The problem of complete evaporation of Schwarzschild black holes raised by Hawking is that one has an explosion of the temperature $T=1/8πM$ for vanishing black hole mass M. We consider the Reissner-Nordstrom black hole and note that if mass $M$ and charge Q<M satisfy the bound Q>M-C M^3, C>0 for small M then the complete evaporation of black holes without blow-up of temperature is possible. We describe curves on the surface of state equations such that the motion along them provides complete evaporation. The radiation entropy follows the Page curve and vanishes at the end of evaporation.
Similar results for rotating Kerr, Schwarzschild-de-Sitter and Reissner-Nordstrom-(Anti)-de-Sitter black holes are discussed.
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 explore the Hawking evaporation of two-dimensional anti-de Sitter (AdS$_2$), dilatonic black hole coupled with conformal matter, and derive the Page curve for the entanglement entropy of radiation. We first work in a semiclassical approximation with backreaction. We show that the end-point of the evaporation process is AdS$_2$ with a vanishing dilaton, i.e. a regular, singularity-free, zero-entropy state. We explicitly compute the entanglement entropies of the black hole and the radiation, as functions of the horizon radius, using the conformal field theory (CFT) dual to AdS$_2$ gravity. We…
We propose a model with multiple qubits that reproduces the thermal properties of 4-dimensional (4-dim) Schwarzschild black holes (BHs) by simultaneously taking account of the emission of Hawking particles and the zero-energy soft hair evaporation at horizon. The results verify that the entanglement entropy between a qubit and other subsystems, including emitted radiation, is much larger than the BH entropy analogue of the qubit, as opposed to the Page curve prediction. Our result suggests that early Hawking radiation is entangled with soft hair, and that late Hawking radiation can be highly e…
One-shot decoupling is a powerful primitive in quantum information theory and was hypothesized to play a role in the black hole information paradox. We study black hole dynamics modeled by a trilinear Hamiltonian whose semiclassical limit gives rise to Hawking radiation. An explicit numerical calculation of the discretized path integral of the S-matrix shows that decoupling is exact in the continuous limit, implying that quantum information is perfectly transferred from the black hole to radiation. A striking consequence of decoupling is the emergence of an output radiation entropy profile tha…
Page curve for the fine grained entropy of the black hole radiation is obtained once we include the effect of a non vanishing quantum Fisher information in the evaporation data. This quantum Fisher information scales as the inverse of the original black hole entropy. The modified fine grained entropy can be naturally related to a relative entanglement entropy with the quantum Fisher information defining the curvature at Page point.
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 o…
We investigate quantum correlations between successive steps of black hole evaporation and investigate whether they might resolve the black hole information paradox. 'Small' corrections in various models were shown to be unable to restore unitarity. We study a toy qubit model of evaporation that allows small quantum correlations between successive steps and reaffirm previous results. Then, we relax the 'smallness' condition and find a nontrivial upper and lower bound on the entanglement entropy change during the evaporation process. This gives a quantitative measure of the size of the correcti…
We show that the Hawking-Page phase transition of a CFT on AdS_{d-1} weakly coupled to gravity has a dual bulk description in terms of a phase transition between a black string and a thermal gas on AdS_{d}. At even lower temperatures the black string develops a Gregory Laflamme instability, which is dual to black hole evaporation in the boundary theory.
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.
We investigate the fine-grained entropy of the de Sitter cosmological horizon. Starting from three-dimensional pure de Sitter space, we consider a partial reduction approach, which supplies an auxiliary system acting as a heat bath both at future infinity and inside the static patch. This allows us to study the time-dependent entropy of radiation collected for both observers in the out-of-equilibrium Unruh-de Sitter state, analogous to black hole evaporation for a cosmological horizon. Central to our analysis in the static patch is the identification of a weakly gravitating region close to the…
In a recent article, we have shown how quantum fluctuations of the background geometry modify Hawking's density matrix for black hole (BH) radiation. Hawking's diagonal matrix picks up small off-diagonal elements whose influence becomes larger with the number of emitted particles. We have calculated the "time-of-first-bit", when the first bit of information comes out of the BH, and the "transparency time", when the rate of information release becomes order unity. We have found that the transparency time is equal to the "Page time", when the BH has lost half of its initial entropy to the radiat…
Motivated by the quantum process of black hole evaporation and its implications for symmetries, we consider a qubit system with a random dynamics as a toy model of black hole. We compute its symmetry-resolved entropies and discuss its implications. We first consider the case where charges are conserved and compute the symmetry-resolved entropies. We derive a symmetry-resolved analogue of the Page curve. We then consider the case where symmetry is explicitly broken and charges are no longer conserved. It serves as a toy model for global symmetry breaking in black hole evaporation. Despite the s…
In a recent paper Osuga and Page have presented an explicitly unitary toy qubit transport model for transferring information from a black hole to the outgoing radiation. Following their idea we propose a unitary toy model which involves (fermionic) Hawking states.
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…
We study the black hole information problem within a semiclassically gravitating AdS$_d$ black hole coupled to and in equilibrium with a $d$-dimensional thermal conformal bath. We deform the bath state by a relevant scalar deformation, triggering a holographic RG flow whose "trans-IR" region deforms from a Schwarzschild geometry to a Kasner universe. The setup manifests two independent scales which control both the extent of coarse-graining and the entanglement dynamics when counting Hawking degrees of freedom in the bath. In tuning either, we find nontrivial changes to the Page time and Page …
In this work new interaction terms between two SYK models are proposed, which allow to define an interaction picture such that it is possible to calculate exactly the vacuum state's time evolution. It is shown that the vacuum evolves as a time dependent $SU(2)$ squezed state. The time dependent entanglement entropy is calculated and it has the same form of the Page curve of Black Hole formation and evaporation.
If the evaporation of a black hole formed from a pure state is unitary, the entanglement entropy of the Hawking radiation should follow the Page curve, increasing from zero until near the halfway point of the evaporation, and then decreasing back to zero. The general argument for the Page curve is based on the assumption that the quantum state of the black hole plus radiation during the evaporation process is typical. In this paper, we show that the Page curve can result from a simple dynamical input in the evolution of the black hole, based on a recently proposed signature of quantum chaos, w…
We develop an microscopic model of the M-theory Schwarzschild black hole using the Banks-Fischler-Shenker-Susskind Matrix formulation of quantum gravity. The underlying dynamics is known to be chaotic, which allows us to use methods from Random Matrix Theory and non-equilibrium statistical mechanics to propose a coarse-grained bottom-up picture of the event horizon -- and the associated Hawking evaporation phenomenon. The analysis is possible due to a hierarchy between the various timescales at work. Event horizon physics is found to be non-local at the Planck scale, and we demonstrate how non…
We consider the black hole information problem in an explicitly defined spacetime modelling black hole evaporation. Using this context we review basic aspects of the problem, with a particular effort to be unambiguous about subtle topics, for instance precisely what is meant by entropy in various circumstances. We then focus on questions of unitarity, and argue that commonly invoked semiclassical statements of long term, evaporation time, and Page time "unitarity" may all be violated even if physics is fundamentally unitary. This suggests that there is no horizon firewall. We discuss how the p…
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 study the generalized entanglement entropy in the higher dimensional two-sided eternal black hole by double holography. By introducing an end-of-the-world ETW brane, which defines the time-dependent effective Hawking radiation region, we find a new type of Ryu-Takayanagi surface besides the Hartman-Maldacena surface and the island Ryu-Takayanagi surface known previously. We study the phase transition among the three Ryu-Takayanagi surfaces at different temperatures and obtain the phase diagram as well as the Page curve.
We present an analytical formulation of the recent one-shot decoupling model of Bràdler and Adami [arXiv:1505.0284] and compute the resulting "Page Information" curves, for the reduced density matrices for the evaporating black hole internal degrees of freedom, and emitted Hawking radiation pairs entangled across the horizon. We argue that black hole evaporation/particle production has a very close analogy to the laboratory process of spontaneous parametric down conversion, when the pump is allowed to deplete.
A new simple qubit model of black hole evaporation is proposed. The model operates on four qubits and is defined in terms of quantum gates and a quantum circuit. The chief features of the model include explicit unitarity and (most notably) causality which is understood as the impossibility of the transfer of information from the interior of the black hole through its horizon. The corresponding von Neumann entanglement entropy yields a crude (four-qubit) approximation of the Page curve.
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 …
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 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…
In this paper, we analyze whether quantum correlations between successive steps of evaporation can open any way to resolve the black hole information paradox. Recently a celebrated result in literature shows that `small' correction to leading order Hawking analysis fails to restore unitarity in black hole evaporation. We study a toy qubit model of evaporation allowing small quantum correlations between successive steps and verify the previous result. Then we generalize the concept of correction to Hawking state by relaxing the `smallness' condition. Our result generates a nontrivial upper and …
A reformulation of the calculation of the semi-classical energy-momentum tensor on a Schwarzschild background, the Bousso covariant entropy bound, and the ER=EPR conjecture of Maldacena and Susskind taken together suggest a scenario for the evaporation of a large spherically symmetric black hole formed in gravitational collapse in which 1) the classical r = 0 singularity is replaced by an initially small non-singular core inside an inner apparent horizon, 2) the radius of the core grows with time due to the increasing entanglement between Hawking radiation quanta outside the black hole and the…
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 …
Hawking's model of black hole evaporation is not unitary and leads to a mixed density matrix for the emitted radiation, while the Page model describes a unitary evaporation process in which the density matrix evolves from an almost thermal state to a pure state. We compare a recently proposed model of semiclassical black hole evaporation to the two established models. In particular, we study the density matrix of the outgoing radiation and determine how the magnitude of the off-diagonal corrections differs for the three frameworks. For Hawking's model, we find power-law corrections to the two-…
Areas of extremal surfaces anchored to sub-regions on screens in Minkowski space satisfy various entanglement entropy inequalities. In 2+1 dimensions where the arguments are simplest, we demonstrate (a) monogamy of mutual information, (b) various versions of (strong) subadditivity, (c) various inequalities involving the entanglement of purification, as well as (e) reflection inequality and (f) Araki-Lieb inequality. Just as in AdS, Linden-Winter and the tower of Cadney-Linden-Winter inequalities are satisfied trivially. All of these are purely geometric (and therefore unambiguous) statements, …
We consider the entanglement island in a deformed Jackiw-Teitelboim black hole in the presence of the phase transition. This black hole has the van der Waals-Maxwell-like phase structure as it is coupled with a Maxwell field. We study the behavior of the Page curve of this black hole by using the island paradigm. In the fixed charge ensemble, we discuss different situations with different charges that influence the system's phase structure. There is only a Hawking-Page phase transition in the absence of charges, which leads to an unstable small black hole. Hence, the related Page curve does no…
We suggest a group-theoretic approach to black holes, which is remotely analogous to the eightfold-way for mesons. As the black hole symmetry group we single out the group SO(2N+1) with N the black hole entropy. The Hilbert space is identified with the spinor irrep of SO(2N+1). Evaporation processes of m-quanta are associated to the breaking SO(2N+1) to SO(2(N-m)+1) X SO(2m). Under these assumptions we get a group-theoretic understanding of the evaporation process and of some typical time scales of black holes, such as Page's and scrambling times. We also discuss from the group theory point of…
We study traversable wormholes by considering the duality between BTZ black holes and two-dimensional conformal field theory on the thermofield double state. The BTZ black holes can be rendered traversable by a negative energy shock wave. Following Gao, Jafferis and Wall [1], we show that the negative energy shock wave is dual to the infinite boost limit of a specific double trace deformation which couples the left and right CFTs. We spell out the mechanism of information transfer through traversable BTZ wormholes, treating the backreaction of the message as a positive energy shockwave. The co…
We calculate the time evolution of entanglement entropy in two dimensional conformal field theory with a moving mirror. For a setup modeling Hawking radiation, we obtain a linear growth of entanglement entropy and show that this can be interpreted as the production of entangled pairs. For a setup, which mimics black hole formation and evaporation, we find that the evolution follows the ideal Page curve. We perform these computations by constructing the gravity dual of the moving mirror model via holography. We also argue that our holographic setup provides a concrete model to derive the Page c…
