A comment on a fine-grained description of evaporating black holes with baby universes

We study a partially fine-grained description of an evaporating black hole by introducing an open baby universe with a boundary. Since the Page's calculation of the entropy of Hawking radiation involves an ensemble average over a class of states, one can formally obtain a fine-grained state by purifying this setup. For AdS black holes with a holographic dual, this purification amounts to introducing an additional boundary (i.e., baby universe) and then connecting it to the original black hole through an Einstein-Rosen bridge. We uncover several details of this setup. As applications, we briefly discuss how this baby universe modifies the semi-classical gravitational Gauss law as well as the gravitational dressing of operators behind the horizon.


Introduction
Ever since Hawking discovered that a black hole has a temperature and emits thermal radiations [1,2], how its time-evolution is consistent with the principle of quantum mechanics is one of the greatest problems in theoretical physics.One of the key points of recent developments in quantum gravity is the role of the Euclidean wormholes, which play a crucial role in resolving the black hole information loss problem through their non-perturbative effects (e.g., [3][4][5]).
The von Neumann entropy of the Hawking radiation can be defined by the entanglement entropy of the bath region R, attached to the asymptotic infinity of the black hole, see figure 1.The island formula [6][7][8] tells us that this entropy is given by where I is a region in the bulk gravitating spacetime.The region which extremizes the above generalized entropy functional is called the island.This formula can be regarded as a natural extention of the RT/HRT formulae and their quantum extentions for holographic entanglement entropy in AdS/CFT [9][10][11][12][13].This island formula is indeed obtained by including so called Euclidean replica wormholes to the gravitational path integral [3,4].The island formula correctly reproduces the Page curve [14,15] for the entanglement entropy of Hawking radiation, thus gives results consistent with the principles of quantum theory within semiclassical regime of gravity.(See e.g., [16,17] for reviews on this topic, and related discussions on the island formula, e.g., .)Y E Y Y j B P F a 9 K F R t D R M S l n F q O 1 S e U p k c R z f w x 1 t X p G m k V 5 H J g 2 0 r d X 6 G e V d Z y 1 G I a K y n x v L E r f / E K S q 6 m x P 0 V 6 n x G a R l q t z y + X S x f L A i C E 8 b D o e T 5 u T E e h 5 D S D 0 P S 1 / M 4 q Y P S S H 0 f U x g F 2 R j r W J 0 l G n p + a Y U K F U d j N G a Q 9 i K F R i W u 4 + a N X F m v B a M s N z 6 d Y C / T l M 7 S K t U g W V V C n 5 1 j b 9 p 4 3 0 j L M j N O G t P S E B p A J U C b p y U M c j V 8 s V j D J G E 3 x 1 N I V W W y A F 1 N V H a S 1 U n b S J Z J F o L l + q p 9 d 2 T M G 8 2 k q q p V 7 E x H A c 1 I p A F Y K C P 5 p g l a r N U H J E o O x b J 3 Z y w P g g U J c a 6 a 0 p P K d 0 M 0 w L e d b F l B F C k X J p O c S + 3 d y r l 6 f + / u 3 L t b M d K W e H f L 3 s t F V 5 y v H d l X x b m 4 f y B x q m G q 6 N i r Y M x C a e b G q 0 8 h G g E 1 E A p B q y j P w u T 8 b H p s p 7 W b z a w s p e U 6 L F o t G C 2 K u r e g l + + J Z f k r 0 X M X 6 S 2 z q q S n Z o C M 8 + z v 5 H H 6 3  The island formula suggests that the entanglement wedge of the Hawking radiation contains not only the radiation region R, but also the island region I. On the other hand, the entanglement wedge of the black hole is its compliment (see figure 1 again).The boundary of the island region ∂I is located just behind the horizon for an evaporating black hole [6-8, 79, 80].However for an eternal black hole, it is located outside the horizon [79][80][81][82][83].
In a recent interesting paper [84], such a form of entanglement wedge disconnected from the asymptotic boundary is argued to be inconsistent with the long range nature of the gravitational force or equivalently diffeomorphism invariance.Let us consider a small local operation on the island I which corresponds to a local excitation of on the radiation Hilbert space H R .In a theory of gravity, such a small operation seems to be detected on the asymptotic boundary of the spacetime using the gravitational Gauss law.This is problematic because the asymptotic boundary belongs to the entanglement wedge of the black hole, see figure 2. This implies that the local operation on the island region I (which is supposed to be an operation on H R ) can actually change the entanglement wedge of the black hole.

< l a t e x i t s h a 1 _ b a s e 6 4 = " F T N c I
o 5 p f x W 5 w P g F X t g U b 3 m 4 D l 5 e I u 0 K u H p n o e W / 0 n i S 6 a i o w z 6 4 H t J w 5 A F w 7 T L h 5 L S I C a f q b + 6 0 R X z 1 A g i 2 P X 9 9 2 P j t J C 7 g 6 x n A c 9 d a Z q d The above apparent inconsistency of the gravitational Gauss law is related to the inconsistency of the gravitational dressing of local operators on the island region [84].In a theory with diffeomorphism invariance, a local operator can not be physical since it is not gauge invariant.Instead, such a local operator φ(P ) needs to be "gravitationally dressed" [85,86] by attaching a Wilson line W (P, P B ) connecting the point P to a point P B on the asymptotic boundary.Then the resulting operator φ(P )W (P, P B ) is gauge invariant.For the gravitational dressing for a local operator on the island region P I ∈ I, which is a part of the entanglement wedge of the Hawking radiation, it is natural to choose a point on the bath region R as the asymptotic boundary point P R ∈ R. In such a case, the Wilson line W (P I , P R ) connecting the two points intersects with the entanglement wedge of the black hole.However the dressed operator φ(P I )W (P I , P R ) should belong only to the entanglement wedge of the Hawking radiation, thus this is problematic again.See the figure 2 again.
In this paper, we address the above paradox by carefully examining how the effects of random fluctuations of an evaporating black hole are geometrized in a semi-classical description of gravity.In principle the black hole evaporation process is described by the bipartite system of the Hilbert space of the black hole H BH and the one for the Hawking radiation H R .Of course, the description of such an entangled state involves a quantum theory of gravity, therefore it seems impossible to study such a system efficiently.However, as was first observed by Page [14], one can obtain a time evolution of the radiation entropy consistent with unitarity, by averaging the entropy over the random fluctuations in the entangled state.This opens up the possibility of having a partially fine-grained description of the evaporating black hole while maintaining its semi-classical nature, to the extent of getting results consistent with the principles of quantum theory.Indeed, in this way, the island formula makes it possible to recover the Page curve in a semi-classical way.Specifically, the Euclidean replica wormholes nicely capture the effects of these random fluctuations and their averaging through a geometric way.
This paper concerns a description of these random fluctuations in a Lorentzian spacetime in the semi-classical regime.We argue that the averaging over the random fluctuations can be purified by introducing an auxiliary system, often called a baby universe.This new piece of the spacetime is connected to the original spacetime with the black hole by an Einstein-Rosen bridge, can be thought of as accommodating partially fine-grained information of the evaporating black hole (see figure 4).See also [87,88] for discussions on the role of baby universes in the information loss paradox.
Motivated by this observation, we then study the gravitational Gauss law in the presence of the baby universe sector.Such an introduction of the baby universe significantly modifies the form of the gravitational Gauss law.For instance, assuming the baby universe part has an asymptotic boundary, the gravitational Gauss law does not exactly hold within the original spacetime as there is a contribution from the baby universe sector.This makes sense, because restricting our attention only to the original black hole spacetime corresponds to a coarsegraining.This is clearly seen from a Schwarzschild black hole solution, which has the horizon area and therefore the Bekenstein-Hawking entropy S BH .However it is just one solution, showing no degeneracy of the states, contradicting the huge degeneracy given by the entropy e S BH .What we expect is that in full quantum gravity, one obtains microstates of the black hole and by counting its degeneracy, one obtains e S BH .However after coarse-graining, all the details of the microstates are lost, and one cannot see the microstate differences in the Schwarzschild solution.Assuming the dynamics of the black hole is sufficiently chaotic, two distinct energy eigenstates can never have the same energy, and the minimal value of the difference is of order e −S BH .Thus the geometric description of such a class of microstates by a single black hole spacetime inevitably involves a coarse-graining, in which the energy differences of order e −S BH are neglected.This suggests that in the black hole spacetime, one can only trust the gravitational Gauss law up to O(e −S BH ) corrections.
The introduction of the baby universe with an asymptotic boundary naturally resolves the paradox of the gravitational dressing as well, because the gravitational Wilson line starting from the island region can now end on the boundary of the baby universe.This is a kind of an expected result because the island region corresponds to fine-grained information of the evaporating black hole, so can not have a simple description within the original black hole spacetime.
The rest of the paper is organized as follows.In section 2, we study wormholes, and the baby universe.In section 3, we present our main idea and explain how we modify the gravitational Gauss law in the presence of the baby universe.We also comment on how the boundary of the baby universe can resolve the gravitational dressing paradox.In appendix A, we give the calculation of the von Neumann entropy of the Hawking radiation and that of the Hawking radiation plus the baby universe in our formalism.
Note added: During the preparation of this paper, the papers [89][90][91] appeared, and discussed extra information coming from the ensemble nature of gravity, which is related to the baby universe degrees of freedom in our paper.

Baby Universe and Ensemble nature of Semi-Classical Gravity
In this section, we clarify the role of the baby universe in the computation of the fine-grained entropy of Hawking radiation through the island formula.
To this end, it is appropriate to begin with the fact that there are two distinct descriptions of a theory of gravity.The first one is the fine-grained description, and the second one is the coarse-grained one.In the first full-fledged fine-grained description of quantum gravity, we have a sufficient number of observables (i.e., the complete set of operators of quantum gravity) to perfectly distinguish quantum states.Note that, in the description, we can perform measurements with arbitrary precision.We are interested in the gravitational system where a black hole keeps emitting Hawking radiations.In a full-fledged fine-grained microscopic description, an actual state in such a system has the following form, where F M iα takes a fixed number.Here we define the orthonormal bases |ψ i BH and |α R of the Hilbert space H BH for microstates for the black hole and the similar Hilbert space H R for the Hawking quanta participating in the entanglement.N and k are their dimensions.
The second description of the system is the coarse-grained one in terms of a semi-classical theory, where we have a restricted number of observables, i.e., a subset of the complete set of observables of quantum gravity, or coarse-grained observables like thermodynamical quantities.The spatial and time resolution of such observables is much larger than the Planck scale.In this description, even by measuring coarse-grained observables precisely, we cannot completely distinguish the underlying full quantum states of the full theory, but at best a set of states with the same expectation values of the coarse-grained observables and the same semi-classical geometries.
Owing to the restricted number of observables and also to the fact that the resolution is much larger than the Planck scale, one is forced to describe the system in a coarse-grained way, in terms of a mixed state, i.e., an ensemble of states {p M , |Ψ M } M1 .This ensemble consists of the class of the states |Ψ M with the random coefficient matrix From the semi-classical gravity point of view, two such states |Ψ M , |Ψ N with different random coefficients C M , C N can not be distinguished.This corresponds that a coarse-grained observer describes the state in terms of the following mixed state, where p M is the Gaussian probability distribution determined by the ensemble of states or random coefficient matrix C M iα as and satisfies M p M = 1.See (A.1)-(A.3) in appendix A. We also note that the coefficients C iα are satisfying the following relationship, where • means the average over the random coefficient matrix C M iα .The randomness of the coefficient in (2.2) is due to the fact that the dynamics of a black hole is highly chaotic.These can be understood as follows: Suppose that an observer tries to experimentally specify the fine-grained state (2.1).Then the observer needs to perform a measurement with the Planck scale precision.However, for coarse-grained observers, the resolution of the measurement is much larger than the Planck scale.Note that during the measurement time-scale, the microscopic state can evolve.Therefore, if the measurement time-scale is much larger than the Planck scale, the microscopic state can evolve to almost all states of the form (2.2).In this way, coarse-grained observers see the black hole state as (2.2).This provides an intuitive way to understand the reason why the random matrices appear in the semi-classical description of the black hole dynamics.
Once we coarse-grain the system, the state is reduced from the pure state (2.1) to the mixed state (2.2), and apparently we lose the microscopic details of the states.However we nevertheless can compute some aspects of the fine-grained entropy of Hawking radiation by purifying this mixed state by introducing an auxiliary system H BU , which we often call the baby universe.For instance, recent progress in understanding the island formula suggests that the purification enables us to capture some part of the fine-grained information of Hawking radiation while maintaining the semi-classical description.Discussions on the relevance of random fluctuations for the physics of black holes can be found for example in [93][94][95].We also note that Gaussian random fluctuations have a geometric interpretation in terms of end of the world branes in two-dimensional JT gravity [3].
Note that to purify the original system with the mixed state (2.3), we need an auxiliary system H BU whose dimension is at least equal to or greater than that of the original system.The dimension of the baby universe Hilbert space depends in particular on the coarse-graining procedure.On this new Hilbert space, the simplest purified state is given by where {|M BU } are orthonormal baby universe states.A fine-grained observer can access this auxiliary system, but coarse-grain observers can not.Let us emphasize that the description using the auxiliary system is not a full fledged fine-grained description of the system.This is because we are artificially adding degrees of freedom, which do not show up in the original Hilbert space H BH ⊗ H R .More concretely, in the quantum gravity description, the actual fine-grained state realized in the system is one of the states in the ensemble, not the one with the baby universe.We nevertheless consider the purified state (2.6), because it has an effective semi-classical description, on the contrary to the full fledged fine-grained state in quantum gravity.Furthermore, as we will show later, if we are only concerned with the averaged property of the fine-grained entropy, such as the Page curve, considering this purified state is good enough.Note that by tracing out the black hole degrees of freedom BH in the mixed state (2.3), the reduced density matrix of the Hawking radiation ρ R gives an approximately thermal mixed state, and the von Neumann entropy S vN [ρ R ] gives the Hawking's result where we have defined, See the appendix A.1 for detailed derivation.Now let us consider the same entropy of Hawking radiation in the fine-grained description.To do so, let us first figure out a geometric description of the purified state (2.6).In this state, the Hawking radiation H R and the black hole H BH are entangled with the auxiliary baby universe H BU .From the viewpoint of ER=EPR [96], we expect that this is realized geometrically by an Einstein-Rosen bridge connecting the baby universe and the original system (see figure 4).The property of the ER bridge depends highly on the choice of the ensemble.If we realize this system within the framework of the AdS/CFT correspondence, the auxiliary universe can be modeled by an additional boundary and its gravity dual involves an Einstein-Rosen bridge connecting the new boundary2 .This purification process is the key in recent studies, especially in the finding of the island formula which captures some aspects of fine-grained information of the quantum gravity states, in the semi-classical description through a non-perturbative way.For instance, in describing an evaporation process of a black hole semi-classically, such non-perturbative contributions are required to get a consistent result.In such a process discreteness of the energy spectrum of the black hole microstates is a crucial ingredient to ensure unitarity of the process.However, in the coarse-grained description, energy differences between black hole micro-states are invisible, since they are typically of order O(e −S BH ), where S BH is the Bekenstein-Hawking entropy [98].A discrete energy spectrum is only after including non-perturbatively small contributions which are provided by Euclidean wormholes [99,100].
What the island formula implies is that one should identify the fine-grained Hilbert space of the Hawking radiation H R with the tensor product of two Hilbert spaces H R ⊗ H BU , after the Page time.On the other hand, before the Page time H R should be identified with just that of the Hawking radiation H R , and correspondingly the Hilbert space of the black hole should coincide with the tensor product of the black hole and the baby universe H BH ⊗ H BU .This difference between the radiation Hilbert spaces before and after the Page time comes from the fact that the inequality for the dimensions of the Hilbert spaces of the Hawking radiation and that of the black hole changes.In fact, before the Page time, since the total state (2.6) is pure, the von Neumann entropy of the union of the black hole and the baby universe BH ∪ BU is equal to the previous von Neumann entropy ( which is consistent with the island formula before the Page time. After the Page time, the reduced density matrix of the Hawking radiation and the baby universe ρ R∪BU in (2.6) gives the the fined grained entropy of the Hawking radiation, which deviates from the entropy (2.7) of the naive density matrix (2.3), (2.9) In appendix A. 2  At the same time, we know that the fine-grained entropy S vN [ρ R ] of Hawking radiation is computed by the island formula (1.1) too.In the entropy calculations using this formula, it was crucial to include the contribution of the island, which typically occupies the region behind the horizon of the black holes.Therefore it is natural to identify the island region behind the horizon with the Einstein-Rosen bridge of the purified state (2.6) connecting the original spacetime and the baby universe, which stores fine-grained information of the original spacetime.
These states {|M BU } in the fine-grained Hilbert space can be naturally identified with so called α states [101][102][103] in the baby universe Hilbert space which diagonalizes the boundary 3 One may also consider the possibility of dividing the baby universe Hilbert space HBU into two parts HBU BH ⊗ HBU R , and then define the radiation Hilbert space as H R = HBU R ⊗ HR, instead of H R = HBU ⊗ HR which we do in the body of the paper.In such a case, the states of the baby universe are given by |M BU BH ⊗ |M BU R .In this case, assuming the orthogonality of the basis of HBU BH , we see that the entropy of ρBU R ∪R is given by SvN where ρ (M )R is the reduced density matrix given by (A.5).Then it is natural to define the fine grained entropy of Hawking radiation S(ρ R ) as a conditional entropy of knowing the probability distribution pM by subtracting the classical Shannon piece H(pM (2.11) However we do not know the natural choice for such a splitting of the baby universe Hilbert space HBU .
in the sense that In other words, LOCCs acting only on the black hole BH and the Hawking radiation R, which can be available to coarse-grained observers, can not distinguish the classically and quantum mechanically correlated states (2.6), (2.12).However one can easily see the entanglement entropy of these two states on R = R ∪ BU are different.Indeed, the entropy of ρ contains a classical Shannon term, whereas the same entropy of (2.6) does not.From another point of view, LOCCs on the sub-system BH ∪ R and the baby universe BU , which can only be available to fine-grained observers, can distinguish the classically and quantum mechanically correlated states, since the equalities in (2.13) do not necessarily hold for operators on BH ∪ R ∪ BU .
In the next section, we discuss several properties of the baby universe and the wormhole connecting the baby universe and the original spacetime.The wormholes may be dependent on the actual geometry of the baby universe.We cannot fully specify the geometry of the baby universe from the first principles of quantum gravity.There is a canonical and minimal choice for such a baby universe; starting from the original system |Ψ M , we prepare a copy of it | ΨM , and regard it as a purifier |M BU = | ΨM Puri. .Then the expression (2.6) becomes The existence of the boundaries in the original system |Ψ M implies that purifier |M BU = | ΨM Puri.should also have boundaries.More generally, there is a possibility that we may choose the multiple copies of the original system as the baby universe and further choose their linear combinations as that.Again from ER=EPR this entanglement between the two spacetimes implies the existence of the wormhole connecting two island regions for two spacetimes.This wormhole will affect the non-perturbative physics of this system.Note that the more the number of copies of the original spacetime increase, the more the effects from wormholes are topologically suppressed.

Gauss Law modified by the Baby Universe
In this section, we discuss the physical consequences of the existence of the baby universe sector introduced in the last section, which accommodates fine-grained information of the system.We are mainly interested in how the baby universe helps to recover information of the black hole interior from Hawking radiation.We will also briefly mention the relation between our discussion and the paradox raised in the recent paper [84].Before doing so, let us present a remark.In the light of AdS/CFT correspondence, the introduction of an additional boundary, i.e., the boundary of the baby universe sounds puzzling, because AdS/CFT is the correspondence between a theory of full quantum gravity in the bulk and a (non gravitating) CFT on the boundary.This means that in principle, all the details of the bulk quantum gravity Hilbert space can be read off from the single CFT Hilbert space.Therefore, we do not need the second copy of the CFT, as we did in the previous section, which results in the baby universe sector.
Nevertheless, we are forced to do so, because we are sticking to a semi-classical description of the system.Then, to restore fine-grained information within the semi-classical regime, we need to introduce an auxiliary system and regard the new degrees of freedom as a part of the radiation degrees of freedom after the Page time.If we do not do this, this restriction amounts to that on the boundary, we are only accessible to a sub-Hilbert space H coarse which characterizes coarse-grained degrees of freedom.To incorporate the rest of the CFT Hilbert space, which we term H fine just because it describes fine-grained degrees of freedom, we need to introduce a second copy of the CFT Hilbert space, and accommodate H fine to it.
The full Hilbert space on the single boundary is obtained by gluing two asymptotic boundaries of the spacetime.In the resulting bulk spacetime, there are two homologically inequivalent paths, both of which connect a point in the interior of the black hole (and belong to the island region) to the boundary of the spacetime (see figure 3).The first path is the trivial one (the blue line in figure 3 ), which entirely lies within the original spacetime.This path necessarily intersects with the entanglement wedge of the black hole.However, in the presence of the baby universe, there is a second path which does not cross the entanglement wedge of the black hole.Instead, it crosses the Einstein-Rosen bridge connecting the original spacetime to the baby universe, and reaches the second asymptotic boundary which accommodates fine-grained degrees of freedom as in the green line in figure 3. Since these two boundaries are in the end glued together, it connects the island region and the conformal boundary, without passing through the entanglement wedge of the black hole.

The modification of Gauss Law
In the presence of the baby universe sector which has its own asymptotic boundary, the gravitational Gauss law is inevitably modified.Let Σ be a time slice of the spacetime, then < l a t e x i t s h a 1 _ b a s e 6 4 = " 3 1 / 4 h P t 0 J e N X t w d n H h 4 H 0 Z I G 5 y 9 m g U / n B K / 6 i e + G + K j b 2 R 8 0 m 8 1 d s / T 5 B w m A w w 3 L 7 5 v N r T M / j s k Q F i l G e y F t I 5 e i P b N n 9 N q H C K H y u p e v m 4 y g t E w E e w 0 3 8 b J l x t z c K g v 3 h f C 7 R p c t q 7 x i U a 5 4 H c Y h p Z s A s j C n q J 9 M 4 p h M D U R y x a j + y B q j I U 4 K W q f K 9 F v t z A 0 r 7 h p L l W 3 v d z m 0 D f t u t d i n / q 8 N X U 0 t S t a S q 1 m m b X K 1 Z 1 e u k e l X N g 8 3 M j + I h 7 r t e / B w P 0 N I p m z / Y 0 J i q F E j I l x F 1 o 3 G I p z i i 6 C k e j j G K R 4 h O M C p S R X S p i m e u H 9 A F Y I p l j I I w T F 2 h b d W N G 8 y 7 V l o F S g 5 M k 5 F 7 a 8 i z y q q X W k 6 j 1  the gravitational Gauss law relates the expectation value of the bulk stress energy tensor T bulk to the boundary energy H ∂ [h] (holographic stress energy tensor) Here the boundary energy H ∂ [h] is explicitly given by the integration of the ADM current J i over the conformal boundary ∂Σ [104], where n i is the normal vector to the boundary ∂Σ, and the ADM current J i is defined by under the ADM decomposition and the expansion from the background metric g ij = g 0 ij + κh ij .More precisely, (3.1) is a perturbative version of the gravitational Gauss law which can be derived from the full Hamiltonian constraint where g ij is the metric on the Cauchy slice, π ij is the conjugate momentum, and H matter is the matter Hamiltonian density.Expanding (3.5) from the background metric, g ij = g 0 ij + κh ij , then look at the second order of the expansion gives (3.1).Details of the derivation can be found, for example in [104].H ∂ [h] should be understood as the change of the mass of the black hole, due to the back reaction from the bulk stress energy tensor, T bulk .
In the paper [84], it was argued that the gravitational Gauss law provides an interesting puzzle on the island formula.Suppose we act a local operation on a state on the island region.Since the information of the island region is encoded in the Hilbert space of Hawking radiation H R , this operation can be regarded as a local operation on H R .This operation changes the expectation value of the bulk stress energy tensor.Then the gravitational Gauss law relates this change of T bulk on the island region behind the horizon to the change of the boundary energy H ∂ .This means that any change on the island region, no matter how it is small, is always detectable from the conformal boundary ∂Σ.However, this sounds troublesome because ∂Σ belongs to the entanglement wedge of the black hole.For instance, this implies that in the bipartite system H R ⊗ H BH , a local operation on H R can change the state of H BH .
The above paradox is naturally resolved, once we take into account the effects of the baby universe sector which admits the new boundary (see figure 3).In the presence of this new part of the spacetime, the gravitational Gauss law must be modified as where we denote H ∂BH [h] by the boundary energy of the original spacetime with the black hole, and similarly H ∂BU [h] is the boundary energy of the baby universe.This form of the gravitational Gauss law immediately implies that, in the presence of the baby universe, operations on the island region need not to be detected on the conformal boundary of the black hole.In other words, T bulk = 0 does not necessarily imply H ∂BH [h] = 0. Rather, it is natural to relate T bulk on the island region to the boundary energy of the baby universe H ∂BU [h] because the island region is encoded to the Hilbert space of finegrained Hawking radiation H R = H R ⊗ H BU .Indeed, the island region encodes fine-grained information of Hawking radiation after the Page time, so from the boundary point of view such bulk operations on this region should be encoded in the fine-grained part of the CFT Hilbert space, which coincides with the boundary Hilbert space of the baby universe.
Another way to put this is the following.Let us consider putting a local operator in the spacetime.The gravitational Gauss law implies that by measuring the total flux for an appropriate closed surface we can know the "mass" of the particle within the closed surface.The non-perturbative gravitational effect from the wormhole makes the measurement of the flux highly non-trivial.The wormhole can release some part of the flux of the original spacetime into the purifier (see figure 5).Here we note that since in our setup the baby universe has boundaries, flux lines can end on the boundaries of the baby universe as figure 5. Namely, in measuring the total flux, we also need to consider the purifier (right spacetime of figure 5) or equivalently the baby universe in addition to the original spacetime (left spacetime of figure 5).By the usual gravitational Gauss law, if we just measure the flux of the original spacetime only (left spacetime of figure 5), then we cannot specify the exact mass.The modification is not visible within the coarse-grained precision.However, without the modification, we may encounter many problems, e.g., violation of the conservation law.
There are several other implications of the generalized Gauss law (3.6) as well.First, the existence of the baby universe boundary energy term indicates that the gravitational Gauss law does not precisely hold within the original black hole spacetime, T bulk = H ∂BH [h] in general.For instance, one way to think about the generalized Gauss law of the form (3.6) is, it relates the spectrum of the fine-grained part H ∂BU [h] to the coarse-grained part H ∂BH [h].We expect that the fine-grained part is discrete, and the typical differences between two nearest energy eigenvalues are of order e −S BH .This forces the coarse-grained part also discrete, which is necessary for unitary time evolution.
Let us estimate the magnitude of the violation of the gravitational Gauss law in the black hole spacetime.In order to obtain a unitary time evolution of an evaporating black hole, we need non-perturbative effects of order e −S BH , where S BH is the entropy of the black hole.This means that we need fine-grained states in a small energy window of order e −S BH , thus H ∂BU is of the same order.This leads us to the conclusion that i.e., the gravitational Gauss law is violated only non-perturbatively.We should emphasize that such a baby universe is different from those appearing by cutting Euclidean wormholes into half, in the semi-classical gravitational path integral.Such a baby universe is always closed and does not have any asymptotic boundary.Such a closed universe, corresponds to an additional factor of the von Neumann algebra of the CFT [106].On the contrary to this, our baby universe has an asymptotic boundary to encode the finegrained information of the state.It would be interesting to further investigate the relation between the two.< l a t e x i t s h a 1 _ b a s e 6 4 = " / m e m K 6 f w t o j 5 s f A N J U k I r x I k Q 9 u q 0 g H k W I Y e I U p 8 S I K K M W Y w h j S m J T 1 / + z u J t r a d 6 P 1 C G 8 D p D S q A 0 y T U G W l R A o 6 0 I Z y 8 A w Y n f 8 z g g x D 8 k J R X 7 8 p r m M g s P z A + z n j e M d 0 q K e f / 2 z U S 6 O 0 n A N E X R 3 a l 8 N P p g T Z u a E W T O c L 3 H C r D o x X D m Y w o h / + B w M 1 d u 5 y p + q H W C d H z 0 J V h z q U k + i M 2 q P Y T t H 4 a 5 a D i Y X S H W Z U v b y z V R V w r C U 7 j I t 1 o S m 9 0 J y R C n R h G p X a e W U 8 k v 7 1 3 D P L L i 3 L G M V 9 8 y 8 e 8 u 0 a t 1 T D F q s n v T 6 G U 8 h 2 z a n Q s A c g 8 q t H E W j C 5 l h r 3 p W s q 5 A G I c K R X 0 O k F N x s B v I 4 T 7 o r t B L U H o Y W o N g w r a O 2 6 3 s k E Y l M u a U e s s q B B p e p U a W A s L U A A z Z V h 9 z U p L T e x W q z f e w x / k e B i v b m p G X T e q g Z i k 8  We also speculate the realization of fine-grained degrees of freedom in terms of a baby universe with a boundary has an interesting application to the physic of a closed universe.Sometimes it is argued that the Hilbert space of such a closed universe is one-dimensional [106,107] 4 , because in the absence of boundary, the left hand of the gravitational Gauss law (3.1) is always zero, so any operations are not allowed at all.However, as we saw above, one way to obtain its fine-grained description is to connect it to an open baby universe with a boundary.Then the generalized Gauss law (3.6)does allow operations on the baby universe boundary only.It would be interesting to explore further implications of the observation.

Comment on gravitational dressing
In a theory with dynamical gravity, a local operator is not physical, since it is not diffeomorphism invariant.One way to make it diffeomorphism invariant is to connect the local point P to a point P ∂ on the boundary, via a gravitational Wilson line, i.e., φ(P ) → φ(P )W gravity (P, P ∂ ).This prescription is called gravitational dressing.In [84] it was argued that such a gravitational dressing of a local operator on the island region leads to an inconsistency of the island prescription.This is because the relevant gravitational Wilson line connects a point on the island to a point on the conformal boundary of the AdS black hole.However, this sounds puzzling, because whereas the island prescription asserts an operator < l a t e x i t s h a 1 _ b a s e 6 4 = " r e u + v 6 b X 9 D / 0 3 H c 4 M p e E i m l r s U 5 l D q J n H + d 4 2 7 3 0 c 2 v I d F G J V / f f O 3 f f e / + D D j + 7 t V T / + 5 N P P P r / / 4 I t X o T c J b P j S 9 h w v e G O Z I X S Q C 1 9 i h B 3 4 x g + g O b Y c + N q 6 / D a a f / 0 W B i H y 3 J / w 1 I e 9 s T l 0 0 Q D Z J i Z D F w 9 q 3 5 x b c I j c G U a X 7 3 x k 4 0 k A 5 9 3 Q N h 1 4 3 F K 0 X r X a k B f X K R x g 8 L 2 N P X 8 S y o V X 9 d z 2 v K C P X B N D 0 M T 2 P j A x a L a U x 1 J 7 / y g 5 a d k t N i t r k q x l Z z U 2 q 5 J J J W d a Z 9 P 8 2 R S u 5 7 8 J R 2 i A l / B q e g U i 8 7 M g o 0 q q o t O F + o F 5 B X z H w 6 A b j j 0 P j y S A o R s 5 k j i m 0 w P L R U e Y a g s s x l j i E / 5 x u u T B c 9 i P W s 5 8 D w K R G v n p i J 8 4 z H n j Z T 1 + v D r m l 5 b 0 m T V n t g 9 s w x 7 U t w 4 j l w m S S s d / k G r P X 3 G w 3 i v 8 l X l 6 0 q z 0 q o 8 q j y t n F R e V F 5 W 7 N q f t b 9 q f 9 f + a f z Y m D Z + b f z G R O / e 4 c 9 8 W U l c j T / + A 3 c u 1 L w = < / l a t e x i t > Black Hole

Hawking Radiation
Black Hole (Purifier) Hawking Radiation (Purifier) on the island region locally acts on the radiation Hilbert space, the gravitational Wilson line attached to it enters the entanglement wedge of the black hole, thus it does change the state of H BH .
In our point of view, the above paradox is naturally resolved, since in the presence of the baby universe with a boundary, the gravitational Wilson line can end on this (see figure 4).Furthermore since this new boundary belongs to the radiation degrees of freedom after the Page time, it is still an operator on the radiation Hilbert space, even after the gravitational dressing.

Discussion
In this paper, we studied a partially fine-grained description of semi-classical evaporating black holes, by introducing an auxiliary system called a baby universe.We argued that in usual consistent long-range gravitational theories, the gravitational Gauss law must be modified by the baby universe connected to the original spacetime, and when there is an island, this modification is crucial to get results consistent with the idea of the entanglement wedge reconstruction.
It would be interesting to study concrete geometric models, to understand further detailed properties of this system.A class of the candidate geometries is the multi-boundary wormhole solution of three-dimensional Einstein gravity with a negative cosmological constant.It is convenient to use the coordinates in which the metric of AdS 3 takes the following form where dΣ 2 2 denotes the metric of two-dimensional hyperbolic space.These multi boundary wormholes are constructed by taking appropriate quotient of the hyperbolic space by the isometry group SL(2, R) × SL(2, R).Such a geometry has multiple conformal boundaries, on each of which we can define a CFT Hilbert space.In each asymptotic region, there is a horizon, whose area counts the number of degrees of freedom in the boundary Hilbert space.For simplicity, below let's consider such a geometry with three asymptotic boundaries.These three boundaries represent the Hilbert space of Hawking radiation H R , the black hole H BH , and the baby universe H BU .Thus, one can identify the horizon area of each asymptotic region with the entanglement entropy of each Hilbert space computed in (A.8), (A.12), (A.16) in Appendix A. The region behind these horizons is identified with the Einstein-Rosen bridge which connects the original black hole with the baby universe, discussed in the body of this paper.The geometric description manifests the following entanglement structure of (2.6).When k = dimH R is small, which models the beginning of the black hole evaporation, this system is almost a bipartite in which H BH and H BU are entangled.As we increase k, the cross section of the ER bridge gets larger, and at sufficiently late times k 1, H BU becomes mostly entangled with the radiation Hilbert space H R .This means that the state (2.6) is reconstructable from the two Hilbert spaces H R and H BU .and in the last line to emphasize the ensemble average of the reduced density matrix ρ (M )R we introduced the notation ρ (M )R M defined by the second line.We note that the average operation is given by the relationship (2.5) explicitly.
Next we consider the Rényi entropy for the reduced density matrix (A.6) where in the third line we distinguished the labels M 1 , • • • , M n , take the ensemble averages for factors, which have the same label M s , and in the forth line we used the relationship (2.5).
Therefore we obtain the von Neumann entropy (A.9) As in the previous case, we consider the Rényi entropy (A.10) where in the fourth line we used the relationship (2.5), and we note that N = e S BH .
From this Rényi entropy, we get the von Neumann entropy of the union of the naive Hawking radiation and the baby universe (2.9) (A.12)

< l a t e x i t s h a 1 _
b a s e 6 4 = " D J G D E G T K 1 n I / F N x 8 e h R / s 5 4 m e 6 w = " > AA A Q v H i c x V f r b t s 2 F F a 7 S 2 t t b d z 1 Z / 8 w i w P E m G R Y T t I L v A x d i g 0 p M G C Z 2 i Q F H C G g K N o m T I k G R S d I j L z O 3 m k v M 4 y U K N u 6 2 M 2 l w A Q Y p n n O + c 7 H c 5 P p j y m J R b v 9 z 4 O H X 3 3 9 z b e P H t f M 7 7 5 / 8 n S t / u y H 4 5 h N O M J H i F H G P / k w x p R E + E g Q Q f G n M c c w 9 C k + 8 U f v l P z k H P O Y s O i j u B x j L 4 S D i P Q J g k J u n T 1 7 + v e p j w c k m g o y u h o T J C Y c X / d i B C n e a 7 d 2 P d M 0 N + 3 C 8 y c n 0 g L S 4 r 5 p 1 k 4 R Y z y Q Q o H B F h V N A A X Y s n e s n W Y X A F C U + z O 5 r R Q K Y q 7 N l 1 h z P x M r Y 7 N W B E c a / L X V V u K 8 r R Z q m V n b z M 4 E W I T N T Z B X d 2 F A X B k l F v 2 R M W 7 t W k 7 r Z R N 0 V y i 7 m u B c t 0 i y D D x X X Q 0 6 1 1 P s f w 0 + g P 0 D c I g j z m I M A g I H H I Y S I + D w I k 2 E b S c B 7 8 4 2 u d 7 k v o 6 e 2 u 4 F G D E u X V p 6 I T 3 u X Z H B F R x 4 c x x p 2 r 2 Z g V / p g w m B A 0 s M C R o t o v o L q H k V v n C A W p K v f S i G 4 N 3 v H 8 t n R K v O i u a u U h w V 2 I Q 0 S M O b E T g n j G K R E R h T J k A v D h k T Q w s I H K l + 2 n M 8 M M 9 RD K a p 8 6 2 G + l q X z b O e y + 9 P W + 3 W t i U / z U a z k P n r 7 v 3 c 8 t Q t L 7 t 1 m 3 b i 1 i 7 7 d V O / e c / e z e h H L M A 9 H 1 N 2 4 Y G p W 6 T v 3 Z B M G U U m 5 L d I w G h A c Y g j A U 5 w M M C A 9

Figure 1 :
Figure 1: The Penrose diagram of the AdS black hole attached to the non-gravitating heat bath.We take the radiation region R (violet solid line) in the heat bath.After the Page time, the island region I (blue solid line) becomes non-empty, and the black hole region BH (orange solid line) is outside the horizon of the black hole.The entanglement wedge of the Hawking radiation R is the union of the domain of dependence of the radiation R and island I regions (violet and blue shaded regions), and that of the black hole BH is the domain of dependence of the black hole region (orange shaded region).

Figure 2 :
Figure 2: The Penrose diagram of the AdS black hole attached to the non-gravitating heat bath with a small local operation φ(P I ) (P I ∈ I, red dot) in the island region I.The entanglement wedge of the black hole BH (orange shaded region) includes the asymptotic AdS boundary.The gravitational Wilson line W (P I , P R ) (green line) connecting two points P I and P R (P R ∈ R, cyan dot) intersects the entanglement wedge of the black hole (orange shaded region).
y 5 J X h a + p a 3 I P U U 0 k r z m x O S S 2 S S Q i Z R Z G p 6 4 p y S t 8 U K b a l B z i k r h O u 6 5 Z x L r Z 6 M e 1 3 3 r G F 1 S q y b 9 N J 2 1 m D a S o d p Z 6 1 l e r D t o s 0 6 6 o y N M r Y N m b y S r r q +

Figure 3 :
Figure 3: Schematic picture of the geometry of the AdS black hole coupled the bath CFT (left Penrose diagram) and the baby universe geometry (right red Penrose diagram) connected by the Einstein-Rosen bridge (transparent green shaded region), corresponding to the state(2.14).After the Page time, the fine-grained Hawking radiation R is the union of the Hawking radiation R (violet region) and the baby universe BU (red region).We regard the above spacetime describing this union by gluing two distinct asymptotic boundary regions BU and R .The island region I is connected to the fine-grained Hawking radiation R ∪ BU through two paths, path 1 and 2. The path 1 (thick blue dotted line) intersects with the entanglement wedge of the black hole BH (orange shaded region), but the path 2 (thick green dotted line) does not intersect with that.

Figure 4 :
Figure 4: Schematic picture of the geometry of the AdS black hole coupled the bath CFT (left Penrose diagram) and their copy (right Penrose diagram) connected to the original spacetime through the wormhole (blue region), corresponding to the state (2.14).The local operator φ in the island (cyan dot) can be gravitationally dressed with a gravitational Wilson line W gravity (P, P puri. ) (green line) which ends on the baby universe (right Penrose diagram) without intersecting the entanglement wedge of the original black hole degrees of freedom (orange shaded region).
4 9 e L i w Z 3 f z / u e P R l D F 9 u O G Y b d a D m p p f p Y s o j u M D j W x 2 O p / x b 5 o W u O Y d i b h d h 0 + 6 b j u X B e P Z + E k N D o S 3 M I Z + Y 4 D K d j a w 4 a Y x O P w v R c N J g 3 1 5 3 g w e P e D L n +

Figure 5 :
Figure 5: Schematic picture of flux lines on the geometry corresponding to (2.14) (, similar to the figure discussed in [105]).The dotted lines are horizons.A local operator φ (red dot) is put at the original spacetime (left region).The spacetimes are connected by the Einstein-Rosen bridge (blue region).Some of flux lines (orange lines) escape into the other spacetime (right red region) through the Einstein-Rosen bridge .

(2. 7 )
S vN [ρ R ] = S vN [ ρ (M )R M ] = − lim n→1 ∂ n tr R ρ n R = log k. (A.7)This von Neumann entropy is coincides with the Hawking's result, and it implies the information paradox.We note that this von Neumann entropy is equal to that of the the union of the black hole and the baby universe BH ∪ BU S vN [ρ BH∪BU ] = S vN [ρ R ] = log k. (A.8) A.2 The entropy of the naive Hawking radiation and the baby universe S vN [ρ R∪BU ] = S vN [ρ BH ] To get the von Neumann entropy of the naive Hawking radiation and the baby universe R ∪ BU (= R), we consider the following reduced density matrix ρ R∪BU = tr BH [|Φ Φ| BH∪R∪BU ] = M N √ p M p N (tr BH |Ψ M Ψ N | BH∪R ) ⊗ |M N | BU = βi |α β| R ⊗ |M N | BU .
vN [ρ R∪BU ] = − lim n→1 ∂ n tr R∪BU ρ n R∪BU = log N = S BH .(A.11)This von Neumann entropy is also equal to that of the black hole BH S vN [ρ BH ] = S vN [ρ R∪BU ] = S BH .

Table 1 :
we provide details of this calculation.The result reproduces the behaviour of the Page curve after the Page time, giving the Bekenstein-Hawking entropy S BH .Therefore by appropriately dividing the total system BH ∪ R ∪ BU , we can get the von Neumann entropy which obeys the Page curve (see table 1) 3 .How to divide the total system BH ∪ R ∪ BU into two sub systems before and after the Page time, and the corresponding von Neumann entropies.
vN [ρ BH∪BU ] = S vN [ρ R ] = log k After the Page time BH R ∪ BU S vN [ρ BH ] = S vN [ρ R∪BU ] = S BH Then each fine-grained state |Ψ M |M belongs to different superselection sector, because each α state does.In particular, this means that off diagonal element of matrix Ψ M | M |(O ⊗ I)|Ψ N |N for any local operator O on the black hole BH and the Hawking radiation R vanishes, therefore any local measurement on them can not distinguish the entangled pure state (2.6) with the mixed state only with classical correlation of the following form ρ