A Model of Black Hole Evaporation and 4D Weyl Anomaly

We analyze time evolution of a spherically-symmetric collapsing matter from a point of view that black holes evaporate by nature. We consider conformal matters and solve the semi-classical Einstein equation $G_{\mu\nu}=8\pi G \langle T_{\mu\nu} \rangle$ by using the 4-dimensional Weyl anomaly with a large $c$ coefficient. Here $\langle T_{\mu\nu} \rangle$ contains the contribution from both the collapsing matter and Hawking radiation. The solution indicates that the collapsing matter forms a dense object and evaporates without horizon or singularity, and it has a surface but looks like an ordinary black hole from the outside. Any object we recognize as a black hole should be such an object.


Introduction and the basic idea
Black holes are formed by matters and evaporate eventually [1]. This process should be governed by dynamics of a coupled quantum system of matter and gravity. It has been believed for a long time that taking the back reaction from the evaporation into consideration does not change the classical picture of black holes drastically. This is because evaporation occurs in the time scale ∼ a 3 /l 2 p as a quantum effect while collapse does in the time scale ∼ a as a classical effect 1 . Here a = 2GM and l p ≡ √ G. However, these two effects become comparable near the black hole. Recently, it has been discussed that the inclusion of the back reaction plays a crucial role in determining the time evolution of a collapsing matter [3,4,5,6,7,8].
We first explain our basic idea by considering the following process. Suppose that a spherically symmetric black hole with mass M = a 2G is evaporating. Then, we consider what happens if we add a spherical thin shell to it. The important point here is that the shell will never go across "the horizon" because the black hole disappears before the shell reaches "the horizon".
To see this, we assume for simplicity that Hawking radiation goes to infinity without reflection, and then describe the spacetime outside the black hole by the outgoing Vaidya metric [9]: where M(u) = a(u) 2G is the Bondi mass. We assume that a(u) satisfies where σ = kNl 2 p is the intensity of the Hawking radiation. Here N is the degrees of freedom of fields in the theory, and k is an O(1) constant.
If the shell comes close to a(u), the motion is governed by the equation for ingoing radial null geodesics: dr(u) du = − r(u) − a(u) 2r(u) (1.3) no matter what mass and angular momentum the particles consisting the shell have 2 .
Here r(u) is the radial coordinate of the shell. This reflects the fact that any particle becomes ultra-relativistic near r ∼ a and behaves like a massless particle [10]. As we will show soon in the next section, we obtain the solution of (1.3): . (1.4) This means the followings (see Fig.1.): The shell approaches the radius a(u) in the time 1 See e.g. [2] for a classical analysis of collapsing matters. 2 See Appendix I in [5] for a precise derivation scale of O(2a), but, during this time, the radius a(u) itself is slowly shrinking as (1.2). Therefore, r(u) is always apart from a(u) by −2a da du . Thus, the shell never crosses the radius a(u) as long as the black hole evaporates in a finite time, which keeps the (u, r) coordinates complete outside "the horizon", r > a(u).
After the shell comes sufficiently close to r = a + 2σ a , the total system composed of the black hole and the shell behaves like an ordinary black hole with mass M + ∆M, where ∆M is the mass of the shell. In fact, as we will see later, the radiation emitted from the total system agrees with that from a black hole with mass M + ∆M.
We then consider a spherically symmetric collapsing matter with a continuous distribution, and regard it as a set of concentric null shells. We can apply the above argument to each shell because its time evolution is not affected by the outside shells due to the spherical symmetry. Thus, we conclude that any object we recognize as a black hole actually consists of many shells. See Fig.2. Therefore, there is not a horizon but a surface at r = a + 2σ a , which is a boundary inside which the matter is distributed 3 . If we see the system from the outside, it looks like an evaporating black hole in the ordinary picture. However, it has a well-defined internal structure in the whole region, and evaporates like an ordinary object 4 5 .
In order to prove this idea, we have to analyze the dynamics of the coupled quantum system of matter and gravity. As a first step, we consider the self-consistent equation (1.5) Here we regard matter as quantum fields while we treat gravity as a classical metric g µν . T µν is the expectation value of the energy-momentum tensor operator with respect to the state |ψ that stands for the time evolution of matter fields defined on the background g µν . T µν contains the contribution from both the collapsing matter and the Hawking radiation, and |ψ is any state that represents a collapsing matter at u = −∞.
In this paper, we consider conformal matters. Then, we show that T µν on an arbitrary spherically symmetric metric g µν can be determined by the 4-dimensional (4D) Weyl anomaly with some assumption, and obtain the self-consistent solution of (1.5) that realizes the above idea. Furthermore, we can justify that the quantum fluctuation of gravity is small if the theory has a large c coefficient in the anomaly.
Our strategy to obtain the solution is as follows. We start with a rather artificial assumption that T t t + T r r = 0. (This is equivalent to T U V = 0 in Kruskal-like coordinates.) By a simple model satisfying this assumption, we construct a candidate metric g µν . We then evaluate T µν on this background g µν by using the energy-momentum conservation and the 4D Weyl anomaly, and show that the obtained g µν and T µν satisfy (1.5). Next, we try to remove the assumption. We fix the ratio T r r / T t t , which seems reasonable for the conformal matter. Under this ansatz, the metric is determined from the trace part of (1.5), G µ µ = 8πG T µ µ , where T µ µ is given by the 4D Weyl anomaly. On this metric, we calculate T µν as before, and check that (1.5) indeed holds. This paper is organized as follows. In section 2 we derive (1.4). In section 3 we construct a candidate metric with the assumption T t t + T r r = 0. In section 4 we evaluate T µν on this metric, and then check that (1.5) is satisfied. In section 5 we remove the assumption and construct the general self-consistent solution. In section 6 we rethink how the Hawking radiation is created in this picture.

Motion of a thin shell near the evaporating black hole
We start with the derivation of (1.4) [3,4,5]. That is, we solve (1.3) explicitly. Putting r(u) = a(u) + ∆r(u) in (1.3) and assuming ∆r(u) ≪ a(u), we have The general solution of this equation is given by where C 0 is an integration constant. Because a(u) and da(u) du can be considered to be constant in the time scale of O(a), the second term can be evaluated as Therefore, we obtain which leads to (1.4): This result indicates that any particle gets close to in the time scale of O(2a), but it will never cross the radius a(u) as long as a(u) keeps decreasing as (1.2) 6 . In the following we call R(a) the surface of the black hole.
Here one might wonder if such a small radial difference ∆r = 2σ a makes sense, since it looks much smaller than l p . However, the proper distance between the surface R(a) and the radius a is estimated for the metric (1.1) as 7 In general, this is proportional to l p , but it can be large if we consider a theory with many species of fields. In fact, in that case we have We assume that N is large but not infinite, for example, O(100) as in the standard model. Then, ∆r = 2σ a is a non-trivial distance. 6 The above analysis is based on the classical motion of particles, but we can show that the result is valid even if we treat them quantum mechanically. See section 2-B and appendix A in [5]. 7 For the general metric, the proper length in the radial direction is given by ∆l = g rr − (gur ) 2 guu ∆r. See [10].

Constructing the candidate metric
The purpose of this section is to construct a candidate metric by considering a simple model corresponding to the process given in section 1 [3,5]. At this stage, we don't mind whether it is a solution of (1.5) or not, which will be the task for the next section.

Single-shell model
As a preliminary for the next subsection, we begin with a simpler model [3]. See Fig.3. Suppose that a spherical null shell with mass M = a 2G comes from infinity, and evapo- rates like the ordinary black hole. Here we consider the shell infinitely thin. We model this process by describing the spacetime outside the shell as the Vaidya metric (1.1) with (1.2). On the other hand, the spacetime inside it is flat because of spherical symmetry, and we express the metric by Now we have two time coordinates (u, U), and we need to connect them along the trajectory of the shell, r = r s (u). This can be done by noting that the shell is moving along an ingoing null geodesic in the metrics of the both sides, (1.1) and (3.1). Therefore, the junction condition is given by This determines the relation between U and u for a given a(u). Generally, connecting two different metrics along a null hypersurface Σ leads to a surface energy-momentum tensor T µν Σ . Indeed, by using the Barrabes-Israel formalism [18,19], we can estimate the surface energy ǫ 2d and the surface pressure p 2d as 8 Note that ǫ 2d is nothing but the energy per unit area of the shell with energy M = a 2G , and that the positive pressure p 2d is proportional to the energy being lost, −ȧ(u) > 0.
Thus, we have obtained the metric without coordinate-singularity that describes the formation and evaporation process of a black hole. Note again that we don't claim yet that this metric satisfies (1.5), but we here construct a candidate metric which formally expresses such a process.

Multi-shell model
Now, we consider a spherically-symmetric collapsing matter consisting of n spherical thin null shells. See Fig.4, where the position of the i-th shell is depicted by r i . We Figure 4: A multi-shell model. assume that each shell behaves like the ordinary evaporating black hole if we look at it from the outside. We postulate again that the radiation goes to infinity without reflection. Then, because of spherical symmetry, the region just outside the i-th shell can be described by the Vaidya metric: The surface tensor is given by T µν Here v = ∂ ∂τ is the 4-vector of a timelike observer with proper time τ who crosses the shell at τ = 0, k is the ingoing radial null vector along the locus of the shell which is taken as k = 2rs(u) rs(u)−a(u) ∂ u − ∂ r for r > r s and k = 2∂ U − ∂ r for r < r s , and σ µν is the metric on the 2-sphere (σ µν dx µ dx ν = r 2 dΩ 2 ). See Appendix F in [5] for the detail.
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, u n = u is the time coordinate at infinity, and a n = a = 2GM, where M is the Bondi mass for the whole system. On the other hand, the center, which is below the 1-st shell, is the flat spacetime (3.1): In this case, the junction condition (3.2) is generalized to This is equivalent to As in the single-shell model, we have the surface energy-momentum tensor on each shell. By generalizing (3.3), we can show that the energy density ǫ (i) 2d and the surface pressure p (i) 2d on the i-th shell are given by [5] 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 shell. Thus, the pressure is induced by the radiation from the shell itself 9 .

The candidate metric
Finally, we take the continuum limit in the multi-shell model and construct the candidate metric [3,4,5]. Especially, we focus on a configuration in which each shell has already come close to R(a i ): where (2.2) has been used 10 . (A more general case is discussed in [8].) 9 See [5] for more detailed discussions. 10 Due to the spherical symmetry, the motion of each shell in the "local time" u i is determined independently of the shells outside it. Therefore, the analysis for (2.2) can be applied to each shell.
We first solve the equations (3.7). By introducing we have Here, at the second line, we have used (3.9); at the third line, we have used (3.11) and assumed a i −a i−1 2σ a i ≪ 1, which is satisfied for a continuous distribution; and at the last With the initial conditions (3.6), we obtain Now, the metric at a spacetime point (U, r) inside the object is obtained by considering the shell that passes the point and evaluating the metric (3.4). We have at where (3.11) and (3.14) have been used. From these, we obtain the metric Note that this is static although each shell is shrinking, and that it does not exist in the classical limit σ → 0. Thus, our candidate metric for the evaporating black hole is given by dU and expressed (3.17) in terms of u. This metric is continuous at the surface r = R(a(u)) = a(u) + 2σ a(u) , where a(u) decreases as (1.2). Next we consider a stationary black hole. Suppose that we put this object into the heat bath with temperature T H = 4πa . Then, the ingoing energy flow from the bath and the outgoing one from the object become balanced each other 11 , and the system reaches a stationary state, which corresponds to a stationary black hole in the heat bath [20]. (See also Fig.5.) The object has its surface at r = R(a), where a =const. Then, the Vaidya metric for the outside spacetime is replaced with the Schwarzschild metric: By introducing the time coordinate T around the origin as we can write the interior metric (3.17) as Thus, by changing T to t through dt = e R(a) 2 4σ dT , we obtain our candidate metric for the stationary black hole: where R(a) = a + 2σ a with a =const. The remarkable feature of (3.22) is that the redshift is exponentially large inside and time is almost frozen in the region deeper than the surface by ∆r σ a .

Summary of the assumptions so far
We start with summarizing the assumptions which we have made to obtain the metric (3.18). Firstly, we assume that the system is spherically symmetric. Then, the time evolution of each shell is not affected by its exterior region after it becomes ultrarelativistic. Secondly, we assume that the radiation coming out of each shell flows to infinity without reflection. Then, the metric of each inter-shell region is given by the Vaidya metric. We consider what these assumptions mean in terms of T µν . Here we discuss in Kruskal-like coordinates (U, V ): U and V are coordinates such that outgoing and ingoing null lines are characterized by U =const. and V =const., respectively. Therefore, the second assumption means that in the inter-shell regions only T U U is nonzero 12 , and in particular, Furthermore, noting the surface energy-momentum tensor (3.10), we find that ǫ Thus, after taking the continuum limit, we have nonzero values for T µν except for T U V . Therefore, the assumption we have made so far are essentially the spherical symmetry and (4.1). We keep the assumption (4.1) within this section, and will remove it in the next section.

Relations among T µν from the energy-momentum conservation
We investigate the relations among the components of T µν obtained from the energymomentum conservation, which will be used to determine T µν . The general spherically symmetric metric can be expressed in Kruskal-like coordinates as We assume that T µν is spherically symmetric, that is, the non-zero components are which depend only on U and V . Here we keep T U V for the convenience of the next section. Then, ∇ µ T µU = 0 and ∇ µ T µV = 0 are expressed as, respectively, 12 We can see this explicitly as follows. Because the Vaidya metric has only G uu , we can expect that only T uu exists in the inter-shell regions. From the definitions of U and V , we have a transformation between (u, r) and (U, V ) such that ∂u ∂V U = 0. Therefore, we evaluate T UU = ∂u ∂U 2 T uu = 0, The other components are satisfied trivially.
On the other hand, because the trace of the energy-momentum tensor is expressed Substituting (4.6) into (4.4) and (4.5), we obtain Once T µ µ is given, we can determine T µν from these equations with some boundary conditions if one of the four functions (4.3) is known [21].

The static case
As a special case, we suppose that the spacetime is static. Then, ϕ(U, V ) and r(U, V ) satisfy ϕ(U, V ) = ϕ(r(U, V )), ∂ V r = −∂ U r. and In this case, the expectation value of the energy-momentum tensor T µν should also be static and satisfy Then, the formulae (4.7) and (4.8) reduce to (4.14)

Evaluation of T µν inside the black hole
Now we can evaluate T µν in the metric (3.21) assuming (4.1) and (4.13). Here we rewrite the metric (3.21) as (4.2) with (4.11) and

Boundary conditions for T µν
We start with the boundary conditions. See Fig.5. We first note that the region around r = 0 is kept to be a flat space. This is because the initial collapsing matter came from infinity with a dilute distribution. Then, the region inside the innermost shell in Fig.4 is flat due to the spherical symmetry, and it is almost frozen in time by the large redshift as in (3.18) 13 . Thus, the boundary conditions for T µν are given by Note that this should be applied to both the evaporating and stationary black holes, because at any rate black holes have been formed by collapse of matters.

Employing
Now, we combine the energy-momentum conservation with the assumption (4.1). Under (4.1), (4.14) becomes Integrating this from 0 to r for √ σ ≪ r ≤ R(a), we have Here, at the first line, we have used (4.11) and (4.15); at the third line, we have assumed that T µ µ (r) does not change as rapidly as e r 2 2σ , which will be checked soon, and used e − 1 2σ (r+r ′ )(r−r ′ ) ≈ e − r σ (r−r ′ ) , since the largest contribution comes from r ′ ∼ r; at the final line, we have omitted the term proportional to e − r 2 σ for r ≫ √ σ. Finally, using the boundary condition (4.16), we have On the other hand, under the assumption (4.1), (4.6) leads to Thus, all the components of T µν are determined by T µ µ .

4.3.3
T µ µ from the 4D Weyl anomaly In the case of conformal matters, T µ µ is provided by the 4D Weyl anomaly once the metric is given [21,22,23,24]: where F ≡ C µναβ C µναβ and G ≡ R µναβ R µναβ − 4R µν R µν + R 2 14 . For the metric (3.21), F and G are calculated as Therefore, only the c-coefficient remains for r ≫ √ σ, and we obtain which is constant and consistent with the assumption made in (4.18). Thus, (4.19) and (4.20) are fixed as, respectively, and which means that the 4D Weyl anomaly provides the angular pressure [4,5] 15 .

The self-consistent equation
Now we can obtain the condition that the self-consistent equation (1.5) holds, as follows. From (4.1), (4.24) and (4.25), we have We note that the dominant energy condition [19] is violated, − T T T ≪ T θ θ , and that the interior is not a fluid in the sense T r r ≪ T θ θ [3,4,5]. We can check the validity of the classical gravity in (1.5). Indeed, in the macroscopic region (r > l p ), all the invariants for (3.21) are of order ∼ 1 σ : (4.29) They are smaller than the Planck scale if is satisfied. Therefore, macroscopic black holes (a ≫ l p ) can be described by the ordinary field theory. We do not need to consider quantum gravity except for the very small region (r ∼ l p ) or the last moment of the evaporation. (3.21) can be trusted for r √ σ.

Evaluation of T µν outside the black hole
In this subsection we investigate T µν in the outside region, r > R(a), for both the evaporating and the stationary black holes.

The evaporating black hole
First we consider the evaporating back hole (3.18). Although we don't assume the static condition (4.13), we use a similar argument to the previous subsection. We first identify the boundary conditions. In the left of Fig.5, no ingoing matter comes after the collapsing matter at U = −∞. Therefore, the boundary condition for the ingoing energy T V V is given by where V out labels the outermost shell. On the other hand, as we have shown in (4.24), the outgoing energy at the surface r = R(a(U)) is given by (4.32) Here we have identified U in (4.2) with u in (1.1) so that A = r 2 −R(a) 2 2σ as in (3.18). U 0 characterizes the time at which the outermost shell gets sufficiently close to R(a(U)) and starts to emit the radiation.
Using these boundary conditions and the conservation laws (4.7) and (4.8) with the assumption (4.1), we obtain (see Appendix A for the derivation.) R(a(U )),U =const. dr(r − a(U)) T µ µ , (4.33) Next, we evaluate T µ µ from (4.21). For the metric (3.18) for r > R(a(u)), we have F = G = 12a(U ) 2 r 6 and obtain where R(a) ≈ a has been used. On the other hand, (4.34) cannot be evaluated explicitly due to the time dependence of a(U). Here, in order to estimate its order, we assume that a(U) is approximately constant. Then, we can have (see Appendix A) Note here that the anomaly leads to particle creation even outside the black hole. The sign of c w − a w depends on the kind of field [23]. For example, it is positive for a massless scalar field, and it is negative for a massless vector field 16 . When c w − a w > 0, (4.36) indicates that the outgoing radiation increases by the amount 3 (cw−aw) 10 1 a(U ) 2 as it goes to infinity from the surface. On the other hand, from (4.37), we can see that the negative ingoing energy is created [21,23,26]. Now we check the self-consistent equation (1.5). First, from (4.35), (4.36) and (4.37), we can see that T µν ∼ 1 a 4 at r ∼ a, which represents the energy-momentum of the radiation around the black hole as in the Stefan-Boltzmann law ∼ T 4 H . The amount of energy in the region around the black hole with the volume V ∼ a 3 is estimated as a , which is much smaller than the mass of the black hole itself, M = a 2G . In this sense, T µν is negligible: T µν ≈ 0, (4.38) and the region outside the black hole is described by vacuum-like solutions such as the Vaidya metric or the Schwarzschild metric.
We have seen so far that the metric (3.18) is the self-consistent solution describing the whole spacetime of the evaporating black hole. There is no horizon or singularity, but this object is the black hole in quantum mechanics (see Fig.6).

The stationary black hole
Next we consider the stationary black hole in the heat bath (3.22). This time we assume (4.13) in addition to (4.1), and use (4.17). We start with examining the boundary condition. See the right of Fig.5. Because the system is stationary, the surface is fixed at r = R(a) =const, and there the ingoing and outgoing energy flows are balanced as (4.39) 16 However, cw 3 + 3(cw−aw) 10 > 0 holds for any kind of massless fields [23], and T UU is always positive at infinity. Here the boundary condition (4.32) plays an important role. Later we will discuss the origin of the radiation more closely.
Here we have used (4.24) and chosen the overall time scale as in (3.22), A(r) = r 2 −R(a) 2 2σ . Then, we calculate T µ µ from (4.20) and obtain the same value as (4.35) except for a =const. We can evaluate T U U from (4.17) with (4.39), and find that T U U = T V V is given by (4.36) with a =const. Now we study the self-consistent equation. Because we have the same order of T µν as in the case of the evaporating black hole, we can follow the same reasoning for (4.38). That is, T µν is negligible, and the metric outside the black hole is close to the Schwarzschild metric.

Generalization
We have assumed so far that the radiation emitted from each shell flows to infinity without reflection, which is expressed by (4.1). For a more realistic description, however, this assumption should be removed.
First we discuss what T U V = 0 means. In the (U, V ) coordinates (4.2), this is equivalent to the nonzero trace in the 2-dimensional part (U, V ): In a (t, r) coordinate system, in which the metric is diagonal, this is expressed as In other words, T U V = 0 is equivalent to − T t t = T r r , which is indeed satisfied by the previous self-consistent solution as in (4.26). Therefore, we characterize T U V = 0 by introducing a function f (t, r) such that Here if we require T r r ≥ 0 and − T t t > 0, f must satisfy |f | ≤ 1. In the following arguments, we assume that the matters are conformal.

Determination of the interior metric
For simplicity, we consider a stationary black hole in the heat bath. More precisely, we describe the exterior by the Schwarzschild metric (3.19), and parametrize the interior metric by (4.10) [4]. Then, we assume that T µν is static and satisfies (4.13). Our program is to fix two functions A(r) and B(r) by two equations.
The first equation comes from (5.3). Once f (r) is given, we rewrite the relation (5.3), by using the self-consistent equation (1.5) for the ansazt (4.10), as In order to build the second equation, we apply the Weyl anomaly formula (4.21) to the trace of (1.5): where we have introduced the notations γ ≡ 8πG c w and α ≡ 8πG a w .
Here, we assume that for r ≫ l p , A(r) and B(r) are large quantities of the same order as expected from (4.15): Then, the first equation (5.4) becomes approximately where A ′ = ∂ r A and we have used B ≫ 1, r∂ r log B. Next, in order to examine what terms dominate in (5.5) for r ≫ l p , we replace A, B, and r with µA, µB, and √ µr, respectively, and pick up the terms with the highest powers of µ. Then, we have Therefore, in the leading order of r, It is natural to expect that the dimensionless function f (r) is a constant for conformal fields [4]: f (r) = const. (5.10) Then, from (5.7), (5.9) and (5.10), we obtain where we have defined (5.12) Thus, the interior metric is determined as Indeed, this is a generalization of (3.21) because (5.12) and (5.13) become (4.28) and (3.21), respectively, if we set f = 0. Redefining the overall scale of time and connecting the metric with the Schwarzschild metric, we reach the generalized metric for the stationary black hole: , − r−a r dt 2 + r r−a dr 2 + r 2 dΩ 2 , for r ≥ R(a) , where R(a) = a + 2σ f a . The metric for the evaporating one is obtained with the outside metric replaced by the Vaidya metric (1.1).

Check of the self-consistent equation
As in section 4, we now evaluate T µν in the metric (5.14), and check the self-consistent equation. Because we assume that T µν is static, we have to determine three functions of r: T U U = T V V , T U V , and T θ θ .

Evaluation of T µν inside the black hole
First we determine T µν in the interior metric (5.13), which can be expressed by (4.2) with (5.11). We assume (5.10) and express the relation (5.3) as where at the second equality we have used (5.12) 17 . Substituting this into (5.21), we obtain which reduces to (4.24) if f = 0. Then, from (4.6), (5.15) and (5.23), we obtain where at the second equality we have used (5.12). On the other hand, we have for the metric (5.13) Comparing (5.24), (5.25) and (5.26) with (5.27), we find that (1.5) is indeed satisfied. Finally, we see that the quantum fluctuation of gravity is small also in the general case. In fact, the invariants of (5.13) are given by where (5.12) has been used. They are small compared with the Planck scale, and therefore the fluctuation is small if (4.30) is satisfied.

Evaluation of T µν outside the black hole
Next we consider the outside region, r > R(a), of the metric (5.14). As we have seen in the previous section, T µν outside the black hole is so small that the modification from the Schwarzschild or Vaidya metric is negligible, although the precise condition to fix T µν is not known. In this subsection, as a simple example, we fix T U U by hand and determine T U V . Then, we show that the region outside the black hole can be described approximately by the Schwarzschild metric. We assume 29) 17 We note that T µ µ is independent of f .
where f is a constant given by (5.10). This means that the total flux emitted from the surface at r = R(a) is kept outside (see (5.23) for A = r 2 −R(a) 2 2(1+f )σ f ) while the other effects (such as particle creation outside the black hole by the anomaly in subsection 4.5) do not contribute to T U U . Furthermore, we take for simplicity T U V | r=R(a) = 0, (5.30) as the boundary condition. We note that (5.29) and (5.30) are not given by some principle but chosen by hand as an example. Then, the first term in the right hand side of (4.14) vanishes while the second term is given through the Weyl anomaly by (4.35) with a =const. Solving (4.14) with the method of variation of constants under (5.30), we obtain 18 This behaves ∼ a 2 r 4 for r ≫ a, which decreases faster than (5.29), and does not contribute to the flux at infinity. Using (4.35) and (5.31), we can evaluate T θ θ through (4.6) as (5.32) Thus, T µν ∼ 1 a 4 around r ∼ a, and we can regard T µν ≈ 0 by the same reasoning for (4.38). Therefore, (1.5) is satisfied by (5.14).

Hawking radiation
In this section we discuss how close the object that we are considering is to the black hole in the conventional picture.

Amount of the radiation
First we show that the object emits the same amount of radiation as the conventional black hole. We prove that the energy flux at r is given by where J is the energy passing through the ingoing spherical null surface at r per unit time. Here the time is "the local time at r" such as u i in (3.4) for the multi-shell model. 18 For given T UU and T µ µ , we solve (4.14) with respect to r 2 T UV and have r 2 T UV (r) = D(r)e ϕ−2 log r = D(r) r−a r 3 , where e ϕ = r−a r has been used. Then, D(r) satisfies ∂ r D = r 3 r−a ∂ r (r 2 T UU (r) ) − 1 2 r 3 T µ µ . Applying (5.29) and (4.35) to this and integrating it from R(a) to r, we obtain (5.31) if (5.30) is considered. [26,27,28,29,30]. It is so strong in the metric (3.21) that the object can be stable against the strong gravitational force 20 21 .

Fate of the incoming matter
Finally we discuss the information problem. In our picture the matter fields simply propagate in the background metric as in the ordinary quantum field theory on curved spacetime, and nothing special happens during the time evolution. Therefore, it is natural to expect that the collapsing matter itself eventually comes back as the radiation.
Indeed, we can get a clue to this by a simple analysis [5]. Suppose that a particle with energy ∼ a comes close to the black hole and becomes a part of it. Then, it starts to emit radiation. As the particle loses energy, its wavelength increases. If the wavelength gets larger than the size of the black hole, then the particle can no longer stay in it. We can estimate the time scale of this process as ∼ a log a √ σ , which is much shorter than that of the evaporation ∼ a 3 σ . Therefore, one of the important future works is to solve the wave equation in the self-consistent metric (3.18) more precisely 22 . If we succeed in it, we should be able to understand how the information of the collapsing matter comes back and especially what happens to the baryon number conservation [5] 23 .

Summary and discussion
Our solution tells what the black hole is. The collapsing matter becomes a dense object and evaporates eventually without forming a horizon or singularity. It has a surface instead of the horizon, but looks like an ordinary black hole from the outside. In the interior the non-trivial structure is formed, where the matter and the Hawking radiation can interact. This can provide a possible solution to the information problem.
There remain problems to be clarified in future. First, as we have mentioned, the important problem is to understand how the information comes back in this picture. To do it, we need to solve the wave equation in the self-consistent metric (3.18).
Second, although we have assumed a constant f to construct the metric (5.14), we don't understand its meaning yet. In principle, f should be determined by the dynamics of matters in the metric (5.14). Therefore, it is interesting to evaluate f concretely by considering a specific theory.
Third, the spherical symmetry has played the important role in our analysis. In the real world, however, we need to consider a rotating black hole, the outside of which is described by the Kerr metric. Although there is a conjecture on the interior metric for 20 We can see explicitly this by constructing the Tolman-Oppenheimer-Volkoff equation with T r r = T θ θ and using − T t t , T r r ≪ T θ θ . 21 See also [31]. 22 See e.g. [32,33] for analysis of matter fields around the black hole. 23 There are many different approaches for the information problem. See e.g. [34,35,36] for one on an infalling observer. a slowly rotating black hole [5], the general form is not known. It would be valuable if we can determine the interior metric by the 4D Weyl anomaly for the general case.
Fourth, we don't know yet how stable the metric (3.21) is for non-spherically symmetric perturbations. When investigating this problem, we need to be careful with the fact that the interior is not a fluid, as we have mentioned below (4.28).
Finally, astrophysics has entered into a new stage by the launch of gravitational wave detectors. For a new physics of black holes it should be exciting to study an observable signal that exhibits some difference between the black holes in our picture and the conventional picture [37,38].
Under (4.1), we integrate (4.7) from V out to V (> V out ) along a fixed U(≥ U 0 ): r(U,Vout),U =const. Here, at the second line (A.2) has been used; at the third line we have used the fact that dr = dV ∂r ∂V U holds along a fixed U (see (A.1)); at the last line we employ (A.1) again. Then, employing the boundary condition (4.32), we obtain (4.33).
Next, we derive (4.34). We integrate (4.8) with the assumption (4.1) and the boundary condition (4.31): where we have used e ϕ(U,V ) = 2 ∂r ∂V U in (A.2). Then, we estimate its order assuming that a(U) varies slowly, a(U) ∼ const. In this case, we can use (4.11) to have dr ′ (r ′ − a) T µ µ .