Island Formula from Wald-like Entropy with Backreaction

We propose a Lorentzian derivation of the generalized entropy associated with the island formula for black holes as a Wald-like entropy without reference to the exterior non-gravitating region or ﬁeld-theoretic von Neumann entropy of Hawking radiation in a ﬁxed curved spacetime background. We illustrate this idea by studying two-dimensional black holes in the Jackiw-Teitelboim gravity and the Russo-Susskind-Thorlacius model in which Hawking radiation is represented by conformal scalars. With some prescriptions assumed, we show that the generalized entropy for the island formula can be reproduced as the Wald-like entropy of the two-dimensional dilaton-gravity theories upon the inclusion of the backreaction from Hawking radiation described by conformal anomaly. We discuss how a similar idea can be applied to higher-dimensional black holes, which may suggest that the island formula is applicable to spacetimes in which gravity is dynamical everywhere and gravitons remain massless.


Introduction
Since the discovery of Hawking radiation [1], the black hole information paradox is considered to be one of the most outstanding problems in understanding the consistency of general relativity and quantum mechanics.Tantalizingly, the black hole behaves as a thermodynamical system [1][2][3][4] and as a consequence, it is as if the information of the pure states that created or fell into the black hole were lost into a mixed thermal state.Over the last few years, there has been a potentially important development towards the resolution of the information paradox that is often colloquially called the island proposal [5].Evidence for the proposal is the so-called Page curve for the radiation entropy [6,7], indicative of the unitary evolution, reproduced by the island formula we are now going to review [8][9][10].
The island formula [8][9][10][11][12] for black holes is built on the (semi-classical) generalized entropy S gen [4].The claim is that the von Neumann entropy S of the black hole, or equivalently, for pure states, of the Hawking radiation is given by1 S = min X ext X S gen (X, Σ X ) where S gen = Area(X) 4G N + S vN (Σ X ) , where X is a codimension-two surface, which is a certain generalization of the black hole horizon, and Σ X is a spacelike region B X bounded by X and an outer surface D collecting the Hawking radiation, or equivalently, for pure states, its complement BX = I X ∪ R. The island I X is the spacelike region bounded by X = ∂I X and extending into the interior of the black hole when the extremal surface X exists.The radiation region R is the spacelike region bounded by D = ∂R and extending to the asymptotic end of the spacetime.The surface X needs to be extremized and the generalized entropy S gen is then minimized with respect to all the extremal surfaces [18,19].Note that the empty set X = Ø (no-island) is included as a part of the definition of extrema.At earlier times, the no-island extremum gives the 2 The Wald-like entropy for islands In general relativity, there exist a family of Noether currents J a associated with the diffeomorphism invariance generated by an arbitrary vector ξ a .Since the Noether current is conserved ∇ a J a = 0, the current J a can be locally expressed as With some choice of ξ a , we define the Wald-like entropy S W as the integral of a Noether potential Q ab over a codimension-two surface X: where dΩ D−2 is the volume-form on X and ǫ ab is binormal to X with the normalization ǫ ab ǫ ab = −2.It is implied that in the most general case, Q ab can, in principle, depend on ξ a and all its derivatives ∇ b 1 • • • ∇ bn ξ a .This is a generalization of the Wald entropy [20][21][22].In Wald's case, X is a spacelike cross-section of the event horizon H for stationary black holes and ξ a is the horizon-generating Killing vector.For Killing vectors, Q ab depends only on ξ a and its first-derivative ∇ b ξ a , and the Wald entropy is independent of the choice of the cross-section X and converges to the value evaluated at the bifurcation surface on which ξ a = 0 and ∇ [a ξ b] = κǫ ab with κ being the surface gravity.For example, in the case of two-derivative gravity theories with the Lagrangian density L, the Noether potential is given by Q ab = − δL δR abcd ǫ cd and in particular, for the Einstein gravity, the Wald entropy yields the Bekenstein-Hawking entropy [20].
When we try to apply Wald's idea to the "derivation" of the generalized entropy S gen for the island formula (1.1), we encounter two immediate issues: (1) The extremal surface X is not a cross-section of the Killing (event) horizon H. (2) There is no timelike or null Killing vector in (evaporating) dynamical black holes.The first issue is present even in the case of islands for the eternal black hole [23] in which the horizon-generating Killing vector exists. 4he second issue arises generically for non-stationary black holes, regardless of islands, as discussed in [21,22].
We propose two prescriptions, both of which yield the identical formula, at least, in twodimensional dilaton-gravity theories: The first is the most obvious prescription which was already proposed in the Wald's original paper [20].When applied to the generalized entropy of our interest, the prescripion is to define the entropy by the same expression as that for the bifurcation surface, while leaving the choice of X arbitrary: where we simply set ξ a → 0 and ∇ [c ξ d] → ǫ cd .To be clear, we stress that X is not fixed and subject to the extremization in contrast to Wald's proposal in which X is a cross-section of the horizon of the dynamical black hole.A caveat is that the entropy so defined is known to be sensitive to the choice of the boundary terms and might not be completely satisfactory [21].However, in a study of the Wald-like entropy in the RST model [34], a similar prescription was implicitly assumed and it was shown that, with X being either a cross-section of the event or apparent horizon, the second law of thermodynamics holds for an evaporating black hole upon the inclusion of the entropy of Hawking radiation.In some sense, our claim and findings below are a refinement of [34] in that we examine the conformal anomaly (Hawking radiation) more carefully and reveal the connection of the Wald-like entropy to the generalized entropy for the island formula. 5he second prescription is to realize (2.2) by a suitable choice of the (non-Killing) vector ξ a that has the properties ξ a = 0 and ∇ [a ξ b] = ǫ ab on the extremal surface X.There indeed exists such a vector: where x denotes the two-dimensional coordinates that specify the location of the extremal surface X, and the normalization function N is adjusted such that ∇ [a ξ b] = ǫ ab on X.To elaborate on it, ξ a is obviously null and vanishes on X by the definition of the extremal surface X.In addition, ξ a is normal to X since ǫ ab is binormal to the hypersurface X.We can then express where t a is tangential to X and κ would be the surface gravity in the case of stationary black holes [22].Now, since ξ a = 0 on X, there exists a choice of N which yields ∇ [a ξ b] = ǫ ab on X.With this choice of the vector ξ a , at least for the gravitational theories in which Q ab does not depend on higher derivatives of ξ a , the generalized entropy is essentially prescribed to be the form identical to (2.2).Moreover, the extremization with respect to X is a requirement on ξ a and built into this definition of the generalized entropy.It may look self-fulfilling since we used what to be defined, S gen (x), to define itself in the form we want.However, this prescription does not, a priori, require the explicit form of S gen (x) and the a posteriori existence of the extremal surface so defined suffices for S gen (X, Σ X ) to be well-defined and self-consistent.The proposed generalized entropy (2.2), supplemented with (2.3), for the island formula would not seem to satisfy the first law of thermodynamics since the proof of the first law relies on the vector ξ a being a Killing vector [20,21].However, since it yields the Page curve [6,7], after the Page time, that approximately follows the thermodynamic entropy of the black hole [8][9][10], it still needs to approximately obey the first law, at least, in the case of quasi-stationary black holes.Empirically speaking, given the resulting Page curves in some examples, that actually seems to be the case, but we do not have proof based on (2.2) and (2.3).We leave this issue for the future.

Two-dimensional dilaton-gravity theories
As an illustration of the above proposal, we study the black holes in two solvable models of two-dimensional dilaton-gravity theories: the Jackiw-Teitelboim (JT) gravity [30,31] and the Russo-Susskind-Thorlacius (RST) model [32], built on the Callan-Giddings-Harvey-Strominger (CGHS) model [25], in which Hawking radiation is represented by conformal scalars.They are defined by the bulk action where φ is the dilaton, f denotes a collection of free massless scalar fields with the total central charge c, and χ is an auxiliary field ("conformalon"), satisfying 2 χ = R, which was introduced to render the Polyakov's non-local anomaly action [40] local.The JT gravity and the RST model correspond to the cases As mentioned earlier, the Wald(-like) entropy in the RST model was studied some time ago in [34], and a more recent work [41] implicitly assumed the Wald(-like) entropy as the black hole entropy in these models.The Noether potential can be found as [21] Note that this final form only comes from the curvature-dependent terms in the action.All the other details of the model are eliminated by the use of the equations of motion.Our proposal (2.2) or the choice of ξ a in (2.3) then yields where X is the extremal surface and we used ǫ ab ǫ ab = −2.
We are now going to demonstrate that our prescribed Wald-like entropy (3.4) correctly reproduces the generalized entropy for the island formula that agrees with the results in [10,12,23,42,43]. 7A point we wish to make is that the field-theoretic von Neumann entropy of Hawking radiation S vN (Σ X ) emerges from a unique solution of χ simply by solving the gravity equations of motion with the choice of an appropriate vacuum.It requires no information about the field-theoretic entanglement entropy.

The JT gravity
The JT gravity [30,31] with a negative cosmological constant has an AdS 2 black hole solution and it is one of the first models studied in the development of the island idea [9][10][11][12]23].So we first discuss the Wald-like entropy for the (eternal) AdS 2 black hole in the JT gravity: where we choose the range of σ ∈ [−∞, 0].This is a solution to the equations of motion following from the action (3.1) with the choice of the set of functionals and parameters given in (3.2).In the conformal gauge defined by (3.5), the equations of motion are given by The first is the φ equation of motion, R + 2 = 0.The second to forth are the Einstein equations.The fifth and sixth are the χ and matter equations of motion, respectively.The solution of our interest is the AdS 2 black hole (3.5) with the static dilaton without classical matter [41]: The most important in this discussion is the "conformalon" solution where the homogeneous part of the solution denoted by the chiral functions h ± satisfy (3.9) These equations follow from the fact that the LHS of the third and fourth equations vanish with the above choice of ρ and φ.The general solution can be expressed as 8 with the integration constants c ± and x ± 0 .Having specified the solution to the gravity equations of motion (3.6), we can now calculate the Wald-like entropy (3.4) as a function of the location (x + , x − ) of the extremal surface X: where s 0 = c 12 − c 6 (c + + c − ) which is an ambiguity in the additive constant [46].The extremization with respect to x ± determines the position (x + , x − ) of the extremal surface in terms of (x + 0 , x − 0 ).This indeed reproduces the results in [9,10,12,23], up to an additive constant, after the maximization in time τ [14] which sets τ = τ 0 eliminating the timedependence.Note that there are no UV cutoffs appearing in the radiation (anomaly) part of the entropy in contrast to the field-theoretic computation and this result does not require regularization or renormalization. 9 The spacelike region Σ X in (1.1) corresponds to the interval Σ X = B X ∈ [σ, σ 0 ], or equivalently, its complement I X ∪ R as defined below (1.1).From the viewpoint of our derivation, it is natural to interpret the position (x + 0 , x − 0 ) as that of an observer collecting Hawking radiation within the AdS 2 boundary, i.e., σ 0 < 0 in our convention, since our spacetime is nothing other than the AdS 2 black hole.However, in the setup of the islands [9,10,12,23], the location σ 0 of the boundary of Σ X is placed, beyond the AdS 2 boundary (σ 0 → 0 − ), in the external bath regions (σ 0 > 0) with a transparent interface condition so that the radiation is transmitted across the AdS 2 boundary rather than reflected back in.Nevertheless, we point out that for the eternal black hole that does not evaporate, the islands (outside the horizon) actually exist for σ 0 < 0, that is, even when the observer at the boundary of Σ X is placed within the AdS 2 boundary, leaving the existence of the baths into oblivion.Since σ 0 is a free parameter, we can choose it to be positive as one pleases, but it is unclear from our derivation that it should be interpreted as an observer in the external bath attached to the AdS 2 black hole.To this end, it is worth emphasizing that we made no reference to the non-gravitating regions to derive the generalized entropy associated with the island formula. 8The solution in [41] corresponds, in our notation, to h ± = ± π β x ± which is a limiting solution with the choice, c ± = Λ + ln(β/(4π)) and x ± 0 = ±Λ with Λ → +∞. 9 It is tempting to interpret this finiteness of the gravitational generalized entropy as a realization of the idea by Susskind and Uglum [47] as recently emphasized by Witten in the context of von Neumann algebra [46].
We now turn to the no-island case which appears in the two-sided eternal black hole.The entropy must be given by where x ± is located in the left Rindler wedge at the mirror point of x ± 0 and in terms of the Kruskal coordinates, The no-island entropy grows linearly in the Rindler time τ at late times.It seems that there is no first-principle derivation of the no-island "extremum" from the Wald-like entropy (3.4).Naively, if X = Ø, the Wald-like entropy (3.4) would have to vanish.One of the issues is that the geometric part Area(X)/(4G N ) and field-theoretic part S vN (Σ X ) in (1.1) are unified into a single geometric entity in the Wald-like entropy (3.4).So, conceptually speaking, it is hard to separate them and single out S vN (Σ X ) from some first principle.However, it is fairly clear that the radiation entropy for the empty set X = Ø is, by definition, extracted as So practically, we can adopt this formula as our prescription for the no-island radiation entropy.
To complete this discussion on the JT gravity example, the minimum of the Wald-like entropy (3.11) and no-island radiation entropy (3.13) reproduces the Page curve in [10,12,23].

The RST model
In the JT gravity we have just seen that the generalized entropy for the island formula can be derived without reference to the external non-gravitating bath attached to the AdS 2 black hole.Moving forward, in the case of asymptotically flat black holes, even though there is no external bath attached in any way, the island formula still assumes the non-gravitating (radiation) region R because the entanglement entropy of Hawking radiation is the fieldtheoretic von Neumann entropy in a fixed curved spacetime background where gravity is treated as non-dynamical [5].Having this point in mind, in this section, our aim is to demonstrate that the generalized entropy can be derived as the Wald-like entropy without assuming the non-gravitating region by studying the asymptotically flat black holes in the RST model [32].The generalized entropy in the RST model was first studied in [48] long time ago.More recently, it was reinvestigated in light of the island proposal [42,43].As we will see, the Wald-like entropy (3.4) reproduces the results in [42,43,48]. 10n addition to the diffeomorphism invariance, the RST model has an extra symmetry associated with the current conservation ∇ a (∇ a (φ − ρ)) = 0.By virtue of this symmetry, working in the conformal gauge we can consistently set Then the equations of motion that follow from the action (3.1) with the choice (3.2) can be expressed as11 where a convenient field redefinition was introduced [49,50] There are exact black hole solutions to these equations of motion [32]: (1) eternal black hole and (2) evaporating black hole.We first discuss the eternal black hole which is given by where Ω c corresponds to the critical value of the dilaton, φ c = − 1 2 ln(c/48) > φ.This is the value at which the determinant of the (ρ, φ)-kinetic terms vanishes and can be thought of as a strong coupling singularity where the spacetime ends.The matter f = 0 and the "conformalon" χ is given by The chiral functions h ± (x ± ) can be found as For the two-sided black hole, the Wald-like entropy (3.4) is thus found to be where we used that 4πU(φ)+ c 6 ρ = 2Ω in the unit 8G = 1 and s 0 = − c 6 (c + +c − ) is an additive constant ambiguity.The extremization with respect to x ± determines the position (x + , x − ) of the extremal surface in terms of the position (x + 0 , x − 0 ) of an observer collecting Hawking radiation.This agrees with [43] up to a constant which does not affect the location of the extremal surface.We note again that there are no UV cutoffs appearing in the radiation part of the entropy in contrast to the field-theoretic computation.Most importantly, once again, we made no reference to the non-gravitating regions in this derivation.
With the prescription for the no-island radiation entropy in (3.13), since the x ± coordinates are Kruskal as implied in (3.18), we find that At late times, this grows linearly in the Minkowski time τ defined via the map x ± = ±e ±(τ ±σ) and agrees with the results in [42,43].
We next discuss the evaporating black hole created by a shockwave where Θ(x + − x + s ) is a step function and h ± will be determined shortly.Before the shockwave is shot in, the spacetime is in the so-called linear dilaton vacuum since φ = ρ = − 1 2 ln(−x + x − ) = −σ for x + < x + s .The metric in the linear dilaton vacuum is flat and given by In this setup, we want to study the radiation entropy in the Minkowski vacuum [42,43,48], meaning that the stress tensor and Liouville field ρ are defined with respect to the Minkowski coordinates (w + , w − ) = (ln x + , − ln(−x − )).After the shock x + > x + s , they are given by where and the Schwarzian derivative {w, x} ≡ w ′′′ /w ′ − 3/2(w ′′ /w ′ ) 2 .The middle expression of the first equation vanishes by using the second equation of motion in (3.16).The field Ω is a scalar and so unchanged since it is a field redefinition of the dilaton. 12From the first equation in (3.2), h± are fixed to be h± (w where χ(w + , w − ) = −ρ(w + , w − ) + h+ (w + ) + h− (w − ).This is the same in form as (3.20).However, it is important that this expression is in the Minkowski coordinates whereas (3.20) is in the Kruskal coordinates.Having specified the evaporating black hole solution, the Wald-like entropy (3.4) yields where we used again that 4πU(φ) + c 6 ρ = 2Ω in the unit 8G = 1 and x ± = ±e ±w ± .As the observer is sent to I + , i.e., for a very large x + 0 for which w + − w + 0 ≈ −w + 0 , this agrees with [43] up to a constant that does not affect the location of the extremal surface.The extremization with respect to w ± determines the position (w + , w − ) of the extremal surface in terms of the position (w + 0 , w − 0 ) of an observer collecting Hawking radiation.We now turn to the no-island case.We are interested in the growth of the radiation entropy with respect to the observer's Minkowski time after the shock x + > x + s .So the coordinates of our interest are ( w+ , w− ) defined by x + = e w+ and M + x − = −e − w− , given the metric (3.23), which are related to the Minkowski coordinates (w + , w − ) on I − by (w + , w − ) = ( w+ , w− −ln(1+Me w− )).Now, the evaporating black hole ends at Ω = Ω c which is timelike before and some time after the shockwave creates the black hole.The location of the endpoint (w + , w − ) depends on the boundary condition at Ω = Ω c .We impose a reflection condition on the timelike boundary after the shock: ( w+ , w− ) = ( w− 0 , w+ 0 ). 13Then the prescription (3.13) adjusted to this case reads At late times the second line dominates and the radiation entropy grows as c w− 0 /12 which agrees with [42,43] even though the details are different.The minimization of the island and no-island entropies yields the Page curve as demonstrated in [42,43].
To summarize our discussion on the eternal and evaporating black holes in the twodimensional dilaton-gravity models, we successfully reproduced the known generalized entropy for the island formula from our proposed Wald-like entropy (2.2) which may be obtained by the choice of ξ a in (2.3).

Discussions
We proposed a Lorentzian derivation of the generalized entropy associated with the island formula as a Wald-like entropy.The derivation is fully gravitational without reference to the exterior non-gravitating region or field-theoretic von Neumann entropy of Hawking radiation in a fixed curved spacetime background.We illustrated how this idea actually works in detail by studying the radiating black holes in the two-dimensional dilaton-gravity theories.
In principle, a similar idea can be generalized to higher dimensions.However, since the conformal anomaly plays an important role, the applicability might be limited to even dimensions in which the conformal anomaly can be expressed by local geometric invariants.Here, we would like to discuss the four-dimensional generalization.First, it is reasonable to assume that Hawking radiation is dominated by conformal matter.So the backreaction of Hawking radiation is incorporated into the gravity theories as the conformally-coupled matter action and their conformal anomaly.In higher dimensions, however, the conformal anomaly may not be all that matters since, for example, the conformally-coupled scalars f have the explicit curvature coupling R f 2 on which the Wald-like entropy can depend. 14ith this caveat in mind, recall that the homogeneous part of the "conformalon" solution to 2 χ = R played very important role in our success in two dimensions.In four dimensions, a similar local form of the anomaly action that serves our purpose would be the one constructed by Mottola [51] which is a covariant version of the so-called Riegert action [52] and quadratic in the conformalon χ similar to (3.1). 15It is then straightforward to run a similar program to that we carried out in two dimensions, but the difficulty will be to solve the equations of motion for the gravity + conformal matter + anomaly system.However challenging it may be, the test is to see if the Wald-like entropy reproduces the salient features of the generalized entropy for four-dimensional black holes found in earlier works [55][56][57][58].It is worth emphasizing that since it is a fully gravitational computation, the spacetime is everywhere gravitating including the radiation region R outside the surface of an observer collecting Hawking radiation.So gravitons remain massless and our findings may partly address the issues of the island proposal raised in [26][27][28].
A natural question is whether and how our Lorentzian derivation is related to the replica wormholes in the Euclidean path integral [11,12].At the moment, we do not have much to say about it except that the Wald-like entropy must correspond to the wormhole saddle, or rather, it is hard to see the "Hawking no-island" saddle in our approach.On that score, it might be useful to study the issue of Lorentzian vs. Euclidean in a more algebraic 2d CFT description of the replica wormholes [59].
his visits where part of this work was done.The work of SH is supported in part by the National Natural Science Foundation of China under Grant No.12147219.