Islands and Page curves of Reissner-Nordström black holes
PPrepared for submission to JHEP
Quantifying islands and Page curves ofReissner–Nordstr¨om black holes for resolvinginformation paradox
Xuanhua Wang, a, ∗ Ran Li, b,c, ∗ Jin Wang ,a,b a Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794, USA, b Department of Chemistry, State University of New York at Stony Brook, Stony Brook, NY 11794,USA c Department of Physics, Henan Normal University, Xinxiang 453007, China
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We apply the recently proposed quantum extremal surface construction tocalculate the Page curve of the eternal Reissner-Nordstr¨om black holes in four dimensions.Without the island, the entropy of Hawking radiation grows linearly with time, which re-sults in the information paradox for the eternal black holes. By extremizing the generalizedentropy that allows the contributions from the island, we find that the island extends out-side the horizon of the Reissner-Nordstr¨om black hole. When taking into the effect of theislands, it is shown that the entanglement entropy of Hawking radiation at late times fora given region far from the black hole horizon reproduces the Bekenstein-Hawking entropyof Reissner-Nordstr¨om black hole with an additional term representing the effect of thematter fields. The result is consistent with the finiteness of the entanglement entropy forthe radiation from an eternal black hole and resolves the information paradox for this case. * Equal contributions Corresponding author a r X i v : . [ h e p - t h ] J a n ontents The information issue of black holes is a fundamental problem in several most importantfields of physics–quantum mechanics, thermodynamics and the theory of general relativity,and is essential for our understanding of quantum gravity [1–4]. Recently, tremendousprogress has been made to provide a quantum description of the information conservationin the process of black hole evaporation [5–10]. This was done without the recourse to acomplete understanding of the quantum dynamics of black holes, which seems to necessarilyinvolve the understanding of quantum gravity.The origin of the information paradox dates back to decades ago. In 1975, Hawkingproposed that the information falling into the black hole would disappear after the evapo-ration of the black hole [1, 4]. However, this proposed process violates the unitarity, whichis one of the foundations of quantum mechanics. According to the principle of unitarity,the evaporation of a black hole from a pure state with zero entropy has to end with thepure-state quantum gas of radiation instead of mixed-state thermal gas which has a largeentropy. This argument incorporating the initial and final state behaviors is representedgraphically by the Page curve [2]. Many other proposals were suggested to resolve theinformation paradox. Some representative ones and their pros and cons were discussed inref. [11]. One approach suggested to include backreaction leading to final pure state. Butit appears to imply that either all of the information has been extracted by the time thefalling matter crosses the horizon or that information escapes acausally from the black hole[12]. Another proposal suggested information release at the end of black hole evaporationat the Planck scale. But this proposal requires the remaining Planck scale energy to carryoff arbitrarily amount of information which would violate the Bekenstein bound [13–15].A different proposal suggested the Planck scale remanent after the evaporation. But the– 1 –emanent is intrinsically unstable [16]. A proposal on including baby universes as a sourceof information loss was suggested. But later studies showed that wormholes only changethe coupling without violation of unitarity [17]. There was also a proposal on a previouslyunexpected mechanism of information release. But suggestion seems to require the viola-tion of causality in the horizon [18]. In 1999, Parikh and Wilczek proposed to address theinformation paradox issue by including the higher order non-thermal effect in the radiationto allow information to leak out from the black hole [19]. However, this effect is negligiblefor massive black holes and not able to compensate for the information loss in this case.Whether black hole dynamics preserve unitarity remained a conundrum until present.One of the breakthrough ideas was made by the discovery of AdS/CFT correspondence[20]. The duality is a mathematical realization of the proposed idea of the black hole com-plementarity [21], and provides a strong evidence for the conservation of information as theblack hole in the anti-de Sitter space (AdS) can be mapped to the boundary CFT. There-fore, the evaporation of the black hole has a dual unitary description using the boundaryCFT. If this argument is true, the evaporation of black holes should roughly follow thePage curve. However, the quest to obtain the Page curve remains unsuccessful until veryrecently. Apart from that, the unitary process was shown to generate a “firewall” (AMPSfirewall) on the black hole horizon which is at odds with the “no drama” principle of thegeneral relativity [21]. For eternal black hole, similar questions on the information paradoxcan be addressed. For a unitary evolution, the corresponding Page curve is expected toreach a bounded value which is the Bekenstein-Hawking entropy of the black hole. Theamount of radiation for a eternal black hole is infinite at the “end stage” or the late times ofthe evaporation. Thus, a thermal spectrum of radiation would produce an infinite amountof entropy. This is contradictory to the unitarity which dictates the maximal entropy pro-duced by the black hole to be the Bekenstein-Hawking entropy. A resolution to all theissues related to the black hole information paradox has been long-yearned.The Page curve of Hawking radiation was recently calculated by using the semi-classicalmethod for the two-dimensional black holes in the asymptotically AdS spacetime in theJackiw-Teitelboim (JT) gravity [6, 7]. Most of the studies on the the black hole informa-tion problem have been concentrated on the two dimensional gravity where the systemshave more symmetries to admit analytic solutions and are easier to analyse [22–29]. Inthe two dimensional systems, islands appear at the later stage of the black evaporation,which is in the entanglement wedge of the radiation, such that the Bekenstein bound of theentanglement entropy is preserved. For a review see ref. [7]. However, whether this islandconstruction can be extended to and resolve the information issue of all black hole solu-tions still remains to be verified. For higher dimensional or “realistic” black holes in fourdimensional asymptotic spacetime, the resolution of the information paradox is much lessstudied due to the difficulty in calculating the entanglement entropy and analysing the dualconformal field theory (CFT). It is argued in [9] that the islands should exist in the higherdimensional black hole spacetimes and the unitary Page curves can also be reproduced iftaking the island’s effect into account. Recently, some interesting phenomenological studiesof the island structure and the Page curves in four dimensional Schwarzschild and dilatonblack hole were performed in ref. [31, 34]. Some other studies of different models in higher– 2 –imensions can be found in refs. [29–38].Our present understanding of the entropy of quantum systems coupled to gravity doesnot necessarily requires holography and AdS space [7], it is nevertheless an essential tool inthe development of the entropy of gravitational systems. The groundbreaking work of Ryuand Takayanag (RT) using AdS/CFT correspondence connects the entanglement entropyof the boundary region to the area of the minimal surface in the bulk space [39]. Laterworks generalized the RT surface to the quantum extremal surface, in which the generalizedentropy includes all the quantum corrections of the bulk fields [9, 10, 40–43]. It is shownthat by applying the extremal surface technique, islands appear at the later stage of blackhole evaporation process, and that the entropy of Hawking radiation obeys the Page curveassuming the unitary [7]. Furthermore, the island formula for the fine-grained entropy ofthe Hawking radiation is proposed to be [10, 23, 43, 44] S ( R ) = min (cid:26) ext (cid:20) Area( ∂I )4 G N + S matter ( R ∪ I ) (cid:21)(cid:27) (1.1)where R is the radiation, I is the island, and S matter is the entropy of quantum fields. It isshown that the island formula can be derived without holography from the Euclidean pathintegral by using the gravitational replica method. The presence of replica wormholes asthe saddle points in the Euclidean path integral leads to the island formula not only forthe eternal black holes but also for the evaporating black holes [7, 45, 46].The black hole information paradox has been addressed mainly in two-dimensionalgravity models. It is equally important, if not more, to resolve the paradox in our realuniverse, which is four-dimensional and reaches Minkowski space asymptotically. In theclassical general relativity and cosmology, there are a few important 4D vacuum solutionsto Einstein’s theory of relativity that deserve particular attention. In this article, we willaddress the information paradox issue in the four dimensional charged black hole solution inthe asymptotic flat spacetime and study the island structure. In this study, we constructthe Page curve for the four dimensional eternal Reissner–Nordstr¨om black holes in theasymptotically flat spacetime and show that the entanglement island in this case saves theentropy of the radiation from exploding at the late times. This quantitatively resolves theinformation paradox for the Reissner–Nordstr¨om black hole.In this work, we will apply the method of quantum extremal surface to study theentropy of Hawking radiation and the corresponding Page curve of the Reissner–Nordstr¨omspacetime in four dimensions. The action is given by the Einstein-Maxwell action I = 116 πG N (cid:90) M d x √− g (cid:18) R − F µν F µν (cid:19) + I matter , (1.2)where G N is the Newton constant, and I matter is the action of the matter fields. If thematter fields are described by the CFT and in the vacuum states, the vacuum solution tothe Einstein-Maxwell action will not be affected by the matter fields. It is straightforwardto generalize the analysis to gravity with higher curvature terms, but we focus only on thedominant contributions. – 3 –he metric of the Reissner–Nordstr¨om black holes is given by ds = − (cid:18) − Mr + Q r (cid:19) dt + (cid:18) − Mr + Q r (cid:19) − dr + r d Ω , (1.3)where we have set the Newton’s constant and the Coulomb constant equal to 1, i.e. G N = K = 1. In the following, these physical constants can be easily restored if nec-essary. Reissner–Nordstr¨om spacetime is one of the most important vacuum solutionsof the Einstein’s field equation representing a charged black hole in the 4D asymptoticMinkowski space. One of the distinctions of the Reissner–Nordstr¨om black hole from theSchwarzschild black hole spacetime is the appearance of the two horizons (event horizonand causal horizon) even though the inner causal horizon is believed to be unstable undersmall perturbations due to the mass inflation phenomenon. The radius of the horizonsare given by r ± = M ± (cid:112) M − Q and the surface gravity at the horizons is given by κ ± = r ± − r ∓ r ± . The Hawking temperature of the Reissner–Nordstr¨om black hole is given by T RN = κ + π . (1.4)In the present work, we only consider the non-extremal black holes.This paper is arranged as follows. In section 2, we present an approximate methodto compute the entanglement entropy for quantum fields in four dimensions. In section3, the entropy of Hawking radiation is computed without islands and the correspondinginformation paradox for the eternal Reissner–Nordstr¨om black hole is sharpened. In section4, we analyze the generalized entropy of Hawking radiation and reproduce the unitary Pagecurve when taking the effect of islands into account. Based on these results, we also discussthe Page time and scrambling time for the Reissner–Nordstr¨om black holes. The discussionand conclusion are presented in the last section. In the following sections, we carry out the calculation of the entanglement entropy in thefour dimensional Reissner–Nordstr¨om geometry without/with involving the islands. Theentanglement entropy for a general four dimensional spacetime is not known. However,the Hawking radiation observed from a distant observer can be properly described by thes-wave approximation. Therefore, we can apply the analysis in the two dimensional caseto obtain the entanglement entropy in the curved four dimensional spacetime under someapproximations.For the one dimensional quantum many-body systems at critiality (i.e. CFT in two-dimensions), it is known that the entanglement entropy ignoring the UV-divergent part (orPlank scale physics) is given as follows, S A = c · log( Lπ(cid:15) sin( πlL )) (cid:39) c · log l , (2.1)where l and L are the lengths of the subsystem A and the total system, (cid:15) is the UV cutoff,and c is the central charge of the CFT. We have assumed that l (cid:28) L and kept only thefinite part. – 4 –s is shown by Ryu and Takayanagi, the entanglement entropy in the boundary (d+1)-dimensional CFT has a dual description in the bulk. It follows a simple area law whenmapped into the bulk, i.e. S A = A G d +2 N , (2.2)where A is the area of the the d-dimensional static minimal surface in the AdS d +2 . For thetwo-dimensional CFT, it is just the length of the minimal curve in the bulk. This formulais applied when no island is formed.For two dimensional systems with multiple disjoint intervals, A = { x | x ∈ [ r , s ] ∪ [ r , s ] ∪ ... ∪ [ r N , s N ] } , the generic formula for the entanglement entropy derived from theRyu-Takayanagi formula is given as S A = (cid:80) i,j L r j ,s i − (cid:80) i
Acknowledgments
R.L. would like to thank Hongbao Zhang and Yuxuan Liu for useful discussions.
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