IIsland in Charged Black Holes
Yi Ling , ∗ , Yuxuan Liu , † , Zhuo-Yu Xian ‡ Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China Institute of Theoretical Physics, Chinese Academy of Science, Beijing 100190, China
Abstract
We study the information problem in the eternal black hole with charges on adoubly-holographic model in general dimensions, where the charged black hole on aPlanck brane is coupled to the bath on the conformal boundary. For a brane with weaktension, its backreaction to the bulk is negligible and the entropy of the radiation isdominated by the entanglement entropy of the matter. We analytically calculate theentropy of the radiation and obtain the Page-like curve with the presence of an islandon the brane. The Page time is evaluated as well. For a brane with strong tension,we obtain the numerical solution with backreaction in four-dimensional space time andfind the quantum extremal surface at t = 0. It turns out that the geometric entropyrelated to the area term on the brane postpones the Page time. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - t h ] S e p ontents t = 0 . . . . . . . . . . . . . . . . . . . . . 164.3 The contribution of the geometric entropy . . . . . . . . . . . . . . . . . . . 18 The black hole information paradox is originated from the problem of whether the informationfalling into the black hole evolutes in a unitary fashion. During the evaporation of a black hole,the information originated from the collapsing star appears to conflict with the nearly thermalspectrum of Hawking radiation by semi-classical approximation. One possible explanationcomes from quantum information theory. If we assume the unitary and chaotic evaporationof a black hole, then the von Neumann entropy of the radiation may be described by thePage curve, which claims that the entropy will increase until the Page time and then decreaseagain [1–3], which is in contrast to Hawking’s earlier calculation in which the entropy willkeep growing until the black hole totally evaporated [4]. Later, a further debate is raisedabout whether the ingoing Hawking radiation is burned up at the horizon owing to the“Monogamy of entanglement”, which is known as AMPS firewall paradox [5]. In order toavoid the emergence of firewall, the interior of black hole is suggested to be part of theradiation system through an extra geometric connection (see “ER=EPR” conjecture [6]).Recently AdS/CFT correspondence brings new breakthrough on the understanding of thePage curve from semi-classical gravity and sheds light on how the interior of the black hole isprojected to the radiation [7–9]. In this context, the black hole in AdS spacetime is coupledto a flat space, where the latter is considered as a thermal bath.Motivated by the Ryu-Takayanagi formula and its generalization [10, 11], it has beenproposed that the fine-grained entropy of a system can be calculated by the quantum extremalsurface (QES) [12]. When applied to the evaporation of black holes, in order to recover the2age curve of Hawking radiation, it is found that an island should appear in the gravityregion such that the fine-grained entropy of the radiation is determined by the formula ofquantum extremal island [13] S R = min I ( ext I " S eff [ R ∪ I ] + Area[ ∂ I ]4 G N , (1)where R represents the radiation system, while the first term is the entanglement entropyof region R ∪ I , and the second term is the geometrical entropy from classical gravity. Thevon Neumann entropy S is obtained by the standard process: extremizing over all possibleislands I , and then taking the minimum of all extremal values.A doubly-holographic model has been considered in [13, 14], in which the matter field in2 D black hole geometry is a holographic CFT which enjoys the AdS /CF T correspondence.By virtue of this the first term in (1) can be calculated according to the ordinary HRT formulain AdS /CF T . At early time, the minimal configuration contains no island and the increaseof the entropy is mainly contributed from the first term in (1) as the accumulation of theHawking particle pairs. But later, with the emergence of the island I , the QES undergoesa phase transition such that a new configuration with island gives rise to smaller entropy.Meanwhile, owing to the shrinking of the black hole, the decrease of the entropy at late timeis dominated by the second term in (1). As a result, the whole process can be describedby the Page curve and the transition time of the configuration determines the Page time.Moreover, the fact that the island I is contained in the entanglement wedge of the radiationrealizes the ER=EPR conjecture.A similar information paradox occurs when black holes and flat baths are in equilibrium.Exchanging Hawking modes entangles black holes and baths, but the entanglement entropyshould be upper bounded by the twice of the black hole entropy according to informationtheory. The authors in [15] consider the 2 D eternal black hole-bath system when the wholeholographic system is dual to Hartle-Hawking state. Its island extends outside the horizonand the degrees of freedom (d.o.f.) on the island are encoded in the radiation system by ageometric connection [15]. A similar conclusion was made independently by only consideringthe d.o.f. of radiation [16, 17].In [18], a nontrivial setup of the doubly-holographic model in higher dimensions wasestablished, where the lower dimensional gravity is replaced by a Planck brane with Neumannboundary conditions on it [19]. The solution at t = 0 is obtained with the DeTurck trick andit was demonstrated that the islands exist in higher dimensions, and the main results in [8,13–15] can be extended to higher dimensional case as well. However, compared to the work inliterature, the geometric entropy contributed by the lower dimensional gravity, was not takeninto account in the discussion [18]. Therefore, it is reasonable to ask whether the contributionfrom the geometric entropy can be ignored. Moreover, in [20], the solution without the3 a) (b) Figure 1: (a): The d -dimensional charged black hole B coupled to flat spacetime R is inequilibrium with radiation. The conformal boundary is located at σ = 0, while the blackhole and radiation are distributed in the region with σ > σ <
0, respectively. The QESat time t is represented by the black dot and the partial Cauchy surfaces are represented bythe dashed blue lines, while the island I is represented by the red curve in the middle. (b):A sketch of the ( d + 1)-dimensional dual of the d -dimensional holographic system. Here the d -dimensional black hole B can be described equivalently by ( d − QM L ( R ) at σ = 0 or the Planck brane in the ( d + 1)-dimensional ambientspacetime. The QES is also replaced by an ordinary HRT surface in the ( d + 1)-dimensionalspacetime.geometric entropy is considered in the weak-tension limit at t = 0, where the difference ofentropy between two configurations might be negative. This peculiar phenomenon impliesthat the Page time is always t = 0, which of course calls for a proper understanding. Inaddition, since the DeTurck trick does not apply to time-dependent case [21, 22], in generalthe standard Page-like curve in higher dimensions is difficult to obtain.Recently, inspired by the doubly-holographic model, a novel framework called wedgeholography was established [23], which manifests an elegant relation between the Newtonconstant in higher dimensional gravity and that in lower dimensions. This relation impliesthat the lower-dimensional Newton constant becomes fairly large for the case of weak tension[23, 24], which may give us a hint to investigate above problems.Therefore, in this paper our purpose are twofold. Firstly, we intend to extend the analysisin [18] to charged black holes in higher dimensions, but more importantly, we will analyti-cally obtain the Page-like curve of charged black holes in the weak-tension limit where thebackreaction of the brane can be ignored. Secondly, we will take the geometric entropy in(43) into account due to the non-zero tension on the brane [15] and find more reasonablesolutions of the entropy difference.The paper is organized as follows. In section 2, we consider the charged matters on the4oundary and build up the doubly-holographic model. In section 3, we explore the Page-likecurve in the weak-tension limit in general dimensions and analyze the evolution at differentHawking temperature. In section 4, we investigate the effects of charge on the island in theback-reacted spacetime, and then discuss how the geometric entropy in (43) affects Pagetime. Our conclusions and discussions are given in section 5. In this section, we will present the general setup for the island within the charged eternalblack hole. We consider a d -dimensional charged eternal black hole B in AdS d coupled totwo flat spacetimes R (bath) on each side, with the strongly coupled conformal matter livingin the bulk, as shown in Fig. 1(a). On each side, the black hole corresponds to the regionwith σ > σ <
0. Moreover, at σ = 0, we gluethe conformal boundary of the AdS d and flat spacetime together and impose the transparentboundary condition on the matter sector. With a finite chemical potential µ , the matter andthe black holes carry charges.There are two possible descriptions of this combined system. The first is the full quantummechanical description, where d -dimensional black holes together with the matter sectors aredual to ( d − σ = 0 on theconformal boundary. While the second description is called the doubly-holographic setup.The matter sector is dual to a ( d + 1)-dimensional spacetime and the d -dimensional blackhole B is described by a Planck brane in the bulk, as shown in Fig. 1(b). In this paper, wewill adopt the second description, namely the doubly-holographic setup.We consider the action of the ( d + 1)-dimensional bulk as I = 116 πG ( d +1) N "Z d d +1 x √− g R + d ( d − L ! + 2 Z B d d x √− h ( K − α ) − Z d d +1 x √− g F . (2)Here the parameter α is proportional to the tension on the brane B and will be fixed later.The electromagnetic curvature is F = d A . Taking the variation of the action, we obtain theequations of motion as R µν + dL g µν = (cid:18) T µν − Td − g µν (cid:19) , with T µν = F µa F ν a − F g µν , (3) ∇ µ F µν = 0 , (4)where T is the trace of energy-stress tensor T µν .5igure 2: A simple setup of Randall-Sundrum brane [19]. Here the Planck brane is anchoredon the conformal boundary at ( z, w ) = (0 ,
0) and penetrates into the bulk with an angle θ . In AdS/CFT setup with infinite volume, the ( d +1)-dimensional bulk is asymptotic to AdS d +1 which in Poincar´e coordinates is described by ds = 1 z − dt + dz + dw + d − X i =1 dw i ! , (5)with the conformal boundary at z = 0. Let θ denote the angle between the Planck braneand the conformal boundary as shown in Fig. 2. Then the Planck brane B can be describedby the hypersurface z + ω tan θ = 0 , (6)near the boundary. One should cut the bulk on the brane B and restrict it in the region with z + ω tan θ < K ij − Kh ij + αh ij = 0 , (7)where h ij is the metric on the brane B . The parameter α is fixed to be a constant α =( d −
1) cos θ by solving (7) near the boundary to concrete the tension on the brane, which is T = α .In addition to (7), we also impose the Neumann boundary condition for gauge field A µ on the brane B [28–30], which is n µ F µν h ν i = 0 , (8)where n µ is the normal vector to the brane and i denotes the coordinates along the brane.6 .2 The Quantum Extremal Surface The von Neumann entropy of radiation R in (1) is measured by the QES in Fig. 1(a). Accord-ing to the doubly holographic prescription, the entanglement entropy of the d -dimensionalmatter is measured by the ordinary HRT surface and the geometric entropy is measured bythe area term on the brane B , namely S R = min I " Area( γ I∪R )4 G ( d +1) N + Area( ∂ I )4 G ( d ) N , (9)where γ I∪R is the HRT surface sharing the boundary with
I ∪ R , as illustrated in Fig. 1(b).Consider a HRT surface anchor at w = w b on the boundary as shown in Fig. 1(b). Itmeasures the entanglement between the combined system at [0 , w b ], which consists of thequantum mechanical system (QMS) at w = 0 and part of the bath at 0 < w ≤ w b , and theremaining bath system. For simplicity, hereafter, we call the combined system at [0 , w b ] asthe black hole system, while the remaining bath as the radiation system.Generally, one will find two candidates of QES, each of which corresponds to a HRTsurface γ I∪R in the bulk, as shown in Fig. 1. One candidate is a trivial surface γ tr anchoredon the left and right baths, and the island I is absent; the other is a surface γ pl anchored onthe Planck brane B with non-trivial island I on the brane. The emergence of island I keepsthe von Neumann entropy (1) from divergence after the Page time [15].In the context of wedge holographic gravity, the relation of d -dimensional and ( d + 1)-dimensional Newton constants was firstly obtained in [23], and then was generalized in [24].In the wedge holography construction [24], for a specific class of solutions ,the relation of theNewton constants becomes 1 G ( d ) N = 1 G ( d +1) N Z ρ cosh d − ( x ) dx. (10)Here, g µν denotes ( d + 1)-dimensional metric of the background geometry with 0 ≤ x ≤ ρ ,while h ij denotes d -dimensional metric, which matches the induced metric on the brane B at x = ρ .In this paper, the solutions we are concerned with are not included in above class, butwe propose that two Newton constants could be related by (10), with the tension T =( d −
1) tanh ρ . Therefore, in the weak-tension limit ρ →
0, the lower-dimensional Newtonconstant G ( d ) N is fairly large compared to the higher-dimensional one G ( d +1) N . While in thelarge-tension limit, the relation becomes L d − /G ( d ) N (cid:29) L d − /G ( d +1) N with the increase of ρ ,which returns to the case as discussed in [13].7 tr γ pl ( z b ) ( W c , z c ) ( W b ,0 ) W z (a) (b) Figure 3: (a): For { d, µ, W b } = { , / , } , the two candidates γ tr and γ pl in the weak-tensionlimit. The black dot near the horizon z = 1 represents the additional geometric entropy. (b):The trivial surface γ tr colored in rose gold at time t . Here z max is the turning point with z | z max = 0. Page curve or Page-like curve plays a vital role in understanding the information paradox. Inthis section, we will explore the Page-like curve in the weak-tension limit, where the lower-dimensional Newton constant G ( d ) N is fairly large compared to the higher-dimensional one G ( d +1) N . In such a limit, the geometric entropy of the black hole is fairly small compared tothe von Neumann entropy of the black hole, and the Planck brane B can be treated as aprobe. That is to say, its backreaction to the background geometry is ignored.Define W = z cot θ + w such that the Planck brane B is located at W = 0 and we onlytake part of the manifold with W ≥ θ → π/
2, thebrane B applies negligible back-reaction to the background and the geometry can be regardedas the planar RN-AdS d +1 , which is ds = L z " − f ( z ) dt + dz f ( z ) + ( dW − cot θdz ) + d − X i =1 dw i , (11) A = µ (cid:16) − z d − (cid:17) dt, (12) f ( z ) =1 − d − d − µ ! z d + d − d − µ z d − , (13)where the horizon is located at z = 1 and µ is the chemical potential of the system on the8oundary. The Hawking temperature is fixed to be T h = d − d + 4 µ − dµ + d µ π − dπ . (14)From (9), the von Neumann entropy of radiation is determined by the minimum S R = 2 min ( S il , S tr ) . (15)Here S il = S pl + Area ( d ) pl G ( d ) N . (16) S pl denotes the entanglement entropy of region R ∪ I , while S tr is the von Neumann entropyof the black hole on one side.Firstly we consider the surface γ pl ended on the Planck brane as the QES. We mayintroduce two different parameterizations in different intervals, just as performed in [18].In ( W, z ) plane, for the curve anchored between point ( W b ,
0) and ( W c , Z c ) (Fig. 3(a)), weintroduce W = W ( z ), while for the curve anchored between ( W c , z c ) and (0 , z b ), we introduce z = z ( W ) instead, with z ( W c ) = W ( z c ) − . Therefore, the corresponding lagrangian andarea term are given as S pl = L d − Ω d − G ( d +1) N Z z c dzz d − vuut f ( z ) (cot( θ ) − W ( z )) + 1 f ( z )+ Z W c dWz ( W ) d − vuut f ( z ( W )) (cot( θ ) z ( W ) − + z ( W ) f ( z ( W )) ! , (17)Area ( d ) pl = L d − Ω d − z (0) d − . (18)Here Ω d − is the volume of the ( d −
2) relevant spatial directions and W b is the location where γ pl is anchored on the conformal boundary z = 0, as shown in Fig. 3(a).Next we consider the trivial surface γ tr penetrated the horizon as the QES. We expressthe area functional in the Eddington-Finkelstein coordinates and the entanglement entropy S tr becomes S tr = L d − Ω d − G ( d +1) N Z dλz ( λ ) d − q − v ( λ ) [ f ( z ( λ )) v ( λ ) + 2 z ( λ )] , (19)where λ is the intrinsic parameter of γ tr (Fig. 3(b)) and v = t − Z dzf ( z ) . h / μ = T h / μ = T h / μ = T h / μ = T h / μ = - - - W b μ S ˜ / μ Figure 4: For d = 3 and θ → π/
2, the relations between the saturated entropy and the endpoint of the HRT surface are plotted at different temperatures, where ˜ S := S .For now, we assume that S tr < S il (which is not always correct as discussed in thefollowing) and hence, γ tr is the genuine HRT surface at t = 0. As time passes by, S il maintains its value due to the static geometry, while S tr keeps growing due to the growth ofthe black hole interior in the ( d + 1)-dimensional ambient geometry [31], which also leads tothe Hawking curve in the entropy of radiation.But the Hawking curve contradicts with the subadditivity of von Neumann entropy. Sincethe total system is in a pure state, we have S R = S B ≤ S il , where S B is the von Neumannentropy of black hole system. In other words, at some moment, the growing S tr ( t ) must reach S il and after that the genuine HRT surface becomes γ pl . S R stops growing at the transition,which determines the Page time t P .We are interested in the growth of the entropy with the time, thus we define ∆ S ( t ) := S R ( t ) − S R (0), which is free from UV divergence. It behaves as∆ S = S tr ( t ) − S tr (0) , t < t P S , t ≥ t P (20)where S = 2 S il − S tr (0) . (21)As shown in Fig. 4, for small endpoint W b , the difference between two candidates is S <
0, which was firstly obtained in [20] with zero tension. At the first sight this lookspeculiar, and our understanding on this result is given as follows. Recall that for zero tensionat θ = π/
2, the Newton constant G ( d ) N is divergent, such that the geometric entropy term10igure 5: The growth of ∆ S ( t ) is shown for µ = 0, where the rate of linear growth approaches KT d − h and S labels the saturation value after the Page time.vanishes and the von Neumann entropy of the black hole is totally contributed from theentanglement of matter fields at (0 , W b ]. Therefore, for small endpoint W b , the negativity ofthe entropy difference S implies that the black hole system lacks d.o.f. to be entangled withthe radiation and no process of the accumulation of the Hawking particle pairs in such cases,which leads to the saturation of the entropy at the beginning.While for large endpoint W b together with a non-zero but weak tension on the brane, theentropy difference is S >
0. In these cases, the backreaction of the brane to the backgroundcan be ignored due to the weak tension, and we expect that a Page-like curve may occur, whichactually portrays the evolution that the von Neumann entropy of the black hole is mainlycontributed by the internal entanglement of the matter fields, as well as a little contributionfrom the geometric entropy. We will focus on these cases and explore its evolution behaviorin the next subsection.
For large endpoints W b , we have S > S ( t ) can be obtainedsimilarly with the method applied in [32]. Since the integrand of (19) does not dependexplicitly on v , one can derive a conserved quantity as C = f ( z ) v + z z d − q − v [ f ( z ) v + 2 z ] . (22)11t is also noticed that the integral shown in (19) is invariant under the reparametrization,hence the integrand can be chosen freely as q − v [ f ( z ) v + 2 z ] = z d − . (23)Substituting (22) and (23) into (19), we have ddt S tr = L d − Ω d − G ( d +1) N q − f ( z max ) z d − max , (24) t = Z z max dz Cz d − f ( z ) q f ( z ) + C z d − . (25)Here z max is the turning point of trivial surface γ tr as shown in Fig. 3(b), and the relationbetween z max and the conserved quantity C is given by f ( z max ) + C z d − max = 0 . (26)At later time, the trivial extremal surface γ tr tends to surround a special extremal slice z = z M , as shown in [31]. Define F ( z ) := q − f ( z ) z d − , (27)we find that C = F ( z max ) keeps growing until meeting the extremum at z max = z M , wherewe have F ( z M ) = (1 − d ) z − dM q − f ( z M ) − z − dM f ( z M )2 q − f ( z M ) = 0 . (28)By solving (28), we finally obtain the evolution of the entropy aslim t →∞ ddt S tr = K F ( z M ) , K := L d − Ω d − G ( d +1) N . (29)Next we discuss the growth rate of entropy for the neutral case and the charged case sepa-rately. • The Neutral Case:
For µ = 0, substituting f ( z ) = 1 − z d into (28), we have thegrowth rate of entanglement entropy at late time aslim t →∞ ddt S tr = c d K T d − h , K := L d − Ω d − G ( d +1) N , (30)where c d = 2 d + d − π d − d − d ( − d ) − dd ( − d ) d − d . For d = 2, c d = 2 π and thegrowth rate is proportional to Hawking temperature of the black hole, which is exactly12 h / μ = T h / μ = T h / μ = T h / μ = T h / μ = T h / μ = t μ Δ S ˜ / μ Figure 6: For { d, W b , L, θ } = { , , , π/ } , the Page-like curves at different Hawking tem-perature are shown in the figure, which are labeled by different rose gold curves. The dashedblue curve represents the Page time tµ = t P µ of each Page-like curve. In the plot, The New-ton constant is fixed by substituting T = ( d −
1) cos θ = ( d −
1) tanh ρ into (10). Therefore,we can set L G (4) N = 1, L G (3) N ≈ .
32 and ˜ S ( t ) := w S ( t ).in agreement with the result in [15, 31].Due to the exchanging of Hawking modes, the entanglement entropy of radiation growslinearly during most of time at a rate proportional to T d − h . If there was no island, theentanglement entropy would keep growing and finally exceeding the maximal entropythe black hole system allowed to contain. It would be an information paradox similarto the version of evaporating black hole. The formation of quantum extremal island atPage time resolves this paradox, since some d.o.f. of black hole system are encoded inthe radiation and the growth will saturate (Fig. 5). • The Charged Case:
Recall that in general dimensions, when turning on the chemicalpotential, the blackening factor becomes f ( z ) = 1 − d − d − µ ! z d + d − d − µ z d − . (31)The late time behavior of entanglement entropy with d ≥ t →∞ ddt S tr = L d − Ω d − G ( d +1) N d − ( d − d − d − µ ) ! − dd vuuut − d − d − − d − d − d − ( d − d − d − µ ) ! d − d µ . (32)Specifically, for d = 3, we havelim t →∞ ddt S tr = L ∆ w G (4) N q µ ) − µ . (33)Here ∆ w is the integral along w direction. Since the saturation occurs approximatelyat S , the Page time is obtained as t P ≈ G (4) N L ∆ w S q µ ) − µ . (34)Some concrete cases are shown in Fig. 6 : in the high temperature limit T h /µ → ∞ ,the Page-like curve recovers the neutral case as mentioned above, while in the lowtemperature limit T h /µ →
0, it becomes the extremal case.For the neutral case, the entanglement grows rapidly and saturates at the highest level.With the decrease of Hawking temperature, the entanglement grows tardily and finallysaturates at a lower level. While for the extremal case, one can easily read from (33),the entanglement entropy never grows.Since the entanglement between the black hole and radiation system is built up bythe exchanging of Hawking modes before the Page time t P , the phenomenon thatentropy increases rapidly at higher temperatures indicates that the higher the Hawkingtemperature is, the higher the rate of exchanging is. In this section we will consider the QES in the presence of island when the back-reaction ofthe Plank brane to the bulk is taken into account for d = 3. We will apply the DeTurcktrick to handle the static equations of motion and find the numerical solution via spectrummethod. Nonetheless, the time dependence in the higher-dimensional geometry is hard toexplore using DeTurck trick. Therefore, in this section we will just derive the solution at t = 0 with the geometric entropy (43) and investigate the effects of the charge on the island.We will also discuss how the geometric entropy affects the Page time.14 .1 The metric ansatz We introduce the Deturck method [22] to numerically solve the background in the presenceof the Planck brane in this subsection, and the numerical results for the QES over suchbackgrounds will be presented in next subsection. Instead of solving (3) directly, we solvethe so-called Einstein-DeTurck equation, which is R µν + 3 g µν = (cid:18) T µν − T g µν (cid:19) + ∇ ( µ ξ ν ) , (35)where ξ µ := [Γ µνσ ( g ) − Γ µνσ (¯ g )] g νσ is the DeTurck vector and ¯ g is the reference metric, which is required to satisfy the sameboundary conditions as g only on Dirichlet boundaries, but not on Neumann boundaries [18].Now we introduce the metric ansatz and the boundary conditions in the doubly-holographicsetup. For W → ∞ , the ambient geometry is asymptotic to 4 D planar RN-AdS black (11)with d = 3. Furthermore, for numerical convenience we define two new coordinates in thesame way as applied in [18], which are x = W W and y = √ − z. (36)The domain of the first coordinate x is compact, while the second coordinate y keeps themetric from divergence at the outer horizon y + = 0.When the back-reaction of the Planck brane is taken into account, the translationalsymmetry along x direction is broken. Therefore, the most general ansatz of the backgroundis ds = L (1 − y ) " − y P ( y ) Q dt + 4 Q P ( y ) dy + Q (1 − x ) (cid:16) dx + 2 y (1 − x ) Q dy (cid:17) + Q dw (37) A = y ψ dt, (38) P ( y ) =2 − y + (1 − y ) −
12 (1 − y ) µ . (39)where { Q , Q , Q , Q , Q , ψ } are the functions of ( x, y ). All the boundary conditions arelisted in Tab. 1. Moreover, the boundary conditions at the horizon y = 0 also imply that Q ( x,
0) = Q ( x, T h = 6 − µ π . (40)The reference metric ¯ g is given by Q = Q = Q = Q = 1 and Q = cot θ . In this15 2 3 4 5 6 y = Q = 1 Q = 1 Q = cot θ Q = 1 Q = 1 ψ = µ y = ∂ y Q = 0 ∂ y Q = 0 ∂ y Q = 0 ∂ y Q = 0 ∂ y Q = 0 ∂ y ψ = 0 x = Q = 1 Q = 1 Q = cot θ Q = 1 Q = 1 ψ = µ x = n µ F µν = 0 n µ ξ µ = 0 Q = cot θ K µν − Kh µν + 2 cos θh µν = 0Table 1: Boundary conditions.case, the corresponding charge density ρ decays exponentially from the Planck brane B tothe region deep into the bath as shown in Fig. 7. t = 0 In this subsection, all free parameters will be fixed for numerical analysis. To regularize theUV divergence of entropy, we should introduce a UV cut-off. Since γ pl and γ tr share thesame asymptotic behavior, their difference should be independent of the choice of the cut-offgiven that it is small enough. We find that y (cid:15) = 1 − /
100 is good enough for numericalcalculation.For the trivial surface γ tr at t = 0, we introduce the parameterization x = x ( y ), whichleads to the corresponding lagrangian˜ S γ tr := S tr ∆ w = L G (4) N Z dy − y ) vuut Q Q P ( y ) + Q (2 y ( x ( y ) − Q + x ( y )) ( x ( y ) − ! . (41)For numerical convenience, we have set L / (4 G (4) N ) = 1 in the following discussion.As for the surface γ pl at t = 0, we also introduce two different parameterizations indifferent intervals. In ( x, y ) plane, for the curve anchored between point ( x b ,
1) and ( x c , y c )(Fig. 8(a)), we introduce x = x ( y ) just as the parameterization of the trivial surface γ tr ,while for the curve anchored between ( x c , y c ) and (0 , y b ), we introduce y = y ( x ) instead, with y ( x c ) = x ( y c ) − . Therefore, the corresponding lagrangian and area term are given as˜ S pl := S pl ∆ w = Z y c dy (1 − y ) vuut Q Q P ( y ) + Q (2 y ( x ( y ) − Q + x ( y )) ( x ( y ) − ! + Z x c dx ( y ( x ) − vuut Q Q y ( x ) P ( y ( x )) + Q Q y ( x ) y ( x ) + 4 Q y ( x ) y ( x )( x − + 1( x − !! , (42)˜ A ( y b ) := Area (3) pl ∆ w = L √ Q − y b , y b := y (0) . (43)16 .0 0.2 0.4 0.6 0.8 1.00.51.01.52.02.53.0 W ρ Figure 7: The charge density ρ ( W ) in the case for µ = 1 / θ = π/ γ pl and γ tr at t = 0 is˜ S = ext y b ˜ S γ pl ( y b ) + ˜ A ( y b )4 G (3) N ! − ˜ S γ tr , (44)which also represents the saturation value of entropy after Page time as mentioned in theprevious section. Notice that for numerical convenience, we also set L/ (4 G (3) N ) = 1 in thefollowing discussion.The numerical result for the difference between two candidates is shown in Fig. 8(b).It indicates the trivial surface γ tr always dominates at t = 0 and ˜ S is obtained when theextremal value is reached. As Hawking radiation continues, the entanglement entropy keepsgrowing until Page time t p . Then, the surface γ pl dominates at t = t p and entanglementsaturates due to the phase transition.After Page time, the island solution dominates and the island I emerges in the entan-glement wedge of the radiation. Therefore, the d.o.f. on the island can be encoded in theradiation by quantum error correction process.By varying the endpoint x b , we illustrate different scenarios black hole for the d.o.f.encoded in the radiation in Fig. 9(a). In detail, the solution with x b = 0 measures theentanglement between the QMS at x = 0 and its complementary, namely the whole bathsystem. While for x b >
0, the island solutions measure the entanglement between QMS at x = 0 together with part of the bath system at 0 < x ≤ x b and its complementary, namely,the remaining bath system at x > x b .Firstly, we find that in most cases, the island will stretch out of the horizon ( y b > x b →
1, which indicates that the whole exterior is encoded in17 a) y b (b) Figure 8: (a): Two candidates of HRT surface colored in rose gold (trivial surface γ tr ) andblue (island surface γ pl ) are anchored at x = x b , while the half of island at t = 0 is colored inred. (b): For µ = 1 and x b = 1 / y b , where ˜ S ≈ . x b →
0, the boundary of islandapproaches y b ≈ . < y < y b )is likely to be encoded in the pure baths without QMS, while the region near the boundary(0 < y < y b | x b → ) is going to be encoded in a system that contains the QMS at x = 0.Moreover, the saturated entropy ˜ S /µ increases with Hawking temperature T h /µ , whichmanifests that the growth of temperature allows more entanglement to be built up betweenthe black hole and radiation system as shown in Fig. 9(b).In addition, the saturated entropy also increases with x b (Fig. 9(c)). As discussed above,the island solutions with x b > x b . As aresult, the more d.o.f. of the bath system we take into account (besides those of the QMS),the larger the entanglement entropy is. Notice that in the model with 2 d/ d duality [13], one requires 1 (cid:28) c (cid:28) Area / (4 G (2) N ) to ensurethat it works in the semiclassical limit in 2d as well as a large-radius dual in 3d, where c is thecentral charge of CFT . A similar relation in higher dimensions is L d − /G ( d ) N (cid:29) L d − /G ( d +1) N .18 .0 0.2 0.4 0.6 0.80.00.10.20.30.40.5 x b ∂ I ( y b ) (a) Saturated EntropyGeometric Entropy T h / μ S ˜ / μ (b) Saturated EntropyGeometric Entropy x b S ˜ / μ (c) Figure 9: (a) For µ = 1, the relation between the location of island boundary ∂ I and x b . (b)For x b = 1 / S /µ and Hawking temperature T h /µ is plotted as the solid curve, while the corresponding geometric entropy is plotted asthe dashed blue curve. (c) For µ = 1, the relation between the saturated entropy ˜ S /µ andthe endpoint x b is plotted as the solid curve, while the corresponding geometric entropy isplotted as the dashed blue curve.According to this relation, it is obvious that after Page time t ≥ t P , we have∆ S ( t ) = S ≈ ( d ) pl G ( d ) N ≈ S BH , (45)where S BH is the Bekenstein-Hawking entropy of the d -dimensional black hole. Nevertheless,as mentioned in Sec. 2.2, we find that (45) is only valid for θ →
0, where the geometricentropy contributes the most part in (21) and (44) .In the case of weak-tension limit, we find S < x = x b (or equiv-alently, W = W b ), which leads to the saturation of the entropy at the beginning. As thegrowth of the tension, the geometric entropy becomes significant. Specifically, in the caseof d = 3 and θ = π/
4, one has the relation
L/G (3) N = L /G (4) N . It turns out that the geo-metric entropy becomes significant in total S and hence can not be ignored as illustrated inFig. 9(b) and Fig. 9(c). Compared with the result in [18], we find that taking the geometricentropy into account will not change the sign of S for the case µ = 0 and θ = π/
4, but itwill increase the Page time considerably as well as the entropy after saturation due to thefinite tension on the brane indeed.
In this paper, we have investigated the black hole information paradox in the doubly-holographic setup for charged black holes in general dimensions. In this setup, the holo- We define the contribution of the geometric entropy as the proportion of Area /G (3) N in (44). W b → W b ,the evolution of the von Neumann entropy in the weak-tension limit still obeys the Page-likecurve. Specifically, for the neutral case in general ( d + 1) dimensions, the entropy growslinearly at the rate proportional to T d − h due to the evaporation and finally saturates at aconstant which depends on the size of the radiation system. In 4 D charged case, we haveplotted the Page-like curves under different temperatures. At high temperatures, the blackhole seems to “evaporate” more rapidly than the cold one and will build more entanglementwith the radiation system. While for the near-extremal black hole, the evaporating processseems to be frozen, since the exchanging of Hawking modes takes place extremely slowly.Moreover, the stationary solution with backreaction is obtained by the standard DeTurcktrick. For the entanglement entropy, the more d.o.f. of baths we take into account, the moreentanglement will be built up and the less region near the black hole will be encoded in theremaining baths . Similarly, with the increase of Hawking temperature, more entanglementwill be built up between the black hole and the radiation system. In addition, the contributionfrom the geometric entropy to the von Neumann entropy is also discussed. Compared to theresult obtained in [18], the Page time is postponed distinctly due to the sufficient geometricentropy.We have obtained the Page-like curve in the weak-tension limit in which the backreactionof the Planck brane to the bulk can be ignored. Notice that in these cases, the final saturatedentropy is not equal to two times of the Bekenstein-Hawking entropy as in the standardprocess. The reason is that in the weak-tension limit, the d.o.f. for the black hole system arefairly few and the entanglement is mainly from the matter fields. Hence, it is very desirable toinvestigate the time evolution of the entanglement entropy with backreaction, which involvesin the dynamics of black holes, thus beyond the DeTurck method. Furthermore, beyond thesimple model with doubly-holographic setup, the information paradox in high-dimensionalevaporating black holes is more complicated and difficult to describe. Therefore, it is quiteinteresting to develop new methods to explore the evaporation of black holes in holographicapproach. A similar result is obtained in literature [15]. cknowledgments We are grateful to Li Li, Chao Niu, Meng-He Wu, Rong-Xing Miao, Cheng-Yong Zhang,Qing-Hua Zhu for helpful discussions. This work is supported in part by the Natural ScienceFoundation of China under Grant No. 11875053 and 12035016. Z. Y. X. also acknowledgesthe support from the National Postdoctoral Program for Innovative Talents BX20180318,funded by China Postdoctoral Science Foundation.