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JHEP07(2020)022 Springer July 3, 2020 May 28, 2020 : June 15, 2020 : : Published Received RST model of dilaton b Accepted N , Published for SISSA by https://doi.org/10.1007/JHEP07(2020)022 and Andrew Strominger a [email protected] , limit of asymptotically ﬂat two-dimensional dilaton gravity cou- . N 3 Edgar Shaghoulian a 2004.13857 The Authors. c 2D Gravity, Black Holes, Models of Quantum Gravity

The large- , free matter fields provides a useful toy model for semiclassical black holes and [email protected] N Department of Physics, HarvardCambridge, University, MA, U.S.A. E-mail: [email protected] Department of Physics, CornellIthaca, University, New York, U.S.A. b a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: classical island rule for entropyentropy has for been proposed. black hole Wegravity define and formation and show compute and that, the in evaporation islandtwo contrast, observations, in rule it and the interesting follows properties the large- of unitary the Page dilaton curve. gravity island The rule, relation are of discussed. these Abstract: pled to the information paradox.entanglement entropy Analyses show of that it theis follows asymptotic destroyed the in Hawking information curve, these flux indicating models. that as information Recently, given motivated by by developments the in AdS/CFT, a semi- Thomas Hartman, Islands in asymptotically flat 2D gravity JHEP07(2020)022 ] for anti- 7 14 ] a remarkable proposal has been 11 6 – 2 4 21 ] showed that the entropy of a black hole can – 1 – 1 8 6 9 11 7 4 13 23 18 21 16 15 action and equation of motion 1 6 N 2.3.1 Entanglement entropy 3.2.1 When does the QES hit the singularity? 2.2.1 Linear2.2.2 dilaton vacuum Eternal black hole A.1 Replica geometries A.2 Derivation of the QES 3.3 Page curve 3.4 Scrambling time 3.1 Quantum extremal3.2 surface Evaporating black hole 2.1 Large- 2.2 Solutions 2.3 Evaporating black hole in RST made invoking semiclassical gravityThe starting to point analyze is the thedeSitter famous information space and well-understood (AdS), flow Ryu-Takayanagi which formula outentropies [ semiclassically of in computes black terms microscopic quantum of holes. formula entanglement areas define of an extremal ‘island rule surfaces. entropy’ for Recent time-dependent proposed evaporating generalizations black of holes this in AdS gral. That calculationthe is now quantum understood, microstate at degeneracyprise least that of in semiclassical the some gravity blacktervening cases, is hole. decades, to smart the correctly enough It unreasonable enumerate todemonstrated efficacy is reproduce in of a this disparate semiclassical wonderful degeneracy. contexts.or gravity and In has Most another deep the been cases — sur- in- however repeatedly such involve as symmetries time of translations. one kind Recently [ 1 Introduction Many years ago, Gibbonsbe and formally Hawking but simply [ calculated in the semiclassical limit from the Euclidean path inte- 4 The eternal black hole A Local analysis of replica wormholes 3 The island rule for an evaporating black hole Contents 1 Introduction 2 Review of the RST model JHEP07(2020)022 ]. ]). 10 – ] can 16 , 8 16 13 – However 13 1 ]. The first . 11 + I , the information flux, or limit of 2D dilaton gravity + ], and the bulk description of the I N 17 dilaton gravity models are not N fully consistent 2D toy model to some ]) analytically computed. The information 14 defined model of 2D gravity. In stringy limits, = 1 matrix model [ It remains possible that some modification of c – 2 – 2 exactly ], originated as a dimensional truncation of the near , suggest that the large- 12 N limit includes Hawking radiation as well as the backreaction N free matter fields. A review can be found in [ limit of an N N ] involves a tower of low-mass states. 19 , limit of any exact fully consistent 2D quantum theory of gravity. follows the Hawking curve, not the Page curve. The models therefore seem to 18 N + I It would certainly be of great interest to find Of special interest in this regard are the near extremal NS fivebranes, which contain In assessing this uncomfortable conclusion it is important to realize that the dilaton In a separate direction, the process of black hole evaporation has been systematically A further reason for doubting the existence of such theories is the exactBlack holes global probably flavor symmetry, cannot which be formed in the 1 2 CGHS plus additional massive fields which survive the scaling limit and are holographically is not expected inwhich quantum collapses gravity. to a It black is hole hard could to ever seeSYK be how model retrieved the [ from information the Hawking about radiation the at flavor of the matter study black hole formation/evaporation. CGHS — perhaps involvingfreedom wormholes, — exists boundary as modes an exact or quantum other theory additional and degrees obeys the of island rule descibed herein. fully succeeded. The apparent destructionknown of exact information, definition together at with finite thethe absence large of any the jury is still outthe concerning information the paradox. best perspective on these models and their relevance to models in their own rightsome without cases recourse fully to soluble a at decouplingHowever, the a limit. quantum faithful level The toy (as 2D model a bulk forboundary null theory black variant is of holes is needed Liouville requires in theory to the [ such dilaton eliminate a field regions to boundary where be condition it positive, and becomes and avoid a negative. naked singularities Efforts in to a implement self-consistent manner never gravity models, despite some efforts, weretheory never derived or either as as a the decoupling large- limittypically of an string infinite tower ofis interacting renormalizable fields (in do fact not equivalent decouple. to Since 2D 2D CFT) quantum one gravity might hope to define and study such on the metric,equivalently and entanglement yet entropy as is asome soluble. function models of retarded (in The time particular quantum canflux the be state at RST numerically on model or [ in destroy information. studied starting in theminimally early 90s coupled in to thesuch context of model, the known large- extremal as NS CGHS fivebrane in [ stringalso theory, be but the studied CGHSinterest as model because toy and the its 2D large- variants black [ hole models in their own right. These models are of The island rule entropyunitary elegantly information flux, reproduces accompanied at byprocess. a leading refined We order semiclassical hope picture the that of Pagea this the significant complete curve evaporation progress microscopic describing will resolution provide of important the clues information to paradox. guide us to — and even for those in flat space — involving a ‘quantum extremal surface’ or QES [ JHEP07(2020)022 + I due M for the evaporating black + I . The explicit expression as a function + I . It implicitly asserts that when an infalling + I – 3 – ]. While these theories are not fully two-dimensional 21 , bulk term with corrections due to the UV cutoff redshift ] but lives at 5 20 , M 12 . This assertion reproduces the Hayden-Preskill scrambling time. to hit the singularity a time of order one before the evaporation + + I I matter fields, they can be semiclassically studied at large black hole N . As they do have a holographic dual, it is plausible that the information flux at M This paper is organized as follows. Section 2 is a lightning review of the needed The island rule supplies a surprising canonical map from a time along the QES curve in More generally, the semiclassical island rule does not agree with the semiclassical In this paper we compare in detail in 2D these old and new semiclassical analyses . This surface magically has zero net quantum entanglement to leading order in + entropy of Hawking quantaexceeds outside the Bekenstein-Hawking the black hole black entropy.assuming This hole paradox the is grows island beautifully rule. without resolved by Inand bound the show and appendix that eventually we the consider localnot the equations construct gravitational of the replica motion wormhole calculation fix saddles the globally. location of the QES, although we will hole. Section 3asymptotic begins entanglement entropy with an island the regiontype definition which gravitational ends of contribution. at the This the formula QESand island is with shown rule then to an have applied area-law entropy, several to which remarkable the properties,the adds scrambling evaporating including time. black reproducing to hole the Section the Page 4 curvesimplified, considers and the if eternal black less hole, physically which has sharp, a version mathematically of the information paradox: the entanglement at its boundaries. This provides another self-consistency check offeatures the of island 2D rule. asymptotically flateternal dilaton black gravity and hole themicroscopic and RST entanglement entropy model. evaporating and the The black information linear hole flux dilaton, at solutions are presented, followed by the The QES is seenendpoint. at The island rulethe entropy then ordinary has quantum no entanglement classicalI of gravitational piece, the and surface reducesto extending to a from cancellation the of QES the endpoint order to is that the 2D context considered here proves athe useful black one hole interior for to resolving aobject this retarded crosses tension. time the on horizon andbecomes reaches the available QES on curve, information about its quantum state information flux ofdifference the and RST try model to learn which something follows from the it. Hawking Hawking calculation curve. for 4D black Weof holes analyze validity. even If though the the they former apparently have is the correct, same one regime would like to find the error in the latter. Our hope of black hole evaporation.entropy in We the analyze RST the model,mimics employing QES the a and ones definition associated of usedof the semiclassical in retarded island island time AdS rule is rule [ entropyconsistency shown which to check closely of reproduce the the island leading-order Page rule. curve, This which of is a course stringent disagrees with the actual measurable and do not have mass follows the Page curve, andthe the information island flux rule measured entropy byderive as an this described asymptotic explicitly. in observer. this It paper would agrees be with very interesting to dual to little string theory [ JHEP07(2020)022 and , (2.3) (2.4) (2.5) (2.6) (2.1) (2.2) 1 x CFT minimally ], largely S ) broken by 14 φ N 0 + ) distinguishes x − = ρ 2.1 ρ . c ( CFT = − µ 3 S N ∂ ∂ + + ln x ∂ 1 4 ) term in ( φ + φR − φR 2 φ ρ N 24 , with ( − − − 6 N ρ R. ) dx 1 2 + + + . − λ φ 2 dx ρ 2 ρ − + 4 CFT 2 e e gR e S 2 effective action becomes ) ]. − − N 12 φ + + 2 N √ ∇ 45 φ = = x , . . 2 − 2 2 d ) 44 anom k φ∂ + 4( = 1 S ds f Z + – 4 – λ R + ∇ ∂ π held fixed. It described by the sum ( 3 ( , χ 8 cl φ N g φ 2 96 S 2 − − N 48 − − ρ e = √ ln Ne − x k = 1 4 ∂ g 2 eff f ]. Indeed for our primary focus of the large mass limit there is no d + S − − with any family of 2D CFTs with central charge − ] to which we refer the reader for further details. For earlier ∂ ∂ ] and was added to restore the conserved current 46 √ 4 with Z k 2 anom φ 43 x 12 f S 2 π CFT φ 1 + + 4 d 2 S φ − x∂ 2 e → ∞ Z =1 N 2 − k X d e π 1 N x 2 − 2 Z N 12 d = = . In these coordinates the large- action and equation of motion =1 N ρ k Z X cl − Ω = S ]. π π N ∂ limit is 1 1 ]. As this work was nearing completion two papers appeared with overlapping CFT 2 2 + S 42 40 ∂ N , – = = 2 41 − 22 eff S = CFT Other recent work addressing islands and information recovery in black holes in- S More generally, replacing − 3 + coupled to gravity would lead tothe few RST changes from in the thethe following CGHS discussion. anomaly. model The The [ existenceaffect of the this basic conserved structure,difference. current see conveniently simplifies e.g. many [ formulae but does not It is convenient to define This action simplifies inR conformal gauge The large- To avoid equation clutter we henceforth choose units in which In the quantum effective action, the conformal anomaly contributes a term following the conventions of [ results on black holes in two dimensions see2.1 [ Large- The classical action of the RST model is cludes [ results [ 2 Review of theThis RST section model contains a lightning review of relevant features of the RST model [ JHEP07(2020)022 ) 2.7 g ) as (2.9) (2.7) (2.8) (2.10) (2.11) (2.12) = Ω + 2.8 χ ]. 48 ) implies , , so 47 are held fixed. 2.9 2Ω − χ quantum matter .( χ 2 K e N ρ CFT ) marks the origin of − S ) with respect to = 2.8 + and 2.1 χ K − 2Ω ∂ χ − + χ ∂ 2 e ], but it has no singularities and . + 16 Ω = limit where Ω and , , ln2) χ − − ∂ 13 N , but certainly spoils the full quantum t − . + 1 N χ∂ − ∂ K is an open question, studied in [ − + ρ . . ( ∂ N 1 4 N 12 f 1 2 T − ln ≥ = Ω Ω = + − 1 2 – 5 – Ω − χ Ω in Kruskal gauge as K − ∂ − ρ ∂ ρ in our large- 2 + and Ω are Ω = Ω K ∂ − ρ e 2 χ + N ∂ ∂ and = χ x φ Ω = 2 d 4 curve turns from spacelike to timelike. This cuts off the / Z π , Ω obeys N as a theory of gravity with black holes, it is essential that the restric- 12 φ eff = S eff Furthermore, without this restriction the collapsing matter configurations ). The residual on-shell conformal diffeomorphism symmetry of conformal S − 4 x be enforced so that the coefficient of the Einstein action is positive. When 2D equations of motion for ( We further impose reflecting boundary conditions on the − φ eff α 5 ]. This condition looks sensible at large S which descends from their reflection through the origin in higher dimensions. k ) + 43 f , + The Note that for real x Some versions of JT gravity have this feature andWhether hence any may such need constraint supplemental restrictions makes to sense provide at finite 14 ( 4 5 + before fixing to conformal gauge: good models for black hole physics. The equation of motion in this gauge is simply There are also constraint equations, which are given by varying ( denote the non-scalar quantities α gauge allows us to choose coordinates in which For reasons which will become apparent we refer to these as Kruskal coordinates and will the point where theevaporation. Ω In = order 1 in [ to have asolubility. theory with blackfields holes, we therefore impose ( the spacetime where thedescribes spheres shrink a soluble, to null zero. variantno If of black we Liouville holes. don’t theory make [ described such below a will restriction, radiate ( foreverwe to can negative motivate infinite a mass, pasting whereas of with the the restriction linear dilaton vacuum at the evaporation endpoint, i.e. In order to identify tion of real dilaton gravity is constructedgravity, as Ω is a the spherical (quantum) area dimensional of reduction the of spheres, and higher-dimensional the condition ( then scales with an overall factor of The resulting action JHEP07(2020)022 ) 2.9 (2.16) (2.14) (2.13) )instead depends + x t ( + 0 x 0, as illustrated in > x , + 0 x and + k ) (2.15) f ∂ 1 4 − x q ∂ ≤ + k 2 + 0 f x − . for notational convenience. 4 x x∂ − π − + + − x∂ 0 N x 12 2 dx x ∂ ln( ]) d + + 1 4 x p 43 Z dx = 0. The physical region of the spacetime with − + – 6 – =1 = N − + k X f t x 2 2 T + 6 ) N ds x + ) can be obtained by integration, although we will only x − = + 0 ∂ f 2.12 Ω = T . In the black hole collapse studied below the ambiguity will )–( = ( − t 0 + t 2.11 ), this implies the 2D Minkowski line element 2.9 shifts by the Schwarzian (see e.g. [ . Linear dilaton vacuum. The physical spacetime is the unshaded region, with a boundary . + . Using ( 1 4 is covered by the coordinate region with t 1 , 1 4 + x ≥ with the classical matter stressΩ tensor figure need the following special cases. 2.2.1 Linear dilatonThe vacuum vacuum of the theory, known asis the given linear by dilaton vacuum, in Kruskal coordinates ( be uniquely fixed by demandingfar the past. absence of incoming particles in inertial2.2 frames in the Solutions The general solution to ( on the choice ofof coordinates prescribed for normal ordering. If we choose with a similar relation for where is the matter stress tensor rescaled by a factor of Figure 1 at Ω = JHEP07(2020)022 1 − W (2.24) (2.22) (2.23) (2.17) (2.18) (2.19) (2.20) (2.21) ) hence 2.15 . Using the − ), where σ 4Ω + − e + − σ ( 1 − 4 is spacelike we choose W / 1 2 = 0. The conformal factor is − , x + x N 12 , . ) is − ln M, . 2 . ) = 2Ω + ln2 + 7 1 4 σ − dσ + . 6 1 K + x π + − 1 e ρ NM 2 σ x ) = 4( dσ + one finds the black hole entropy + = 0 in the Minkowski frame and ( = = − − − x – 7 – = σ π ( − T − = = t BH σ + 12 NM t 2 x S in the standard Minkowski vacuum. t = ds 2 1 Ω = k f E = , we conclude that 2 φ ) and 1 x 1 T 4( ) we find that = ). The standard Minkowski vacuum is annihilated by Minkowski + Ω = σ BH 2.14 2 ∂E ∂S ∂ − ) in Kruskal coordinates as − = 0. To make sure the singularity at Ω = 1 . Normal ordering in these coordinates leads to an energy observed by σ 2.6 − ( 1 2 t dσ + + = χ dσ 4. The event horizon of the black hole is at f / − 1 T = 2 Using the first law This arises from outgoing thermal radiation flux at temperature with M > obtained from ( is the product logarithmds function. In Minkowskiasympotically coordinates, inertial observers, the metric near infinity is The solution for the eternal black hole (see figure modes which are negativetransformation frequency with law respect ( tocorresponds Minkowski to time the quantum fields 2.2.2 Eternal black hole the dilaton is linear and Ω = in which the line element takes the standard flat form We will also employ “Minkowski coordinates” defined by From the fact that JHEP07(2020)022 (2.28) (2.25) (2.26) (2.27) is on the is the event AH EP . 1) . ), and the conformal 2 ) − 2.9 . 1 + x x 1) which is the outer boundary 4( 6 − − 1)Θ( M = 4 turns spacelike, and is interpreted − e ( 1 4 Ω = 0, + x + 1 4 1 M ( ∂ . The dotted line prior to 4 , t − M = EP = 0 − ) = ) + EP f − M −− x – 8 – + + , x x − in Kruskal coordinates ( ,T 1 4 x 2 ( 1) − − with + − x M M . There is an incoming matter shockwave taken to be ln( 4 EP 1 4 + 4 − x P e ( − − = Mδ x + − EP = ), the matter stress tensor is x x − f ++ 2.12 T = 0 in Minkowski coordinates) that creates the black hole. Using the is the edge of the spacetime. This curve is initially timelike, as in the Ω = also defines the event horizon. + 1 4 σ − x 1. The evaporation ends where the apparent horizon meets the singularity. This . Evaporating black hole in Kruskal coordinates. The apparent horizon > = 1 (or + The solution has an apparent horizon defined by + x Interpreting Ω as the generalized quantum area, this is the boundary of the region of trapped spheres. x 6 This value of for occurs at the endpoint denoted as the black holeit singularity. becomes After a the timelike black boundary hole again. completely evaporates atof the the evaporating endpoint black hole. This is the curve The curve Ω = linear dilaton vacuum; here weWhen impose the reflecting shockwave boundary hits the conditions boundary, on the the curve matter Ω fields. = The spacetime is illustrateddiagram in appears figure below inat figure constraint equation ( A beautiful feature ofhole can the be RST described model exactlyendpoint is and of analytically the that in evaporation the the process. formation semiclassical The limit, and corresponding at evaporation solution least of is up to a the black Figure 2 dashed line and the evaporationhorizon. endpoint is marked 2.3 Evaporating black hole in RST JHEP07(2020)022 1 4 − dσ + (2.35) (2.29) (2.30) (2.31) (2.32) (2.33) (2.34) with an dσ − M = 2 free scalars on ds N which returns the EP P . , . , 2 , ··· ) . To leading order the evapo- − ) 1 + . However, the details of this do + ˜ σ . + M x − σ 4 ( ADM − EP e M − − x e e − . M ln ( ADM π O . = − EP O − N x 12 + πM σ M M ) 1 + = 4 e + = 2 M ∼ defined in terms of Kruskal coordinates by − − ). ++ + – 9 – ˜ σ 1 = T 6 1 d − + σ NM − x + 2.22 , x . In particular we wish to understand how much x ( ˜ σ M dσ + M = + d ˜ σ 4 EP in ( x e e − Z P 1 4 BH − = = = S = = 2 + = ln x M + K ds t ρ − EP , the singularity emerges from behind the horizon, the radiation 2 ˜ σ e EP P matches the time it takes to radiate away a total mass takes the manifestly flat form ) we note that the endpoint occurs at the (affine) retarded time + M I 2.28 is the canonical ADM mass. The initial black hole entropy is limit includes information about the quantum state of the , the geometry is in the linear dilaton vacuum, so the metric is − N I ADM , the conformal factor in Kruskal coordinates is + M to the future light cone of I Near After the endpoint . This is a pure state on any everywhere spacelike or null slice which stretches from -independent energy flux + 2.3.1 Entanglement entropy The large I spatial infinity to the timelike portion of the singularity — the ‘origin’ — where Ω = where the correctionsration are time suppressed of by 4 M powers of the metric near Comparing to ( In the outgoing Minkowski coordinates ˜ Near The (rescaled) initial mass ofwave. the In black Minkowski hole coordinates is equal to the energy of the incoming shock- where information is returned before the retarded time in the incoming Minkowski coordinates defined by continues and, problematically, theRST mass propose a of new, the modifiedsystem “thunderbolt” spacetime to boundary becomes the condition linear arbitrarily at dilatonnot negative. vacuum, concern matching us along hereprior because all of our considerations address the behavior of the theory JHEP07(2020)022 . O A P used . (2.38) (2.37) (2.36) − I σ of such a slice is A ] 43 . for the image point obtained ¯ + + O σ σ uv − . Such slices are all prior to the k + O f σ A ρ is at spatial infinity, the other at an . ln A A ) + ln ln4 ρ is the value of − O − tr + P , σ ), which obeys σ − − O + O σ 3 σ – 10 – ( = ) = in ¯ A + ρ O ( σ ), where ent 6 + P N S , σ (see figure + P ) = is the incoming coordinate of the null line which reflects off σ O ] when one end of A P ¯ ( + O ) and the quantum fields are in their vacuum state on 43 σ − O ent S , σ + O σ obtained by tracing over the quantum state on the slice outside of A = ( ρ O P is the conformal factor evaluated in the incoming Minkowski coordinates . Points are labeled by ( in ρ Here to define the vacuum, and the boundary and arrives at can be computed exactlyinterior point [ will often be taken to be the location of an Observer. The result is [ and reflecting boundary conditionsfuture are light imposed cone on of the a the density endpoint. matrix TheThe quantum entanglement state entropy on a subregion Figure 3 by reflecting across the timelike boundary. JHEP07(2020)022 will (3.1) (2.40) (2.39) A , it is the bosons. They + I by the central N with that after N O P . ) is the entropy of the ¯ + O σ 2.37 . uv − ) becomes + O )] appears because one wants the σ R in ln ρ 2.37 ∪ . , precisely as expected for a thermal 6 I N which involves an extremization of a ) with quantum mechanical unitarity. . In terms of the outgoing retarded − O ( ˜ ) result. This is clearly impossible in the RST σ 3 uv O gen N ) + NM N 12 P 2.40 S − O is chosen to be a point on [ ˜ σ Q ), indicating the evaporation process is not O ) = P P , equation ( Me − EP O 2.31 – 11 – σ P limit we need a sequence of theories labeled by (˜ . At the evaporation endpoint the net entropy in π 1 c 2 ent ln(1 + S )) of = ∆ N 12 ) = min ext T O 2.33 P ) = ( of the quantum extremal surface. The formula for the island I − O Q S σ (˜ (eq. ( P − O ent Hence we seek some kind of modified semiclassical theory with a σ S , the second term contains both IR and UV divergences. These are 7 + ] on the spacelike slice. The factor of I 4 .) A more involved formula for more general choices of the region – P c 2 on ) in fact applies to any CFT with the simple replacement of is chosen to be a point on the apparent horizon, ( scalars at temperature → ∞ 2.37 . (Of course to take a large- O c + O N P σ For the usual reasons, it is hard to reconcile ( If Eq. ( is a UV regulator required by the divergent entanglement of short-wavelength modes In general there is a concern (or hope) that off-diagonal terms in the Hawking radiation density matrix . This entanglement entropy is what is physically measured by an inertial asymptotic 7 O uv could conspire to correct themodel leading — semiclassical (or which large- isare by strictly definition decoupled on and aof therefore fixed the the topology density flavor — matrix. information due cannot to be the encoded flavor symmetry in of the the off-diagonal elements 3.1 Quantum extremalThe surface island rule associatesfunction an over entropy the to location arule point entropy is [ It appears that,destroys to information. the extentdifferent that entropy it flux. is This brings well-defined, us the to RST the next model3 section. (and its cousins) The island rule for an evaporating black hole This is exactly twiceadiabatic. the initial entropy ( independent of the retardedHawking flux. time At of late the timesflux the observer of entropy grows and as sothe are Hawking not radiation, associated which is to the outgoing finite change in the entanglement entropy, is Minkowski coordinate ˜ Since be introduced below when it is needed in thequantum definition fields of outside the islandentanglement the rule entropy black entropy. of hole. theP If portion of thedetector that outgoing cannot state detect prior modes to shorter than across the point UV regulator to have the same fixed proper lengthcharge at any slice endpointarbitrarily in large the spacetime. JHEP07(2020)022 to ] (3.2) (3.3) B . The 43 gen S is a complete R ∪ B , ∪ ). We also use I uv 4 ) is a natural proposed ln . 3.1 is an “island” region which 6 collects the radiation in region N ) is the region from spacelike when computing I ) will cancel against the UV R O Q R P ∪ 3.3 P 1) + I ( − ) is the quantum extremal surface. , and ent S Q 3.1 defined in ( (ln4 P I term in ( ) + N S 24 Q uv + P ( – 12 – K 2 grav ρ S The ln − . We restrict the points so that 8 ) = K O ρ P R 2 is a Cauchy slice. The endpoint of the island is the quantum − ∪ e R I ). and ( − ∪ 6 Q σ N gen B P − S = ∪ + σ I ( 1 2 grav is the Bekenstein-Hawking entropy. In the RST model this is [ S + . The dashed lines are the apparent horizon and the curve of extremal surfaces. , and K Q grav I ρ P S through which Hawking radiation passes (see figure = O in ρ P is in Kruskal coordinates. which extremizes the right hand side of ( . Island in the evaporating black hole. The Observer at K Q ρ P can consist of several points in general, but in this section We note 8 . The island is Q point If there are multiple extremawhich the minimizes generalized the entropy generalized isgeneralization entropy. computed to by the The picking asymptotically the flat solution 2D context of formulae which can be derived in where divergence of the matter entropy coming from the point denote the region between everywhere spacelike or null Cauchy surface. The generalized entropy is The first term P extends from the timelikeinfinity origin to to a single point Figure 4 R extremal surface JHEP07(2020)022 (3.9) (3.6) (3.7) (3.8) (3.4) (3.5) ) and ]. We . The ), since Q 50 3.5 P 3.5 ) to ( ) 3.3 ], so this formula ¯ + O 43 # σ ) ) ¯ − Q O P ¯ . + Q ( ,P σ in remain. However they do ) Q ρ . + ) O P − O ) ( O σ ¯ + Q P d P Q σ ( ) P − . ( ¯ − ) + ln( . in O ) ρ e + ¯ in Q + Q ρ M −   σ ,P σ 1, this can be approximated by ) − + − ¯ + O )( O Q M e − Q σ − ) P ¯ x + P O ( M ¯ ( O 4( + σ Q ) ) in d e ¯ ρ σ Me ] Q O ) , ( ,P − 4 − Q − = 4 Q ,P ,P 1 2 + 1 4 e 49 O Q x ¯ ¯ ¯ + Q O Q + Q − σ P P P ,P σ σ 2 uv ]. (1) radius) to avoid subtleties with the zero from the point ( + ( ( = 1 ( O 2 − d d O ) ) ) = 1 4 e = 0 are d − + O − O P ) is given by a formula which is more complicated q ¯ uv 49 – 13 – O O ( , σ x 2 uv M 2 B ln − ,P ,P uv ( d gen 43 ) Q Q ) = 6 + N S " P P ¯ ) the Minkowski distance. When the conformal cross Q + O ent ( (   + Q − Q d d − P P σ S ln σ x σ ln ∂ ( ) = − 6 Ω( N 6 + Q − N R ¯ = ) = + Q 0 x σ − ∪ P 6 R + N ( ) = σ I gen ∪ ( R )( S M 0 I ¯ + ) = Q ( ∪ + ent P σ does not terminate at spatial infinity. Assuming the scalars are R σ S I ∂ ent ( ∪ − S B I ent + P ( one finds S σ ( gen + S ≡ ) 0 → I ) because O ) is a physical observable. This would be a major achievement. P P,P ( 2.37 2 3.1 d This is the CFT entropy in the vacuum state. To describe a shockwave we may use a The matter term In Kruskal coordinates, Note that the UVnot and affect IR divergences the ln extremization.extremality conditions In fact only the first line depends on the point In this subsection we computehole. the We island will rule focus entropyit on for the can the regime case be of where checkedsending an the that evaporating entanglement the black entropy is this given is by always ( valid for the island. Adding ( mode that would modify this formula [ coherent state of thealso matter applies fields to which the does evaporating not black affect hole. 3.2 the entropy [ Evaporating black hole In fact, this latterare equation assuming applies the to scalar any compact is CFT compact by (with an OPE argument [ with ratio of the four endpoints satisfies the standard formulation ofthat the ( RST model. One may seek tothan modify the ( RST modelcompactified so at the self-dual radius, one finds [ some AdS contexts, but it has not been identified as an asymptotic observable quantity in JHEP07(2020)022 ) ) to . 2.35 (3.12) (3.13) (3.14) (3.11) grav M S , namely 1 4 , and the event 4 1 . − k ) becomes a function ;. (3.10) at retarded time ) = ) ) 3.1 − Q − Q + M x x I + + + 1 1 ) in ( − ··· O x M M P ( ( . Comparing to expression ( 4( 4( + I 2 + x , M S gravitational contribution ( 4 = = − , . − M Q + Q + Q e 4 M σ ln3 ln3 − ) hits the singularity Ω = 2 e M − − ) between QES time and retarded time and 3.8 2 M , x , x − 1 + 3.9 ln – 14 – − = − − O − O − = that the island rule entropy vanishes when the QES M M x x 4 4 σ 4 4 M e e . The explicit expression however involves the product ∆˜ M 4 − QE − O = 0 . Therefore the QES sits inside the event horizon, x 0 1 ≈ σ − W M W 4 + M − OQE 4 ˜ M σ − = . Using the map ( x = − Q − EP x − Q x x ) explicitly associates a retarded time to the QES inside the black hole. 3.9 to be large, the collision ‘appears’ to an observer on M ) is the product logarithm function with branch cut indicated by One may check directly at order Recall that the apparent horizon is on the curve Because all time foliations are on equal footing in general relativity, one generically Once we have found the QES, and solved for x ( k The conclusion is that evenbefore for the large endpoint. mass, thethis We breakdown will do occurs not not only prevent a know us time how from of to reproducing order extend the one Page thehits curve the analysis to singularity. beyond leading Interestingly, this there order is time, at no large but order where the corrections arewe suppressed see by that powers the of collison appears (for large mass) before the endpoint by an amount which is greater than taking Eventually the QES always collideshow with to the define singularity. the After islandthe this rule, QES point, and curve we the given do calculation by not breaks the know down. first The equation collision in occurs ( where point in the interiorexpectation, — especially ( if theSuch interior an is association inside might the seemthe required black black in hole. hole. any Contrary mechanism which to pulls this information out3.2.1 of When does the QES hit the singularity? horizon is slightly before and lies on a curve obtained by reflecting theexpects apparent that horizon across there the is event no horizon. canonical way to associate a boundary or retarded time with a W only of the retarded observer timelogarithm ˜ function and is unilluminating. A simple formula for large mass is given below. are two solutions depending on the chosen branch cut: A solution to these equations defines a quantum extremal surface (QES). In this case there JHEP07(2020)022 . ). ), is λ − I 3.10 2.39 (3.15) (3.16) (3.17) = 1 = negative uv back to B factors would not factors at the two is obtained by first ) ) ) in equation ( σ − σ O , dropping subleading ( − O ( ). σ 6 in 6 where the QES is seen σ in M (˜ Nρ Nρ + . This provides a stringent O 3.10 ent uv σ ) . S − OQE M R σ ln − O ˜ σ )), each of which give a factor ∪ 6 N − I e does not change the entanglement ( would have become exponentially 3.6 + 1 3 B − gen ≈ − O I ˜ σ ,S M ) − ]: it grows linearly until an O(1) fraction of − O + σ − EP also has zero entanglement entropy since it is − Q 52 (˜ ). This projected region is a narrow sliver of ˜ , σ R , ent 51 – 15 – 3.6 S ∪ − O σ B 2˜ , x we can solve to leading order for the location of the ), the entropy as a function of ˜ − ∪ O ˜ σ I e N M 24 3.1 4 3 ) = min ) term in ( ≈ − O ¯ + O σ Q + (˜ σ = min I x . The entanglement of the projection of region I S − of order ). One finds B S to the entanglement entropy. This is possible because we have ¯ (the third and last factors in ( + Q − O σ 3.9 σ B 3 . NM , due to the tremendous blueshift. Therefore this term gives a 2 is the evaporation endpoint. M M − 4 ) and ( M − 4 e 3.8 large and ˜ ≈ or equivalently M − EP , has the form anticipated by Page [ σ . R 5 ∪ The change in the entropy (which subtracts the IR and UV divergences), plotted in For According to the island rule ( Of course the combined region 3 NM I where ˜ figure QES using ( Plugging into the island rule one finds Therefore the island rule entropy is where the second term is evaluated at the extremum given by ( finding all surfaces obeying theminimal quantum extremality entropy. condition, One then choice choosing isin the the one this trivial with case surface, simply with equalswhich no the island. follows quantum The entanglement the entropy island Hawking rule entropy curve. For a single island, the only other choice is ( consistency test of the island rule. 3.3 Page curve In this subsection we derive the Pageterms curve of to leading order order ln at large a Cauchy slice. It isentropy. curious This that removing is the a region to very special hit the property of singularity.have the changed Had exact much, we point but waitedsmaller, the a resulting in projected time a interval of negative entanglement on order entropy of one order the subtracted the cutoff dependence, whichThe is (subtracted) operationally entanglement equivalent of to aHowever taking region this smaller negative than contribution theends is cutoff of exactly then becomes cancelled the negative. by interval of the is small. Therefore theof entire expression can be interpretedrepresented as by CFT the entanglement ln( entropy affine length contribution of the island rule entropy, because the end of the island is on the singularity where the area JHEP07(2020)022 ]. − O x 3 , 2 (3.18) , when 3 M . 4 M = P t and N 1). be the retarded time − OQ M σ ) for large in the radiation. Therefore the M δS y . ln( − ∼ t S N = ln − π β ˜ σ 2 . After the Page time, an object is dropped into at the canonically associated retarded time ∼ – 16 – 6 y, + scr I t + t = + ˜ σ ]), namely 55 is available at . Page curve (assuming a large initial mass , − Q 54 x ) to the object crossing the QES curve and entering the island. Then be the retarded time when the object reaches a few Schwarzschild radii − 0 Figure 5 3.9 σ of the black hole produces an entropy +2 ] (see also [ δS − . Let ˜ 53 + ). The dropped information hence becomes available when it hits the QES curve. I The experiment is illustrated in figure 3.9 associated via ( The observer should beeffectively stationary for far the whole from experiment.along the The observer black collects the hole, Hawking radiation from but the close black enough hole so (the precise the definition black is hole given is below) and ˜ the black hole by a distant observer, who is at rest in the inertial coordinates near infinity, reappear in the Hawking radiation.in The the island island rule at effectively time in asserts that ( the information The main difference in asymptoticallytrack flat of space the compared initial to timePreskill AdS in [ is the that scrambling we experiment. need The to result keep agrees with Hayden and 3.4 Scrambling time We now compute the scrambling timeThe associated scrambling to the time island is rule entropy the following [ time it takes for information dropped into a black hole to the entropy starts to decrease, isThis at is the point because where black 1/3rddecrease of hole the evaporation black is hole hasblack non-adiabatic, evaporated. hole and entropy is in equal to the the RST radiation entropy model, after 1/3rd entropy of the black hole evaporates. the black hole has evaporated, then decreases back to zero. The Page time JHEP07(2020)022 (3.19) (3.20) (3.21) to be large so A to return from the fall t . In fact that difference equals − OQ σ and ˜ , H − . AH . . σ y )Ω − 0 (1) constant. We want fall ˜ , then it is scrambled, then the Hawking σ A t − O − fall t + 2 fall t starts the timer to define the scrambling time. (1 + − OQ scr σ + t – 17 – ≡ + 2 I 0 is an 0 = ˜ = Ω obj t . Scrambling experiment. t scr t ∆ . A > = Ω 9 − 0 to be the retarded time when the object reaches the curve σ − 0 ] for discussion. Figure 6 σ 56 , 4 – 2 ) upon using ˜ 3.19 . Thus we define ˜ is Ω at the horizon, and M H . This is when the observer along 0 or Now we need to define ‘near’ the black hole. In higher dimensions this would be a region Note that light rays emitted by the object when it crosses the apparent horizon hover near the black hole N 9 ln3 and is negative! See [ to Ω = Ω for a long timetime and suffer with a the great difference− delay between before this arriving retarded at crossing infinity. time Hence ˜ we cannot equate the scrambling to define the near region as where Ω that we can neglect black-hole-related redshifts, but to be parametrically small compared Since the black hole is effectively stationary,time we implies can see ( that this definition of the scrambling within a few Schwarzschild radii. In two dimensions, the coordinate-invariant analogue is the object falls toradiation near carrying the information black about hole the innear object time region takes the to same the time observer.the object Thus the to total seeing proper it return time in elapsed the at Hawking infinity, from radiation, dropping should be the scrambling time is defined by To see that this definition makes sense, note that the experiment has three stages: first JHEP07(2020)022 ], on 54 (4.1) O , (3.23) (3.24) (3.25) (3.26) (3.22) P 53 , so we have BH is its reflection N S 6 R ], the eternal black on the left to 57 and , O . + . We can treat the black P 56 I , ) , i.e. Ω = ]. ) + obj − O is O x 43 . This agrees with [ 56 x x M (¯ π − = − . K − . ρ at + dx is near = − O − = 2 x ) 0 + .) Both intervals are of equal size, x BH . − O R N S 7 )(¯ β x x dx , 6 ( BH . ! + O K K ¯ N x ρ ρ 2 BH AS + obj BH N − ) + 3 e + obj − 8 S x e − 4 + O M , where AS ln 2 uv to enter the island at the retarded time Nx x ]. 6 = R π ( + 2 β + 2 2

∪ = ln – 18 – − I + ln x ds ln = R ( − 0 − OQ 6 σ + N = ln scr ˜ σ x t = ˜ − 0 . The entanglement entropy of a conformal field theory ˜ σ ) = + . The horizon is at ≈ − − OQ I R ˜ σ M Ω ∪ = R ( BH N ent S 6 S ] in the spacetime O ), the object is seen at correction also found in [ , x . The object reaches the curve Ω = Ω O N BH x N 3.16 S 6 ) A ) term is subleading, so for the scrambling time we find to leading order A Consider the entropy in the region We take the object to fall on a null trajectory with = (1 + 0 complement, which is athe single right, interval with extending both from pointson the near an point interval [¯ Our discussion here follows a closely related AdS versionon in the [ other sideand of as the we Penrose make diagram. thembe bigger (See seen the figure as entanglement follows. entropy increases Since without the bound. global This quantum can state is pure we consider the entropy on the havior of the quantum extremal surfacesby in those the in evaporating black the hole eternalhole is black well-approximated furnishes hole. a Moreover, simple asin discussed version the in of Hartle-Hawking [ the vacuum informationHawking seems radiation, paradox: to despite have at the an fact late that arbitrarily times, it large the has entanglement black a with fixed, hole the finite Bekenstein-Hawking entropy. 4 The eternal blackIn hole this section weaddition apply to being the mathematically island simpler than rule the evaporating to black the hole, the eternal late time black be- hole in the RST model. In The ln(8 where the inverse temperatureincluding the of ln the black hole is Therefore According to ( hole as stationary, with where as we have seen Ω JHEP07(2020)022 ) is 4.3 B O (4.2) (4.3) (4.4) (4.5) , y 1 , where O B , t 1 that this be less , the endpoints of ∪ + 10 B I M ]. ) , to be the region from the O 57 , P uv − O ( B 43 x , K . ln ρ + O − )) x 6 uv N e B − ( uv − ln ent 3 O N S 6 ln − Ny + = 0, taking 6 O N t t + , expand in the regime grav 3 + N O S 6 y H – 19 – ≈ NM 2 ( Ω : ) O R . ≈ ≈ R t ) ) ∪ (1). Considering the interval of interest in the black = R B O ( R O ∪ ( σ ent R ent S ( ) could only be measured by metaobservers who access both asymptotic S S + 4.2 grav . Then S O y 1, and drop terms of . Island in the eternal black hole. With the radiation collected on 1 uv The entanglement entropy ( M, 10 regions of the eternal geometry.analogy Hence to, the the paradox one here encountered is in less the physical previous than, section though or a discussed close in mathematical [ plus bath, with ais factor time-independent, of so 2 webifurcation can to point calculate account to it for at the two sides. The right-hand side of ( the region between the black hole and with the gravitational term evaluated at the horizon. This is the entropy of the black hole This increases linearly with time.than On the the entropy other of hand, the unitarity black suggests hole together with the thermal bath on the island are on the horizon. In the rest of this section,ln we set hole background, the entanglement entropy becomes Figure 7 JHEP07(2020)022 ) 3.3 . (4.9) (4.6) (4.7) (4.8) (4.10) (4.11) ! 1 4 ) clearly − 4.2 is the island. I ) up to subexten- ) + ln2 O 4.6 P where . ( # R K ) ρ ) ∪ − O Q . I + x P . In this approximation we ( # ∪ . + O uv K − ) ρ x R − Q O − − uv ) ln . Thus the expected unitarity x x O O )( 3 P ln − N y ( + Q K Q − 3 x N − ρ x or at large − − O uv O )( − e . y 3 ∓ O + O Q + 1 M 2 uv O x − O x x + Ny ( 3 σ − " Ny + M = ). Hence the explicit expression ( ≈ − O + ) shows that the island saddle dominates for ln – 20 – + & + x Q 3 ( 4.2 3 Q σ N 4.2 O x " NM t BH S . 2 ) = ) R ≈ gives ) + ln R ∪ ent Q ∪ Q S I gen P x . At late times these two intervals are far apart and the total S R ∪ + ( Ω( B S R . The generalized entropy is

( ∪ Q grav 3 x N S ent B S = ) = R ∪ I ). ∪ ) is 4.3 R 4.3 ( gen Now we consider the island rule entropy. We are looking for a quantum extremal S useful conversations. The work ofSimons ES Collaboration is supported on by the Simons NonperturbativeDOE Foundation Bootstrap. grant grant 488643, The DE-SC0014123. the work The ofand TH work is by of supported AS the by is Gordongrants supported via and by the Betty DOE Black grant and Hole DE-SC/0007870 Initiative. Moore Foundation and John Templeton Foundation sive terms. Acknowledgments We thank Ahmed Almheiri,Raghu Mahajan, Raphael Juan Bousso, Maldacena, Kanato Suvrat Goto, Raju, Dan Douglas Harlow, Stanford Dan and Amir Jafferis, Tajdini for Comparing to the case with no island ( After the Page transition, the entropy saturates the unitarity bound ( Extremizing with respect to find that the endpoint of the island lies on the horizon at The entropy becomes evaluated at the point entanglement entropy becomes twice the single interval answer. We then have The gravitational entropy also has two contributions, so is given by twice the formula ( surface which is twoThen points the bounding matter an entanglement is islandBy computed purity containing on of the the the surface interior globalwhich of state is we the two can black intervals again hole. consider the entropy on the complement region, bound ( There may bebe subleading overtaken corrections by to theviolates this ( linear bound, growth but in in ( any case eventually it will up to subleading terms not proportional to JHEP07(2020)022 (A.3) (A.1) (A.2) ]. 59 matter sector, we , and the associated N ¯ R is independent of the 1, on the right side of ∪ ¯ R R ∪ R O ρ y The saddlepoint is simply the . 11 ). = Tr =1 1 n ), with = 0. Its reflection across the Penrose , by continuing the solution itself, and Z 2.21 . t n ∞ ) n n , n ) ¯ n R ) 1 O ∪ A Z y Z [ R ρ ( ρ ∈ = 0. The same results apply (locally near the ]). By assuming a large- n y t – 21 – ∂ 60 = Tr ( − = Tr ( n n Z Z ) = A ρ ] and the general analysis of quantum extremal surfaces ( S 58 ] (see also [ 59 , the island rule has been derived directly from the gravitational 2 = 1. The partition function n ]. The path integral is used to calculate the replica partition functions 6 , 5 . We are interested in the von Neumann entropy of to be the semi-infinite region , and the dilaton asymptotics given by ( ¯ π R R Consider first Take The local analysis closely follows the derivation of the Ryu-Takayanagi formula by The canonical ensemble is ill-defined in the RST model, since the temperature is not a free parameter. of size 2 11 1 We therefore fix both theare leading needed and to subleading make the terms action in finite, Ω. but As we usual, will boundary not terms need and the counterterms details. region. The path integralS has asymptotic boundary conditionsEuclidean specifying black that hole, we i.e., have the an cigar geometry: diagram is replica partition functions calculated from the gravitational path integral. We will focus on theQES) eternal to black any hole spacetime at a with Euclidean a path time-reflection integral. symmetry, such that it canthe be Penrose prepared diagram by forobserver the collects eternal the black Hawking hole. radiation, but This it is is at analogous to the region where an Lewkoywycz and Maldacena [ by Dong and Lewkowyczsidestep [ some of the difficulties in definingA.1 the matter entanglement entropy Replica in geometries [ discuss replica wormhole geometries forthe the eternal solutions black locally hole in nearmotion the the RST enforce quantum model, the extremal construct QES surface,globally, condition. so and this show However is we that not will the a not equations complete construct of derivation of the the wormhole island saddles rule. used to obtain the von Neumann entropy, It is not obviousgravity whether is these dynamical results everywhere. carry over In to this asymptotically appendix flat we spacetime, make where a step in this direction. We For black holes inpath AdS integral [ This can be analytically continued to non-integer A Local analysis of replica wormholes JHEP07(2020)022 - n Z . The n in copy 1 is R is half way around and the appropriate ¯ R 1 S , because it is in a different boundaries. To find the O 1 P S The interior of the manifold, : ¯ R 12 . ¯ O P and R and O P copies of the cigar. The geometry of a ). 2 n 1) , and the gluing accounts for the matrix multi- ¯ – 22 – R − ∪ n R (( ρ O in copy 2, and vice-versa. Note that conical excesses. R π conical excesses at , we can in principle consider all topologies at each π n . There are two distinct asymptotic R . The gluing is defined cyclically, so the top of region 2 ) -fold twist defects: ¯ n R = 2. The standard replica manifold has the same topology as two copies ∪ 1 as it only appears at R n ρ = 1 there is an ambiguity in where we place the point ∼ = 2, we can also consider higher topology, contributions to the path integral, n n n , where there are 2 cyclic symmetry permuting the ¯ O n P Z Proceeding to higher Still at and Away from 12 O discussion near metric. This ambiguity would need to be fixed to fully specify the replica saddles, but it will not affect our corresponding saddles, if they exist, arewith the replica a wormholes. We willsymmetric restrict to replica manifolds wormhole can bean represented even as number a of branched cover of a single cigar, with We impose the same boundary conditionsdilaton as asymptotics, before with — 2 two copiesincluding of the wormhole connecting theof two replicas, motion. is completely smooth by the equations connected in the interior of the geometry: glued to the bottom ofthe region thermal cycle from saddle, the equations ofP motion should be solved everywhere except at the branch points Each cigar is associated toplication a in factor Tr ( of of the Euclidean black hole, glued along the regions Next we turn to JHEP07(2020)022 ) ), = 1) ¯ z z − 2 A.5 ( (A.7) (A.8) (A.4) (A.5) (A.6) n ds . Here ( ¯ z/ i π z dzd and the behavior i z to sit at extrema of i ), near the dynamical z = 0. The full manifold , where it has 2 ¯ , O z A.4 . P i . . This is the advantage of z = 0 i n z , and → zz ) ρ → O z 2 is the matter stress tensor on the πT P . that covers a neighborhood of the z ∂ = 1) is at − − flat Ω) as /n zz ) n 1 − T φ as ) χ i 2 ¯ . z z ∂ 2( ¯ z | . According to ( e 2 i ∂ρ∂ρ ¯ − O ( z dzd − = z P 2 π dzd − − N 12 z ¯ ∂χ = ( | – 23 – branch points /n and 2 ∂ − | ln ∂ρ∂φ w i O z (4 P flat 1 zz − Ω = is the full stress tensor, including the anomaly. The T ¯ ∂ − z | . It is therefore the same stress tensor that would appear ∂ N original 48 1. We will see that this fixes the zz = 1 2 n ¯ z 1 T n + ∼ zz , and the horizon (for T φ dzd ≈ n πi ≈ 2 2 2 − ρ ) at the defects, with the Euclidean black hole boundary condi- e ze ds A.8 → in the coordinate -symmetric replica wormhole is a solution to the equations ( z n ¯ w Z dwd -plane. As we will see, both the positions of the defects ≈ is the stress tensor in the metric z 2 copies of this picture, glued cyclically along all of the cuts. This full manifold is ), subject to ( ds n are flat zz ¯ T z A.7 near these points are fixed by the EOM. has Note that this formulation makes sense for non-integer The Euclidean constraint equations of the RST model in conformal gauge dzd n φ is regular near the original defects ρ 2 A.2 Derivation of theWe will QES not attempt to solvelocally the near full the replica equations defects, inthe for this generalized paper, entropy. but we will solve them tions at infinity. The fieldsi.e., are on also the required toof be single-valued on the quotient manifold, working on the quotient space. defects, it obeys To summarize, a and ( in a purely CFT calculationmotion of are the Renyi entropy, without gravity. The other equations of ρ where replica manifold with metric e and similarly with bars.metric dependence Here can be extracted by writing That is, defect, so the full manifold is smooth at this point, as required. the thermal cycle is M smooth everywhere except at the conical excesses. In particularthe it equations is of smooth motion at impose the new, dynamical branch points Here we have switched to a planar coordinate, so the metric is asymptotically JHEP07(2020)022 zz (A.9) (A.17) (A.13) (A.14) (A.15) (A.16) (A.10) ], . 62 ) and expand ). This would ¯ z flat µν z ) A.5 δT ln( = 0. We will solve ··· z 1) z 0 to leading order in + − 2 constraint and from the ··· ¯ n z ∼ ¯ . z 02 + z z b + ( . Note that it depends im- 2 ¯ z ¯ z + flat µν ]. It satisfies a conformal Ward 2 02 ... . ) = 0 z a dzd T . 62 20 + reg.+ reg. (A.11) (A.12) . + ) , ) + . The simple pole term in the i b → ], and CFT z 0 fix the position of the defect. The 2 00 S 61 /n z . + b 1 63 flat flat flat ¯ + const CFT CFT z z µν ¯ QES , ) → z 20 + ¯ z z a P − φ z ¯ 43 z ( ∂S ∂S 11 BH n ρ + N b 12 ( S − − ln( ¯ z 6 ( N + z 1 2 − ¯ N 24 δφ, T ∂ ¯ z 12 12 11 φ 2 2 + – 24 – 1) 2 ¯ z z a = 01 − b ∼ − ) = N/ N/ − n e + flat ¯ h CFT n + ¯ z = = δρ S CFT z = = 2 01 S + ( ¯ z = a 10 n flat flat ¯ z zz b φ h + BH + S + z CFT → πδT πδT ) (and its barred version) near ) near the defect is BH 2 2 00 S 10 S b . The we plug the results into the ¯ a ( are not allowed, except in the combination A.5 A.8 )( ∂ 10 + ¯ z b z is the same stress tensor that appears in the CFT calculation of z 00 ln δρ, φ a z ln( flat 1) and µν , and all of the dynamical defects = ¯ T O − 01 P b n δφ , O + ( P term we used the chiral scaling dimension of the twist operator, ρ 2 → /z 0. The leading quadratic singularity fixes ρ is the matter entanglement entropy in the metric → Now we will show that the constraints near Let us shift the coordinates so that a dynamical defect sits at 1. Expand ¯ z − flat CFT Here is the gravitational contribution to the entropy [ constraint fixes simple pole find the conditions Note that terms like not be single valuedz, on the replica manifold. Plug this expansion into ( plicitly on the globallocations structure of of the replica manifold.dilaton In has particular an it expansion depends on the In the 1 S identity that relates its singular behavior to the CFT entanglement entropy [ The stress tensor entanglement entropy in flat space, without gravity [ the constraint equationn ( The boundary condition ( JHEP07(2020)022 , 394 , . The ¯ z 05 dzd ]. ρ 2 , ]. e NATO JHEP , → SPIRE ]. ¯ z Nucl. Phys. B IN SPIRE , ][ IN dzd ][ (2019) 063 ]. SPIRE IN 12 Evanescent black holes ][ ]. SPIRE IN ][ JHEP ]. 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