Black Hole as a Quantum Field Configuration

We describe 4D evaporating black holes as quantum field configurations by solving the semi-classical Einstein equation $G_{\mu\nu}=8\pi G \langle \psi|T_{\mu\nu}|\psi \rangle$ and quantum matter fields in a self-consistent manner. As the matter fields we consider $N$ massless free scalar fields ($N$ is large). We find a spherically symmetric self-consistent solution of the metric $g_{\mu\nu}$ and state $|\psi\rangle$. Here, $g_{\mu\nu}$ is locally $AdS_2\times S^2$ geometry, and $|\psi\rangle$ provides $\langle \psi|T_{\mu\nu}|\psi \rangle=\langle0|T_{\mu\nu}|0 \rangle+T_{\mu\nu}^{(\psi)}$, where $|0\rangle$ is the ground state of the matter fields in the metric and $T_{\mu\nu}^{(\psi)}$ consists of the excitation of s-waves that describe the collapsing matter and Hawking radiation with the ingoing negative energy flow. This object is supported by a large tangential pressure $\langle0|T^\theta{}_\theta|0 \rangle$ due to the vacuum fluctuation of the bound modes with large angular momenta. This describes the interior of the black hole when the back reaction of the evaporation is considered. The black hole is a compact object with a surface (instead of horizon) that looks like a conventional black hole from the outside and eventually evaporates without a singularity. If we count the number of self-consistent configurations $\{|\psi\rangle\}$, we reproduce the area law of the entropy. This tells that the information is carried by the s-waves inside the black hole. $|\psi\rangle$ also describes the process that the negative ingoing energy flow created with Hawking radiation is superposed on the collapsing matter to decrease the total energy while the total energy density remains positive. As a special case, we consider conformal matter fields and show that the interior metric is determined by the matter content of the theory, which leads to a new weak-gravity conjecture.

(ψ) µν , where |0 is the ground state of the matter fields in the metric and T (ψ) µν consists of the excitation of swaves that describe the collapsing matter and Hawking radiation with the ingoing negative energy flow. This object is supported by a large tangential pressure 0|T θ θ |0 due to the vacuum fluctuation of the bound modes with large angular momenta l ≫ 1. This describes the interior of the black hole when the back reaction of the evaporation is taken into account. In this picture, the black hole is a compact object with a surface (instead of horizon) that looks like a conventional black hole from the outside and eventually evaporates without a singularity. If we count the number of configurations {|ψ } that satisfy the self-consistent equation, we reproduce the area law of the entropy. This tells that the information is carried by the s-waves inside the black hole. |ψ also describes the process that the negative ingoing energy flow created with Hawking radiation is superposed on the collapsing matter to decrease the total energy while the total energy density remains positive. Finally, as a special case, we consider conformal matter fields and show that the interior metric is determined by the matter content of the theory, which leads to a new weak-gravity conjecture.

Introduction
Black holes evaporate [1]. This property defines the black hole itself. The Penrose diagram of the space-time should have the same topology structure as the Minkowski space-time, where there is no event horizon or singularity. The black hole should be formed by a collapsing matter and evaporate in a finite time. This view is an accepted consensus in the context of quantum theory [2]- [27] 1 .
To examine the interior of the black hole, we first suppose that we throw a test spherical shell or particle into an evaporating spherically symmetric black hole. If it evaporates completely in a finite time without any singular phenomenon, the particle should come back after evaporation. Let's take a closer look at this process. As the particle comes close to the Schwarzschild radius, it becomes ultra-relativistic and behaves like a massless particle [39]. Then, if the back reaction of evaporation and the motion of the particle are considered together, the particle does not enter the Schwarzschild radius but moves along an ingoing null line just outside it, as we will see below [9]. Eventually, it returns to the outside through the center after evaporation. Now, let us consider a process in which a spherical matter collapses to form a black hole. We can focus on the motion of each of the spherical layers that compose the matter because of the spherical symmetry. As a layer approaches the Schwarzschild radius that corresponds to the energy of itself and the matter inside it, it moves at the speed of light. At the same time, the time-dependent space-time causes particle creation, and the energy begins to decrease [40,41]. Applying the above result, we can see that the layer keeps falling just outside the Schwarzschild radius. As this occurs for all the layers, the entire of the collapsing matter just shrinks to form a compact object. It has (instead of horizon) a null surface that evaporates eventually. This is the black hole [9,14,16,24]. In this paper, we show that this story can be realized in field theory.

Basic idea
We first explain the above idea of the black hole more precisely [9,14,16,24].
Step1. Suppose we throw a spherical shell with an infinitely small energy to an evaporating spherical black hole. As we see in the following, the black hole evaporates to disappear before the shell catches up with "the horizon", and thus the shell will never enter "the horizon".
Because of the spherical symmetry, the gravitational field that the shell feels is determined by the energy of itself and the matter inside it no matter what is outside the shell. Therefore, the metric which determines the motion of the shell near the black hole is given by the time-dependent Schwarzschild metric: where M(t) ≡ a(t) 2G is the energy inside the shell at time t (including the energy of the shell itself) 2 . This metric has G t t = 0, which means that there is approximately zero energy density near the black hole 3 , while it has G r t =˙a (t) r 2 , which corresponds to the total outgoing energy flux through a sphere with radius r, J = 4πr 2 −T r t 4 . Therefore, we assume that a(t) decreases according to the Stefan-Boltzmann law of Hawking temperature T H = 4πa(t) : (1.2) Here, σ is the intensity of the Hawking radiation, which is determined by dynamics of the theory 5 . In general, it takes the form of σ = kNl 2 p , where N is the degrees of freedom of quantum fields in the theory, k is an O(1) constant, and l p ≡ √ G is the Planck length.
Suppose that the shell consists of many particles. If a particle of them comes close to a(t), the motion is governed by the equation for ingoing radial null geodesics, no matter what mass and angular momentum the particle has 6 . Here, r(t) is the radial coordinate of the particle. At this point, we can see a general property of (1.3): Once a particle starts from a position outside r = a(t), the r.h.s. of (1.3) is always negative as long as a(t) is nonnegative. This means that the particle keeps approaching r = a(t) without passing it. Therefore, when a(t) decreases to zero in a finite time, the particle will reach the center (r = 0) and return to r → ∞. See the left of Fig.1. Note that the time coordinate t describes the outside spacetime region (r > a(t)) globally.
Let us examine more specifically where r(t) will approach when a(t) evolves according to (1.2). We are interested in the difference ∆r(t) ≡ r(t) − a(t), which is much smaller than a(t), ∆r(t) ≪ a(t). Then, r(t) in the denominator of (1.3) can be replaced with a(t) approximately, and (1.3) becomes (1.4) 2 Note that the mass M (t) may be different from the usual ADM mass, generically. Outside the shell, there exists dilute radiation with energy density −T t t ∼ T 4 H ∼ a 4 , where T H ∼ a . Then, the outside region may have energy ∼ a 4 × 4πa 3 3 ∼ a , which makes the difference. In the following, however, we use the name "ADM energy" as usual to represent the mass we are considering. 3 It is well known that there is an ingoing negative vacuum energy near the evaporating black hole [40,42,43], which has −T t t ∼ O 1 Ga 2 < 0 as shown in section 8. We will include this contribution into the interior of the black hole. Therefore, we consider as the exterior the region with −T t t ≪ O 1 Ga 2 . 4 Later, we will give a more proper definition of the energy flux. See (2.32). 5 σ may depend on the size a, but the dependence should be small and we assume here for simplicity that it is a constant. Furthermore, σ may take a different value far away from the black hole. See [14,16] for a more general case. 6 See Appendix I in [16] for a precise derivation is the position of the particle and a(t) is the Schwarzschild radius. Left: The black hole evaporates before the particle reaches it, and the particle comes back after evaporation. Right: The particle cannot catch up with the Schwarzschild radius a(t) due to the back reaction of the evaporation.
The first term is negative, which is the effect of collapse, and the second one is positive due to (1.2), which is the effect of evaporation. The second term is negligible when ∆r(t) ∼ a(t), but it becomes comparable to the first term when the particle is so close to a(t) that ∆r(t) ∼ l 2 p a(t) . Then, the both terms are balanced so that the r.h.s of (1.4) vanishes. We have ∆r(t) = −a(t) da(t) dt . This means that any particle moves asymptotically as , (1.5) and so does the shell. By solving (1.4) explicitly 7 , we can check that this approach occurs exponentially in the time scale ∆t ∼ O(a) (see Appendix A) 8 . Note that instead of (1.1) we can also use the outgoing Vaidya metric [44] and obtain the result (1.5) [9,16]. In this sense, the asymptotic behaviour (1.5) is not sensitive to the choice of the metric near the black hole. This behavior can be understood as follows. See the right of Fig.1. The particle approaches the radius a(t) in the time ∆t ∼ a. During this time, the radius a(t) itself is slowly shrinking as (1.2). Therefore, r(t) cannot catch up with a(t) completely and is always apart from a(t) by −a da dt = 2σ a 9 .
Step2. Now, suppose that we add a spherical shell with a small but finite energy ∆M to the evaporating black hole with mass M. When the shell comes close to r = a ′ ≡ 2G(M + ∆M), the motion of the shell becomes lightlike, and particle creation occurs 7 By a numerical method, we can also solve (1.3) exactly and show (1.5). See Appendix B in [16]. 8 The above analysis is based on the classical motion of particles. We can also show that the result is valid even if we treat them quantum mechanically. See section 2-B and appendix A in [16]. 9 One might think that this result is strange because a typical time scale of evaporation ∆t ∼ a 3 /l 2 p is much larger than that of of collapse ∆τ ∼ a. Here, t is the time coordinate at infinity and τ is that of a comoving observer along the collapsing matter. It doesn't make sense to compare these two time scales, which are measured by different clocks. In the above we have considered the both time evolutions of the spacetime and particle in a common time t to reach the result (1.5).
around the shell. As we will see later, the sum of the radiations from the shell and the black hole is equal to the radiation from a larger black hole with mass M + ∆M. Then, the total system composed of the shell and the original black hole emits radiation as (1.2) where a is replaced with a ′ . Using (1.1) for a ′ (t) and applying the same analysis for the motion of the shell as above, we can see that the shell will approach to r = a ′ (t) + 2σ a ′ (t) . Thus, the total system evaporates as usual and has the size of r = a ′ (t) + 2σ a ′ (t) . Step3. At last, we consider time evolution of a spherical collapsing matter with a continuous distribution (instead of starting from a black hole). Due to the spherical Figure 2: A spherical collapsing matter becomes a dense object with a surface at r = a(t) + 2σ a(t) .
symmetry, we can regard it as consisting of many spherical shells with a small mass ∆M (see the left of Fig.2). Then, we can apply the result of the step 2 to each shell recursively because its time evolution is not affected by the outside due to the spherical symmetry. That is, the i-th shell approaches to r = a i + 2σ a i , where a i is the Schwarzschild radius corresponding to the mass inside the i-th shell (including the mass of the shell itself). In this way, many shells pile up and form a dense object with mass M (see the right of Fig.2.) The object has, instead of a horizon, a surface as the boundary between the high-density interior and the low-density exterior at If we see this object from the outside, it looks like a conventional black hole because 2σ a ≪ a. However, it is not vacuum and has an internal structure.
Step4. One might wonder if the above idea can be realized in field theory or not. To see this point simply, we examine the distance ∆r = 2σ a more because it is the typical length scale in this picture. The proper length ∆l is evaluated as where we have used (1.1). We here assume that N is large but finite, for example, O(10000): Then, ∆l is larger than l p , and ∆r = 2σ a is sufficiently long in order for the effective local field theory to be valid. In particular, the position of the surface (1.6) is meaningful physically.
A way to show the validity of the field theory more precisely is to construct a concrete solution and confirm its self-consistency. In this paper, we solve the semi-classical Einstein equation in a self-consistent manner to find the metric g µν and state |ψ which represent the interior of the black hole. Here, we treat gravity as a classical metric g µν while we describe the matter as N massless free quantum scalar fields. ψ|T µν |ψ is the renormalized expectation value of the energy-momentum tensor operator in g µν that contains the contribution from both the collapsing matter and the Hawking radiation. The solution tells that the interior has no singularity or trapped region if (1.8) is satisfied. Eventually, the object will evaporate like an ordinary object with the lifetime ∆t ∼ a 3 σ 10 . Therefore, the Penrose diagram has the same topology as the Minkowski spacetime, and the spacetime region with ∆r ∼ a and ∆t ∼ a 3 σ should be identified as the black hole in quantum theory (see Fig.3). In Appendix B, we check the spacetime of the black hole in various coordinate systems. Figure 3: The Penrose diagram of the quantum black hole. The matter becomes ultrarelativistic in the final stage of the collapse, and particle creation occurs inside it. Then, a dense object is formed with the size ∆r ∼ a and evaporates in the time ∆t ∼ a 3 σ .
Finally, one might wonder why the matter doesn't collapse. The solution answers: a large tangential pressure is produced inside and supports the matter against the gravity. The pressure is consistent with 4D Weyl anomaly and so strong that the interior is 10 (1.2) cannot describe the final stage of the evaporation because it diverges as a(t) → 0. A small black hole with a = O(l p ) should be described by some bound state in string theory and it may decay in a finite time. Therefore, we postulate that the remaining small part disappears in a finite time which is much smaller than ∆t ∼ O( a 3 σ ).
anisotropic locally (that is, the interior is not a fluid) and the dominant energy condition breaks down. Therefore, this object doesn't contradict Buchdahl's limit [45].

Strategy and result
The above idea has been checked partially by a simple model, a phenomenological discussion and the use of conformal matters [9,14,16,24]. Furthermore, by a thermodynamical discussion, the entropy density inside the object was evaluated and integrated over the volume to reproduce the area law [16]. Therefore, it seems that this picture is plausible and works universally for various black holes. However, there still remain several questions about this picture. What is the selfconsistent state |ψ ? What configurations do the matter fields take inside? How is the large pressure produced inside? Can we reproduce the entropy area law by counting microscopic states of fields? How does the energy of the collapsing matter decrease? These are crucial for understanding what the black hole is and how the information of the matter comes back after evaporation. In this paper, we solve directly (1.9) together with N massless free scalar fields to answer these questions.
We here explain our self-consistent strategy and the results. In section 2, we describe the formation and evaporation of the black hole as in Fig.2 explicitly by a simple model to construct a candidate metric. The exterior metric (for r ≥ R(a(t))) is given by (1.1), and the interior metric is shown to be static, which can be expressed as We write down two functions A(r), B(r) in terms of two phenomenological functions: one is the intensity of Hawking radiation σ and the other is a function η that provides the ratio between the radial pressure and energy density. We also show that this metric is locally AdS 2 × S 2 geometry. We are interested in the black holes most likely to be formed in gravitational collapse. As shown in section 5, the statistical fluctuation of the mass is evaluated as ∆M ∼ m p , where m p ≡ G is the Planck mass. Therefore, from a macroscopic perspective, all black holes with mass ∈ [M, M + m p ] are the same. We consider the set of states {|ψ } that represent the interior of such statistically identified black holes.
In section 3, we examine the potential energy of the partial waves of the matter fields in the interior metric. Modes with angular momenta l ≥ 1 are trapped inside, and they emerge in the collapsing process even if they don't exist at the beginning. We show that if such bound modes are excited, the energy increases by more than O mp √ N , which means that the number of excited bound modes is at most order of O( √ N) in the set {|ψ }. Therefore, those modes can be regarded as the ground state because O( √ N) is negligible compared to the number of total modes O( a 2 l 2 p ) (which is shown in section 5). On the other hand, s-wave modes can enter and exit the black hole and represent the collapsing matter and Hawking radiation. Thus, the state |ψ provides where |0 is the ground state in the interior metric, and T (ψ) µν is the contribution from the excitations of the s-waves.
In section 4, we evaluate 0|T µν |0 . We first solve the equation of motion of the scalar fields in the interior metric. We calculate the regularized energy-momentum tensor 0|T µν |0 reg in the dimensional regularization. Then, we renormalize the divergences and obtain the finite expectation value, 0|T µν |0 ′ ren . This contains contributions from the finite renormalization terms (α 0 , β 0 ), which correspond to the renormalized coupling constants of R 2 and R µν R µν in the action, respectively.
In section 5, we combine Bekenstein's discussion of black-hole entropy [46] and our picture of the interior of the black hole to determine the form of T (ψ) µν , which is fixed by a parameter a ψ .
In section 6, we solve (1.9) by using the ingredients obtained so far: g µν , 0|T µν |0 ′ ren , and T (ψ) µν . We determine the self-consistent values of (σ, η, a ψ ) for a certain class of the finite renormalization terms (α 0 , β 0 ). We then check the various consistency. Especially, we see that there is no singularity for a large N, and that the vacuum fluctuation of the bound modes with l ≫ 1 leads to the large tangential pressure 0|T θ θ |0 ′ ren . In section 7, we consider the stationary black hole which has grown up adiabatically in the heat bath. We count the number of the states of the s-waves inside the black hole to evaluate the entropy, reproducing the area law. Thus, the information is carried by the s-waves.
In section 8, to understand the mechanism by which the energy of the collapsing matter decreases, we use a 2D dynamical model in the outermost region of the black hole to study the time evolution of s-waves. We see that |ψ describes the process that the negative ingoing energy flow created with Hawking radiation is superposed on the collapsing matter to decrease the total energy while the total energy density remains positive.
Thus, three independently conserved energy-momentum flows appear in this solution: that of the bound modes of the vacuum 0|T µν |0 , that of the collapsing matter, and that of the pair of the Hawking radiation and negative energy flow, where the last two contribute to T (ψ) µν . These three form the self-consistent configuration of the black hole. In section 9, as a special case, we consider conformal matter fields and show that the parameters (σ, η) are determined by the matter content of the theory. Interestingly, the consistency of η provides a constraint to the matter content. For example, the Standard Model with right-handed neutrino satisfies the constraint but a model without it doesn't. Therefore, this can be regarded as a new weak-gravity conjecture which is required in order for the black hole to evaporate.
In section 10, we conclude and discuss future directions. Especially, we consider how the information comes back after evaporation if there are interactions between the collapsing matter, Hawking radiation and negative energy flow.

Construction of the candidate metric
In this section, we use a simple model which represents our basic idea as in Fig. 2, and construct a candidate metric for the interior [9,16,24] 11 . At this stage, we do not mind if it is the solution of (1.9) or not, which will be examined in section 6.

A multi-shell model
As in Fig.2, we consider a spherical collapsing matter consisting of n spherical thin null shells 12 . See Fig.4, where r i represents the position of the i-th shell. We assume that each shell behaves like the ordinary evaporating black hole if we look at it from the outside. Here, some part of the radiation emitted from a shell can be scattered by the other shells or the gravitational potential, but we neglect the effect for simplicity. (We will introduce it in subsection 2.2.1.) Then, because of spherical symmetry, for i = 1 · · · n, the region just outside the i-th shell can be described by the time-dependent Schwarzschild metric: 13 11 Some of the discussion below overlaps with the previous paper [24], but we give it to make the argument self-contained. We can also employ a thermodynamical discussion, which is universal and phenomenological [14]. 12 Here, we consider the shells infinitely thin. The use of the null shell is motivated by the fact that any particle behaves like ultra-relativistic one near r ∼ a as we have seen below (1.3). 13 Here, the description of the region outside each shell by (2.1) means that t i is the local time between r i and r i+1 . Therefore, dai dti is the derivative along an ingoing null line between r i and r i+1 .
Here, a i = 2Gm i ≫ l p , and m i is the energy inside the i-th shell (including the contribution from the shell itself). For i = n, t n = t is the time coordinate at infinity, and a n = a = 2GM, where M is the total mass. On the other hand, the center, which is below the 1-st shell, is the flat spacetime, because of the spherical symmetry. Therefore, we can regard that To complete the setup, we need to connect time coordinates, t n = t, t n−1 , · · · , t 0 = T . The point is that the trajectory of each shell, r = r i , is null from both sides of the shell. Therefore, we have the connection conditions: (2.6) determines r i (t i ) for a given a i (t i ), and then the solution r i (t i ) and (2.7) give the relation between {t i }. Thus, the coordinates are connected smoothly. When two different metrics are connected along a null hypersurface, a surface energymomentum tensor exists on it generically. By using the Barrabes-Israel formalism [48,47], we can calculate the surface energy ǫ 2d expresses the energy density of the shell with energy m i ≡ a i −a i−1 2G . In the expression of p (i) 2d , the first term corresponds to the total energy flux observed just above the shell, and the second one represents the energy flux below the shell that is redshifted due to the mass of the shell. Thus, the pressure is induced by the radiation from the shell itself. As we will see later, this pressure is positive, which means that, as the shell contracts, the pressure works to resist gravity, and Hawking radiation is dissipated.

The continuum limit
Let us take the continuum limit by m i ≡ a i −a i−1 2G → 0 and construct a candidate metric for the interior of the object. Especially, we focus on a configuration in which each shell has already come close to R(a i ) 15 : where (1.6) has been used 16 . We first solve the equations (2.5). By introducing we have Here, at the second line, we have used (2.7); at the third line, we have used (2.9) and assumed a i −a i−1 2σ a i ≪ 1, which is satisfied for a continuous distribution; and at the last line, we have approximated 2a i ≈ a i + a i−1 . With the condition (2.4), we obtain Now, the metric at a spacetime point (T, r) inside the object is obtained by considering the shell that passes the point and evaluating the metric (2.1). We have at 4σ , (2.14) 15 A more general case is discussed in [21]. 16 Due to the spherical symmetry, the motion of each shell in the local time t i is determined independently of the shells outside it. Therefore, the analysis for (1.5) can be applied to each shell.
where (2.9) and (2.12) have been used. From these, we can obtain the metric Note that this is static although each shell is shrinking physically 17 . Thus, our candidate metric for the evaporating black hole is given by under the assumption that the scattering effects are neglected. See Fig.5. This metric is continuous at the null surface located at r = R(a(t)) = a(t) + 2σ a(t) , where the total Schwarzschild radius a(t) decreases as (1.2). Here, we have converted T in (2.15) to t in (1.1) by the relation dT = e − R(a(t)) 2 4σ dt, which can be obtained by (2.10) and (2.12). As shown in Fig.5, this object evaporates emitting Hawking radiation through the null surface although the interior is static. Thus, the whole system is time-dependent. This object has the surface at r = R(a(t)), instead of a horizon. We can also check that there is no trapped region inside. However, the redshift is exponentially large inside, and time is almost frozen in the region deeper than the surface by ∆r σ a . Therefore, this object looks like the conventional black hole from the outside.
In the construction, we have started from the vacuum spacetime and piled up many shells. Then, the center region around ∆r ∼ √ σ is extremely delayed due to the large redshift, and is still flat because of the spherical symmetry. In fact, around r ∼ √ 2σ, the interior metric of (2.16) is approximately the same as the flat metric (2.3) because of dT = e − R(a(t)) 2 4σ dt. We assume in this paper that the region 0 ≤ r √ 2σ is flat.

Generalization of the metric
The interior metric of (2.15) doesn't contain the effect of scattering which has been mentioned above (2.1). Actually, the metric has −G t t = G r r = 1 r 2 for r ≫ l p , which is, through (1.9), equivalent to − T t t = T r r . Here, − T t t and T r r are the energy density and radial pressure, respectively. This indicates that from a microscopic point of view, the collapsing matter and radiation move radially in a lightlike way without scattering. In this subsection, we introduce the effect of the scattering and generalize the candidate metric (2.16) [14,16,24].

Another phenomenological function f
We first examine the energy-momentum flow in the general static metric (1.10). The self-consistent energy-momentum flow must be time-reversal, which can be characterized by Here, κ is a function. As we have seen that the interior of the object is very dense, we assume that η = η(r) is a function of O(1) which varies slowly 18 . l and k are the radial outgoing and ingoing null vectors, respectively: where T kk stands for T µν k µ k ν , and so on. Furthermore, this can also be expressed in terms of the ratio between − T t t and T r r : Therefore, η must satisfy 1 ≤ η < 2. (2.21) Here, the first inequality is required by the fact that in (2.19), η − 1 plays a role of the ratio between two energy flows, which must be positive. The second one is needed in order for the pressure to be positive under − T t t > 0. Now, we discuss the physical meaning of η from a microscopic point of view. See Fig.6. We focus on one of the null shells that make up the black hole and consider the moment when the size is r, which is represented by S in Fig.6. The vector P µ ≡ T µk expresses the energy-momentum flow through the shell, which moves lightlike inward along k. If the radiated particle is massless and propagates outward along the radial direction without scattering, P µ should be parallel to l µ , which means η = 1. Therefore, Figure 6: The meaning of a phenomenological function η. η − 1 represents the deviation from this ideal situation. η − 1 can become non-zero if the massless particle is scattered in the ingoing direction by the gravitational potential or interaction with other matters 19 . This is because such a scattered particle comes back to the surface in the time scale of O(a) according to (1.3), and produces an energymomentum flow along k direction. In this way, η is a phenomenological function that depends on the detail of microscopic dynamics 20 .

The candidate metric
We determine A(r) and B(r) by considering (2.20) for a given η. Here, we assume for simplicity that η is a constant satisfying (2.21). We can expect that the introduction of η( = 1) should not change the functional forms of A(r) and B(r) in a drastically different way from those of (2.15) 21 . Therefore, we can put where C 1 and C 2 are some coefficients such that C 1 , C 2 → 1 in η → 1 22 . Then, using (2.20) and the Einstein equation (1.9), we have for r ≫ l p that is, C 1 = 1 η C 2 . Therefore, (2.22) becomes (2.24) 19 If the radiated particle is massive, P µ is timelike, and we have η > 1. In this paper, however, we consider massless particles basically. 20 The form of η in the surface region where time flows may be different that in the deeper region where time is frozen. See section 2-D of [16]. 21 We can justify this expectation by a thermodynamical discussion [14]. 22 In A of (2.22), we have dropped a term proportional to R(a(t)) 2 , which corresponds to considering A in ds 2 = − e A B dT 2 + · · · . This does not affect the calculation (2.23) because G t t = G T T .
At this stage, the intensity σ is arbitrary, and we can redefine it and remove C 2 without losing generality to obtain Thus, introducing the scattering effect by η, (2.16) has been generalized to This metric is again continuous at the surface at r = R(a(t)) ≡ a(t) + 2σ a(t) , where a(t) decreases as (1.2). The center is assumed to be flat, which requires that η = 1 there. The center flat metric (2.3) is expressed in terms of t approximately as to O(r −2 ) for r ≫ l p . As mentioned at the end of section 1, a large pressure exits in the angular direction, which is almost Planckian. Next, we calculate the curvatures in √ 2σ r ≤ R(a(t)): This means that if (1.8) and (2.21) are satisfied, the geometry has no singularity, which leads to the Penrose diagram shown in Fig.3. We here discuss Hawking radiation in this picture. We consider the total energy flux through an ingoing null shell along k in Fig.6: (2.32) 23 We consider this metric as a first approximation metric in that we have neglected the effect of dilute radiation near the black hole and connected the interior and exterior metrics directly at r = R(a(t)).
To obtain a more physical description, we would need to consider such an effect and construct a "interpolation" metric connecting the two metrics in a smooth manner. However, the most dominant effect of the back reaction of the evaporation is incorporated into the interior one of (2.26).
where (2.18) has been used, and u ≡ e − A 2 ∂ t is the vector of the local time (like t i in Fig.4) 24 . Applying this formula to the interior metric of (2.26) and using (1.9) and (2.28), we have J = 4πr 2 −B −1 1 8πG G t t = σ Gr 2 → σ Ga 2 , where in the limit we have considered the case of r = R(a(t)) ≈ a(t). For the exterior metric of (2.26), on the other hand, we have J = 4πr 2 Thus, these agree each other, and (2.32) represents to the total flux of Hawking radiation from each shell.
Finally, we argue that the interior metric of (2.26) is AdS 2 × S 2 locally. First, we can check that in general, a spherically symmetric metric Then, (2.33) is satisfied for l = 2ση 2 around the point r we are focusing. Thus, the interior metric can be approximated locally as AdS 2 × S 2 geometry: 26 . This means that the local symmetry for a fluid before the matter collapses becomes that for AdS 2 after the black hole is formed. It would be interesting to study this more in future.

Field configurations
We consider N massless free scalar fields in the candidate metric (2.26) and (2.27) and study their configurations to find a candidate state. This action leads to the equation of motion 27 24 This definition of the energy flux is consistent with the comment in footnote 13. 25 Comparing g tt in the both metrics, we put f (r) = l 2 z 2 , from which we have ∂ r f dr = −2 l 2 z 3 dz. Then, we can obtain dr = − 2 z∂r log f dz and see that h(r)dr 2 = l 2 z 2 dz 2 holds if (2.33) is satisfied. 26 In fact, the Ricci scalar (2.29) is approximately − 1 η 2 σ , which is negative and constant. 27 For simplicity, we write φ a as φ in the following.

Classical effective potential and bound modes
Before going to quantum fields, we study the classical behaviour of matter to see intuitively what happens in the spacetime (2.26). To do that, we analyze (3.2) in the classical approximation to draw the effective potential for the partial waves of the fields.
We first consider the general static metric (1.10) and set where i = (ω, l) and the normalization factor N i will be fixed in subsection 3.2. Then, the field equation (3.2) becomes This equation (3.4) takes the same form as the Schrödinger equation with energy E = 0 and the potential V (r) = −p 2 i (r). Therefore, the classically allowed region is determined by the condition p 2 i (r) ≥ 0. In general, when one studies a field equation in a curved spacetime, he needs to take the curvature effect into account, which is expressed here as the "mass term" M, the two derivative term of the metric 28 However, M should not significantly affect the purely classical motion of matter because a classical particle equation (Hamilton-Jacobi equation) does not include derivative terms of the metric. Therefore, in this subsection, we ignore M for a while. Let us draw the classical effective potential p This is continuous 30 . Note that the frequency ω is measured at r ≫ a. 28 For example, in studying cosmological particle creation, an equation of the same form as (3.4) is used to analyze an adiabatic calculation [40,41]. 29 In fact, M is the same order as R inside the black hole. See (2.29). 30 In particular, it is continuous at r = √ 2σ because the center region is flat, which means that η = 1 for r √ 2σ, as we have mentioned below (2.26).
For l = 0, we have the left of Fig.7, which shows that the whole region is allowed classically; s-waves can enter the inside from the outside, pass the center, and come back to the outside, which takes an exponentially long time because of the large redshift. We refer to such modes as continuum modes in the following. For l ≫ 1, on the other hand, the classically allowed region consists of two disconnected domains as in the right of Fig.7. The outer one indicates that such modes coming from the outside are reflected by the barrier l(l+1) r 2 while the inner one shows that they are trapped inside, which we call bound modes. i (r) 2 > 0 is allowed classically. Now, we discuss the behavior of each mode in the formation process of the black hole. We first study the condition for a mode with (ω, l) to enter the black hole with a from the outside. From (3.7), for r ≫ a we have p (cl) i (r) 2 ≈ ω 2 − l 2 r 2 , which becomes zero at r = l ω . This means that the mode is reflected at the point and returns to the outside. Therefore, the condition is r = l ω < a, that is, l < ωa. From this, we can also see that a mode with (ω, l) composing i-th shell of the multi-shell model in Fig.4 can enter inside if l < ωa i . As we will see in section 5, most of such modes have the energy ω ∼ a i . Then, the condition becomes l < ωa i ∼ 1. Thus, we conclude that only the continuum modes of s-wave can enter the black hole from the outside.
Although modes with l ≥ 1 cannot enter from the outside, they emerge inside the black hole in the formation process. To see it, we study how the potential p (cl) i (r) 2 changes for a given (ω, l) as a increases from zero. See Fig.8. Initially, there is no mass, a = 0, so the spacetime is flat and the potential is given by the purple line. When the mass becomes larger than m p , the allowed region appears inside, which is described by the blue line. Then, the bound modes emerge there. As the mass increases, the potential grows up and the allowed region is broaden (see the green and yellow lines). Then, the outer zero point (which is depicted by the arrows) moves inward, which means that as the mass increases, the gravitational attraction increases, allowing the mode to overcome the centrifugal repulsion and penetrate more inside. In this way, many bound modes emerge inside the black hole independently of the initial state.
Finally, we note that if the curvature term M is considered in (3.5), the bound mode of s-wave can also emerge inside the black hole. In fact, for This means that the s-waves with ω < O( 1 R(a) ) are trapped inside r = R(a). See Fig.9. Thus, the s-wave in the interior metric can be in a continuum mode as in the left of Fig.7 or in a bound mode as in Fig.9.

WKB approximation
We now consider quantum fields and try to solve (3.2) by WKB method. In the interior metric of (2.26), the leading WKB solution is given, through (3.3), (3.4) and (3.5), by , B(r) = r 2 2σ , i = (α, l, m), and θ i is a phase factor, where α labels the frequencies for each l and m. For the bound modes, the frequencies are quantized, ω i = ω nl (n ∈ Z), and are approximately given by 32 where p nl (r) vanishes at r = r + nl , r − nl . The normalization is fixed as (see Appendix C) On the other hand, the s-waves have continuum modes (as in the left of Fig.7), which we will discuss more in section 5.
We also have the commutation relation and 14) The ground state |0 for all modes {u i } in the interior metric is characterized by

A candidate state
We show that the bound modes are in the ground state while the continuum modes of s-wave are in an excited state. If |ψ is the ground state for the bound modes {u i } i∈B , it satisfies a i |ψ = 0 for i ∈ B, (3.16) 31 Note again that we are writing φ a just by φ, and u i (x) depends on the kind of fields. 32 does not appear in the l.h.s. of (3.10) because p i is not momentum but wave number.
where B stands for the set of the bound modes. From Fig.8, (3.16) is independent of the initial state of the system because the bound modes emerge in the region disconnected from the outside. In general, the ADM energy M inside radius r in a spherically-symmetric spacetime is given by [39] (3.18) Then, we can define the ADM-energy increase (∆M) i of the first-excited state of the i-th bound mode (of a component of fields where |ψ satisfies (3.16) and the integration interval is the allowed region of the bound mode u i . This is finite because the UV divergence is subtracted by the second term.
We estimate the order of this by using the WKB approximation. We first check the condition for the bound mode to exist, which is determined by the quantization condition (3.10). That is, the quantum number of the i-th bound mode must be at least greater than 1: (3.20) where we have considered only the most dominant term of (3.5) in the WKB approximation and evaluated the integration as 33 . Therefore, the frequency must satisfy Similarly, we can calculate from (C.9) The function r 2 2σ e − r 2 4ησ is like a Gaussian with width ∼ √ σ, and the integration has the dimension of the length. Therefore, we can have .8) and (3.9). Using (3.13) and (3.16), we can see easily i; ψ|φ 2 |i; ψ = 2ω 2 i |u i | 2 + ψ|φ 2 |ψ . We check the form Then, we evaluate Here, at the first line, we have applied ∂ r only to cos drp i in u i (x); at the second line, we have used (3.9), cos 2 drp i ≈ 1 2 and |Y lm | 2 ≈ 1 4π ; and at the last line, we have picked up only the most dominant terms because we are interested in the order estimation.
Thus, we estimate (3.19) as Here, at the first line, we have used (3.24); at the second line, we have expressed (1); at the third line, we have employed (3.22); and at the last line, we have applied the condition (3.21). That is, we have 34 Here, as we will see in section 5, the statistical fluctuation of the mass of the black hole is O (m p ). Thus, (3.26) means that when the number of excited bound modes exceeds O( √ N), the excitation energy becomes larger than O (m p ), and the object becomes different from the black hole. Therefore, we can regard that the bound modes are in the ground state because O( √ N) is negligible compared to the number of total modes, which is order of O( a 2 l 2 p ) (see section 5). 34 In this sense, the spectrum of the black hole is quantized and gapped.
On the other hand, the continuum modes of s-wave are not restricted by the condition (3.21) because they are not trapped inside. Therefore, those modes can enter and exit the black hole as an excitation that represents the collapsing matter and Hawking radiation. Thus, the candidate state |ψ is a state in which the bound modes are in the ground state and the s-waves are excited, leading to (1.11). (In section 5, we will characterize |ψ more specifically.) 4 Energy-momentum tensor in the ground state |0 In this section we evaluate 0|T µν |0 , where |0 is the ground state (3.15).

Solutions of the bound modes
We construct the solutions of the bound modes in the interior, which will be used in subsection 4.3.

Breakdown of the WKB approximation
We first examine the validity of the WKB approximation in subsection 3.2. If we want to evaluate the energy-momentum tensor at a point r = r 0 inside the object, we need u i (x) at r = r 0 with various values of i = (ω, l, m). Then, the proper frequency ω and the proper angular momentum L at r = r 0 are physically important: 35 In terms of these, (3.5) is expressed as This means that the WKB approximation is good at r = r 0 for ω 2 ≫ L because the semi-classical treatment is valid for a large wave number: p i (r 0 ) ≫ 1 [49]. On the other hand, the approximation is bad for ω 2 ∼ L; r = r 0 becomes the turning point. Although the WKB approach has a potential to reproduce the UV-divergent structure of the energy-momentum tensor properly, it cannot determine the finite values without O(1) errors 36 .
For later analysis, we investigate more precisely how the approximation breaks down. In oder for the WKB analysis to be valid, the wave length λ i (r) ≡ 1/p i (r) must change 35 Here, we have dropped R(a) in A(r), which will not affect the followings. 36 Note that in subsection 3.3 we have used the WKB approximation only to evaluate the order of (∆M ) i in the range where the approximation is good.
slowly [49]. We pick up only the first term of (4.2) because the exponential factor changes most drastically as a function of r. We then evaluate at r = r 0 − ∆r where the derivative has applied only to the exponential. If ∆r = 0, the approximation at r = r 0 is good for ω ≫ 1/ ση 2 and bad for ω ∼ 1/ ση 2 , which is consistent with the fact that the curvature radius is ∼ ση 2 from (2.29). The evaluation (4.3) tells more: even when ω ∼ 1/ ση 2 , the approximation is good at

A perturbative method and the leading exact solution
The finite values of the energy-momentum tensor are important for our self-consistent discussion. We here develop a new perturbation method to solve exactly the field equation (3.4) in the interior metric of (2.26). Suppose that we want to determine ϕ i (r) at a point r = r 0 . Motivated by (4.4), we set and expand (4.2) as where ω and L are considered as O (1), and s is an expansion parameter. P Now, we can determine ϕ i (r) around r = r 0 by the perturbative expansion with respect to 1/r 2 0 . We put it as 37 Note that ση 2 ∼ √ N l p , which is near the Planck scale, and the proper frequency should be smaller than it: ω 1/ ση 2 . Therefore, log(1/ ω ση 2 ) > 0.
Combing this and (4.6), then (3.4) becomes This gives an iterative equation: By construction, ϕ and rewrite (4.9) as This is the Bessel equation, whose solutions are Bessel functions J A (ξ) and J −A (ξ).
Using the boundary condition that the mode is bounded, we can choose J A (ξ) properly. Considering the evaluation (4.4) and the region where the WKB solution (3.9) is valid, we can fix the normalization. See Appendix D. Thus, we obtain the leading solution (4.14) Substituting this and (3.11) for (3.3), the leading bound-mode functions in the interior metric of (2.26) without R(a) (which will not affect the following analysis) are given by Note again that ξ depends on ω nl and A on l.
The subleading solution ϕ (1) i (x) can be determined from (4.10) and (4.14), for example, by Green-function method. We leave it as a future task.
Finally, we make a comment on the origin of the leading potential P i (x). We can also use the AdS 2 × S 2 metric (2.34) and focus a region around r = r 0 to show that (3.2) becomes the Bessel equation. In this sense, the local AdS 2 × S 2 structure makes the field equation (3.2) solvable. More generally, the condition (2.33) is the origin.

Dimensional regularization and renormalization
In general, when one considers composite operators such as T µν (x), he needs to regularize them. We use the dimensional regularization technique, which has an advantage that it is covariant, so is the UV-divergent part. Before going to the explicit evaluation of 0|T µν (x)|0 ren , we here give the general discussion about the dimensional regularization and renormalization of the energy-momentum tensor.
The bare action of our theory on a d-dimensional spacetime is where the index B expresses the bare quantities. First, we don't consider quantization of gravity, and the quantum scalar fields are free. Therefore, the bare fields are the same as the renormalized ones: (4.17) Next, we introduce a renormalization point µ. Because there is no mass in the theory, the renormalization of the Newton constant does not appear. Therefore, by dimensional analysis, we can put where d = 4 + ǫ, and G is the 4D physical Newton constant, which is independent of µ. G B agrees with G in ǫ → 0. On the other hand, α B , β B and γ B need to be renormalized. In the limit ǫ → 0, they take the form (4.19) Here, we choose the counter terms with 1 ǫ as those which are required to renormalize the UV-divergent part of the effective action of N massless free scalar fields by the minimal subtraction scheme [40,41,50]. α(µ), β(µ) and γ(µ) are the renormalized coupling constants at a renormalization point µ.
Then, the equation of motion for g µν , δS B δg µν =0, is given by Here, T µν is the regularized energy-momentum tensor operator, which is formally given by (3.18) with S M replaced by the d-dimensional matter action S matter B of (4.16). The other tensors are proportional to the identity operator, which are given by Here, we have defined where the 4D renormalized energy-momentum tensor at energy scale µ is This is finite in the limit ǫ → 0, as we will see explicitly below. The second term in the r.h.s. of (4.25) is a finite renormalization which plays a role in choosing the theory. The point is that the µ-dependence of the renormalized coupling constants α(µ), β(µ), γ(µ) must be chosen so that the bare coupling constants α B , β B , γ B are independent of µ. From in ǫ → 0. We can do the same procedure for the other couplings. From these, we obtain Here, α 0 and β 0 fix a 4D theory at energy scale µ 0 while we have chosen γ 0 = 0 because of the 4D Gauss-Bonnet theorem. We will see later that (4.25) with (4.27) is independent of µ, which is consistent with the l.h.s. of (4.24).

Leading terms of the energy-momentum tensor
Now, we evaluate the leading value of the renormalized energy-momentum tensor (4.25) for |0 , 0|T µν |0 ′(0) ren . Generically, the continuum modes of s-wave also appear in the mode expansion of a field φ(x). However, as we will see in subsection 4.3.2, the leading value 0|T µν |0 (0) reg becomes O(1) by integrating both ω and l, which means that the sum of such s-wave modes (integration only over ω) can contribute to at most O(r −2 ). Therefore, in this subsection we keep only the bound modes (4.15) in the expansion of the leading solution of φ(x).

Fields in the dimensional regularization
To use dimensional regularization, we consider the (4 + ǫ)-dimensional spacetime manifold M ×R ǫ , where M is our 4D physical spacetime and R ǫ is ǫ-dimensional flat spacetime [51]. That is, we take In this metric, the leading bound mode function (4.15) becomes The plane wave part e ik·y makes a shift M → M + k 2 in (3.5), and the definition of the label A changes from (4.13) to In terms of (4.29), we express the leading solution of φ(x) as where i = n,l,m d ǫ k and a i satisfies (3.15).

Renormalized energy-momentum tensor
From (4.31), we can obtain the leading values of the regularized energy-momentum tensor (see Appendix E for the detailed calculation) 38 : where the components are in the order of (t, r, θ, φ) and Here, γ is Euler's constant and c is the non-trivial finite value for |0 : Note here that, as shown in Appendix E, this leading value of O(1) is obtained as a result of the integration over ω and l, and the 4D dynamics is important. Next, we can check that the poles 1 ǫ of (4.32) are cancelled by the counter terms in (4.26), and 0|T µ ν |0 (0) ren (µ) is indeed finite. Here, we have used the explicit form of H µν , K µν and J µν for the interior metric of (2.26): Finally, we use (4.25) with the running coupling constants (4.27) to obtain the renormalized energy-momentum tensor with a finite renormalization: Here, the µ-dependence disappears. In section 6, we will see that α 0 and β 0 should be tuned properly in order to have a self-consistent solution of the form of (2.26). The point is that the leading value of the trace is fixed independently of (α 0 , β 0 ): (4.38) We can see that this comes from the UV-divergent structure and is essentially the 4D Weyl anomaly [40,41,52] before taking the limit ǫ → 0. However, the trace of the counter terms in (4.26) makes a non-trivial contribution:  Here, the last equation has been evaluated by using the interior metric of (2.26). The first term of (4.40) agrees with (4.38) 39 .

Sub-leading terms of the energy-momentum tensor
The sub-leading bound-mode solution ϕ (1) i (x) of (4.10), which we have not found yet, determines the subleading value µ −ǫ 0|T µν |0 (1) reg completely 40 . In this subsection, we instead use the conservation law to show that 0|T µν |0 ren is expressed in terms of two parameters.
where T µ ν is assumed to be static and spherically symmetric: T µ ν (r) and T θ θ = T φ φ 41 . H µν , K µν and J µν are conserved, and their explicit forms are given by (4.35), which are negative power polynomials in r 2 starting from a constant. Also, the leading value 0|T µ ν |0 ren is constant. Motivated by these facts, we can set the ansatz as (4.45) Then, we substitute this into (4.44) and solve it for each order of r to get In fact, (4.36) satisfies the first equation of (4.46). From (4.25), we put where we have used (4.27) and (4.35). This means By construction, a 1 should be a µ-independent finite number of O(1), as we have seen in (4.37). We suppose that such a a 1 is given. By using (4.36), (4.37), the second equation of (4.46) and (4.48), we can obtain Finally, we write the subleading value of the trace part 0|T µ µ |0 where τ 1 is a finite value independent of the finite renormalization in (4.25) because of (4.35). We suppose that such a τ 1 is obtained. This together with (4.47) and (4.49) determines 0|T θ θ |0 ren . Thus, we can express all the components of 0|T µ ν |0 ′ ren to O(r −2 ) in terms of ( a 1 , τ 1 ).

Contribution of s-wave excitations to the energymomentum tensor
In this section we first consider the generic configuration of excitations of the continuum modes of s-wave from a statistical point of view and obtain a candidate state |ψ . Then, we study the excitation energy in the interior of the black hole and find the energy density −T (ψ)t t . Finally, we determine T (ψ) µν of (1.11).

Fluctuation of mass of the black hole
We consider the black holes with mass M = a 2G that are most likely to be formed statistically. Such black holes are created in the most generic manner. When making a black hole in several operations, the more energy given per operation, the more specific and less generic the formed black hole is. Therefore, the most generic black hole should be created by using a collection of quanta with as small energy as possible. According to Bekenstein's idea [46], in order for a wave to enter the black hole, the wavelength λ must be smaller than a: λ a. This means that the energy of the wave, ǫ ∼ λ , must satisfy ǫ a . As discussed in subsection 3.1, only continuum modes of s-wave can play a role in the black-hole formation. Therefore, a quantum required is such a s-wave with the minimum energy, Suppose that we form a black hole of size a by injecting such quanta many times. Because the wavelength of each quantum is almost the same as the size a ′ of the black hole at each stage (λ ∼ a ′ ), the wave enters the black hole with a probability of 1/2 and is bounced back with a probability of 1/2. Here, we model the formation as a stochastic process according to binomial distribution. Then, the average number of trials N is given by dividing M = a 2G by the energy ∼ a that can enter at one time: Therefore, the statistical fluctuation of mass M is evaluated as [56] ∆M This means that all the black holes with mass ∈ [M, M + m p ] are not statistically distinguished, and they are considered as the same macroscopically. In this way, our candidate state |ψ is one of the states {|ψ } that represent microscopically different continuum s-wave configurations and have the energy-momentum tensor in the form of (1.11). The number of those states reproduces the area law of the entropy, as shown in section 7.

S-wave excitation inside the black hole and T (ψ)t t
To determine the functional form of T (ψ)t t (r), we consider how the s-wave excitations used to form the black hole are distributed in the interior metric of (2.26). First, suppose that an s-wave having a proper wavelength λ local is excited at a certain point r inside the black hole. It has the proper energy ǫ local ∼ λ local . Here, the local Hamiltonian is given by ∆H local = 4πr 2 ∆l T (ψ)τ τ , where ∆l is the proper length of a width ∆r around r, T (ψ) µν is the energy-momentum tensor from the contribution of the excitation, and τ is the proper time of the local inertial frame at r. Therefore, by considering the number of fields N, we can set ∆H local = Nǫ local and ∆l = λ local and obtain Here, T (ψ)τ τ = −T (ψ)τ τ holds because we have g τ τ = −1 in the local inertial frame. Furthermore, in the static metric (2.26), we have (5.5) Using this and (3.17), the ADM-mass contribution from this excitation can be expressed as Here, at the first line, we have divided it by N because the s-wave is an excitation of one kind of field and T (ψ) µν contains all the contribution from N kinds of fields; and at the last line, we have used λ local = ∆l = √ g rr ∆r = r √ 2σ ∆r for the interior metric of (2.26). According to the formation process in the previous subsection, each s-wave excitation corresponds to the ADM energy of (5.1) 42 . Therefore, the condition (∆M) (ψ) ∼ a indicates λ local ∼ √ σ, and then (5.5) becomes where we have used σ ∼ Nl 2 p . This should be the self-consistent form. In section 8, we will discuss how to obtain this functional form from the field equation. 42 One might wonder that bound modes of s-waves with ω ∼ O( 1 a ) also contribute to this excitation as discussed in Fig.9. As shown in subsection 3.3, however, such modes cannot be excited inside the black hole, and only the continuum modes of s-wave are responsible for (5.8).
We here discuss the entropy from this point of view. We consider a unit with width ∆l ∼ λ local ∼ √ σ inside the black hole. It contains N waves with ǫ ∼ a because (5.7) corresponds to ∆r = ∆l √ grr ∼ σ r and the ADM mass of the unit is evaluated from (5.8) as 4πr 2 ∆r(−T (ψ)t t ) ∼ σ Gr ∼ N a . Therefore, the entropy per proper radial length, s, can be evaluated as which means that O( √ N ) bits of information are packed into such a unit. On the other hand, the proper length of the interior of the black hole is evaluated from (2.26) as which is much longer than a. Thus, the entropy is evaluated as (5.11)

The form of T (ψ) µν
Using the form of T (ψ)t t (5.8) and the conservation law inside the black hole (4.44), we express all the components of T (ψ) µν in terms of a single parameter a ψ . First, by the same reason as 0|T µ ν |0 ′ ren , we can set the ansatz for T (ψ)µ ν as (4.45): Putting this into the conservation law (4.44), we can obtain (like (4.46)) , · · · . (5.13) Now, (5.8) shows that T (ψ)t t is at most order of O(r −2 ), which means a (0)′ ren appears as a result of the integration over (ω, l). Therefore, T (ψ)µ ν , which represents the contribution of the s-wave excitation, should be smaller than O(1). This requires c (ψ) 0 = 0, and then the second one of (5.13) leads to b Thus, all the components are at most order of O(r −2 ), which can be expressed as Here, κ t , κ r , κ θ are the O(1) ratios between terms of r −2 and r −4 which are determined by dynamics of the s-waves including the contribution from the negative energy flow (see section 8).
Then, the third one of (5.13) determines c In section 6, we will determine a ψ by the self-consistent equation (4.24).

Self-consistent solution
We have obtained all the ingredients for the self-consistent analysis: the candidate metric (2.26), which is written in terms of (σ, η), the energy-momentum tensor for the ground state (4.36), (4.47) and (4.50), which depend on (α 0 , β 0 ), and the energy-momentum tensor for the s-wave excitation contribution (5.14), which is controlled by a ψ . In this section, we solve the Einstein equation (4.24) and determine the self-consistent values of (σ, η, a ψ ) for a certain class of (α 0 , β 0 ). We then examine the consistency.
6.1 Self-consistent values of (σ, η, a ψ ) and (α 0 , β 0 ) First, we examine the trace part of (4.24), G µ µ = 8πG( 0|T µ µ |0 ′ ren +T (ψ)µ µ ), to O(r −2 ). It is obtained from (2.29), (4.38), (4.50), and (5.14) (with (5.15) and (5.16)) as Equating the terms of O(1) on the both sides, we have 1 This is indeed proportional to Nl 2 p , which is consistent with the assumption we have put just below (1.2). Also ,η appears in the denominator, and σ with η > 1 is smaller than that with η = 1, which is consistent with the meaning of η in Fig.6. Then, the terms of O(r −2 ) in (6.1) together with (6.2) bring − 2 . From (2.28), (4.37), (4.47), and (5.14) (with (5.16)), we have Here, β 0 should be chosen so that (2.21) is satisfied. Note that η is actually constant, as we have assumed in subsection 2.2.2. Thus, we have determined the self-consistent values of (σ, η, a ψ ) in terms of ( a 1 , τ 1 , κ t , κ r ) that can be fixed in principle by dynamics (see (4.47), (4.50) and (5.14) for their definitions). Therefore, the interior metric of (2.26) and the state |ψ are the self-consistent solution of the Einstein equation (1.9). Note that this is a non-perturbative solution w.r.t. because we cannot take → 0 in the solution metric (2.26) with (6.2) and (6.7). which is smaller than the Planck scale if N is large, (1.8) 43 . Thus, no singularity exists inside the black hole 44 and the Penrose diagram is actually given by Fig.3. Note that this result is so robust because (6.2) is not affected by the values of (α 0 , β 0 ). 43 See e.g. [53,54] for a Planckian-energy scale correction to the geometry. 44 See e.g. [55] for another semi-classical resolution of singularity.

Consistency check
Second, we study the energy density. From (4.37), (4.47), (5.14) (with (5.16)), (6.7) and (6.8), we have − ψ|T t t |ψ ′ ren = 1 8πGr 2 , (6.10) to which both 0|T t t |0 ′ ren and T (ψ)t t contribute. Integrating this over the volume in (3.17), we can reproduce the total energy M. We also see from (2.32) that (6.10) also leads to the Hawking radiation consistently. Therefore, both the bound modes and s-waves play a role in the energy inside the black hole.
Third, we discuss the tangential pressure. The self-consistent solution has which can be checked by (4.36), (5.14), (6.2) and (6.5). From this, we can see that this near-Planckian pressure comes from the vacuum fluctuation of the bound modes in the ground state. The origin is the same as the 4D Weyl anomaly because the value (6.11) comes from the second term of (4.36), which appears from the pole 1 ǫ as shown in (E.60). Indeed, twice of (6.11) is equal to the trace (4.38). Therefore, the pressure is very robust in that it is independent of the state. Note also that this large pressure supports the object against the strong gravity, which can be seen by noting that in (4.44) T θ θ and ∂ r log √ −g tt are balanced for the self-consistent solution.
This large tangential pressure associated with the anomaly should be universal for any kind of matter fields. In the previous work [24] only conformal matter fields were considered, and the 4D Weyl anomaly was used to obtain the self-consistent solution with the large tangential pressure. In the present work, however, we reach the same picture of the black hole without using the conformal property, which means that the assumption of conformal matter is not essential. It is related to the fact that any matter behaves ultra-relativistic near the black hole, as we have discussed below (1.3). Therefore, our picture of the black hole should work for any kinds of matter fields. Even for a massive field it should if the mass is smaller than O(m p ).
The surface pressure p (i) 2d in (2.8) also comes from this tangential pressure. When a shell in Fig.4 approaches so that (1.5) is satisfied, the interior metric becomes (2.26) and the shell itself forms a new surface, whose position is represented by the point R in Fig.7. Then, the vacuum fluctuation of the bound modes which reach the point R produces the tangential pressure (6.11), which corresponds to the continuum version of p (i) 2d . Note that the mode that makes the pressure is different from the mode that constitutes the shell while (2.8) has been obtained by a geometrical analysis. Therefore, the two modes are combined to form the consistent picture.
Finally, we discuss the finite renormalization. In this solution, there are no quantities with dimension other than σ. ). Therefore, the R 2 , R αβ R αβ , R αβµν R αβµν terms cannot be much larger than the R term, which means that the three terms would not change the picture of the black hole drastically even if we consider them from the beginning 45 . Thus, our argument based on the Einstein-Hilbert term is self-consistent.
As we have seen, the coupling constants α 0 and β 0 should be related by (6.5). However, there is a possible way to relax it. Instead of the interior metric in (2.26), we consider B(r) = br 2+ζ and A(r) = ar 2+ ζ 2 , where ζ is a small number. Because this ansatz satisfies (2.33), we can solve the wave equation (3.2) locally as discussed in the end of subsection 4.1.2. Then, we can evaluate the renormalized energy-momentum tensor 0|T µ ν |0 ′ ren , which should contain ζ-dependent terms. Therefore, we should be able to choose ζ so that the terms satisfy the Einstein equation (4.24) for a given value of (α 0 , β 0 ). It would be exciting to find another class of solutions in this direction.

Black hole entropy
We count the number of states of the s-waves inside the black hole to evaluate the entropy more quantitatively than the discussion in subsection 5.2.

Adiabatic formation in the heat bath
Let us consider the adiabatic formation of the black hole in the heat bath [14]. Specifically, suppose that we grow a small black hole to a large one very slowly by changing the temperature and size of the heat bath properly 46 . Then, we obtain the stationary black hole, whose metric is given by (2.26) with a =const., and consider it as consisting of excitations of the s-waves, as discussed in section 5.
This black hole is not in equilibrium in an exact sense [16]. Let us see the above formation process from a microscopic view. See Fig.10. At a stage where the black hole has the radius a ′ , s-wave ingoing radiation from the bath comes to the black hole, balancing s-wave Hawking radiation of T H = 4πa ′ with the s-wave intensity σ s . Here, σ s is the intensity of only s-waves which is obtained by a 2D dynamical model (see (8.20)). The ingoing radiation approaches to r = a ′ + 2σ s a ′ (7.1) 45 Such terms may change each numerical coefficient up to O(1), but they will not make a significant change to the basic picture. 46 We can discuss the relaxation time, which is estimated as (10.1). See section 2-E of [16]. as we have shown (1.5). Then, the local temperature that the radiation feels there is which is near the Planck temperature ∼ √ N lp because of (8.20). After this, the subsequent radiations pile up and cover the radiation. Then, it is redshifted exponentially, almost frozen in time t, and suspends there. From the local point of view, however, the radiation keeps falling with the initial information. Therefore, the black hole is not in equilibrium in the usual sense. Rather, the ingoing radiation from the bath and the Hawking radiation are just balanced, and therefore the system is stationary.
This property allows us to use (7.2) as the local temperature at each radius r inside the black hole. Because of the exponentially large redshift, the outgoing Hawking radiation there is frozen in time t. Even if we wait for an exceptionally long time which is longer than the lifetime of the universe, the Hawking radiation of this temperature (7.2) will not go outside to exchange energy with the radiation from the heat bath. This indicates that Tolman's law [56] doesn't hold inside the black hole 47 . As we will see below, this is consistent with the area law of the entropy.

Micro-counting of the s-wave states
We count the number of possibilities of these s-wave configurations to find the entropy densityŝ(r) per radius r inside the black hole. First, we evaluate the number ∆n(r) of the s-waves with frequency ≤ ω in the width ∆r near the surface when the black hole of a ′ is formed. In the semi-classical approximation [49], using (3.10) for l = 0, M = 0, 47 If we apply Tolman's law to the interior metric of (2.26) naively, the local temperature would be √ Nlp , which is exponentially large (for r ≪ R − 2ση R ) compared to the Planck scale. This is not allowed physically. A = 0, and B = r r−a ′ , we can evaluate 48 is the blueshifted frequency when approaching the surface, and we have considered the number of fields N and approximated 1 near (7.1). Thus, the number of modes for [ ω, ω + ∆ ω] is given by Next, we review the entropy of a harmonic oscillator with frequency ω in the heat bath of temperature β −1 [56]. The canonical distribution is given by Therefore, the entropy is evaluated as where we have defined a function From (7.4) and (7.6), we obtain the entropy density: where β local = 1/T local , which is given by (7.2). The reason of the factor 1 2 is that because of the time-freezing effect only ingoing modes carry the information, and the integration only over ingoing direction momenta should be performed. This can be evaluated aŝ The s-waves are trapped in the heat bath with a finite size, and we can use approximately the condition (3.10) to count the quantum number n [57].
where at the third line we have used (8.20) 49 . Integrating this over the interior reproduces the area law: where R(a) 2 = a + 2σ a 2 = a 2 + 4σ + 4σ 2 a 2 and A ≡ 4πa 2 have been used 50 . This shows that the information is stored inside the black hole as excitations of the s-waves.

Mechanism of decrease in energy of the collapsing matter
In this section, we examine how the energy of the collapsing matter decreases and discuss how T (ψ)µ ν , (5.14), is obtained by dynamics. The point is to consider the contribution from the vacuum fluctuation of s-waves.

Setup
Suppose that a classical matter collapses to the black hole which is described by the self-consistent metric. To analyze the time evolution of this matter, we can focus just on the part of the matter around the surface, since the deeper region is frozen in time due to the exponentially large redshift. We consider the matter as consisting of many shells like the multi-shell model of Fig.4 and study the energy of the outermost shell along r = r n (u) ≡ r s (u) as in Fig.11. For convenience, we use, instead of (1.1), the outgoing Vaidya metric [44] in the 49 From this, we can evaluate the entropy per the proper length ∆l = √ g rr ∆r = r √ 2σ ∆r as s(r) = s(r) lp , which is consistent with (5.9) and a thermodynamical discussion [16]. Therefore, the assumption that (7.2) is kept at each r inside the black hole is consistent. 50 See [58] for another approach to the black-hole entropy from the interior region.
(u, v) coordinate to describe the outside region: where each function is given by The Bondi mass a(u) 2G decreases as which is the uu-component of the Einstein equation because of G uu = −˙a r 2 51 . This means that Hawking flux for the Bondi mass is identified as 52 We label the outermost shell by v s so that its location is given by r s (u) = r(u, v s ).

2D model of quantum fields
So far we have considered the collapsing matter as an excitation of s-waves, and we have also seen that only s-waves can enter and exit the black hole. Therefore, in order to study quantum fields propagating in this time-dependent background spacetime, we can use a 2D model of N massless free scalar fields for convenience 53 . In this subsection, we explain the general setup. We consider s-waves of a scalar field, φ = φ(x a ), in a spherically symmetric metric, where x a is the 2D part of the coordinate. The 4D action can be reduced to the 2D one: where r is the areal radius, and the derivative ∂ a is applied only to φ(x a ) as the approximation. Here, we have defined the 2D field as The 2D energy momentum tensor is given by which is related to the 4D energy-momentum tensor T ab through (8.6) as Classically, this 2D part is traceless, τ a a = 0. However, the 2D Weyl anomaly contributes to the trace part [40,59]: where R 2d is the 2D Ricci scalar. We here consider a 2D metric of the form ds 2 2d = −e ϕ(u,v) dudv. Then, the 2D action (8.5) becomes S = dudv∂ u Φ∂ v Φ, and the equation of motion δS δΦ = 0 is Each component of the energy-momentum tensor is given from (8.7) and (8.9) by where we have used R 2d = 4e −ϕ ∂ u ∂ v ϕ and considered N scalar fields. The conservation law ∇ a (2d) τ ab = 0 is expressed as In the following, we consider τ ab as the expectation value with respect to a state |ψ . The information of |ψ is introduced as a boundary condition of τ ab .

Hawking radiation
Now, let us find τ uu in our setup of Fig.11. Initially (u = −∞), there was only the collapsing matter with τ (0) vv (v). Therefore, τ uu is induced as a quantum effect. First, we can use the third of (8.11) to express (8.12) as ∂ v τ uu +2γ∂ v ∂ 2 u ϕ − 1 2 (∂ u ϕ) 2 = 0. From this, we obtain Next, we determine τ (0) uu (u) by the boundary condition that the quantum fields have started in the Minkowski vacuum state |0 M from a distance 54 , come to and passed the center as in Fig.11 [1]. This corresponds to the condition that τ (0) U U (U) = 0, where U is the outgoing null time for the center flat region. Now, by requiring τ uu to transform covariantly under u → U(u), we have [59] where U = U(u) and {u, U} ≡Ü ... U U . Thus, (8.14) becomes The shell we are now considering corresponds to the outermost one in the multi-shell model of section 2.1. Therefore, we can use the result (2.12), that is, 55 From this and (8.2), we construct the formula (see Appendix F for the derivation) Here, we ignore the effect of reflection with η > 1 for simplicity. On the other hand, τ uu vanishes near the surface at r = a + 2σ a : which is consistent with the literature [40,42]. By a similar manner in the eikonal approximation, we can show that the time-evolving fields which lead to (8.19) create particles, and the distribution takes the same form as the Planck distribution with the time-dependent temperature T BH = 4πa(u) [9,16] 57 . Thus, the Hawking radiation of the s-waves is the result of particle creation at r ∼ (1 + k)a (k = O(1)) by excitation of the quantum fields that propagate in the time-dependent spacetime. 54 Note that this vacuum is different from the ground state |0 in the interior metric (3.15). 55 In (2.12), we have used the time-dependent Schwarzschild metric, but we can do the same procedure in the Vaidya metric to obtain the same result [24]. 56 This is consistent with the result obtained by using the eikonal approximation [9,16]. 57 See also [60,61] for a general discussion of a Planck-like distribution

Ingoing negative energy flow near the surface
In turn, we determine the ingoing energy flow τ vv by using (8.13). From (8.2), we have ∂ v ∂ u ϕ = a r 3 ∂ v r = a 2r 3 e ϕ , which gives τ uv = γ a r 3 e ϕ through the third of (8.11). From this and (8.2), we can express (8.13) as ∂ u τ vv = −e ϕ ∂ v γ a r 3 = 3γa 2r 4 e 2ϕ . Therefore, we obtain the general expression Let us focus τ vv at v ∼ v s . Then, the boundary term τ (0) vv (v) represents the energy of the outermost shell that comes from a distance at u = −∞. The shell is classical, and the configuration Φ (cl) (v) is an ingoing solution of (8.10) so that τ is non-zero only around v ∼ v s and zero otherwise. We then put this classical ingoing energy flow as an idealization where W is a positive constant. This indicates that the energy of the matter is kept along a line of v = v s . Now, suppose that this shell comes close to r = a + 2σ a at a time u = u * . Then, we can evaluate , for u * u, (8.24) and obtain for u u * where a * = a(u * ) (see Appendix G for the derivation). The first negative term has the same absolute value as the outgoing energy flux at infinity (8.19) [40,42]. Also we can check that the 4D energy density of this negative part is order of O(a −2 ) (see (8.35)). The mechanism for reducing the energy of the system is as follows (see Fig.11). First, the collapsing matter comes with the positive ingoing energy τ (0) vv . As it approaches to the black hole, the negative energy flow is created from the vacuum fluctuation of the swaves and superposed on the matter. At the same time, the same amount of the positive energy flow is emitted to infinity. Thus, the negative energy cancels the positive energy of the matter locally, and the energy of the system decreases. We can also see how this occurs in the interior metric (see section 5 of [14] and section 4-E of [16]) 58 .

Configuration of s-wave and T (ψ)µ ν
Here, by considering the above mechanics of the energy decrease, we discuss how to obtain −T (ψ)t t ∼ 1 r 2 in (5.14) directly from dynamics of fields.
We first point out that the energy of the outermost shell in the multi-shell model decreases exponentially in the time ∆u ∼ 2a [16]. See the right of Fig.11. The region between the n-th and the (n − 1)-th shells is described by where a ′ 2G is the Bondi mass inside the (n − 1)-th shell. When the n-th shell comes close to r = a + 2σ a , the two time coordinates u, u ′ are connected like (2.7): ∆a, (8.27) where ∆a ≡ a − a ′ corresponds to the mass of the outermost shell, which is assumed to be small here. From this, we can see the time evolution of ∆a(u): which means that the mass of the shell decreases exponentially in ∆u ∼ 2a. This is the self-consistent time evolution of the energy of a matter that has come from the outside and become a part of the evaporating black hole. The behavior (8.28) can be understood as a combination of the increase of the energy of the classical matter and the decrease of the energy by the negative energy flow: (8.29) where the first and second terms are the classical and quantum contributions, respectively. We explain these terms below. First, it is natural that, when we throw a particle to the evaporating black hole, the energy of the particle itself increases if we don't consider any quantum effect. This is because when viewed from a distant, the approaching particle is pulled by the gravitational force of the "escaping" surface of the black hole and thus gains the extra kinetic energy just like gravity assist for spacecraft. In fact, we can consider the contribution only from the ingoing energy τ where we have used (8.3). This gives the first term in (8.29). Second, we evaluate the contribution of the negative energy flow that directly reduces the energy of the shell. As we have seen in the previous subsections, the amount of the negative energy flow near the black hole is the same as that of the positive energy flow going out. Thus, the contribution we should take is equal to the difference ∆J between the energy flows τ uu and τ u ′ u ′ exiting the systems with mass a 2G and a ′ 2G , respectively, where u ′ is the time coordinate of the metric (8.26). See Fig.11. More precisely, it is given by where we have considered the two redshift factors in order to translate τ u ′ u ′ in terms of u. This can be evaluated as ∆a. (8.33) Here, at the first line, (8.19) has been applied to both τ uu and τ u ′ u ′ because (8.17) holds for both u and u ′ ; at the second line, (8.27) has been used; and at the last line, σ has been assumed to be the same as (8.20). This provides the second term of (8.29). Now, we discuss the energy density. As shown in Appendix H, the energy density corresponding to (8.30) is given by while the energy density corresponding to the negative energy flow in (8.25) is evaluated as where we express these in the (u, r) coordinate of (8.1) 60 . These are not consistent with −T (ψ)t t ∼ 1 r 2 in (5.14).
However, the above observation (8.29) implies that there should be a way to include the effect of the negative energy flow in solving the field equation of the s-wave. Such an analysis could be similar to "Lorentz friction", a recoil force on an accelerating charged particle caused by electromagnetic radiation emitted by the particle [39]. While the radiation is emitted directly from the particle in the case of the Lorentz friction, the negative energy flow in our case is induced through the change of the metric by the motion of the shell. In this sense, these might be qualitatively different. A single shell model in [9] could be useful for clarifying what happens to the shell in the time-dependent analysis. Then, we should be able to obtain the proper functional form T (ψ)t t ∼ 1 r 2 , and the form of T (ψ)µ ν (5.14) will be determined together with κ t , κ r , κ θ . We will consider this problem in future (see also the end of section 10).

Conformal matter fields
As a special case, we consider conformal fields as the matter fields in (1.9) 61 . In the previous work [24], we have investigated this case and constructed the self-consistent metric. Here, we perform further analysis to completely determine (σ, η).

The interior metric
The key equation is the trace part of the Einstein equation (1.9). The trace of the energy-momentum tensor is given by the 4D Weyl anomaly, independently of the state |ψ [40,41,52]: 2) where C αβγδ is the Weyl curvature and G is the Gauss-Bonnet density. The coefficients c W and a W are fixed only by the matter content of the action, as we will give explicitly below, while b W also depends on the finite renormalization.
We can use the interior metric of (2.26) because the multi-shell model is independent of the matter content. Using (2.29), (2.30), (2.31) and (4.43), the self-consistent equation (9.1) becomes to O(r −2 ). Note that b W does not contribute here because of R = O(r −4 ) from (4.43).
which is similar to (6.2). Next, comparing the terms of O(r −2 ) and using (9.4), we can obtain easily which is a new result 62 . Note here that this result is not influenced by a finite renormalization because H µ µ , K µ µ = O(r −4 ) from (4.41) and (4.43). Thus, the interior metric has been determined completely only by the matter content, independently of the state and the theory of gravity.

A new weak-gravity conjecture
We discuss the meaning of (9.5). First, the explicit values of c W and a W are given by (9.6) where N S , N F and N V are respectively the numbers of scalars, spin-1 2 Dirac fermions and vectors in the theory [52] 63 . Using this, (9.5) can be expressed as For example, we can see 9 4 Dirac ferminon, 9 14 vector. (9.8) The consistency condition of η is (2.21): 1 ≤ η < 2. The lower limit has come from the positive definiteness of the ratio of the scattered part of the radiation as in Fig.6 [14]. Therefore, we can compare the both sides of (9.3) properly to O(r −2 ). 63 The graviton effect should be included here, but we ignore it because there seems no available result that satisfies the consistency condition of the trace anomaly [40,67,68,69]. 64 If we relax the condition of η more, at least η must be non-negative: η > 0. If η ≤ 0, the redshift would make no sense. 65 Here, we assume that the Standard Model is a kind of conformal field theory and study what happens to (9.7). N V = 12, and then (9.7) becomes η = 438 433 ≈ 1.012, (9.9) which satisfies (2.21). On the other hand, if we remove right-handed neutrino, N F = 24 changes to N F = 24 − 1 2 × 3, and then (9.7) gives η = 849 854 ≈ 0.994, (9.10) which is outside (2.21). This result suggests that right-handed neutrino should exist. Next, we consider N = 4 supersymmetric theory. In this case, we have N S = 6, N F = 2 and N V = 1, and (9.7) lead to η = 6 5 = 1.2, (9.11) which satisfies (2.21). Instead, if we apply (9.7) to N = 1 supersymmetric theory with vector-multiplet, which has N S = 0, N F = 1 2 , N V = 1, we get η = 3 4 = 0.75, (9.12) which does not meet (2.21). In this way, the condition (2.21) with the formula (9.7), which is a result of the selfconsistent solution of the evaporating black hole, can classify effective field theories. The weak-gravity conjecture tells that black holes should evaporate without global symmetry charge [70,71]. Therefore, (2.21) and (9.7) may play a role of a new weak-gravity condition 66 . It would be exciting to study phenomenology using them.

Conclusion and discussions
In this paper, we have solved time evolution of the collapsing matter taking the full dynamics of the 4D quantum matter fields into account in the semi-classical Einstein equation. Then, we found the self-consistent metric g µν and state |ψ . This solution tells that the black hole is a compact dense object which has the surface (instead of horizon) and evaporates without a singularity 67 . It consists of three parts. One is the vacuum fluctuation of the bound modes in the ground state, which produces the large tangential pressure supporting the object. Another one is the excitation of the s-waves composing the collapsing matter, which carries the information responsible for the area law of the entropy. The other one is the excitation of the s-waves producing the pair of the outgoing 66 See section 3-D of [16] for a discussion on the non-conservation of baryon number in our picture. 67 Even if we consider the evaporating black hole as a closed trapped region, which is the conventional model e.g. [7], the collapsing matter is below the timelike trapping horizon just by a proper Planckian distance [72,73]. This is similar to our picture in that the matter is above the would-be horizon by the Planckian distance (1.7). These imply that such a conventional model is eventually very close to ours.
Hawking radiation and the ingoing negative energy flow, which decreases the energy of the matter. These three energy-momentum flows are conserved independently if there is no interaction among them. Now, we would like to ask: How is the information |ψ of the collapsing matter reflected in the Hawking radiation and recovered in this picture? There are two remarkable points to answer this. One point is interaction between the outgoing Hawking radiation and the ingoing collapsing matter. As discussed below (6.10), the Hawking radiation is emitted outwards from each region inside the black hole. Then, the collapsing matter may collide with the outgoing radiation coming from below, go outwards, and exit from the black hole together with the information. See Fig.12. Because the energy scale Figure 12: Time evolution of the evaporating black hole with interaction. The matter with the information |ψ and energy-momentum p µ comes and is scattered with a certain probability by the radiation with energy-momentum q µ . Then, the ingoing radiation should approach to a vacuum state through interaction with the negative energy flow.
there is near the Planck scale (from (6.9), (6.12) and (7.2)), some string-theory effect may be relevant and any kind of matter may interact with the radiation universally 68 . Actually, we can describe this idea by a classical φ 4 -model and evaluate the scattering time scale as ∆t ∼ a log a λNl p , (10.1) where λ is the dimensionless coupling constant (see section 3 of [16]). Then, the scattering is an elastic collision in the radial direction, and the two particles swap their momenta (the right of Fig.12). Therefore, we expect that the energy distribution of the coming-back matter follows the Planck distribution of T H = 4πa(t) with a small correction that depends on the initial state. This scattering process is stochastic. If an outgoing matter in the deep region goes out while the other remains inside, the matter would collide many times before going out, so the probability is very small. Therefore, the information returns in order from the surface region during the evaporation as if one peels off an onion 69 (left of Fig.12).
The other point is the role of the negative energy induced by the vacuum. If the black hole evaporates and the information comes out by the above mechanism, the state along an ingoing line should approach to a vacuum state (see the right of Fig.12). When the negative energy is superposed on the matter and decreases the energy, interaction between them should play a role in this evolution of the state. This point may be related to the problem of finding the correct configuration of the s-waves discussed in subsection 8.5. We will study this scenario in future.

B Black-hole spacetime region in various coordinates
We describe the black-hole spacetime region in various coordinate systems. Here, we don't consider the effect of interaction, which is discussed in section 10. First, we consider the (t, r) coordinate of (1.1). See Fig.13. The dashed line represents the time evolution of the Schwarzschild radius a(t) of the total mass of the collapsing matter, and the shaded region corresponds to the interior of the black hole. Other lines are null geodesics. The line 3 is drawn to match the trajectory of the outermost part of the collapsing matter. Suppose the matter comes close to the Schwarzschild radius at P . The line 2 represents the propagation of quantum field which starts from a distance in the Minkowski vacuum state, passes the center (r = 0) and crosses a(t) around P . Then, particle creation occurs around r = a(t), Hawking radiation is emitted along the line 2, and a(t) begins to decrease as (1.2) [1]. At the same time, the line 3 approaches to the asymptotic position as (1.5), and then it is kept outside r = a(t), where the distance between the line 3 and r = a(t) is ∆r = 2σ a(t) . From P the line 3 plays the role of the surface of the evaporating black hole. Eventually, a(t) becomes zero, and then the line 3 passes the center and comes back to infinity. The line 1 starts to go inward before the black hole is formed, and the line 4 starts after the evaporation. They come back soon.
Second, we study the (u, r) coordinate of the first line in ( coordinate, outgoing null geodesics are described by vertical lines. Third, we describe the spacetime in the (v, r) coordinate. Here, v is an ingoing Eddington-Finkelstein-like coordinate that describes an ingoing null geodesic by a vconstant line. See Fig.15. In this coordinate, a(v) changes suddenly and becomes zero Figure 15: The black-hole spacetime in the (v, r) coordinate. in a short time ∆v ≪ 1. Then, the line 3 propagates just outside r = a(v), reaches r = 0, and comes back. This is a picture in the v coordinate showing that an infalling particle will not enter into the evaporating black hole.
Finally, we discuss the spacetime in the (u, v) coordinate of the second line of (8.1). See Fig.16. Here, an ingoing and outgoing null geodesic are represented by a line with v =const. and one with u =const., respectively.

C Normalization factor
We derive the normalization for the bound modes, (3.11). This can be obtained from the relation (3.12): First, (3.8) and (3.14) give and then (C.1) is equivalent through (3.13) to We here note that (C.5) is equivalent to We use (C.4) and the leading WKB solution Therefore, the normalization condition is given by where we have used dθdφ sin θY lm Y * l ′ m ′ = δ ll ′ δ mm ′ and replaced cos 2 drp i to the average 1 2 because the WKB approximation is good for a high frequency. Now, we take derivative of (3.10) with respect to n, use the property p i (r ± i ) = 0 and obtain Thus, (C.8) and (C.9) provide (3.11).
D Derivation of the leading solution (4.14) First, two Bessel functions J A (ξ), J −A (ξ) satisfy (4.12). Because we are now interested in the bound modes, the solution should vanish as r − r 0 = −x → ∞. From (4.11), this corresponds to ξ → 0, in which J ±A (ξ) behave as Because A is positive, we can find Next, we fix the normalization k. (4.4) means that the WKB approximation becomes better as r − r 0 = −x → −∞, which corresponds through (4.11) to ξ → ∞. Then, the leading behavior of J A (ξ) is 3) where (4.11) has been used. In (3.9), on the other hand, we have for x → ∞ We first summarize the results about the leading solution: The metric is From these, we can make easily the formulae we are using below. We have at r = r 0 where we have used ξ Next, we take the derivative of (E.2) with respect to r: where we have used the second one of (E.3) and a formula of Bessel function: Therefore, we have where in the second line we have used (E.4) and (E.5) and in the third line (E.7) and (E.8). In evaluating the leading contribution, we can neglect the correction from the Euler-Maclaurin formula in n,l,m 70 and use Then, we have where we have introduced Instead of considering directly the index A, which is the first of (E.3), we can insert the identity and consider This integration is performed to the extent that the square root is not negative. K-integration: We first perform the K-integration as is the area of a ǫ-dimensional unit sphere, and Y -integration: We next integrate it over Y as (E.19) We have discarded the contribution from the top of the integral because in the dimensional regularization we drop 0 ǫ and ∞ ǫ by choosing ǫ properly. Then, (E.17) becomes Ω-integration: To evaluate the Ω-integration of h t (Ω, A), we introduce a damping factor e −sΩ (s > 0), integrate it and then expand the result around s = 0: Here, H n is Harmonic number: which has useful formulae: Using these properly, we can rewrite Note here that any term of the form A×(polynomial in A 2 ) can be dropped by choosing ǫ properly in the A-integration. Therefore, we can drop the first line of (E.22) and the second and third terms of the r.h.s. in (E.26) and have in the limit s → 0 .

(E.27)
A-integration: Next, we consider the A-integration. Using the integral representation of Harmonic number Here, in the third line we have discarded the term proportional to a polynomial in A and K s (x) is the modified Bessel function of the second kind. Introducing y = − 1 2 log x, we have which is difficult to evaluate analytically. Numerical evaluation: From (E.20) and (E.30), thus we obtain the expression for an energy scale µ Let us evaluate this numerically. Because the integration diverges around y = 0, we divide it into the pole part and the finite one. We first expand f ǫ (y) around y = 0: (E.33) Here, we have where we have discarded the contribution from the bottom of the integral. This means that n = 1 provides a pole −1/ǫ. Therefore, we obtain the pole part: which also includes the finite contribution from y −1 in the range 0 ≤ y ≤ 1. Next, we consider the finite part in the range 0 ≤ y ≤ 1. The contributions from y −5 and y −3 are given by Subtracting the divergent part, we can evaluate numerically These v 0 , v 1 , v 2 , v 3 determine the value from 0 ≤ y ≤ 1 although v 0 diverges. On the other hand, all the contribution from the range 1 ≤ y ≤ ∞ is evaluated numerically as v 4 ≡ C ǫ=0 E.2 Evaluation of 0|g rr (∂ r φ) 2 |0 (0) reg We next go to the evaluation of the r-derivative. . (E.42) Here, in the third line we have used (E.11) and in the forth line we have taken the same procedure as in the previous subsection. We can do the same calculation for K and Y as in (E.18) and (E. 19) and obtain 0|g rr (∂ r φ) 2 |0 (0) reg | r=r 0 = − 16πσ 2 η 4 (8πση 2 ) . Note that this is the same as (E.27), which means 0|g rr (∂ r φ) 2 |0 (0) reg | r=r 0 = 0|g tt (∂ t φ) 2 |0 (0) reg | r=r 0 . (E. 46) E.3 Evaluation of 1 2 0|g θθ (∂ θ φ) 2 + g φφ (∂ φ φ) 2 |0 (0) reg We do almost the same calculation again.
where we have used (E.7), introduced the same variables and performed the K-integration (E.18). Now, we integrate it over Y as .
where we have performed a similar integration to (E.18). The Y -integration is done as .
(E.54) Then, we have Comparing this to (E.50), we find h y = ǫh θ , which means through (E.52) 0| ǫ a=1 (∂ y a φ) 2 |0 (0) reg | r=r 0 = ǫ Combing the results we have obtained so far, we can evaluate each component of 0|T µ ν |0 (0) reg . Because each free field φ a (x) gives the same contribution, we can just multiply N to obtain through (3.18) where we have used (E.46), (E.52) and (E.57). Similarly, we calculate Again, we have This average value gives 0|T θ θ |0 reg because of the spherical symmetry. Note here that ǫ 2 picks up the contribution from the pole 1 ǫ of N 0|g tt (∂ t φ) 2 |0 (0) reg , which is completely determined by the UV structure. In this sense, the term is anomalous.
Thus, these and (E.40) provide (4.32). G Derivation of τ vv (u, v s ) (8.25) First we derive (8.24). For simplicity, we assume that for u u * the shell falls approximately in the static Schwarzschild metric with the initial mass a 0 2G . This is motivated by the fact that the time scale in which the shell approaches from, say, r = 2a to r = a + 2σ a is ∆u ∼ a while the time scale in which the energy of the system changes significantly is ∆u ∼ a 3 σ . Then, we can evaluate for u u * where in the third line we have changed the variable fromũ to r s by using the second of (8.2) with a 0 . Now, suppose that the black hole starts to evaporate from u ∼ u * , which means that a 0 ≈ a(u * ) ≡ a * . Similarly, we can calculate for u u * where we have used r s (u) = a(u) + 2σ a(u) , and in the second term of the third line we have changed the variable fromũ to a by using (8.3).
Using these, we consider for u u * Here, the suffixū stands for the u-component in the (u, r) coordinate of (8.1). Then, we express Tūū in terms of the (u, v) coordinate. From dū = du and dr = ∂r ∂u v du+ ∂r ∂v u dv in (8.1), we have the coordinate transformation between (ū, r) and (u, v): du = dū, dv = 2e −ϕ dr + 1 2 r − a(ū) r dū , (H.2) where we have used (8.2). We calculate which is (8.30). Here, in the fifth line we have used the part for u u * of (8.24) and r s ≈ a + 2σ a ; and in the last one we have made an approximation a(u) ≈ a * , since the difference does not contribute to the time evolution, compared to the exponential factor.
Then, we check the energy density (8.34). We can use (8.24) to evaluate the classical contribution (H.4) for u u * as where we have used a ≈ r approximately. Finally, we derive the energy density induced by the vacuum near the surface for u u * , that is, (8.35). We first evaluate (H.7) where we have used the third one of (8.11) and (8.2). Putting this and the first term of (8.25) into (H.3), we have where we have used σ = σ s (8.20) and kept only the second term as the leading. This gives (8.35) because of a ≈ r.