Firewall From Effective Field Theory

For an effective field theory in the background of an evaporating black hole with spherical symmetry, we consider non-renormalizable interactions and their relevance to physical effects. The background geometry is determined by the semi-classical Einstein equation for an uneventful horizon where the vacuum energy-momentum tensor is small for freely falling observers. Surprisingly, despite the absence of large curvature invariants, the contribution of a class of higher-dimensional operators to the probability amplitudes of particle creations from the Unruh vacuum grows exponentially with time after the collapsing matter enters the near-horizon region. The created particles have high energies for freely falling observers. As a result, the uneventful horizon transitions to a firewall, and the effective field theory breaks down.


Introduction
The information loss paradox [1][2][3] has been puzzling theoretical physicists since the discovery of Hawking radiation [4]. Nowadays, most people, including Hawking [5], believe that the answer is affirmative, at least for a consistent theory of quantum gravity such as string theory. But a persisting outstanding question is how string theory (or any theory of quantum gravity) ever becomes necessary during the evaporation of black holes. 1 That is, how does the low-energy effective field theory break down in the absence of high-energy events [2]?
If there is no high-energy event around the horizon (i.e. "uneventful"), the effective field theory is expected to be a good approximation. But it is incapable of describing the transfer of the complete information inside an arbitrary collapsing matter into the outgoing radiation.
For example, the information hidden inside a nucleus in free fall cannot be retrieved unless there are events (e.g. scatterings) above the scale of the QCD binding energy. This conflict between an uneventful horizon and unitarity has been emphasized in Refs. [2,9] and it has motivated the proposals of fuzzballs [10] and firewalls [9,11].
We shall explain in this paper how an effective field theory predicts the creation of highenergy particles (firewall) around the horizon, which leads to its breakdown.
In the modern interpretation of quantum field theories (see e.g. §12.3 of Ref. [12]), the effective Lagrangian (see eq.(3.2) below) includes all higher-dimensional local operators which are normally assumed to be negligible at low energies because they are suppressed by powers of 1/M p , where M p is the Planck mass (or the cut-off energy). It is well known that, when there are Planck-scale curvatures, the higher-dimensional terms cannot be ignored, and the effective field-theoretic description fails. However, no rigorous proof has been found to show that a non-trivial spacetime geometry without large curvature cannot introduce significant physical effects through these non-renormalizable interactions. In this paper, we show that there are indeed higher-dimensional interactions with large physical effects around the horizon of the evaporating black hole where the curvature is small, and that this eventually leads to the formation of a firewall and the breakdown of the effective field theory.
We first construct the spacetime geometry for a dynamical and uneventful horizon, including the back-reaction of the vacuum energy-momentum tensor in Sec.2. "Uneventful" means that the energy-momentum tensor is small for freely falling observers. In Sec. 3, we show that, after the collapsing matter enters the near-horizon region, a class of higherdimensional interaction terms, which are naively suppressed by powers of 1/M 2n−2 p , leads to an exponentially growing probability of particle creation from the Unruh vacuum in a time scale ∆t ∼ O 1 n−1 a log a p (for n ≥ 2). Here, t is the time for distant observers, a is the Schwarzschild radius of the black hole, and p = 1/M p is the Planck length. Eventually, the effective field theory breaks down. Furthermore, we show that the created particles have 1 Other outstanding questions about the paradox include whether Hawking radiation is thermal, and how its entanglement entropy should be computed. There is significant recent progress in these directions [6][7][8].
high energies as a firewall for freely falling observers. We conclude in Sec.4 with comments on potential implications of our results.
We use the convention = c = 1 in this paper.
2 Back-reacted geometry A hint at the invalidity of low-energy effective field theories around the horizon was the recent finding [13,14] that, until the black hole is evaporated to a tiny fraction of its initial mass, the proper distance between the trapping horizon and the surface of the collapsing matter is at most a few Planck lengths, although the curvature is still small. Such a near-horizon geometry of the dynamical black hole has a Planck-scale nature which is not characterized by the curvature invariants. In this section, we describe the geometry around the near-horizon region by reviewing and extending the results of Refs. [13,14].
We consider the gravitational collapse of a null matter of finite thickness from the infinite past. The spacetime geometry is determined by the expectation value T µν of the energymomentum tensor through the semi-classical Einstein equation Assuming spherical symmetry, the metric can be written in the form We shall consider an asymptotically flat spacetime and adopt the convention that C(u, v) → 1 at large distances.
In the classical limit, T µν = 0 for the space outside the matter, and the geometry is described by the Schwarzschild metric: where a is the Schwarzschild radius.
The vacuum energy-momentum tensor T µν leads to a quantum correction to this solution via eq.(2.1). While the classical solution has a curvature tensor ∼ O(1/a 2 ), the vacuum energy-momentum tensor is κ T µν ∼ O( 2 p /a 4 ) (see eqs.(2.7) -(2.10) below). Therefore, in the Einstein equation (2.1), we can take 2 p /a 2 as the dimensionless parameter to treat the quantum correction perturbatively well outside the horizon where C(u, v) O( 2 p /a 2 ). Such treatment has been widely applied to the study of black-hole geometry in the literature. On the other hand, the geometry close to the horizon could be modified more significantly. Following recent progresses [13][14][15], we give in this section the approximate solution to the semi-classical Einstein equation in the near-horizon region for an adiabatic process. It is characterized by two (generalized) time-dependent Schwarzschild radii a(u) andā(v) (see eq.(2.18) for their definitions). 2 Both a(u) andā(v) agree with the classical Schwarzschild radius a in the limit p /a → 0.

Near-horizon region and uneventful condition
We start by reviewing the definition of the near-horizon region. Roughly speaking, it is defined to be the region near and inside the trapping horizon, but outside the collapsing matter [14]. The surface of the collapsing matter is the inner boundary of the near-horizon region. The outer boundary is slightly outside the trapping horizon where the Schwarzschild approximation is valid. We will restrict our consideration to the early stage of black-hole evaporation when the trapping horizon is timelike in the near-horizon region. (See Fig.1.) The definition of the outer boundary of the near-horizon region is clearly not unique.
Nevertheless, since the quantum correction is small when C(u, v) O( 2 p /a 2 ), or equivalently, when r(u, v) − a 2 p /a according to eq.(2.3), it is reasonable to define it by the condition where u out (v) is the u-coordinate of the outer-boundary of the near-horizon region for a given value of v. 3 The number N should be so large that the Schwarzschild metric with 2 The solution is consistent with previous studies on special cases [13,[16][17][18][19]. 3 It is equally natural to use the condition instead of eq.(2.5). This different choice would not make any essential difference in the discussion below.
the Schwarzschild radiusā(v) is a good approximation around the outer boundary, but so small that the approximation (2.21) given below is good. (This range of N exists because the second condition only requires N a 2 / 2 p .) For a given value of u, the v-coordinate of the outer boundary of the near-horizon region will be denoted by v out (u). It should be the In the conventional model of black holes, the horizon is assumed to be "uneventful" [20][21][22][23][24]. This means that the vacuum energy-momentum tensor is not larger than O(1/a 4 ) for freely falling observers comoving with the collapsing matter. After the coordinate transformation to the light-cone coordinates (u, v), the conditions for uneventful horizons are given by [21,22] T uu ∼ O(C 2 /a 4 ), (2.7) This can be computed either by solving the geodesic equation for freely falling observers, or by computing the transformation factor dU/du between the coordinate u and the light-cone coordinate U of the flat space inside the collapsing matter (see eq.(3.12)).
From the viewpoint of distant observers, the positive outgoing energy T uu (2.7) is nearly vanishing around the horizon because C 1 there, otherwise there would be a huge outgoing energy flux for freely falling observers. 4 On the other hand, T uu approaches O(1/a 4 ) > 0 in the asymptotically flat region where C → 1, corresponding to Hawking radiation at large distances, and the energy of the system must decrease. 5 This means that the ingoing energy flux T vv must be negative around the horizon for energy conservation.
This negative ingoing energy is also the necessary condition for the appearance of a timelike trapping horizon (see e.g. Ref. [15]). The outer boundary of the near-horizon region, which stays outside the trapping horizon, is also time-like. Hence, any point (u, v) inside the where v ah (u) and u ah (v) are the v and u coordinates of the trapping horizon at given u or v, respectively. (See Fig.(1).) In this paper, we will only consider the range of near-horizon region in which (2.12) 4 Using eq.(3.12) below, we obtain T U U C −2 T uu , which would become very large for C 1 unless T uu ∝ C 2 as in eq.(2.7). 5 In the large distance limit r → ∞, only the uu-component remains: For our conclusion about the breakdown of the effective field theory, we will only need the knowledge of the spacetime geometry in a much smaller neighborhood.
The energy-momentum tensor (2.7)-(2.10) for an uneventful horizon corresponds to the Unruh vacuum and is often viewed as an implication of the equivalence principle. However, we will see in Sec.3 that an uneventful horizon always evolves into an eventful horizon at a later time for a generic effective theory soon after the collapsing matter enters the nearhorizon region.

Solution of C(u, v)
In this subsection, we review the solution of C(u, v) in the metric (2.2) [13,14]. Two of the semi-classical Einstein equations G uv = κ T uv and G θθ = κ T θθ can be linearly superposed as [14] ∂ where Σ is defined by (2.14) For the Schwarzschild solution in the near-horizon region.
We shall carry out our perturbative calculation in the double expansion of 2 p /a 2 and C(u, v). The red-shift factor C(u, v) is of order O( 2 p /a 2 ) around the trapping horizon, but C(u, v) gets exponentially smaller as one goes deeper into the near-horizon region. (See eq.(2.21) below.) With more focus on the deeper part of the near-horizon region, every quantity is first expanded in powers of C(u, v), and then the coefficients of each term in powers of 2 p /a 2 . We expand Σ as , etc. At the leading order, eq.(2.13) indicates where we have used eqs.(2.8), (2.10) to estimate T µ µ and T θ θ . Eq.(2.17) can be easily solved by Σ 0 = B(u) +B(v) for two arbitrary functions B(u) and B(v). Without loss of generality, we can define a(u) andā(v) by ,  [14] for more discussion. At the leading order,ā(v) agrees with the mass parameter in the special case of the ingoing Vaidya metric (see App.A). In the classical limit 2 p /a 2 → 0, both a(u) andā(v) approach to the Schwarzschild radius a.
More precisely, since ∂ u Σ 0 is independent of v, it can be identified with ∂ u Σ at the outer boundary of the near-horizon region (where the Schwarzschild solution is a good approximation). Similarly, ∂ v Σ is independent of u and it can also be determined this way. We can think of a(u) andā(v) as the Schwarzschild radii for the best fit of the Schwarzschild metric on constant-u and constant-v slices in a small neighborhood around the boundary of the near-horizon region. For a larger N (see eq.(2.5)), the Schwarzschild approximation is better at the outer boundary of the near-horizon region, hence there should be a smaller difference between a(u out (v)) andā(v). In App.B, we derive the relation between a(u) andā(v) at the boundary of the near-horizon region. The functional forms of a(u) andā(v) are determined by differential equations (2.37), (2.39) to be derived below.
It is then deduced from eqs.(2.14), (2.16), and (2.19) that the solution of C(u, v) can be approximated by [14] C where C * ≡ C(u * , v * ) and r * ≡ r(u * , v * ) for an arbitrary reference point (u * , v * ) in the near-horizon region. For given u, since v < v out (u) inside the near-horizon region (2.11), eq.(2.21) implies that C(u, v) < C(u, v out (u)), where C(u, v out (u)) can be estimated by the Schwarzschild approximation (2.3) to be ∼ N 2 p /ā 2 , using eq.(2.5). Due to the exponential form of C(u, v) (2.21), the value of C is exponentially smaller as we move deeper inside the near-horizon region, i.e. for larger u − u * or larger v * − v.

Solution of r(u, v)
The solution of r(u, v) in the metric (2.2) can be readily derived using the solution of C(u, v) (2.21). We start by estimating the orders of magnitude of ∂ u r and ∂ v r. From the definition of the Einstein tensor G uu for the metric (2.2): the semi-classical Einstein equation G uu = κ T uu and eq.(2.7), we derive which can be integrated as In a similar manner as App.C, we can use C(u, v) (2.21) again to derive from G vv = κ T vv with eq.(2.9) where we chose v * = v ah (u) so that the reference point (u, v ah (u)) is located on the trapping horizon, and used the condition ∂ v r(u, v ah (u)) = 0 on the trapping horizon.
As the linear combination (2.13) of the semi-classical Einstein equations G uv = κ T uv and G θθ = κ T θθ is already satisfied by C(u, v) (2.21), only one more independent linear combination of them is needed. We choose to look at Using eqs.(2.8), (2.21), (2.25), and (2.26) to estimate the order of magnitude of each term in this equation, we find it to be dominated by the two terms C/2r 2 and 2∂ u ∂ v r/r, so that To integrate this, we suppose that the black hole evaporates in the time scale of ∆u, ∆v ∼ O(a 3 / 2 p ) as usual [4]. Hence, the u and v derivatives of a(u) andā(v) introduce additional factors of order O( 2 p /a 3 ) because the two radii are approximately the Schwarzschild radius. Also, from eqs.(2.25) and (2.26), the u and v derivatives of r(u, v) lead to extra factors of . On the other hand, with C(u, v) given by eq.(2.21), its u and v derivatives provide only factors of −1/2a and 1/2ā, respectively. Thus, the functions a(u),ā(v) and r(u, v) are approximately constant in comparison with C(u, v), and eq.(2.28) can be solved by for arbitrary functions f 1 (u) and f 2 (v). However, comparing the first equation (2.29) with eq.(2.25), we see that f 1 (u) has to vanish, because a function of u cannot go to 0 as fast as . The consistent solution to the two equations above is where the function r 0 (v) can be determined as follows. First, in the classical limit, 3) can be rewritten as r = a + rC a + aC near r ∼ a, which resembles eq.(2.31). Since both a(u) andā(v) coincide with the Schwarzschild radius a in the classical limit, we have r 0 = a in the limit as well. Therefore, turning on quantum effect, we expect r 0 (v) to be approximately equal toā(v). To estimate the order of magnitude of the difference we plug the solution r(u, v) (2.31) into the condition (2.5) on the outer boundary of the nearhorizon region for u = u out (v). Then we find the relation where we used eqs.(2.3), (2.5) to evaluate C(u out (v), v). Using eq.(2.32) in eq.(2.31), we find in the near-horizon region. Let us now determine the time-evolution of the functions a(u) andā(v). Plugging eqs.(2.21) and (2.33) back into the semi-classical Einstein equations G uu = κ T uu (with eq.(2.22)), we can check that this equation is trivially satisfied at the leading order in the 2 p /a 2 expansion and does not provide any condition to a(u). Similarly, we can see that As the left-hand side of this equation is u-independent, T vv (u, v) is u-independent at the leading order in the near-horizon region. Recall the the uneventful condition (2.9) that T vv (u, v) must be negative and of the order of O(1/a 4 ). It can thus be expressed as Now, we consider an adiabatic process [25] of Hawking radiation for which |ā /ā| |ā |.
which determines the functional form ofā(v). The function a(u) is approximately equal tō where we used eq.(B.4). Using eq.(2.20) on the right-hand side of eq.(2.37), we find

Near-horizon geometry
The solution for C (2.21) can now be further simplified using the solution for r (2.33) as In the following, we will also need the Christoffel symbol of the metric (2.2): Finally, note that the characteristic length scale for all curvature invariants is stillā, e.g.
As we will see below, nevertheless, the metric (2.33) and (2.40) together with the quantum effect of non-renormalizable operators lead to a non-trivial physical effect.
3 Breakdown of effective theory and formation of firewall For the low-energy effective theory of, say, a 4D massless scalar field φ, we have an action with a Lagrangian density given as a 1/M p -expansion: (Assuming the symmetry φ → −φ, we omit terms of odd powers of φ for simplicity.) The dimensionless parameters a 1 , a 2 , a 3 , b 1 , b 2 , · · · are conventionally treated as coupling constants in a perturbation theory. Higher-dimensional terms are suppressed by higher powers of 1/M 2 p . For a given physical state, the validity of the low-energy effective description relies on the assumption that all higher-dimensional (non-renormalizable) interaction terms, which are suppressed by powers of 1/M 2 p , only have negligible contributions to its time evolution. We will show below that there are in fact higher-dimensional operators in the effective Lagrangian (3.1) that contribute to large probability amplitudes of particle creation from the Unruh vacuum in the near-horizon region. We will see that this particle creation makes the uneventful horizon "eventful" (or even "dramatic").

Free-field quantization in the near-horizon region
In this subsection, we introduce the quantum field theoretic formulation for the computation of the amplitudes mentioned above. It is essentially the same as the standard formulation for the derivation of Hawking radiation (see e.g. Ref. [26]). The difference is that we shall consider the background geometry given in Sec.2, instead of the static Schwarzschild background.
For a massless scalar field φ in the near-horizon region, we shall focus on its fluctuation modes with spherical symmetry. It is convenient to define for the s-wave modes. For the metric (2.2), the free-field equation ∇ 2 φ = 0 is equivalent to According to eqs.(2.28), (2.32) and (2.33), it becomes in the near-horizon region. The free-field equation is thus well approximated by deep inside the near-horizon region where C is exponentially small. Therefore, the general solution there is given by Here, U (u) and V (v) are arbitrary functions of u and v, respectively. The creation and with the rest of the commutators vanishing.
In principle, we can use any functions U (u) and V (v) as the outgoing and ingoing lightcone coordinates. We shall choose the light-cone coordinates U and V so that the vacuum |0 defined by is the Minkowski vacuum of the infinite past before the gravitational collapse starts. This is the vacuum which evolves into Hawking radiation at large distances after it falls in from the past infinity, passes the origin, and then moves out [4]. We assume that this vacuum |0 is the quantum state of the near-horizon region. It is equivalent to the Unruh vacuum -the vacuum state for freely falling observers at an uneventful horizon [27].
The relation between the coordinates U and u can be derived easily by considering the special case when the collapsing matter is a spherical thin shell at the speed of light, and identifying U with the retarded light-cone coordinate of the flat Minkowski spacetime inside the collapsing shell [20,28] as follows. 6 The trajectory of the areal radius R s (u) = r(u, v s ) of the thin shell (where v s is the v-coordinate of the thin shell) satisfies and hence the conditions (2.7)-(2.10) simply mean that We decompose the field φ = ϕ/r (3.7) into the outgoing and ingoing modes. In the near-horizon region, the outgoing modes can be expanded in two bases: (3.14) The two expressions above are related by the coordinate transformation (3.12) and the creation and annihilation operators They are related to {a ω , a † ω } via a Bogoliubov transformation The equality between eqs.(3.13) and (3.14) determines the coefficients A ωω and B ωω as For the vacuum state |0 defined by eq.(3.9), it is natural to define a 1-particle state On the other hand, it is also possible to define a 1-particle state which is a superposition of the 1-particle states |ω a . When this state |ω c propagates to large distances, it would be naturally identified as a 1-particle state (on top of the Hawking radiation) for distant observers. But it can also be defined for a fiducial observer in the near-horizon region. The normalization factor N should be chosen such that In the calculation below, we will need to evaluate the quantity c ω|φ|0 , and hence we have to estimate the matrix A * ωω appearing in eq.(3.20). As we will see, only a short time scale ∆u ∼ O(a log a/ p ) is relevant to our calculation below. Within this time scale, the black-hole mass does not change much so that a(u) ≈ a(u * ), where u * is the time around the region under investigation. Therefore, from eq.(3.12) and eqs.(2.39)-(2.40), we have (3.22) for arbitrary constants U 0 and c 0 , and the Bogoliubov coefficients can be approximated by 7 One then deduces from eqs.(3.23) and (3.24) that For the ingoing modes, we have and there are counterparts of the equations shown above for the outgoing modes. In particular, we can define the 1-particle states But we will not need the operatorsc ω ,c † ω defined with respect to the light-cone coordinates (u, v) for the ingoing modes.

Large matrix elements and firewall
In general, the effective Lagrangian (3.2) includes all local invariants. We consider a special class of higher-dimensional (non-renormalizable) local observables of dimension M 2n+m+2 for all n ≥ 2 and m ≥ 0. The fields φ 1 , φ 2 , and φ 3 are all massless scalars, and all equations for φ in Sec.3.1 apply to φ 1 , φ 2 and φ 3 . (The calculation below will be essentially the same The corresponding term in the action (3.1) is where λ mn is the coupling constant of O(1). We focus on the matrix element where the spacetime integral is done in the near-horizon region. This is related to the probability amplitude of the creation of particles from the Unruh vacuum |0 . We will show below that M mn becomes exponentially large inside the near-horizon region for certain particle states |particles . One might naively think that such an amplitude must be small. Naively, the integral of More precisely, the matrix element (3.33) we shall focus on is The Hilbert space of the perturbative quantum field theory is the tensor product of the Fock spaces of the 3 fields φ 1 , φ 2 and φ 3 . The initial state is the tensor product of the Unruh vacuum for each field, The final state is the multi-particle state |f ≡ |ω c ⊗ |ω ã ⊗ |ω 1 , · · · , ω m ã . (3.36) Here, |ω c is the 1-particle state (3.20) for the outgoing modes of φ 1 , |ω ã the 1-particle state (3.30) for the ingoing mode of φ 2 , and |ω 1 , · · · , ω m ã the m-particle state of the ingoing modes of φ 3 , respectively.
We shall choose so that the outgoing particle has a frequency of the same order of magnitude as the dominant frequency in Hawking radiation. This frequency range must be allowed; otherwise, the existence of Hawking radiation would be dubious. The values of ω , ω 1 , ω 2 , · · · , ω m will not play an important role in showing the matrix element (3.34) to be large. We shall simply choose ω ω 1 · · · ω m 0 (3.38) for simplicity.

Estimation of M mn
Let us now estimate the order of magnitude of the matrix element M mn (3.34). Due to the s-wave reduction, all the spacetime indices µ i , ν i are either u or v, so that each factor of g µν contributes a factor of g uv = −2/C(u, v). The covariant derivatives ∇ u , ∇ v involve derivatives ∂ u , ∂ v , which contribute factors of frequencies ω, ω , in addition to the Christoffel symbol (2.41)-(2.42). Hence, the matrix element is the integral of a polynomial in ω and ω up to the factor e iωu e iω v e i i ω i v . To show that the matrix element (3.34) is large, it is sufficient to focus on a term with given powers of ω and ω , as ω and ω are independent free parameters. We shall focus on the terms with the largest power of ω but independent of ω . It is where we used eq.(3.38) to ignore the factor e iω v e i i ω i v and introduced the coordinates Note that, at large distances, (t, x) are the Minkowski coordinates. We estimate the order of magnitude of eq.(3.39) in the following two steps. for ∆t a and the frequency (3.37). Therefore, unless the time-dependence of the dynamical background is taken into consideration, there can be no particle creation and the matrix element (3.34) vanishes.
When the back-reaction of Hawking radiation is included and the background is timedependent, the matrix element no longer has to vanish. Eq.(2.39) says that the typical time scale of the geometry is ∆t ∼ O(a 3 / 2 p ). Therefore, around a time t = t 0 after the collapsing matter has entered the near-horizon region, the weak time-dependence of the integrand can be approximated by the first-order term in the Taylor expansion in powers of (t − t 0 ): The normalization factor N (3.27) is ignored in the expression above because it is of O(1) and its time-dependence only contributes a small correction. We will consider the integral over a finite spacetime region (x 0 , x 1 ) × (t 0 , t 1 ) in the near-horizon region. (We will check later how large (t − t 0 ) is allowed in order for this perturbative analysis to be valid. See eq.(3.49).) One has to be careful with the time dependence of the back-reaction of the vacuum energy-momentum tensor. If we do not take care of the time-dependence of the metric (2.3) properly, we could mistakenly find the matrix element (3.42) to be negligibly small. Via a straightforward calculation (see App.D), we have for the Schwarzschild metric, so that ∂ ∂t ∂ ∂t can both be large due to the large factors 1/C n−1 in the near-horizon region. Note that the Taylor expansion in powers of (t − t 0 ) carried out in eq.(3.42) is justified only if the 0th order term is much larger than the 1st order term: Thus, C (2.40) can be approximated by which is exponentially small as we move deeper inside the near-horizon region. 8 We shall choose the reference point (t * , x * ) to be located somewhere inside the near-horizon region so Now, consider local observers occupying a certain neighborhood (x 0 , x 1 ) × (t 0 , t 1 ) inside the near-horizon region, with t 0 = t * and x 1 = x * (see Fig.2). For definiteness, we take where the first one is consistent with the condition (3.49). For these local observers, we can estimate the order of magnitude of the probability of observing the final state |f (3.36) by 8 As we go deeper inside the near-horizon region, x becomes smaller and x * − x gets larger. See eq.(3.40) and Fig.2. Thus, eq.(3.51) becomes exponentially small as we get deeper inside the near-horizon region. 9 We can see where such a reference point can be. At the outer boundary of the near-horizon region, we evaluate C(x out ) N 2 p /a 2 0 by applying the conditions (2.5) and (3.50) to the metric (2.3). Here, x out is the x-coordinate of the outer-boundary of the near-horizon region. According to eq.(3.51), we have computing the matrix element M mn (3.42) with the integral defined over this neighborhood (x 0 , x 1 ) × (t 0 , t 1 ). Using eq.(3.47), we estimate it as To derive the second line above, we used where we dropped factors of order 1 (e.g. e iωt 1 and e iωt 0 ) and kept only the first term, as the second term is at most of the same order because of eqs.(3.37) and (3.53). Also, to derive the 3rd line above, the integral in the 2nd line is estimated as

Transition to firewall
Finally, we show that the matrix element (3.57) becomes large for certain domains (x 0 , x 1 ) × (t 0 , t 1 ). In the calculation above, we have assumed the region (x 0 , x 1 ) × (t 0 , t 1 ) to be in the near-horizon region, outside the collapsing matter. Hence, denoting the x-coordinate trajectory of the surface of the collapsing matter as x s (t) and that of the outer boundary of the near-horizon region x out (t), we need x s (t 0 ) x 0 and x 1 x out (t 0 ) (see Fig.2). The spatial range (x 0 , x 1 ) is thus restricted by the condition We choose x 0 close to x s (t 0 ) and x 1 close to x out (t 0 ) so that (x 1 − x 0 ) takes the maximal value consistent with eq.(3.58), (3.59) to maximize the value of the matrix element M mn (3.57). Since t 1 is assumed to be related to t 0 by eq.(3.53), our task is to find t 0 so that (x 1 − x 0 ) (3.59) is sufficiently large. Let t c denote the time when the surface of the collapsing matter enters the outer boundary of the near-horizon region, i.e. x s (t c ) = x out (t c ) (see Fig.2). For t 0 > t c , the value of is expected to be of the same order as t 0 − t c since classically the matter is falling close to the speed of light and x out (t) changes slowly with t, From eqs.(3.59) and (3.60), we deduce and so the matrix element (3.57) is which increases exponentially with t 0 . This implies an exponentially higher probability of the creation of the particles (3.36) at a later time, including both outgoing and ingoing particles.
As exponentially more particles are created at a later time, there is an exponentially larger energy flux. The horizon will eventually be "eventful" and even "dramatic". Let us examine more specifically when the matrix element becomes large or when the firewall is formed. We parametrize the time t 0 (> t c ) by a parameter k as Thus, the expression (3.62) is simply which is huge for k > m + 4. In other words, the matrix element M mn becomes large when t 0 > t c + m+4 n−1 a 0 log(a 0 / p ). Therefore, within a period of time after the matter enters the near-horizon region, the firewall starts to form (for n ≥ 2 and Notice that the time scale ∆t ∼ m+4 n−1 a 0 log(a 0 / p ) is shorter for larger n with m fixed. That is, a higher-dimensional operatorÔ mn with a larger n induces a large probability of particle creation within a shorter time. In other words, higher-dimensional operators are more important at a given time. The 1/M p -expansion of the low-energy effective action (3.2) becomes ill-defined, and the low-energy effective theory loses its predictive power. In fact, ∆t ∼ m+4 n−1 a 0 log(a 0 / p ) → 0 as n → ∞ according to eq.(3.65). This means that, right after the matter enters the near-horizon region, the effective theory starts to break down. 10 As we will see in the next subsection, the created particles have trans-Planckian energies for freely falling observers. The eventful horizon is actually a firewall.

Viewpoint of freely falling observers
In this subsection, we discuss the same physics from the perspective of freely falling observers when the matrix element M mn becomes large. We follow the conventional interpretation [27] that |ω U a (3.19) is the natural outgoing 1-particle state for freely falling observers, where the frequency ω U is the eigenvalue conjugate to the coordinate U . 11 First, we consider the role of the states |ω U a in the matrix element M mn . The final state |f (3.36) contains |ω c with ω ∼ 1/a, which corresponds to the outgoing 1-particle state for observers at infinity. It is a superposition of |ω U a (see eq.(3.20)), where the distribution of ω U is given by |A ωω U | ∝ 1/ √ ω U (see eq.(3.23)). The profile 1/ √ ω U does not approach 0 fast enough at large ω U to guarantee that the integral in the matrix element M mn is not dominated by the accumulative effect of large ω U . Therefore, there is a possibility that such high-frequency modes |ω U a contribute significantly to large matrix elements.
To investigate this issue, we evaluate the matrix element M mn with |ω c in |f replaced by |ω U a . By a calculation similar to the previous subsection, we obtain (see App.(F)) (3.66) For ω U ∼ 1/a 0 , this matrix element is extremely small because dU du (u 0 ) C(u 0 , v s ) 1 according to eq.(3.12). However, if we take 67) 10 The precise time when the low-energy effective theory breaks down depends on the couplings λ mn in the large n limit. In fact, if λ mn → 0 sufficiently fast as n → ∞, it could take a longer time forÔ mn with a larger n to induce a large matrix element, so that the 1/M p -expansion of the effective theory is still valid. Even so, for finite n, it takes only a finite time ∼ O(a 0 log(a 0 / p )) forÔ mn to induce a large probability of particle creation, and the perturbation theory breaks down. 11 Note that U is originally defined as the coordinate of the flat space before the matter collapses from past infinity (see around eq.(3.9)). the matrix element (3.66) is This is essentially identical to M mn (3.57), although the coefficients (n−2) and (n−1) in the exponent are different. Note here that because of eq.(3.12), the frequency (3.67) observed by freely falling observers is exponentially large, i.e., "trans-Planckian". Hence, there is an exponentially large probability for freely falling observers to observe outgoing particles at trans-Planckian energies created out of the Unruh vacuum through the higher-dimensional operators in the near-horizon region, which leads to the firewall.
One might find it strange that such trans-Planckian modes are involved in the low-energy effective theory calculation. There are reasons why the trans-Planckian modes should not be removed. First, the frequencies defined with respect to light-cone coordinates u, v have no locally invariant meaning due to the local Lorentz boost Therefore, assuming Hawking radiation, we cannot remove the trans-Planckian modes |ω U a (3.67), and the assumption of the uneventful horizon leads to the breakdown of the low-energy effective theory.
As an effort to resolve this trans-Planckian problem, there have been proposals of alternative derivations of Hawking radiation which assume non-relativistic dispersion relations such that the energy is bounded from above to be cis-Planckian [30]. They reproduce the same spectrum of Hawking radiation, but this does not completely resolve the trans-Planckian problem [31] as the wave numbers can still be infinite. In the context of this paper, it is reasonable to expect that, since the wave number is still allowed to go to infinity, there are higher-dimensional operators (which are no longer required to be Lorentz-invariant) that produce large matrix elements, and the low-energy effective theory still breaks down. While this remains to be rigorously proven, what we have shown is at least that, for relativistic low-energy effective theories, Hawking radiation (which necessarily includes trans-Planckian modes) is in conflict with the assumption of an uneventful horizon.

Conclusion and discussion
In this work, we showed that the higher-dimensional operators in the effective action change the time evolution of the Unruh vacuum in the near-horizon region of the dynamical black hole so that it evolves into an excited state with many high-energy particles for freely falling observers. The uneventful horizon transitions to an eventful horizon (the firewall), and ultimately the effective field theory breaks down.
We emphasize that we have only used the semi-classical Einstein equation and the con- However, we found that these operators have exponentially large matrix elements related to the creation of particles from the Unruh vacuum, which is in contrast with renormalizable operators. A lot of the high-energy particles would be created for freely falling observers, resulting in a firewall. This invalidates the conditions (2.7)-(2.10) for an uneventful horizon.
Note that no local curvature invariants of the dynamical background are found to be large in the near-horizon region. The high-energy events only arise from the higher-dimensional terms in the effective action, and their origin is a joint effect of the quantum fluctuation and the peculiar geometry of the near-horizon region.
A frequently asked question about such a firewall scenario is this: If freely falling observers see anything extraordinary near the horizon, does it imply the violation of the equivalence principle? The simple answer is that, due to the higher-dimensional operators, the Unruh vacuum evolves to an excited state with many particles for freely falling observers, and the equivalence principle does not apply to the state just like any classical non-gravitational background. 12 We have demonstrated explicitly in this paper how the higher-dimensions operators lead to the large matrix element that corresponds to the probability of knocking virtual particles out of the vacuum into real particles. Freely falling observers are exposed to these high-energy particles. Thus, general covariance, including the transformation (3.69), is not inconsistent with the creation of high-energy particles around the horizon.
While our calculation shows that the low-energy effective theory is inconsistent with an uneventful horizon, it does not tell us with certainty what really happens to an evaporating black hole, or how the information comes out. As high-energy particles are created in the near-horizon region, the background geometry should be modified by the back-reaction of the created particles, and the change of the background should in turn change the rate of particle creation.
It is possible that there is a consistent low-energy effective theory capable of describing an evaporating black hole if the energy-momentum tensor T µν violates the conditions (2.7) -(2.10) so that the horizon is no longer uneventful. A self-consistent scenario is perhaps one that would have no horizon or trapped region, such as the model proposed in Ref. [32][33][34] (see also [35]). 13 It is also recently argued that a consistent quantum theory of gravity should always admit the VECRO [36], which should modify the conventional energy-momentum tensor.
To conclude, we have shown that the horizon cannot remain uneventful (it might not even exist). While the low-energy effective theory eventually breaks down as a result of time evolution from the Unruh vacuum through higher-dimensional operators, it suggests the creation of outgoing high-energy particles in the near-horizon region. Because the proper distance between the collapsing matter and the horizon is of the order of a Planck length [14], the trans-Planckian scattering between the created outgoing particles and the collapsing matter cannot be ignored. It is possible that, through such trans-Planckian scatterings, the information of the collapsing matter is transferred into the outgoing particles, and information loss is no longer a necessary consequence of black-hole evaporation -not until one examines this problem with a Planck-scale theory such as string theory. How information is preserved is still a problem, but it is no longer a paradox.

A Ingoing Vaidya Metric
We consider the ingoing Vaidya metric as an example to demonstrate the meanings of the generalized Schwarzschild radii a(u) andā(v). The ingoing Vaidya metric where a 0 (v) is proportional to the mass parameter of the black hole, is a spherically symmetric solution to the Einstein equation for the energy-momentum tensor with all other components (T vr , T rr , T θθ etc.) vanishing. For a 0 (v) ∼ O( 2 p /a 2 0 ), the energymomentum tensor satisfies the uneventful-horizon condition (2.7) -(2.10), hence the metric (A.1) should be a special case of the general solution (2.21), (2.33) in the near-horizon region.
To put the metric (A.1) in the form of eq.(2.2), we plug r = r(u, v) into the metric (A. 1) and demand that it agrees with eq.(2.2). It is which means that there.

B Relation between a(u) andā(v)
Here we derive the relation (2.20) between the Schwarzschild radii a(u) andā(v) on the outer boundary of the near-horizon region. Take the v-derivative of eq.(2.5), which defines the location of the outer boundary of the near-horizon region, we find ∂r ∂u Use eqs.(2.3), (2.4), (2.5) to estimate ∂r/∂u and ∂r/∂v as Then, together with eq.(2.39), the equation above becomes assuming that N a 2 / 2 p . Next, we take the v-derivative of C(u out (v), v) according to eq.(2.21); which should agree with the Schwarzschild approximation of the same quantity This agreement at the leading order of the 2 p /a 2 expansion means where we used eq.(B.1) and dropped the last term of eq.(B.5) as a higher-order term.
C Order-of-magnitude of the first term in eq.(2.24) Using eqs.(2.7), (2.21), and r/a ∼ O(1), the first term in eq.(2.24) can be estimated as where we assumed that the range (u − u * ) O(a 3 / 2 p ) so that the Schwarzschild radius a remains the same order of magnitude. (This assumption is consistent with the range (2.12).) The integral above can then be estimated as In the evaluation of eq.(2.24), we have taken u * = u out (v) so that (u * , v) lies on the outer boundary of the near-horizon region. Then we can use eqs.(2.3) and (2.5) to evaluate C(u out (v), v) N 2 p /a 2 1. Following eq.(C.1), the first term in eq.(2.24) is estimated as On the other hand, the second term in eq.(2.24) is of O(C). Therefore, the first term is negligible in comparison.

D Derivation of Eq.(3.43)
We derive eq.(3.43) here. For the Schwarzschild background, we have Hence, we can calculate It is straightforward to check explicitly that eq.(3.43) holds for n = 1 and n = 2. By mathematical induction, assuming that eq.(3.43) is correct for a given n, we can show that it also holds for n → n + 1: First, for n = 1 and n = 2, we can use eqs.(2.32), (2.33), (2.37), and (2.42) and check As we can see, a v-derivative leads to an extra factor of O( 2 p /a 2 ) when it acts on a function ofā, so the covariant derivatives are dominated by the 2nd term. Similar to the derivation of eq.(3.43) in App.D, we can use mathematical induction to reach (E.1) for any n. .
The t-derivative at t = t 0 can be estimated as ∂ ∂t where we used eqs.(2.37) and (2.40). A few comments are now in order.
First, as we mentioned in Sec.3 below eq.(3.42), the normalization factor N (3.27) is ignored because it is of order 1 and its time dependence is small. If we had kept N in the time derivative, we would have another piece ∂N /∂t ∼ O(ωȧ) ∼ O( 2 p /a 3 ) in the bracket [· · · ] of the expression above, which is of the same order as the last term in the bracket. Hence, ignoring the contribution of the t-derivative of N does not change the order-of-magnitude estimate of the matrix element.
Second, note that the bracket (n − 1) a(u)−ā(v) a(u)ā +ā (v) a(v) + (m + n + 2) σ 2 p a 3 (v) above contributes to the integral (3.42) in the form of a linear combination An + Bm + C of n and m with some coefficients A, B, and C, where B = σ 2 p /ā 3 (v). It is clear that, for any given value of n, at most one value of m would satisfy An + Bm + C = 0. It is impossible for the bracket to vanish for all m, n. which is analogous to eq.(3.34), but with the final state |f replaced by |f ≡ |ω U a ⊗ |ω ã ⊗ |ω 1 , · · · , ω m ã .

(F.2)
The only difference between this state |f from |f (3.36) is the first factor |ω a versus |ω U c .
The calculation below is in parallel with the calculation of M mn in the main text. Hence we will skip some of the details.
In Sec.3.1, we defined the coordinate U by eq.(3.12). To emphasize the fact that our calculation is invariant under the local Lorentz boosts (3.69), we introduce a class of lightcone coordinates (U , V ) including the U coordinate as a special case. We define this class of coordinate systems by where the free parameter λ corresponds to the freedom of the choice of initial velocity for freely falling observers. The coordinate U defined by eq.