Dilaton gravity with a boundary: from unitarity to black hole evaporation

We point out that two-dimensional Russo-Susskind-Thorlacius (RST) model for evaporating black holes is locally equivalent - at the full quantum level - to Jackiw-Teitelboim (JT) gravity that was recently shown to be unitary. Globally, the two models differ by a reflective spacetime boundary added in the RST model. Treating the boundary as a local and covariant deformation of quantum JT theory, we develop sensible semiclassical description of evaporating RST black holes. Nevertheless, our semiclassical solutions fail to resolve the information recovery problem, and they do not indicate formation of remnants. This means that either the standard semiclassical method incorrectly describes the evaporation process or the RST boundary makes the JT model fundamentally inconsistent.

causally equivalent to empty two-dimensional spacetime, see Fig. 1a. This precludes formation of authentic black holes with horizons and singularities. Introducing N matter fieldsf i (x), one can unitarily transfer information between the past and future JT infinities, but learn nothing about the black hole physics.
In this paper we consider the celebrated Russo-Susskind-Thorlacius (RST) model [18] which is specifically designed to describe evaporating black holes in two dimensions. We point out that this model is locally equivalent, at the full quantum level, to the unitary JT gravity. To establish the equivalence, one promotes the one-loop RST model to a full quantum theory: one adds RST counter-term [18] and N matter fields f i (x) to the action of dilaton gravity [19], and then quantizes the resulting theory in a consistent way suggested by Strominger [20]. After that the path integrals of JT and RST models are related by the local Weyl transformation 1 , where g µν (x), φ(x) and f i (x) are the RST metric, dilaton and matter fields, whilê g µν (x),φ(x) andf i (x) are the fields in the JT model. Note that the change of variables (1.1) causes anomalous transformation of the functional measure which is important for the equivalence. In what follows we denote Q 2 ≡ N/(48π).
Unfortunately, the transformation (1.1) is not defined globally. Indeed, it is noninvertible near the critical value of the dilaton field φ cr = − 1 2 log(Q 2 /2) corresponding to dφ/dφ = 0. This is a problem because all finite-energy JT/RST solutions, e.g. the linear dilaton vacuum, have position-dependent dilaton reaching φ cr at some critical curve (dashed line in Fig. 1b). The transformation from the healthy JT theory to the RST variables is singular at this curve. Early works observed [18,[22][23][24] that the semiclassical solutions describing evaporating RST black holes develop curvature singularities at φ = φ cr , and this impedes quantization of the theory [25]. We show that these singularities disappear in full quantum theory after transforming to the JT terms.
It would be natural to equate the RST and JT models discarding singular parametrization by the RST fields. But that would degrade the spectacular RST black holes to fictitious objects 2 hiding information under the fake singularities of the RST fields, cf. [27].
In this paper we explore another possibility suggested in the original works [18,28,29]. Namely, we deform the RST model: we introduce a reflective spacetime boundary along the line of constant dilaton φ(x) = φ 0 < φ cr and restrict all fields in the path integral to the submanifold φ ≤ φ 0 (white region in Fig. 1b). This excludes the Weyl singularity from the physical domain and makes the model causally similar to the spherically-symmetric multidimensional gravity.
Unlike in the earlier studies, we have a solid tool for selecting sensible boundary conditions at φ = φ 0 . Indeed, we first add the boundary to the healthy JT theory, making it a local, covariant, self-consistent, and weakly coupled deformation, and then transform to the RST terms. Moreover, since (1.1) is valid at the quantum level, we compute one-loop effective action [30] with correct boundary terms [31] in the quantum JT theory and then perform Weyl transformation. This gives one-loop RST boundary conditions which automatically satisfy all self-consistency criteria [32]. Our reflection laws at φ = φ 0 are similar to those in [29,33] but differ from the laws in [28,[34][35][36].
Once the self-consistent boundary conditions are found, we re-inspect information loss problem in the RST model. To this end we study, both analytically [37] and numerically, the semiclassical solutions extremizing the one-loop effective action. A typical high-energy solution is shown in Fig. 1c. It still displays some of the undesirable features observed in the earlier studies [38]. In particular, φ(x) equals φ 0 at three distinct lines: the timelike boundaries i − S and S i + , and a spacelike curve SS . We cannot impose reflective boundary condition at SS , as it would imply strong violation of causality, but we still have to trim the spacetime along this line. Thus, SS is an analog of black hole singularity in the RST model. The incoming matter irreversibly disappears behind this line, cf. [18].
We find that the spacetime of evaporating black hole can be continued into the future beyond the last ray S L. Indeed, the "singularity" line φ = φ 0 generically becomes timelike after some point S -the endpoint of evaporation. Imposing the boundary conditions in that region, we obtain the branch S i + of reflective boundary in Fig. 1c. This makes the spacetime flat in the asymptotic future i + . The price to pay is the jump of second metric derivatives at the last ray S L leading to a small δ-burst of negative energy along this ray ("thunderpop" in [18,38]). The latter inconsistency, however, appears due to sharp change of boundary conditions across the point S . Once "smearing" near S is allowed, the "thunderpop" becomes smooth.
Despite our effort to construct a sensible model, the semiclassical solution in Fig. 1c still indicates an apparent loss of quantum coherence by the evaporating black hole. The effect is caused by the singularity SS irreversibly "eating" the pure quantum state of the incoming matter. We support this intuition by computing the entanglement entropy [39][40][41] of the final state at the future null infinity i 0 i + . We demonstrate that large entropy of Hawking radiation at i 0 L cannot be compensated by the non-thermal radiation at Li + . Thus, initial pure state of matter transforms into a density matrix with nonzero entanglement, and unitarity is broken.
The same mechanism that ruins quantum coherence implies apparent non-conservation of global charges by the evaporating black holes [15][16][17]. Indeed, our model possesses a global shift charge, and we explicitly demonstrate that it disappears behind the black hole singularity SS .
To summarize, in this paper we try to make sense of the evaporating RST black holes by appealing to the healthy JT theory deformed by a boundary. Nevertheless, we observe apparent information loss inside these objects. Possible reasons for this failure will be discussed in Sec. 5.
2 From RST to JT

Weyl transformation
Two-dimensional Russo-Susskind-Thorlacius (RST) model [18] describes interaction of N matter fields f j (x) with non-dynamical gravitational sector: metric g µν (x) and dilaton φ(x). The action of the model includes the classical part representing CGHS dilaton gravity [19] and a quantum counter-term [18] -the last term in the integrand -providing exact solvability at one-loop level. The parameter Q 2 ≡ N/(48π) is proportional to the number of scalar fields, while λ sets the energy scale of the model.
It will be important for us that the above model has a family of classical black hole solutions extremizing the classical action (2.1) without the counter-term, see [19]. These objects evaporate once quantum corrections are taken into account. However, naive description of the evaporation reveals a pathology [20] that has to be cured by choosing a proper quantization procedure.
Usually, one quantizes (2.1) by fixing the conformal gauge, and introducing the standard Faddeev-Popov ghosts [31] where all functional measures depend on the metric scale factor ρ(x) due to the Weyl anomaly. Now, we can evaluate Eq. (2.3) in the one-loop approximation. To this end we integrate out quadratic fluctuations of all fields in some fixed backgroundsay, the black hole metric. We obtain Z naive RST ≈ e iS RST +iS 1−loop , where the correction is given by the non-local Polyakov action [30], In Eq. (2.4) we introduced the Green's function −1 of the d'Alembertian and denoted the total central charge of all fields by c = N − 24. Extremizing the effective action (2.1), (2.4) with respect to the background metric and fields, one obtains one-loop semiclassical equations, with solutions describing evaporating black holes, see [19,42,43] and [18,29,35]. The problem is that the Hawking flux from these objects is always proportional [19] to the factor in front of the one-loop action (2.4) -the total central charge c = N − 24. The latter, however, receives contributions from the entire field content of the model: N from matter fields, +2 from non-dynamical fields φ and ρ, and −26 from ghosts. This means that at N < 24 the black holes emit mainly ghosts and the total energy flux is negative. At larger N the total flux, though positive, is not proportional to the number of dynamical fields f j .
One may wonder, why Hawking evaporation of unphysical Faddeev-Popov ghosts is included in Eq. (2.4). But this effect is unavoidable! Indeed, in Heisenberg picture the ghost operators b µν and c µ satisfy causal equations in the black hole background, just like the ordinary fields. As a consequence, their positive-and negative-frequency components get mixed during evolution between the horizon and asymptotic infinity. Then the same Heisenberg vacuum that is ghost-free at the horizon, automatically contains a flux of ghosts at infinity. This suggests that Eq. (2.3) is physically inconsistent.
Strominger proposed [20] a simple way to avoid this pathology. Namely, let us use the Weyl-transformed metricĝ µν = e −2φ(x) g µν with the scale factor ρ − φ in the functional measures of all non-dynamical fields ρ, φ, b, c, (2.5) Below we will see that any classical solution in the model (2.1) has flatĝ µν . Then the unphysical fields are not emitted from the black holes. Indeed, loop contributions from these fields still produce the Polyakov action (2.4), but with the curvatureR of g µν replacing R. The effect of these contributions vanishes atR = 0 implying that the Hawking flux in Eq. (2.5) is strictly proportional to the number N of dynamical fields.
In what follows we use Eq. (2.5) as a natural quantization of the RST model and do not consider alternative prescriptions.
Let us now argue that the quantum RST model is locally equivalent to the quantum JT theory [1,2]. To this end we rewrite the path integral (2.5) in variables (1.1) taking into account Weyl transformation law of the functional measure [31], where the new metricĝ µν with scale factorρ ≡ ρ − φ is used everywhere in the right-hand side. We obtain, where is the action of the JT model with metricĝ µν , dilatonφ and matter fieldsf j , see [1,2]. The equivalence (2.7) is very natural. One can check [21] that the classical part of the RST action is related to the classical JT gravity by the transformation 3 (1.1). Thus, any quantization on the RST side can be performed in JT terms and vice versa. Moreover, the RST counter-term in Eq. (2.1) was originally introduced [18] to extend the symmetryφ →φ + const to one-loop level. This shift symmetry is manifest in (2.8), but nonlinearly realized in the RST terms. It is not a wonder that a consistent quantization preserving the symmetry reproduces the quantum JT model.
We summarize that the quantum RST model defined in (2.5) is a disguised version of JT gravity. Note that the JT metricĝ µν is flat on the classical field equations because variation of (2.8) with respect toφ givesR = 0. This property guarantees that the Hawking flux of unphysical fields is zero.
In what follows we will strongly rely on the fact [3,4] that the flat-space JT gravity is a healthy quantum theory with unitary S-matrix. This implies, in particular, that the RST model remains local and diffeomorphism-invariant after distortion of the functional measures performed in (2.5). Indeed, inverse Weyl transformation to Eq. (2.6), gives path integral with canonical functional measures and new local counter-terms in the action.

Adding the boundary
Let us explicitly show that the transformation between the JT and RST models cannot be performed globally. Consider the classical JT vacuum in flat light-cone The dilatonφ takes arbitrary values in this two-dimensional spacetime. On the other hand, the functionφ(φ) in Eq. (1.1) is bounded from below by the critical valueφ cr given in the Introduction:φ(φ) ≥φ cr . Thus, the transformation (1.1) cannot be performed atφ <φ cr i.e. in the region between the dashed lines in Fig. 1b. The latter lines are the true singularities of the RST fields.
To remedy the RST model, we introduce a reflective spacetime boundary along the line of constant dilaton field φ = φ 0 , φ 0 < φ cr . To this end we restrict all fields in the path integral (2.5) to the submanifold φ ≤ φ 0 (the rightmost region in Fig. 1b) and add the boundary term to the action, where τ is the proper time at φ = φ 0 and K = ∇ µ n µ is the extrinsic curvature computed with the outer normal n µ . Expression (2.11) includes the Gibbons-Hawking term [44,45] for the curvature part of the action (2.1) and a negative "mass" µ = −2λ(2e −2φ 0 + Q 2 ). We will see that the latter parameter stabilizes the position of the boundary in vacuum, cf. [37]. As before, Q 2 ≡ N/(48π) appears in front of all counter-terms. Note that the boundary makes the RST model weakly coupled, and the parameter e 2φ 0 1 controls semiclassical expansion. Indeed, the change of variables φ ≡ φ − φ 0 and f j ≡ e φ 0 f j brings e −2φ 0 in front of the action (2.1), (2.11) and makes all counter-terms proportional to Q 2 e 2φ 0 ∼ N e 2φ 0 . Below we are interested in the regime e −2φ 0 Q 2 1 (2.12) where the quantum corrections are small and dominated by loops of matter fields.
To describe evaporating black holes in the RST model, we need to derive one-loop Polyakov action with correct boundary terms. It is uniquely fixed by the path integral (2.5). Note first that the boundary introduces local and covariant deformation of the JT action: performing the transformation (1.1) in Eqs. (2.1), (2.11) and adding the correction 4 (2.6), one obtains Eq. (2.8) with the boundary term whereφ 0 is related to φ 0 by Eq. (1.1). Next, we recall that Weyl transformation of the Polyakov action is fixed by the Wess-Zumino condition [31]. Namely, δ Wĝµν = 2w(x)ĝ µν should lead to in any consistent quantum theory, e.g. in the JT gravity. From now on, we use c = N because non-dynamical fields will not contribute into the final Polyakov action anyway. Solution of Eq. (2.14) can be conveniently written in terms of an auxiliary field χ(x) satisfyingˆ χ = −QR ,n µ∇ µχ = 2QK .
The Polyakov action is then One can explicitly check that it satisfies Eq. (2.14) and coincides with Eq. (2.4) in the bulk. Equations (2.15) simply mean that the action is extremal with respect toχ. Since we are going to solve the semiclassical field equations anyway, we will treatχ on equal grounds with other fields.
To summarize, we have defined the quantum RST model, Eqs. (2.5), (2.1), (2.11), in the most sensible way. First, we related it locally to the quantum JT theory [1,2] with unitary S-matrix [3,4]. Second, we regularized the singularity of the Weyl transformation with a dynamical boundary. This gave us the unique one-loop action (2.17). In what follows we describe black hole evaporation using this action.

Semiclassical equations
Given the equivalence between the quantum RST and JT models, it is worth reanalyzing semiclassical evaporation of the RST black holes, cf. [18,28,29,32,33]. To this end we consider the one-loop effective action where the bulk action S RST , boundary term S RST, b , and one-loop correction S RST, 1−loop are given by Eqs. (2.1), (2.11), and (2.17), respectively. In what follows we use only one classical field f = f 1 keeping f j = 0 at j ≥ 2. Recall that nevertheless, all fields fluctuate and contribute into χ. Let us review the semiclassical solutions extremizing the effective action; we leave details of their derivation to Appendices A.1 and A.2. As usual, equations for the propagating fields f , χ simplify in the conformal frame (2.2) with ds 2 = −e 2ρ dudv, where we use the light-cone coordinates u and v. General solution to these equations has the form, where f in,out are the incoming and outgoing wave packets, while χ in,out are their quantum counterparts. It is worth noting that Eq. (2.2) does not completely fix the reparametrization invariance leaving residual conformal symmetry u →ũ(u), v →ṽ(v) [32]. Quantum wave packets transform nontrivially under this symmetry, Of course, Eq. (2.17) satisfies Wess-Zumino condition by itself, and one can derive it without resorting to the JT model. We imposed consistency requirement in explicitly sane terms to avoid confusion. see Eq. (3.2). To explain the transformation law, we introduce the energy fluxes: the incoming flux (3.4) and the outgoing flux T uu (u) -by the same expression with v → u and "in" → "out," see Appendix A.1 for details. According to Eq. (3.3), the fluxes have the standard conformal transformation laws 6 , e.g.
where the term with the Schwarzian comes from the quantum field χ in . As expected, this term is proportional to the total central charge Q 2 ∝ N . Now, recall the correspondence (1.1) with the JT model. Since the JT metric is flat,R = 0, one can introduce flat coordinates withĝ µν = η µν . In RST terms this corresponds to choosing the "Kruskal" gauge with in Eq. (2.2). In the model with a boundary the "Kruskal" coordinates u and v have semi-infinite ranges −∞ < u < 0 and 0 < v < +∞, see Fig. 1b and Appendix A.1. Importantly, equations for the gravitational sector simplify in these coordinates giv- where are the second primitives of the energy fluxes. Let us define the RST vacuum as a solution with f in = f out = 0, This configuration does not look like a vacuum in the "Kruskal" coordinates. However, transforming it via Eqs. (3.3), (3.5) to the uniformly accelerating frame with respect to u and v,ū In what follows we describe the RST boundary φ = φ 0 by a function u = U (v). We derive the reflection laws on this boundary in Appendix A.2 by extremizing the effective action (3.1) with respect to the boundary values of the matter fields. This gives, implying that the boundary acts precisely like a conformal transformation from v to The reflection laws relate the energy fluxes, cf. Eq. (3.5). Note that the boundary condition (3.13) generically appears in all models with moving mirrors, cf. [29,46,47], where the first term conserves classical energy in the rest frame of the mirror and the Schwarzian represents quantum particle production. Equation of motion for the boundary u = U (v) is obtained by recalling that φ = φ 0 along this line, (3.14) where q = (1 + e 2φ 0 Q 2 /2) −1 , see Appendix A.2. We stress that the derivation of this equation heavily relies on the reflection law (3.13); modification of the latter changes U (v) as well. Terms with Q 2 in Eq. (3.14) represent quantum corrections. They are small in the semiclassical regime Q 2 e −2φ 0 because after rescaling f = e −φ 0 f , χ = Qχ , and U = e −2φ 0 U equation (3.14) involve Q and φ 0 in the combination Q 2 e 2φ 0 1. Solving Eq. (3.14) for the RST vacuum (3.10), one obtains, Thus, the vacuum boundary is static, In what follows we solve the Cauchy problem in the semiclassical RST model. Namely, we prepare the finite-energy incoming wave packets f in (v) keeping the initial quantum field in vacuum, χ in = Q log(λv). Computing the incoming flux (3.4), (3.8), we solve the equation of motion (3.14) for U (v). Then the reflection law (3.13) determines the outgoing flux T uu , while Eqs. (3.9), (3.7) fix the spacetime geometry ρ = φ.
a r c t a n ( λ v ) a r c t a n ( λ u ) a r c t a n ( λ v ) a r c t a n ( λ u ) a r c t a n ( λ u )

General properties of solutions
To warm up, we explore the semiclassical solutions numerically. We fix the incoming wave packet, keeping its quantum counterpart χ in in vacuum (3.10). Here the parameters A, σ, and v 0 represent respectively the amplitude, width, and initial position of the wave packet. Numerical integration of Eq. (3.14) gives the boundary trajectories U (v) at low and high energies of f in , see the leftmost solid lines in Figs. 2a and 2b, respectively. The low-energy solution in Fig. 2a describes trivial reflection of matter. Indeed, the respective boundary starts at i − close to the vacuum solution U ∝ v −1 (dashed line), departs from it due to matter pressure and approaches it again at i + . As a direct consequence, the reflected energy flux T uu in Eq. (3.13) tends to the vacuum value at u → ±∞, and the the spacetime (3.7) becomes flat in the asymptotic past and future. We argue in Appendix A.3 that the total energy of the outgoing matter in this case coincides with the energy of f in .
The solution changes completely if the incoming energy exceeds certain value, see Fig. 2b. In this case the boundary crosses the axis u = 0 and continues growing. Numerically computing the dilaton in Eq. (3.7), we discover another anomaly. Namely, the line φ(u, v) = φ 0 has two intersecting branches: the original timelike boundary U (v) and the additional line Si + with spacelike part SS . The semiclassical boundary conditions (3.12) are not satisfied along Si + , and one cannot enforce them without strongly violating causality. As a consequence, this line plays the role of a singularity in the one-loop RST model. Recall that we restrict the spacetime to the region φ < φ 0 , below the line Si + . Yet, the matter freely crosses this line and goes away, see Eq. (3.2). At late times the singularity becomes timelike (part S i + in Fig. 2b). We will consider this region in Sec. 3.4. Let us show that the salient features of solutions in Figs. 1c and 2a,b are, in fact, generic. First, the RST vacuum is indeed stable, and all low-energy solutions correspond to reflection. Adding small perturbation δū λ −1 to the vacuum solution, , one arrives to the linearized equation, where the asymptotic RST coordinatev is used, Eq. (3.11). Both fundamental solutions δū ∝ exp(ikv) of the left-hand side in this equation have Im k > 0 and therefore die off exponentially in the asymptotic future if Q 2 e 2φ 0 < 2, cf. (2.12). This means that the boundary shifted by the matter source in the right-hand side always returns back to the vacuum position, describing reflection. This regime holds if the incident energy is below some threshold. Second, we prove that the spacelike "singularity" φ = φ 0 always forms in the limit of high matter flux, cf. Fig. 2b. In this case the function u = U (v) grows fast and crosses the axis u = 0 due to large right-hand side in Eq. (3.14). Define the apparent horizon 7 u a (v) as the boundary of the region where the curves of constant φ become space-like. Taking the derivative of Eq. (3.7) along these curves and finding the point du/dv = 0, one obtains u a = ∂ v g(v)/λ 2 . Notably, the apparent horizon approaches the axis u = 0 at v → +∞, see Eq. (3.8). Then the growing boundary u = U (v) intersects u a (v) at some point S: U (v s ) = u a (v s ), and this is where the singularity appears. Indeed, ∂ u φ and ∂ v φ are zero at S, see Eqs. (3.7) and (A.14). Hence, Taylor series expansion for φ(u, v) − φ 0 starts from quadratic terms near S. It will be convenient to relate the unknown coefficients of this expansion to the apparent velocity of the boundary u 1 = ∂ v U (v s ) using φ(U (v), v) = φ 0 . Equation φ = φ 0 in the vicinity of the point S takes the form, where T vv is the incoming flux at v = v s and we omitted higher-order terms in v − v s or u − u s . Equation (3.18) has two solutions: the time-like boundary u − u s = u 1 (v − v s ), where u 1 > 0 due to Eq. (3.14), and the spacelike branch . This proves that the space-like singularity φ = φ 0 generically appears in high-energy solutions. Note that the region behind the light-like "horizon" in Fig. 2 can be interpreted as the black hole interior, since matter in this region cannot escape the singularity. 7 Here we treat exp(−2φ) as an analog of the sphere area in the multidimensional gravity.

Solvable deformation
Unfortunately, Eq. (3.14) is not exactly solvable. One can avoid this obstacle by imposing Dirichlet boundary conditions 8 on the auxiliary field χ, This modification is healthy at the one-loop level, but not in the full quantum approach: recall that the boundary condition for χ is fixed by the Wess-Zumino consistency condition [31]. Nevertheless, we will see that solutions in the deformed model (3.19) approximate well the original ones. The condition (3.19) gives the simplified reflection law for the quantum field, χ out (U (v)) = −χ in (v) + const, cf. Eq. (3.12). As a consequence, the naive energymomentum tensors T uu reflect from the boundary classically, cf. [18], Physically, this means that the modification (3.19) switches off direct particle production by the accelerating boundary but leaves the field χ carrying the Hawking flux.
In Appendix A.4 we derive equation for U (v) in the solvable model, where the incoming vacuum χ in = Q log(λv) for the quantum field is assumed. Comparing Eqs. (3.21), (3.14), one finds that the modification replaces ∂ 2 v U/∂ v U in the original equation with −2/v. This is trustworthy in the limit of high energies when the boundary crosses the horizon before colliding with f in . Indeed, the solution before the collision is With this behavior, equations (3.21) and (3.14) coincide everywhere outside the horizon. Apart from the high-energy limit, we expect that the model (3.19) will be useful at the qualitative level for any parameters. Importantly, Eq. (3.21) can be solved exactly. Namely, changing the variables, one obtains a Riccati equation for the unknown W (v). Alternatively, one can fix W (v) and compute U (v), f in (v) by Eqs. (3.23), (3.24). The nontrivial problem, however, is to find the W -ansatz that gives localized and positive-definite (∂ v f in ) 2 . It was shown in Ref. [37] that rational functions from a wide class satisfy the necessary requirements. The simplest ansatz is where a and b are free real parameters and the constant  Fig. 3. It has the same qualitative features as the original diagram in Fig. 2b, with main distinctions related to different choice of the incoming flux, cf. Eqs. (3.24), (3.25) and (3.16). Note that using the method of Ref. [37] one can construct an infinite number of multi-parameter solutions with different shapes of the incoming wave packets. The latter solutions can be classified using the Gaudin spin chain as an auxiliary tool.

Beyond the endpoint
At late times the singularity in Fig. 2b becomes timelike, cf. Fig. 3b. The endpoint S = (u e , v e ) where this happens satisfieŝ where we used Eq. (3.7). Let us compute u e and v e in the quasi-stationary limit (2.12) when evaporation takes large asymptotic timet due to small Q 2 . In this case v e ∝ e λt is exponentially large and u e is small 9 , see Eq. (3.11). Thus, the incoming flux T vv (v e ) at the endpoint is already in vacuum implying where we expressed the integral in Eq.
which confirms that v e is indeed exponentially large at small Q 2 . Indeed, at large energies when Eq. (3.22) is approximately valid, 2λg(v × ) ≈ M cr , where M cr = 2λe −2φ 0 is the minimal black hole mass at Q 2 → 0, see [19,37]. We will see that M cr coincides with the black hole mass at the endpoint; it is smaller than E in . We have just shown explicitly that the endpoint (3.29) exists in generic quasistationary spacetimes of evaporating black holes. Beyond the endpoint the line φ = φ 0 is a naked timelike singularity. A natural way to cure this pathology is to impose reflective boundary conditions (3.13) at the timelike branch S i + of the line φ = φ 0 . In this case one solves Eq. (3.14) for the second boundary U 2 (v) starting from U 2 (v e ) = u e with some apparent velocity ∂ v U 2 (v e ), see Fig. 2c. Since the incoming flux is already at vacuum, U 2 (v) trivially approaches the equilibrium trajectory U 2 → −e −2φ 0 /(λ 2 v) at v → +∞, see discussion in Sec. 3.2. Nevertheless, the resulting spacetime is special in two respects. First, it depends on the arbitrary parameter ∂ v U 2 (v e ). Second, it is non-analytic at the last ray S L due to the jump of the boundary condition at S . Importantly, the initial velocity ∂ v U 2 (v e ) of the new boundary is bounded from above. Indeed, the endpoint is a meeting place of U 2 (v) with the apparent horizon ∂ vφ = 0, see Eq. (3.27). In Sec. 3.2 we learned that the line φ = φ 0 generically 9 The value of λu e is visibly large in Fig. 2b where Q 2 ∼ e −2φ0 for visualization purposes. Typical black hole spacetimes at small Q 2 have λu e 1 like in Fig. 3.
bifurcates at such points, with reflective boundary conditions broken along the additional "singular" branch. We therefore require that the singularity is not naked like in Fig. 4a, but hides under U 2 (v). This gives where we solved Eq. (3.18) and substituted the vacuum flux in the approximate equality. We will see that Eq. (3.30) ruins the "remnant" explanation of the RST information paradox, cf. [27]. Now, consider the energy balance of the complete solution in Fig. 2c. Using Appendix A.3, we compute the total energy of black hole emission before the last ray, In terms of the asymptotic coordinateū this vicinity is a large region, sinceū ≈ −λ −1 log(λv × − e λv ) + const is singular at v → v × . Using thisū(v), one computes the reflected energy flux via Eq. (3.13), where the second term comes from the Schwarzian. With time, the stimulated emission in the first term dies off leaving the stationary thermal flux λ 2 Q 2 = λ 2 N/(48π).
Recall that the right-moving Bose gas with N species and Hawking temperature T H = λ/(2π) has precisely the same as in Eq. (3.32). Note that although h(u) is continuous at u = u e , its derivative is not. Indeed, differentiating (3.7) along the boundaries U (v) and U 2 (v) at u = u e ± 0, we find,  In Sec. 4.2 we will remind that the thunderpop is an inevitable artifact of nonanalytic sewing of two classical spacetimes along the last ray. It gives negligible energy contribution at small Q 2 , so one can disregard it altogether. In Appendix A.3 we show that the total energy is conserved for the complete solution in Fig. 3c which starts from the RST vacuum in the past and arrives to it in the future. This means that the remaining energy is emitted 10 in a non-thermal way at u > u e . Expression (3.35) is the mass of the critical black hole at the endpoint of evaporation, decaying afterwards. In Fig. 4b we plot the outgoing energy flux for the numerical solution in Fig. 2c. It displays all features observed above. Notably, it is not strictly positive-definite beyond the last ray, which is expected [48][49][50].

Information loss revisited 4.1 Endpoint singularity
At first glance, the semiclassical RST solutions look smooth -even their "singularities" are just the spacelike branches of the line φ = φ 0 . Nevertheless, one runs into a trouble trying to apply these solutions to computation of the effective action (3.1). Indeed, the left border of the spacetime in Fig. 2b is not smooth but constructed from three time-and spacelike pieces (lines i − S, SS , and S i + ). It is not clear how to evaluate the extrinsic curvature K and therefore the Gibbons-Hawking term (2.11) at the angles S and S of these pieces.
One can try to regularize S with a smooth curve U reg (v) going between the spacelike and timelike branches of φ = φ 0 . Then the curve becomes light-like at we evaluate the extrinsic curvature of U reg (v) in this region, where τ is a proper distance along U reg at v < v 0 and a proper time at v > v 0 , with τ = τ 0 corresponding to v = v 0 . We see that the Gibbons-Hawking integral (2.11) diverges at the light-like point τ = τ 0 , and there is no apparent way to regularize it, cf. [51,52]. As a second try, we can leave the angles S and S as they are and simply ignore their Gibbons-Hawking contributions. But then arbitrary local counter-terms S S and S S depending on all RST fields at the respective points can be added to the effective action. Importantly, the point S is visible to the distant observer. As a consequence, its counter-term sets boundary conditions for future evolution: the apparent initial velocity of U 2 (v) and quantum state of matter going along the last ray S L.
To summarize, the point S is a naked singularity of the complete solution. It adds arbitrary parameters to the problem and therefore limits our understanding of evaporating black holes.

Thunderpop
In Sec. 3.4 we have found the thunderpop -an outgoing δ-flux carrying small negative energy (3.34) along the last ray, cf. [18]. It is harmless and can be ignored. There is a belief in the literature [38,52], however, that the last ray in causal oneloop models always turns into an infinite-energy "thunderbolt" singularity. Let us show, why this is not the case in our model.
For a start, we reproduce the argument of [38]. Consider the quantum correlator of one RST field, say, f N . Initially, this field is in vacuum, The outgoing two-point function then follows from the reflection law (3.12), where we subtracted the vacuum correlator for regularity and introduced the boundaryv(ū) in asymptotic coordinates. If the boundary is smooth, the regularized correlator (4.2) is finite in the limit u 1 →ū 2 due to cancellation between the first and second terms. However,v(ū) has a jump at the last rayū =ū e , see Fig. 2c. As a consequence, the first term stays finite whenū 1 andū 2 approachū e from different sides, and the second term is divergent. This produces a strong non-integrable singularity in the energy flux atū =ū e , where we used regularization by point separation.
We have already argued, however, that the endpoint S is a naked singularity in the one-loop model, and the quantum state of the fields f i leaving this point is not under control. Let us smear this point into a tiny timelike singularity φ = φ 0 , see Fig. 5a. Then f N, out is not fixed atū L <ū <ū L . One can therefore continue the regularized correlator (4.2) into this region in an arbitrarily smooth way, matching it together with its first two derivatives to the correlators atū L andū L . This will give finite energy-momentum tensor inside the interval LL . There is a restriction, however: the jump of the primitive ∂ u h betweenū L andū L is fixed by the equations for the boundary, see Eq. (3.33). This gives the total energy E th−pop ≈ −λQ 2 inside the "quantum" region LL . Notably, E th−pop in vanishingly small in the quasistationary limit (2.12) that we consider.

Absence of remnants
Let us make the information paradox explicit in our model. To this end we introduce geometric entropy Σ reg (ū 1 ,ū 2 ) characterizing entanglement of the outgoing fields {f i, out } inside the intervalū 1 <ū <ū 2 with anything outside it [39]. In Appendix B we evaluate the entropies of the Hawking quanta Σ − ≡ Σ reg (−∞,ū e − 0) before the last ray and of non-thermal radiation beyond the endpoint Σ + ≡ Σ reg (ū e + 0, +∞), see Fig. 2c. For N = 48πQ 2 fields, wherev(ū) is the boundary in the asymptotically flat coordinates (3.11); atū <ū e andū >ū e it represents U (v) and U 2 (v), respectively. If the state of the outgoing radiation was pure at −∞ <ū < +∞, the geometric entropies Σ − and Σ + would coincide because their intervals are complementary 11 . This equality is automatic in Eq. (4.3) for any smoothv(ū). Physically, Σ + = Σ − would mean that the Hawking radiation is entangled with the remnant [27] -a state of quantum fields beyond the endpoint. However our boundary has an unavoidable jump atū =ū e , and we are going to demonstrate that it makes the two entropies essentially different.
Start with Σ − for the Hawking radiation. To make the estimate transparent, we assume that the incoming matter is well localized, has large energy, and therefore collides with the boundary after crossing the horizon in Fig. 2c. Then U (v) is approximately given by Eq. (3.22), and we obtain, where we introduced the crossing time U (v × ) = 0, applied Eq. (3.29), and ignored the terms suppressed by Q 2 . Expression (4.4) coincides with the thermal entropy of one-dimensional gas with energy E in − M cr and temperature T H = λ/(2π). Thus, the entanglement entropy of the Hawking radiation has the maximal possible value. It can be arbitrarily large at high E in . This reproduces the standard result. The "remnant" entropy Σ + in Eq. (4.3) depends on the apparent initial velocity ∂ v U 2 (v e ) of the second boundary, which is not fixed. In Sec. 3.4 we demonstrated, however, that this velocity satisfies the inequality (3.30), or the spacetime would be singular beyond the endpoint. We thus obtain, where Eq. (3.29) was used in the last equality. Thus, non-thermal radiation beyond the endpoint is almost pure. Parametric difference between Eqs. (4.4) and (4.5) is the essence of the Hawking information paradox. One can try to store the entanglement entropy into the thunderpop which emanates from the endpoint singularity and can be in any quantum state. For example, introducing tanh-like smoothing ofv(ū) atū =ū e , one automatically obtains Σ − = Σ + in Eq. (4.3). The entropies remain equal even in the limit when the smoothing region becomes small, δū → 0. However, the same smoothing introduces an infinite-energy thunderbolt going along the last ray. Indeed, the Schwarzian reflection law (3.13) gives, where we ignored all the terms regular at δū → 0, used the Cauchy inequality, and expressed the result via the entropy mismatch between Eqs. (4.4) and (4.5). We see that the energy of the "thunderbolt" diverges at δū → 0. Moreover, "Planckian" thunderbolt with δū ∼ λ −1 has energy E 2 in /(λQ 2 ) which parametrically exceeds the black hole mass. Thus, our small innocent thunderpop cannot recover the information. To do that, radical large-distance modification of the semiclassical geometry is needed.
There is another curious property of the entanglement entropy which is apparent in our calculations. Recall that the black hole mass at the endpoint takes finite critical value M ≈ M cr = 2λe −2φ 0 . Euclidean methods show [53,54] that the thermodynamical entropy of such black hole is finite, where φ 0 is the value of the dilaton field at the apparent horizon. Due to some reason, the entanglement entropy (4.5) of the same black hole is parametrically smaller, i.e. this object is in almost pure state. Importantly, similar effects were observed in other dynamical calculations where critical black holes behaved as if they had zero entropy [55]. To explain what happens, consider the process with E in ≈ M cr at the threshold of black hole formation. At E < M cr the transition is classical. Then by continuity one does not expect production of large entropy (4.6) at infinitesimally higher energies, even if the critical black hole was formed for a short period during the process. This means that the critical black holes decay into classical small-entropy final states similar to the ones for the processes with E < M cr . Our beyond-the-endpoint solutions reproduce this "classical" decay.

Non-conservation of a global charge
It has long been believed that quantum gravity does not tolerate any conserved global charges. Indeed, by causality black holes evaporate into all sorts of particles regardless to what they were made of, violating all global quantum numbers.
The RST action (2.1), (2.11) has a global shift symmetry f → f + const and the respective conserved current 12 j µ = ∇ µ f . This gives the asymptotic conservation law, where we used the Gauss theorem with boundary terms coming from the line φ = φ 0 and two light-like infinities. Recall that n µ and τ are the outer normal and proper time of the boundary, whereas (ū,v) represent asymptotically flat coordinates. If the spacetime has simple topology like in Fig. 2a, the matter fields satisfy Neumann conditions at the boundary, and the conservation law holds. But once the black hole appears in Fig. 2c, this law gets broken because n µ ∇ µ f = 0 at the spatial sections of the line φ = φ 0 . Figure 5b illustrates violation of the charge (4.7) by the evaporating black hole. In this figure the incoming wave packet f in (v) approaches f atv → ±∞ taking the value f in between (grey region). Its global charge is zero. But reflection from the boundary gives f out = f at the endpoint S of black hole evaporation, and by continuity 13 -in the entire region beyond the last ray S L. Then the global charge of f out is Q out ≡ f − f = 0.
One can again try to pack the compensating charge inside the thunderpop, to save the conservation law. But this requires energy, like in the case with entropy. The energy of the thunderpop in asymptotic coordinates is where δū is the size of the uncontrollable region, we used the Cauchy inequality and denoted Q th−pop = f out (ū e + δū) − f out (ū e − δū). On the other hand, the incoming energy is E in ∼ (f − f ) 2 /∆v = Q 2 out /∆v, where ∆v is a typical width of the incoming energy flux. Thus, Q th−pop Q out , if we want to keep E th−pop < E in and δū ∆v. The other (unphysical) options would be to introduce a large naked singularity with δū ∼ ∆v or give the thunderpop energy exceeding E in .
We conclude that charge non-conservation in our model is robust against quantum corrections at the last ray.

Discussion
In this paper we demonstrated local equivalence between the quantum RST and JT models, and applied it to describe evaporating black holes. We regularized the singularities of the RST fields with the reflective boundary and derived one-loop effective action satisfying all self-consistency requirements. Still, our semiclassical black hole solutions are not satisfactory in three respects. First, they violate the boundary conditions at the spacelike line φ = φ 0 behind the horizon and therefore do not extremize the effective action. This deprives the solutions from their original role of saddle-point configurations for the path integral and therefore makes all further physical interpretations speculative. Second, the endpoint of evaporation corresponds to a naked singularity with divergent action. Regularizing the action, one introduces arbitrary counter-terms which determine the subsequent evolution. Third and as a consequence of the first two, the final state of Hawking radiation has large entanglement entropy indicating unitarity loss.
One can search for the root of the above problems in two directions. The first suspect is the semiclassical method. There are many situations in semiclassical physics where the "naive" solutions do not exist: they are either singular [56] or do not satisfy necessary boundary conditions [57], just like the solutions in the RST model. In this case one can apply the method of constrained instantons [56], i.e. enforce correct behavior of the solutions with additional constraint and then integrate over the constraint in the path integral. This procedure was proposed for the black hole evaporation [55], but has never been used beyond the simplest thin shell models.
Also, one can make the semiclassical method work by computing a different quantity. For example, consider the unitarity relation, where |a , |b , and |c are the coherent Fock states of matter fields in flat spacetime with amplitudes a k , b k , and c k , whileŜ is the S-matrix of the RST model. At a = b Eq. (5.1) involves exponentially suppressed coherent amplitudes of black hole formation and decay into a prescribed final state. The processes of this kind are described by complex semiclassical solutions with distinct properties, cf. [58] and [55,59]. Using 14 the latter in Eq. (5.1), one can directly test unitarity of the RST Smatrix. As a second possibility, one notes that the "regularization" boundary may ruin unitarity of the JT model. Indeed, the boundary action (2.11) describes point particle stiffly coupled to the dilaton field, with trajectory following the line φ = φ 0 . However, models with first-quantized relativistic particles are generically non-unitary due to Klein paradox [60]: computation of transition probabilities in these models gives unphysical results at energies exceeding the threshold for particle-antiparticle production. Our RST solutions demonstrate similar behavior. To remedy this inconsistency, one can try to second-quantize the boundary, promoting it to a quantum field. This may restore unitarity due to boundary-antiboundary production.
14 The integral over c k and c † k can be evaluated in the saddle-point approximation.
A Deriving the semiclassical equations

A.1 Solution in the bulk
We obtain the semiclassical field equations by extremizing the effective action (3.1) with respect to the background fields. In particular, the matter fields f = f 1 and χ satisfy, while variation with respect to g µν gives, In the last equation we introduced the energy-momentum tensors, where expression for the (vv) component can be obtained by replacing u → v. In Eq. (A.7) we separated the contributions T uu (u) and T vv (v) of the incoming and outgoing matter wave packets defined in Eq. (3.4). By themselves, T vv and T uu are not tensors. However, they coincide with the energy fluxes at infinity in the asymptotically flat (ρ = 0) coordinatesū andv. Using Eqs. (A.7) in the "Kruskal" gauge ρ = φ, we rewrite Eq. (A.2) as where the primitives g and h are given by Eqs. (3.8), (3.9).
In the main text we consider solutions starting from the RST vacuum (3.10) in the asymptotic past. This means that the spacetime is flat, R = 0, and φ = −λx in the beginning, see Eq. (3.11). On the other hand, the coordinates u and v cover semi-infinite intervals in Fig. 1b. We can shift them to the domains −∞ < u < 0 and 0 < v < +∞. Then the past time infinity i − is reached at u → −∞, v → 0, and the dilaton fieldφ → −λ 2 uv − Q 2 log(−λ 2 uv)/2 + c 1 + c 2 u should be a finite function ofx in this limit, see Eq. (A.9). Thus, c 2 = 0 and uv is finite at i − . Moreover, we compute the curvature plugging the past asymptotics ofφ into Eq. (A.5) and find,

A.2 Reflection laws
We obtain boundary conditions by varying the effective action (3.1) with respect to the boundary values of all fields. Due to reparametrization invariance we can consider only particular variations preserving the coordinate position of the boundary: δφ = 0 there. Then [37,45] where we left only the boundary terms, introduced the outer normal derivative ∇ n ≡ n µ ∇ µ and the induced metric h µν ≡ g µν − n µ n ν . As before, we keep only one matter field f = f 1 . Then the semiclassical boundary conditions are, where the value of µ was taken from Sec. 2.2. Note that the flat RST vacuum (3.10), (3.11) with the static boundary (3.15) satisfies Eqs. (A.12): in this case K = 0, χ = 0 and ∇ n φ = λ.
In "Kruskal" coordinates (u, v) with ρ = φ the boundary is described by the function u = U (v). The outer normal has components where ∂ v U > 0 for the timelike boundary. Substituting the extrinsic curvature into the first two of Eqs. (A.12), we obtain the boundary conditions Solutions of these equations are the reflection laws of matter fields (3.12) and of their energy fluxes (3.13). The latter reflection laws together with the bulk constraint give equation for U (v). Indeed, full derivative ofφ(U (v), v) along the boundary is zero, hence, where Eq. (3.7) was used. We take derivative of Eq. (A.14) along the boundary and then express ∂ 2 u h from Eqs. (3.9), (3.13). We find, Integrating this equation, one arrives at Eq. (3.14) with arbitrary constant in front of the right-hand side. The last boundary condition in Eqs. (A.12) fixes the value of the multiplicative constant in Eq. (3.14). Indeed, we have already shown that the RST vacuum (3.10), (3.15) satisfies the entire set of semiclassical equations and boundary conditions. On the other hand, U (v) in Eq. (3.15) agrees with Eq. (3.14) only if the constant in that equation equals e 2φ 0 q 2 /λ 2 . One can explicitly check that once this value is fixed, the third of Eqs. (A.12) is equivalent to Eq. (3.14).

A.3 Energy conservation
We introduce the energies E in and E out of the incoming and outgoing matter by recalling that the fluxesTūū andTvv coincide with the respective components of the energy-momentum tensor T µν in the asymptotically flat coordinates (ū,v) with ρ = 0, see Eq. (A.7). Thus, 16) Transforming to the "Kruskal" frame by Eqs. (3.5), (3.11), one gets, where g(v) and h(u) are the second primitives of T vv and T uu in Eqs. (3.8), (3.9). Let us demonstrate that the energy is conserved, E in = E out , for the semiclassical solutions starting from the RST vacuum in the past and arriving to it in the asymptotic future. A black hole in the intermediate state, if it evaporates completely, does not affect this conservation law.
Using the outer normal (A.13), dilaton (3.7), Eq. (A.14), and reflection law for χ in the gauge ρ = φ, we obtain equation, We take the incoming quantum field in vacuum, χ in = Q log(λv), and arrive to Eq. (3.21) from the main text.

B Entanglement entropy
In this Appendix we review the geometric entropy Σ(v 1 ,v 2 ) measuring entanglement of the quantum field inside the interval [v 1 ,v 2 ] with the rest of the world. In the particular case of one massless scalar field in the vacuum state this entropy equals [39], where we restricted attention to the left-moving sector and introduced UV cutoffs δv 1 , δv 2 at the borders of the interval. In quantum theory (B.1) is a divergent quantity which should be renormalized. In thermodynamics, δv 1 and δv 2 are the parameters of coarse-graining. Importantly, the entanglement entropy is conformally invariant [39]. Indeed, general conformal transformation is local. It may distort the state of the quantum field but never changes the degrees of freedom inside the interval. Selecting one RST field -say, f N -one recalls that it is initially in vacuum and its reflection from the boundary is a conformal transformation (3.12). Thus, the outgoing entropy of this field is, where we introduced the trajectoryv =v(ū) of the boundary in asymptotic coordinates and related the cutoffs δv(ū i ) to δū i ; primes are theū-derivatives. We stress that the outgoing field f N, out with entropy (B.2) is not in vacuum. Finally, we regularize the entropy (B.2) by subtracting its vacuum value, This quantity measures entanglement of the outgoing field within the interval [ū 1 ,ū 2 ] relative to the vacuum. In the main text we use half-infinite intervals (−∞,ū) and (ū, +∞). Sendingū 2 → ±∞ and recalling that in this limitv(ū 2 ) →ū 2 − 2φ 0 /λ, we obtain Σ reg (−∞,ū) = Σ reg (ū, +∞) = − 1 12 logv (ū) .