections. A Page curve is also obtained for a semi-classical eternal black hole, where energy loss due to Hawking evaporation is balanced by an incoming energy flux.
Quantum extremal islands reproduce the unitary Page curve of an evaporating black hole. This has been derived by including replica wormholes in the gravitational path integral, but for the transient, evaporating black holes most relevant to Hawking's paradox, these wormholes have not been analyzed in any detail. In this paper we study replica wormholes for black holes formed by gravitational collapse in Jackiw-Teitelboim gravity, and confirm that they lead to the island rule for the entropy. The main technical challenge is that replica wormholes rely on a Euclidean path integral, while the qua…
Recent work has established a route towards the semiclassical validity of the Page curve, and so provided evidence that information escapes an evaporating black hole. However, a protocol to explicitly recover and make practical use of that information in the classical limit has not yet been given. In this paper, we describe such a protocol, showing that an observer may reconstruct the phase space of the black hole interior by measuring the Uhlmann phase of the Hawking radiation. The process of black hole formation and evaporation provides an invertible map between this phase space and the spac…
We give a quantum circuit interpretation of evaporating black hole geometry. We make an analogy between the appearance of island for evaporating black hole and the transition from two-sided to one-sided black hole in the familiar example of perturbed thermofield double. If Alice perturbs thermofield double and waits for scrambling time, she will have a one-sided black hole with interior of her own. We argue that by similar mechanism the radiation gets access to the interior (island forms) after Page time. The growth of the island happens as a result of the constant transitions from two-sided t…
We study the entanglement entropy of a one-parameter family of exactly solvable gravities in the 2-dimensional asymptotically-flat space. The islands and Page curves of eternal, evaporating and bath-removed black holes are investigated. The different theories in this parameter class are identified through field redefinitions which leave the island invariant. The Page transition is found to occur at the first a third of the black hole life time in the evaporating case for this family of solutions. In addition, we consider gluing the equilibrium black hole and the evaporating one along a null tr…
We study the Page curve and the information paradox for the BTZ black hole coupled to two thermal baths by applying the island paradigm. We prove that as the island locates outside the event horizon, the entanglement entropy of Hawking radiation for finite temperature black holes obeys the Page curve. However, for extremal rotating black holes, the Page time and the scrambling time become divergent. To avoid the ill-defined of the Page time, we consider the contribution of superradiance, which is a process that can extract rotational energy from black holes. The superradiance continues in a pe…
Defect extremal surface is defined by minimizing the Ryu-Takayanagi surface corrected by the defect theory, which is useful when the RT surface crosses or terminates on the defect. Based on the decomposition procedure of a AdS bulk with a defect brane, proposed in arXiv:2012.07612, we derive Page curve in a time dependent set up of AdS$_3$/BCFT$_2$, and find that the result from island formula agrees with defect extremal surface formula precisely. We then extend the study to higher dimensions and find that the entropy computed from bulk defect extremal surface is generally less than that from …
We study the Page curve in the four dimensional asymptotically flat eternal Garfinkle-Horowitz-Strominger dilaton black holes by applying the "quantum extremal surface" prescription. The results demonstrate that without island, the entanglement entropy of Hawking radiation is proportional to time and divergent at late times. While taking account of the emergence of the island extending slightly outside the event horizon, the entanglement entropy of Hawking radiation stops growing and eventually reaches a saturation value, which is twice of the Bekenstein-Hawking entropy and consistent with the…
We show that the canonical purification of an evaporating black hole after the Page time consists of a short, connected, Lorentzian wormhole between two asymptotic boundaries, one of which is unitarily related to the radiation. This provides a quantitative and general realization of the predictions of ER=EPR in an evaporating black hole after the Page time; this further gives a standard AdS/CFT calculation of the entropy of the radiation (without modifications of the homology constraint). Before the Page time, the canonical purification consists of two disconnected, semiclassical black holes. …
Variants of the black hole information paradox are studied in Type IIB string theory setups that realize four-dimensional gravity coupled to a bath. The setups are string theory versions of doubly-holographic Karch/Randall brane worlds, with black holes coupled to non-gravitating and gravitating baths. The 10d versions are based on fully backreacted solutions for configurations of D3, D5 and NS5 branes, and admit dual descriptions as $\mathcal N=4$ SYM on a half space and 3d $T_ρ^σ[SU(N)]$ SCFTs. Island contributions to the entanglement entropy of black hole radiation systems are identified th…
We study the islands and the Page curve in the 1+1-dimensional eternal dilaton black hole models. Without islands, the entanglement entropy of the radiation grows linearly at late time. However with an island, its growth stops at the value of almost twice of the black hole entropy. Therefore an island emerges at the late time, and the entanglement entropy of the radiation shows the Page curve.
We argue that the `island conjecture' and the replica wormhole derivation of the Page curve break monogamy of entanglement through allowing black hole interior states to be non-classically correlated while also pairwise entangled with radiation states. The reason is that quantum degrees of freedom (present in any half of a Hawking pair) cannot all be identified with the environment at semi-classical pair production, and can only be fixed relative to a subsystem, as required for the Page curve, by correlations equivalent to entanglement - regardless of what those correlations are attributed to.…
We consider the Hawking radiation emitted by an evaporating black hole in JT gravity and compute the entropy of arbitrary subsets of the radiation in the slow evaporation limit, and find a zoo of possible island saddles. The Hawking radiation is shown to have long range correlations. We compute the mutual information between early and late modes and bound from below their squashed entanglement. A small subset of late modes are shown to be correlated with modes in a suitably large subset of the radiation previously emitted as well as later modes. We show how there is a breakdown of the semi-cla…
We study the Page curve of an evaporating black hole using a toy model given by Jackiw-Teitelboim gravity with Fateev-Zamolodchikov-Zamolodchikov-Teschner (FZZT) antibranes. We treat the anti-FZZT branes as dynamical objects, taking their back-reaction into account. We construct the entanglement entropy from the dual matrix model and study its behavior as a function of the 't Hooft coupling $t$ proportional to the number of branes, which plays the role of time. By numerical computation we observe that the entropy first increases and then decreases as $t$ grows, reproducing the well-known behav…
Recently, remarkable progress in recovering the Page curve of an evaporating black hole (BH) in Jackiw-Teitelboim gravity has been achieved through use of Quantum Extremal surfaces (QES). Multi-boundary Wormhole (MbW) models have been crucial in parallel model building in three dimensions. Motivated by this we here use the latter models to compute the subregion complexity of the Hawking quanta of the evaporating BH in AdS$_{3}$ and obtain the Page curve associated with this information theoretic measure. We use three- and $n$-boundary wormhole constructions to elucidate our computations of vol…
We compute the entanglement negativity for various pure and mixed state configurations in a bath coupled to an evaporating two dimensional non-extremal Jackiw-Teitelboim (JT) black hole obtained through the partial dimensional reduction of a three dimensional BTZ black hole. Our results exactly reproduce the analogues of the Page curve for the entanglement negativity which were recently determined through diagrammatic technique developed in the context of random matrix theory.
We revisit the model of an AdS$_2$ black hole in JT gravity evaporating into an external bath. We study when, and how much, information about the black hole interior can be accessed through different portions of the Hawking radiation collected in the bath, and we obtain the corresponding full quantitative Page curves. As a refinement of previous results, we describe the island phase transition for a semi-infinite segment of radiation in the bath, establishing access to the interior for times within the regime of applicability of the model. For finite-size segments in the bath, one needs to inc…
The role played by the mutual information of subsystems on the Page curve is explored in this paper. With the total system consisting of the black hole and radiation, together with the inclusion of island, we observe that the vanishing of mutual information between $B_+$ and $B_-$ which in turn means the disconnected phase of the entanglement wedge corresponding to $B_+\cup B_-$, yields a time scale of the order of scrambling time. This results in a time independent expression for the fine grained entropy of Hawking radiation consistent with the correct Page curve. We also find corrections to …
By applying the island rule proposed recently, we compute the entanglement entropy of Hawking radiation and study the Page curve for the eternal black holes in massive gravity. We investigate for both the neutral and charged black holes which the corresponding results of Schwarzschild and Reissner-Nordström black holes are restored in the limit of massless graviton. We show for the neutral and non-extremal charged black holes that the island is not formed at the early times of the evaporation and hence the entanglement entropy increases linearly in time. However, for the extremal charged black…
Page transition for the complexity of an evaporating black hole
University of California, Berkeley（加州大学伯克利分校）；Leinweber Institute for Theoretical Physics and Department of Physics, University of California, Berkeley（加州大学伯克利分校莱因韦伯理论物理与物理系）；Theoretical Physics Group, Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室理论物理组）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN（理化学研究所跨学科理论与数学科学中心（iTHEMS））；Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, The University of Tokyo（东京大学宇宙物理学与数学 Kavli 研究所（WPI））；Sandia National Laboratories（桑迪亚国家实验室）
Great breakthrough in solving black hole information paradox took place when semiclassical island rule for entanglement entropy of Hawking radiation was proposed in recent years. Up to now, most papers which discussed island rule of asymptotic flat black hole with $D \ge 4$ focus on eternal black hole. In this paper, we take one more step further by discussing island of "in" vacuum state which describes one-sided asymptotically flat black hole formed by gravitational collapse in $D \ge 4$. We find that island $I$ emerges at late time and saves entropy bound. And boundary of island $\partial I$…
We use quantum extremal island method to study the information paradox on certain cosmological setups known as three dimensional Kerr-de Sitter spacetimes. To do so, we couple an auxiliary flat bath system to this spacetime in timelike singularity and measure entropy of hawking radiation in its asymptotic regions where the gravity is weak. We show that adding island regions to the entanglement wedge of radiation causes its entropy to obey Page curve. The boundary of island i.e. quantum extremal surface is located outside the cosmological horizon in the region connected directly to the bath. Ta…
We study the Page curve for asymptotically flat eternal Schwarzschild black holes in four and higher spacetime dimensions. Before the Page time, the entanglement entropy grows linearly in time. After the Page time, the entanglement entropy of a given region outside the black hole is largely modified by the emergence of an island, which extends to the outer vicinity of the event horizon. As a result, it remains a constant value which reproduces the Bekenstein-Hawking entropy, consistent with the finiteness of the von Neumann entropy for an eternal black hole.
We consider the setup of an evaporating black hole in AdS$_{4}$ coupled to an external bath, embedded in type IIB string theory. We study quantum extremal islands in these backgrounds, in relation to the existence of a massive graviton. Using explicit results of the microscopic embedding of AdS$_{4}$ massive gravity in string theory, we investigate whether it is possible to achieve backgrounds with extremal islands, in which the lowest lying graviton is only slightly massive. For certain regions of the microscopic parameters, the graviton mass can be computed explicitly, and we explain how it …
We apply the recently proposed quantum extremal surface construction to calculate the Page curve of the eternal Reissner-Nordström black holes in four dimensions ignoring the backreaction and the greybody factor. Without the island, the entropy of Hawking radiation grows linearly with time, which results in the information paradox for the eternal black holes. By extremizing the generalized entropy that allows the contributions from the island, we find that the island extends to the outside the horizon of the Reissner-Nordström black hole. When taking the effect of the islands into account, it …
We study the information paradox for the eternal black hole with charges on a doubly-holographic model in general dimensions, where the charged black hole on a Planck brane is coupled to the baths on the conformal boundaries. In the case of weak tension, the brane can be treated as a probe such that its backreaction to the bulk is negligible. We analytically calculate the entanglement entropy of the radiation and obtain the Page curve with the presence of an island on the brane. For the near-extremal black holes, the growth rate is linear in the temperature. Taking both Dvali-Gabadadze-Porrati…
Reflected entropy and islands in JT gravity from partial dimensional reduction
Shing-Tung Yau Center and School of Physics, Southeast University（东南大学物理学院）；Department of Physics, Indian Institute of Technology Kanpur（坎普尔印度理工学院物理系）；MAEER’s Maharashtra Institute of Technology（马哈拉施特拉技术学院）
