Semiclassical S-matrix for black holes

We propose a semiclassical method to calculate S-matrix elements for two-stage gravitational transitions involving matter collapse into a black hole and evaporation of the latter. The method consistently incorporates back-reaction of the collapsing and emitted quanta on the metric. We illustrate the method in several toy models describing spherical self-gravitating shells in asymptotically flat and AdS space-times. We find that electrically neutral shells reflect via the above collapse-evaporation process with probability exp(-B), where B is the Bekenstein-Hawking entropy of the intermediate black hole. This is consistent with interpretation of exp(B) as the number of black hole states. The same expression for the probability is obtained in the case of charged shells if one takes into account instability of the Cauchy horizon of the intermediate Reissner-Nordstrom black hole. Our semiclassical method opens a new systematic approach to the gravitational S-matrix in the non-perturbative regime.


Introduction
Almost forty years of intensive research leave the black hole information paradox [1,2] as controversial as ever. Although the argument based on the AdS/CFT correspondence [3,4] indicates that quantum gravity is dual to a healthy CFT and therefore unitary, the process of black hole evaporation still presents an apparent mismatch between the principles of low-energy gravity and those of quantum theory. In particular, recent AMPS version of the paradox [5,6] suggests that certain measurements of Hawking quanta reveal a firewall around the black hole which destroys infalling observers and violates the equivalence principle (see [7][8][9] for related works). Thus, a systematic approach to the processes of black hole formation and evaporation is needed.
A plausible source of confusion is perturbative expansion around the classical black hole background. This expansion is certainly valid at short time scales but has been argued [10] to give inappropriate quantum state of Hawking radiation at late stages of black hole evaporation when the information is released. Indeed, the classical black hole does not correspond to a well-defined asymptotic state of quantum gravity; at best it can be regarded as a metastable state. Used as a zeroth-order approximation for quantum calculations, it is likely to introduce inconsistencies.
A consistent approach to black hole unitarity considers a two-stage scattering process involving collapse and black hole evaporation, see Fig. 1. The initial and final states Ψ i and Ψ f of this process represent free matter particles and free Hawking quanta in flat space-time. Unlike the black hole, these are the true asymptotic states of quantum gravity related by an S-matrix [11][12][13]. The latter must be unitary if black hole formation does not lead to information loss. To realize that the scattering setup is natural for unitarity tests, one imagines a gedanken experiment at a future trans-Planckian collider where collision of a few energetic particles forms a micro black hole, the latter evaporates and its decay products are registered. Experimentalists analyse the scattering amplitudes in various channels and verify if they obey relations imposed by unitarity.
The importance of collapse stage for the resolution of information paradox was emphasized before [13][14][15][16][17][18]. However, no working scheme for calculating the black hole S-matrix from first principles has been formulated so far. Interesting approaches to the gravitational S-matrix have been developed in Refs. [19,20] (see also references therein). Based on perturbative calculations, they demonstrate that scattering of two trans-Planckian particles is accompanied by an increasingly intensive emission of soft quanta as the regime of black hole formation is approached. However, the validity of perturbative expansion in the black hole-mediated regime is not fully understood. To circumvent the obstacles, we focus on the case when both the final and initial states of the scattering process are made of a large number of soft particles. We assume that the total energy of particles exceeds the Planck scale, so that the intermediate black hole has mass well above Planckian. Then the overall process is expected to be described semiclassically in low-energy gravity. Below we develop a systematic semiclassical method to calculate the gravitational S-matrix elements.
A straightforward application of the semiclassical approach to scattering through an intermediate black hole is problematic. The reason is traced to the mismatch between the asymptotic states of classical and quantum gravity. We want to evaluate the amplitude of transition between the initial and final asymptotic states with wave functionals Ψ i [Φ i ] and Ψ f [Φ f ]. The path integral in Eq. (1.1) runs over all fields Φ of the theory including matter fields, metrics and ghosts from gauge-fixing of the diffeomorphism invariance; S is the action. In the asymptotic past and future the configurations Φ in Eq. (1.1) must describe a collection of free particles in flat space-time. However, this condition is not satisfied by the saddle-point configuration Φ cl saturating the integral (1.1) in the semiclassical limit → 0. Indeed, Φ cl extremizes S i.e. solves the Einstein-Hilbert equations and classical equations for matter fields. Since black holes are stable asymptotic states in classical gravity, the solution Φ cl starts with matter in flat space-time and arrives to a black hole in the asymptotic future. It fails to describe the second part of the process -the evaporation of the black hole -and as such, does not satisfy the final-state boundary conditions in Eq. (1.1). One concludes that the amplitude (1.1) cannot be computed with the standard saddle-point technique even when the conditions for the semiclassical approximation are fulfilled.
To overcome this obstacle, we use the modified semiclassical method of Refs. [21][22][23] (see [24,25] for the seminal ideas and [26,27] for field theory applications). The key idea is to constrain integration in the path integral (1.1) to scattering configurations Φ where the mass is concentrated in a compact volume for a fixed time T 0 as measured by the asymptotic observer. Since T 0 is finite, this constraint explicitly eliminates configurations with eternal black holes from the domain of integration. The resulting constrained path integral is saturated by the saddle-point solution with the correct asymptotic behavior corresponding to free particles in the past and future flat space-times. One can say that the constraint forces the intermediate black hole to decay. At the final step of the computation one recovers the original amplitude by integrating over T 0 , i.e. one-parameter family of saddle-point configurations corresponding to different values of black hole lifetime 1 .
Two points must be emphasized. First, in our approach one works with the saddlepoint configurations satisfying the asymptotic boundary conditions and thus encapsulating the black hole decay in the leading order of the semiclassical expansion. This is a crucial difference from the fixed-background semiclassical methods where the black hole evaporation is accounted for only at the one-loop level. Second, the saddle-point configurations saturating the scattering amplitudes are in general complex and do not admit a straightforward interpretation as classical geometries. In particular, they are meaningless for an observer falling into the black hole. Indeed, the latter observer measures local correlation functions given by the path integrals in the in-in formalism -with different boundary conditions and different saddle-point configurations as compared to those in Eq. (1.1). This distinction lies at the heart of the black hole complementarity principle [29].
Our approach is completely general and can be applied to any gravitational system with no symmetry restrictions. However, the task of solving nonlinear saddle-point equations is rather challenging. Below we illustrate the method in several exactly tractable toy models describing spherical gravitating dust shells. We consider neutral and charged shells in asymptotically flat and anti-de Sitter (AdS) space-times. Applications to field theory that are of primary interest are postponed to future.
Although the shell models involve only one collective degree of freedom -the shell radius -they are believed to capture some important features of quantum gravity [30][31][32][33]. Indeed, one can crudely regard thin shells as narrow wavepackets of an underlying field theory. In Refs. [33][34][35] emission of Hawking quanta by a black hole is modeled as tunneling of spherical shells from under the horizon. The respective emission probability includes back-reaction of the shell on geometry, where B i and B f are the Bekenstein-Hawking entropies of the black hole before and after the emission. It has been argued in [36] that this formula is consistent with unitary evolution.
In the context of shell models we consider scattering processes similar to those in Fig. 1: a classical contracting shell forms a black hole and the latter completely decays due to quantum fluctuations into an expanding shell. The initial and final states Ψ i and Ψ f of the process describe free shells in flat or AdS space-times. Our result for the semiclassical amplitude (1.1) has the form The probability is P f i exp(−2Im S reg / ). We show that for neutral shells it coincides with Eq. (1.2), where B i is set equal to the entropy of the intermediate black hole and B f = 0. This is consistent with the result of Refs. [30][31][32][33] since the first stage of the process, i.e. formation of the intermediate black hole, proceeds classically. For charged black holes the same result is recovered once we take into account instability of the inner Cauchy horizon of the intermediate Reissner-Nordström black hole [37][38][39][40][41][42]. Our results are therefore consistent with the interpretation of Hawking radiation as tunneling. However, we obtain important additional information: the phases of the S-matrix elements which explicitly depend, besides the properties of the intermediate black hole, on the initial and final states of the process.
The paper is organized as follows. In Sec. 2 we introduce general semiclassical method to compute S-matrix elements for scattering via black hole formation and evaporation. In Sec. 3 we apply the method to transitions of a neutral shell in asymptotically flat spacetime. We also discuss relation of the scattering processes to the standard thermal radiation of a black hole. This analysis is generalized in Sec. 4 to a neutral shell in asymptotically AdS space-time where scattering of the shell admits an AdS/CFT interpretation. A model Figure 2. The contour used in the calculation of the S-matrix elements. Quantum transition from t i to t f is preceded and followed by the free evolution.
with an electrically charged shell is studied in Sec. 5. Section 6 is devoted to conclusions and discussion of future directions. Appendices contain technical details.
2 Modified semiclassical method 2.1 Semiclassical S-matrix for gravitational scattering The S-matrix is defined as whereÛ is the evolution operator; free evolution operatorsÛ 0 on both sides transform from Schrödinger to the interaction picture. In our caseÛ describes quantum transition in Fig. 1, whileÛ 0 generates evolution of free matter particles and Hawking quanta in the initial and final states. The time variable t ∈ [t i , t f ] is chosen to coincide with the time of an asymptotic observer at infinity. Using path integrals for the evolution operators and taking their convolutions with the wave functionals of the initial and final states, one obtains the path integral representation for the amplitude 2 (2.1), where Φ = {φ, g µν } collectively denotes matter and gravitational fields 3 along the time contour in Fig. 2. The interacting and free actions S and S 0 describe evolution along different parts of the contour. The initial-and final-state wave functionals Ψ i and Ψ f depend on the fields Φ ∓ ≡ Φ(t = 0 ∓ ) at the endpoints of the contour. In the second equality of Eq. (2.2) we combined all factors in the integrand into the "total action" S tot [Φ]. Below we mostly focus on nonlinear evolution from t i to t f and take into account contributions from the dashed parts of the contour in Fig. 2 at the end of the calculation.
To distinguish between different scattering regimes, we introduce a parameter P characterizing the initial state [43] -say, its average energy. If P is small, the gravitational interaction is weak and the particles scatter trivially without forming a black hole. In this regime the integral in Eq. (2.2) is saturated by the saddle-point configuration Φ cl satisfying the classical field equations with boundary conditions related to the initial and final states [44]. However, if P exceeds a certain critical value P * , the classical solution Φ cl corresponds to formation of a black hole. It therefore fails to interpolate towards the asymptotic out-state Ψ f living in flat space-time. This marks a breakdown of the standard semiclassical method for the amplitude (2.2).
To deal with this obstacle, we introduce a constraint in the path integral which explicitly guarantees that all field configurations Φ from the integration domain have flat space-time asymptotics. Namely, we introduce a functional T int [Φ] with the following properties: it is (i) diff-invariant; (ii) positive-definite if Φ is real; (iii) finite if Φ approaches flat space-time at t → ±∞; (iv) divergent for any configuration containing a black hole in the asymptotic future. Roughly speaking, T int [Φ] measures the "lifetime" of a black hole in the configuration Φ. Possible choices of this functional will be discussed in the next subsection; for now let us assume that it exists. Then we consider the identity where in the second equality we used the Fourier representation of the δ-function. Inserting Eq. (2.3) into the integral (2.2) and changing the order of integration, we obtain, The inner integral over Φ in Eq. (2.4) has the same form as the original path integral, but with the modified action This implies that Φ has correct flat-space asymptotics. The integral over T 0 is saturated at = 0. Importantly, we do not substitute = 0 into the saddle-point equations for Φ , since in that case we would recover the original classical equations together with incorrect asymptotics of the saddle-point solutions. Instead, we understand this equation as the limit → +0 (2.7) that must be taken at the last stage of the calculation. The condition Re > 0 is required for convergence of the path integral (2.4). We obtain the saddle-point expression (1.3) for the amplitude with the exponent 4 where the limit is taken in the end of the calculation. To summarize, our method breaks computation of the S-matrix elements into two steps. First, one modifies the action according to Eq. (2.5), where Re > 0, and solves the corresponding classical equations of motion. The modified solutions Φ automatically approach flat space-time in the asymptotic past and future. Second, one evaluates the action on the modified solutions and sends → +0 obtaining the leading semiclassical exponent of the S-matrix element, see A remark is in order. Since the modification adds complex terms to the action, the modified saddle-point configurations Φ are also complex. Typically, the space of complex saddle-point solutions is complicated and selecting the physical solution poses a non-trivial challenge. To this purpose we use the method of continuous deformations. Namely, we pick up a real classical solution Φ 0 describing scattering at a small value of the parameter P < P * . By construction, Φ 0 approaches flat space-time at t → ∓∞ and gives the dominant contribution to the integral (2.4). Next, we modify the action and gradually increase from = 0 to the positive values constructing a continuous branch of modified solutions Φ . At → +0 these solutions reduce to Φ 0 and therefore saturate the integral (2.4). We finally increase the value of P to P > P * assuming that continuously deformed saddlepoint configurations Φ remain physical 5 . In this way we obtain the modified solutions and the semiclassical amplitude at any P . We stress that our continuation procedure cannot be performed with the original classical solutions which, if continued to P > P * , describe formation of black holes. On the contrary, the modified solutions Φ interpolate between the flat-space asymptotics at any P . They are notably different from the real classical solutions at P > P * .

The functional T int [Φ]
Let us construct the appropriate functional T int [Φ]. This is particularly simple in the case of reduced models with spherically-symmetric gravitational and matter fields. The general spherically-symmetric metric has the form where dΩ is the line element on a unit two-sphere and g ab is the metric in the transverse two-dimensional space 6 . Importantly, the radius r(y) of the sphere transforms as a scalar 4 Below we consider only the leading semiclassical exponent. The prefactor in the modified semiclassical approach was discussed in [21][22][23]. 5 In other words, we assume that no Stokes lines [45] are crossed in the course of deformation. This conjecture has been verified in multidimensional quantum mechanics by direct comparison of semiclassical and exact results [21-25, 46, 47]. 6 We use the signature (−, +, . . .) for the metrics gµν and g ab . The Greek indices µ, ν, . . . are used for the four-dimensional tensors, while the Latin ones a, b, . . . = 0, 1 are reserved for the two-dimensional space of the spherically reduced model. under the diffeomorphisms of the y-manifold. Therefore the functional is diff-invariant. Here w(r) and F (∆) are non-negative functions, so that the functional (2.10) is positive-definite. We further require that F (∆) vanishes if and only if ∆ = 1. Finally, we assume that w(r) significantly differs from zero only at r r w , where r w is some fixed value, and falls off sufficiently fast at large r. An example of functions satisfying these conditions is To understand the properties of the functional (2.10), we consider the Schwarzschild frame where r is the spatial coordinate and the metric is diagonal. The functional (2.10) takes the form, Due to fast falloff of w(r) at infinity the integral over r in this expression is finite. However, convergence of the time integral depends on the asymptotics of the metrics in the past and future. In flat space-time g 11 = 1 and the integrand in Eq. (2.10) vanishes. Thus, the integral over t is finite if g ab approaches the flat metric at t → ±∞. Otherwise the integral diverges. In particular, any classical solution with a black hole in the final state leads to linear divergence at t → +∞ because the Schwarzschild metric is static and g 11 = 1. Roughly speaking, T int can be regarded as the Schwarzschild time during which matter fields efficiently interact with gravity inside the region r < r w . If matter leaves this region in finite time, T int takes finite values. It diverges otherwise. Since the functional (2.10) is diff-invariant, these properties do not depend on the particular choice of the coordinate system. The above construction will be sufficient for the purposes of the present paper. Beyond the spherical symmetry one can use the functionals T int [Φ] that involve, e.g., an integral of the square of the Riemann tensor, or the Arnowitt-Deser-Misner (ADM) mass inside a large volume.
3 Neutral shell in flat space-time

The simplest shell model
We illustrate the method of Sec. 2 in the spherically symmetric model of gravity with thin dust shell for matter. The latter is parameterized by a single collective coordinate -the shell radius r(τ ) -depending on the proper time along the shell τ . This is a dramatic simplification as compared to the realistic case of matter described by dynamical fields. Still, one can interprete the shell as a toy model for the evolution of narrow wavepackets in field theory. In particular, one expects that the shell model captures essential features of gravitational transition between such wavepackets. 7 7 Note that our approach does not require complete solution of the quantum shell model which may be ambiguous. Rather, we look for complex solutions of the classical equations saturating the path integral.

The minimal action for a spherical dust shell is
where m is the shell mass. However, such a shell always collapses into a black hole and hence is not sufficient for our purposes. Indeed, as explained in Sec. 2.1, in order to select the physically relevant semiclassical solutions we need a parameter P such that an initially contracting shell reflects classically at P < P * and forms a black hole at P > P * . We therefore generalize the model (3.1). To this end we assume that the shell is assembled from particles with nonzero angular momenta. At each point on the shell the velocities of the constituent particles are uniformly distributed in the tangential directions, so that the overall configuration is spherically-symmetric 8 . The corresponding shell action is [49] where L is a parameter proportional to the angular momentum of the constituent particles.
Its nonzero value provides a centrifugal barrier reflecting classical shells at low energies.
Decreasing this parameter, we arrive to the regime of classical gravitational collapse. In what follows we switch between the scattering regimes by changing the parameter L ≡ P −1 . For completeness we derive the action (3.2) in Appendix A.
Gravitational sector of the model is described by the Einstein-Hilbert action with the Gibbons-Hawking term, Here the metric g µν and curvature scalar R are defined inside the space-time volume V with the boundary 9 ∂V. The latter consists of a time-like surface at spatial infinity r = r ∞ → +∞ and space-like surfaces at the initial and final times t = t i,f → ∓∞. In Eq. (3.4) σ are the coordinates on the boundary, h is the determinant of the induced metric, while K is the extrinsic curvature involving the outer normal. The parameter κ equals +1 (−1) at the time-like (space-like) portions of the boundary. To obtain zero gravitational action in flat space-time, we subtract the regulator K 0 which is equal to the flat-space extrinsic curvature of the boundary [50]. For the sphere at infinity K 0 = 2/r ∞ , while the initial-and final-time hypersurfaces have K 0 = 0. The Gibbons-Hawking term (3.4) will play an important role in our analysis. Let us first discuss the classical dynamics of the system. Equations of motion follow from variation of the total action with respect to the metric g µν and the shell trajectory y a (τ ). In the regions inside and outside the shell the metric satisfies vacuum Einstein equations and therefore, due to Birkhoff theorem, is given by the flat and Schwarzschild solutions, respectively, see Fig. 3a.
Introducing the spherical coordinates (t − , r) inside the shell and Schwarzschild coordinates (t + , r) outside, one writes the inner and outer metrics in the universal form The parameter M is the ADM mass which coincides with the total energy of the shell. In what follows we will also use the Schwarzschild radius r h ≡ 2M . For the validity of the semiclassical approach we assume that the energy is higher than Planckian, M 1. Equation for the shell trajectory is derived in Appendix B by matching the inner and outer metrics at the shell worldsheet with the Israel junction conditions [51,52]. It can be cast into the form of an equation of motion for a particle with zero energy in an effective potential,ṙ This potential goes to −∞ at r → 0 and asymptotes to a negative value 10 1 − M 2 /m 2 at r = +∞, see Fig. 4. At large enough L the potential crosses zero at the points A and A -the turning points of classical motion. A shell coming from infinity reflects from the point A back to r = +∞. When L decreases, the turning points approach each other and 10 Recall that the shell energy M is always larger than its rest mass m. coalesce at a certain critical value 11 L = L * . At even smaller L the turning points migrate into the complex plane, see Fig. 5 (upper left panel), and the potential barrier disappears. Now a classical shell coming from infinity goes all the way to r = 0. This is the classical collapse. Now, we explicitly see an obstacle for finding the reflected semiclassical solutions at L < L * with the method of continuous deformations. Indeed, at large L the reflected solutions r = r(τ ) are implicitly defined as where the square root is positive at r → +∞ + i 0. The indefinite integral is performed along the contour C running from r = +∞ − i0 to r = +∞ + i0 and encircling the turning point A -the branch point of the integrand (see the upper left panel of Fig. 5). As L is lowered, the branch point moves and the integration contour stays attached to it. However, at L = L * when the branch points A and A coalesce, the contour C is undefined. It is therefore impossible to obtain reflected semiclassical solutions at L < L * from the classical solutions at L > L * .

Modification
To find physically relevant reflected trajectories at L < L * , we use the method of Sec. 2 and add an imaginary term i T int to the action. We consider T int of the form (2.10), where the function w(r) is concentrated in the vicinity of r = r w . The radius r w is chosen to be large enough, in particular, larger than the Schwarzschild radius r h and the position r A of the right turning point A. Then the Einstein equations are modified only at r ≈ r w , whereas the geometries inside and outside of this layer are given by the Schwarzschild solutions with masses M and M , see Fig. 3b. To connect these masses, we solve the modified Einstein equations in the vicinity of r w . Inserting general spherically symmetric metric in the Schwarzschild frame, into the (tt) component of Einstein equations, we obtain, The solution reads 12 , This gives the relation Here˜ > 0 is the new parameter of modification. As before, the ADM mass M of the system is conserved in the course of the evolution. It coincides with the initial and final energies of the shell which are, in turn, equal, as will be shown in Sec. 3.3, to the initial-and final-state energies in the quantum scattering problem. Thus, M is real, while the mass M of the Schwarzschild space-time surrounding the shell acquires a positive imaginary part 13 . The shell dynamics in this case is still described by Eq. (3.8), where M is replaced by M in the potential (3.9). Below we find semiclassical solutions for small˜ > 0. In the end˜ will be sent to zero. Let us study the effect of the modification (3.14) on the semiclassical trajectories r = r(τ ) in Eq. (3.10). At L > L * the complex terms in V eff are negligible and the reflected trajectory is obtained with the same contour C as before, see the upper left panel of Fig. 5. The modification of V eff becomes important when L gets close to L * and the two turning points A and A approach each other. Expanding the potential in the neighborhood of the maximum, we write, where V max , µ and r max depend on L and M . For real M = M the extremal value V max is real and crosses zero when L crosses L * , whereas the parameters µ 2 > 0 and r max remain approximately constant. The shift of M into the upper complex half-plane gives a negative imaginary part to V max , where the last inequality follows from the explicit form (3.9). Now, it is straightforward to track the motion of the turning points using Eq. (3.15) as L decreases below L * . Namely, A and A are shifted into the lower and upper half-planes as shown in Fig. 5 (upper right panel). Importantly, these points never coalesce. Physically relevant reflected solution at L < L * is obtained by continuously deforming the contour of integration in Eq. (3.10) 12 The functionf is time-independent due to the (tr) equation. 13 In this setup the method of Sec. 2 is equivalent to analytic continuation of the scattering amplitude into the upper half-plane of complex ADM energy, cf. [25]. while keeping it attached to the same turning point 14 . As we anticipated in Sec. 2, a smooth branch of reflected semiclassical solutions parameterized by L exists in the modified system.
If L is slightly smaller than L * , the relevant saddle-point trajectories reflect at Re r A > r h and hence never cross the horizon. A natural interpretation of the corresponding quantum transitions is over-barrier reflection from the centrifugal potential. However, as L decreases to L → 0, the centrifugal potential vanishes. One expects that the semiclassical trajectories in this limit describe complete gravitational transitions proceeding via formation and decay of a black hole.
We numerically traced the motion of the turning point A as L decreases from large to small values, see Fig. 5 (lower panel). It approaches the singularity 15 r = 0 at L → 0. This behavior is confirmed analytically in Appendix C. Thus, at small L the contour C goes essentially along the real axis making only a tiny excursion into the complex plane near the singularity. It encircles the horizon r = r h from below. 14 In the simple shell model we can take˜ = 0 once the correspondence between the solutions at L > L * and L < L * is established. This may be impossible in more complicated systems [21,22,24,25] where the relevant saddle-point trajectories do not exist at = 0 and one works at nonzero till the end of the calculation. 15 For the validity of low-energy gravity the turning point should remain in the region of sub-Planckian curvature, R µνλρ R µνλρ ∼ M 2 /r 6 1. This translates into the requirement rA M 1/3 which can be satisfied simultaneously with L L * provided the total energy is higher than the Planck mass, M 1. Figure 6. The time contour corresponding to the semiclassical solution at small L. Solid and dashed lines correspond to interacting and free evolution respectively, cf. Fig. 2.

S-matrix element
The choice of the time contour. The action S reg entering the amplitude (1.3) is computed along the contour in complex plane of the asymptotic observer's time t ≡ t + .
Since we have already found the physically relevant contour C for r(τ ), let us calculate the Schwarzschild time t + (r) along this contour. We write, where the indefinite integral runs along C. In Eq. (3.17) we used the the definition of the proper time implying and expressedṙ 2 from Eq. (3.8). The integrand in Eq. (3.17) has a pole at the horizon r = r h , f + (r h ) = 0, which is encircled from below, see Fig. 5, lower panel. The halfresidue at this pole contributes iπr h to t + each time the contour C passes close to it. The contributions have the same sign: although the contour C passes the horizon in the opposite directions, the square root in the integrand changes sign after encircling the turning point. Additional imaginary contribution comes from the integral between the real r-axis and the turning point A; this contribution vanishes at L → 0. The image C t of the contour C is shown in Fig. 6, solid line. Adding free evolution from t + = 0 − to t + = t i and from t + = t f to t + = 0 + (dashed lines), we obtain the contour analogous to the one in Fig. 2. One should not worry about the complex value of t f in Fig. 6: the limit t f → +∞ in the definition of S-matrix implies that S reg does not depend on t f . Besides, the semiclassical solution r = r(t + ) is an analytic function of t + and the contour C t can be deformed in complex plane as long as it does not cross the singularities 16 of r(t + ). Below we calculate the action along C t because the shell position and the metrics are real in the initial and final parts of this contour. This simplifies the calculation of the Gibbons-Hawking terms at t + = t i and t + = t f . 16 In fact, Ct is separated from the real time axis by a singularity where r(t+) = 0. This is the usual situation for tunneling solutions in quantum mechanics and field theory [24,25]. Thus, Sreg cannot be computed along the contour in Fig. 2; rather, Ct or an equivalent contour should be used.
Interacting action. Now, we evaluate the action of the interacting system S(t i , t f ) entering S reg . We rewrite the shell action as An important contribution comes from the Gibbons-Hawking term at spatial infinity r = r ∞ → +∞. The extrinsic curvature reads, (3.20) The first term here is canceled by the regulator K 0 in Eq. (3.4). The remaining expression is finite at r ∞ → +∞, where we transformed to integral running along the contour C using Eq. (3.17). Note that this contribution contains an imaginary part Finally, in Appendix D we evaluate the Gibbons-Hawking terms at the initial-and finaltime hypersurfaces. The result is where r i,f are the radii of the shell at the endpoints of the contour C. The latter radii are real, and so are the terms (3.23). Summing up the above contributions, one obtains, This expression contains linear and logarithmic divergences when r i,f are sent to infinity. Note that the divergences appear only in the real part of the action and thus affect only the phase of the reflection amplitude but not its absolute value.
Initial and final-state contributions. The linear divergence in Eq. (3.24) is related to free motion of the shell in the asymptotic region r → +∞, whereas the logarithmic one is due to the 1/r tails of the gravitational interaction in this region. Though the 1/r terms in the Lagrangian represent vanishingly small gravitational forces in the initial and final states, they produce logarithmic divergences in S(t i , t f ) when integrated over the shell trajectory. To obtain a finite matrix element, we include 17 these terms in the definition of the free action S 0 . In Appendix E the latter action is computed for the shell with energy M , where r 1,2 are the positions of the shell at t + = 0 ∓ and are the initial and final shell momenta with 1/r corrections. The path integral (2.2) for the amplitude involves free wavefunctions Ψ i (r 1 ) and Ψ f (r 2 ) of the initial and final states. We consider the semiclassical wavefunctions of the shell with fixed energy E, where p i,f are the same as in Eq. (3.26). In fact, the energy E is equal to the energy of the semiclassical solution, E = M . Indeed, the path integral (2.2) includes integration over the initial and final configurations of the system, i.e. over r 1 and r 2 in the shell model. The condition for the stationary value of r 1 reads, It is straightforward to check that this expression is finite in the limit r i,f → +∞. In Fig. 7 we plot its real and imaginary parts as functions of L for the case of massless shell (m = 0).  In the most interesting case of vanishing centrifugal barrier L → 0 the only imaginary contribution to S reg comes from the residue at the horizon r h = 2M in Eq. (3.29), recall the contour C in Fig. 5. The respective value of the suppression exponent is This result has important physical implications. First, Eq. (3.30) depends only on the total energy M of the shell but not on its rest mass m. Second, the suppression coincides with the Bekenstein-Hawking entropy of a black hole with mass M . The same suppression was obtained in [33,34] for the probability of emitting the total black hole mass in the form of a single shell. We conclude that Eq. (3.30) admits physical interpretation as the probability of the two-stage reflection process where the black hole is formed in classical collapse with probability of order 1, and decays afterwards into a single shell with exponentially suppressed probability. One may be puzzled by the fact that, according to Eq. (3.29), the suppression receives equal contributions from the two parts of the shell trajectory crossing the horizon in the inward and outward directions. Note, however, that the respective parts of the integral (3.29) do not have individual physical meaning. Indeed, we reduced the original twodimensional integral for the action to the form (3.29) by integrating over sections of constant Schwarzschild time. Another choice of the sections would lead to an expression with a different integrand. In particular, using constant-time slices in Painlevé or Finkelstein coordinates one obtains no imaginary contribution to S reg from the inward motion of the shell, whereas the contribution from the outward motion is doubled. The net result for the probability is, of course, the same. 18 The above result unambiguously shows that the shell model, if taken seriously as a full quantum theory, suffers from the information paradox. Indeed, transition between the only two asymptotic states in this theory -contracting and expanding shell -is exponentially suppressed. Either the theory is intrinsically non-unitary or one has to take into consideration an additional asymptotic state of non-evaporating eternal black hole formed in the scattering process with probability 1 − P f i .
On the other hand, the origin of the exponential suppression is clear if one adopts a modest interpretation of the shell model as describing scattering between the narrow wavepackets in field theory. Hawking effect implies that the black hole decays predominantly into configurations with high multiplicity of soft quanta. Its decay into a single hard wavepacket is entropically suppressed. One can therefore argue [36] that the suppression (3.30) is compatible with unitarity of field theory. However, the analysis of this section is clearly insufficient to make any conclusive statements in the field theoretic context.
As a final remark, let us emphasize that besides the reflection probability our method allows one to calculate the phase of the scattering amplitude Re S reg . At L = m = 0 it can be found analytically, (3.31) It explicitly depends on the parameter r 0 of the initial-and final-state wavefunctions.

Relation to the Hawking radiation
In this section we deviate from the main line of the paper which studies transitions between free-particle initial and final states, and consider scattering of a shell off an eternal preexisting black hole. This will allow us to establish a closer relation of our approach to the results of [33,34] and the Hawking radiation. We focus on the scattering probability and thus consider only the imaginary part of the action. The analysis essentially repeats that of the previous sections with several differences. First of all, the inner and outer space-times of the shell are now Schwarzschild with the metric functions where M BH is the eternal black hole mass and M denotes, as before, the energy of the shell. The inner and outer metrics possess horizons at r − h = 2M BH and r + h = 2(M BH +M ), respectively. The shell motion is still described by Eq. (3.8), where the effective potential is obtained by substituting expressions (3.32) into the first line of Eq. (3.9). Next, the global space-time has an additional boundary r = r ∞ → +∞ at the second spatial infinity of the eternal black hole, see Fig. 8. We have to include the corresponding Gibbons-Hawking term, cf. Eq. (3.21), 18 Note that our semiclassical method is free of uncertainties [53][54][55] appearing in the approach of [33].
shell r ∞ r ∞ Finally, the eternal black hole in the initial and final states contributes into the free action S 0 . We use the Hamiltonian action of an isolated featureless black hole in empty spacetime [56], 35) where the integration contour C is similar to that in Fig. 5 (lower panel), it bypasses the two horizons r − h and r + h in the lower half of complex r-plane. In the interesting limit of vanishing centrifugal barrier L → 0 the imaginary part of the action is again given by the residues at the horizons, where B ± = π(r ± h ) 2 are the entropies of the intermediate and final black holes. This suppression coincides with the results of [33,34].
At M BH = 0 the process of this section reduces to reflection of a single self-gravitating shell and expression (3.36) coincides with Eq. (3.30). In the other limiting case M M BH the shell moves in external black hole metric without back-reaction. Reflection probability in this case reduces to the Boltzmann exponent where we introduced the Hawking temperature T H = 1/(8πM BH ). One concludes that reflection of low-energy shells proceeds via infall into the black hole and Hawking evaporation, whereas at larger M the probability (3.37) includes back-reaction effects.

Space-time picture
Let us return to the model with a single shell considered in Secs. 3.1-3.3. In the previous analysis we integrated out the non-dynamical metric degrees of freedom and worked with the semiclassical shell trajectory (t + (τ ), r(τ )). It is instructive to visualize this trajectory in regular coordinates of the outer space-time. Below we consider the case of ultrarelativistic shell with small angular momentum: L → 0 and M m. One introduces Kruskal coordinates for the outer metric, We choose the branch of the square root in these expressions by recalling that M differs from the physical energy M by an infinitesimal imaginary shift, see Eq. (3.14). The initial part of the shell trajectory from t + = t i to the turning point A (Figs. 5, 6) is approximately mapped to a light ray V = V 0 > 0 as shown in Fig. 9. Note that in the limit L → 0 the turning point A is close to the singularity r = 0, but does not coincide with it. At the turning point the shell reflects and its radius r(τ ) starts increasing with the proper time τ . This means that the shell now moves along the light ray U = U 0 > 0, and the direction of τ is opposite to that of the Kruskal time U +V . The corresponding evolution is represented by the interval (A, t f ) in Fig. 9. We conclude that at t + = t f the shell emerges in the opposite asymptotic region in the Kruskal extension of the black hole geometry. This conclusion may seem puzzling. However, the puzzle is resolved by the observation that the two asymptotic regions are related by analytic continuation in time. Indeed it is clear from Eqs. (3.38) that the shift t + → t + − 4πM i corresponds to total reflection of Kruskal coordinates U → −U , V → −V . Precisely this time-shift appears if we extend the evolution of the shell to the real time axis (point t f in Fig. 6). At t + = t f the shell emerges in the right asymptotic region 22 with future-directed proper time τ . The process in Fig. 9 can be viewed as a shell-antishell annihilation which is turned by the analytic continuation into the transition of a single shell from t i to t f . Now, we write down the space-time metric for the saddle-point solution at m = 0 and L → 0. Recall that in this case the shell moves along the real r-axis. We therefore introduce global complex coordinates (r, t + ), where t + belongs to C t and r is real positive. The metric is given by analytic continuation of Eqs. (3.6), (3.7), where we changed the inner time t − to t + by matching them at the shell worldsheet r = r shell (t + ). Importantly, the metric (3.39) is regular at the origin r = 0 which is never reached by the shell. It is also well defined at r h = 2M due to the imaginary part of M ; in the vicinity of the Schwarzschild horizon r h the metric components are essentially complex. Discontinuity of Eq. (3.39) at r = r shell (t + ) is a consequence of the δ-function singularity in the shell energy-momentum tensor. This makes the analytic continuation of the metric ill-defined in the vicinity of the shell trajectory. We expect that this drawback disappears in the realistic field-theory setup where the saddle-point metric will be smooth (and complex-valued) in Schwarzschild coordinates.

Reflection probability
In this and subsequent sections we subject our method to further tests in more complicated shell models. Here we consider a massless shell in 4-dimensional AdS space-time. The analysis is similar to that of Sec. 3, so we will go fast over details. The shell action is still given by Eq. (3.2) with m eff = L/r, while the Einstein-Hilbert action is supplemented by the cosmological constant term, Here Λ ≡ −3/l 2 , l is the AdS radius. The Gibbons-Hawking term has the form (3.4), where now the regulator at the distant sphere is chosen to cancel the gravitational action of an empty AdS 4 . The metric inside and outside the shell is AdS and AdS-Schwarzschild, respectively, where M is the shell energy. The trajectory of the shell obeys Eq. (3.8) with the effective potential given by the first line of Eq. (3.9), The -modification again promotes M in this expression to M = M + i˜ . Repeating the procedure of Sec. 3.2, we start from the reflected trajectory at large L. Keeping˜ > 0, we trace the motion of the turning point as L decreases 23 . The result is a family of contours C spanned by the trajectory in the complex r-plane. These are similar to the contours in Fig. 5. In particular, at L → 0 the contour C mostly runs along the real axis encircling the AdS-Schwarzschild horizon r h from below, as in the lower panel of Fig. 5. Calculation of the action is somewhat different from that in flat space. First, the space-time curvature is now non-zero everywhere. Trace of the Einstein's equations gives 24 R = 4Λ. The Einstein-Hilbert action takes the form, The last term diverging at r ∞ → ∞ is canceled by the similar contribution in the Gibbons-Hawking term at spatial infinity, Second, unlike the case of asymptotically flat space-time, Gibbons-Hawking terms at the initial-and final-time hypersurfaces t + = t i,f vanish, see Appendix D. Finally, the canonical momenta 25 of the free shell in AdS, are negligible in the asymptotic region r → +∞. Thus, the terms involving p i,f in the free action (3.25) and in the initial and final wavefunctions (3.27) are vanishingly small if the normalization point r 0 is large enough. This leaves only the temporal contributions in the free actions, (4.8) 23 Alternatively, one can start from the flat-space trajectory and continuously deform it by introducing the AdS radius l. 24 In the massless case the trace of the shell energy-momentum tensor vanishes, T Summing up Eqs. (4.5), (4.6), (4.8) and the shell action (3.2), we obtain, where the integration contour in the last expression goes below the pole at r = r h . The integral (4.9) converges at infinity due to fast growth of functions f + and f − . In particular, this convergence implies that there are no gravitational self-interactions of the shell in the initial and final states due to screening of infrared effects in AdS. The imaginary part of Eq. (4.9) gives the exponent of the reflection probability. It is related to the residue of the integrand at r h , We again find that the probability is exponentially suppressed by the black hole entropy.
Remarkably, the dependence of the reflection probability on the model parameters has combined into r h which is a complicated function of the AdS-Schwarzschild parameters M and l.

AdS/CFT interpretation
Exponential suppression of the shell reflection has a natural interpretation within the AdS/CFT correspondence [3,57,58]. The latter establishes relationship between gravity in AdS and strongly interacting conformal field theory (CFT). Consider three-dimensional CFT on a manifold with topology R × S 2 parameterized by time t and spherical angles θ. This is the topology of the AdS 4 boundary, so one can think of the CFT 3 as living on this boundary. Let us build the CFT dual for transitions of a gravitating shell in AdS 4 . Assume the CFT 3 has a marginal scalar operatorÔ(t, θ); its conformal dimension is ∆ = 3. This operator is dual to a massless scalar field φ in AdS 4 . Consider now the composite operator where G M (t) is a top-hat function of width ∆t 1/M . This operator is dual to a spherical wavepacket (coherent state) of the φ-field emitted at time t 0 from the boundary towards the center of AdS [59,60]. is proportional to the amplitude for reflection of the contracting wavepacket back to the boundary. If the width of the wavepacket is small enough, ∆t l, the φ-field can be treated in the eikonal approximation and the wavepacket follows a sharply defined trajectory. In this way we arrive to the transition of a massless spherical shell in AdS 4 , see Fig. 10.
Exponential suppression of the transition probability implies respective suppression of the correlator (4.12). However, the latter suppression is natural in CFT 3 because the state created by the composite operatorÔ M (0) is very special. Submitted to time evolution, it evolves into a thermal equilibrium which poorly correlates with the state destroyed bŷ O + M (πl). Restriction of the full quantum theory in AdS 4 to a single shell is equivalent to a brute-force amputation of states with many soft quanta in unitary CFT 3 . Since the latter are mainly produced during thermalization, the amputation procedure leaves us with exponentially suppressed S-matrix elements.

Elementary shell
Another interesting extension of the shell model is obtained by endowing the shell with electric charge. The corresponding action is the sum of Eq. (3.5) and the electromagnetic contribution where A µ is the electromagnetic field with stress tensor F µν = ∂ µ A ν − ∂ ν A µ and Q is the shell charge. This leads to Reissner-Nordström (RN) metric outside the shell and empty flat space-time inside, Other components of A µ are zero everywhere. Importantly, the outside metric has two horizons r Figure 11. Motion of the turning points and the contour C defining the trajectory for (a) the model with elementary charged shell and (b) the model with discharge.
at Q < M . At Q > M the horizons lie in the complex plane, and the shell reflects classically. Since the latter classical reflections proceed without any centrifugal barrier, we set L = 0 henceforth. The semiclassical trajectories will be obtained by continuous change of the shell charge Q. The evolution of the shell is still described by Eq. (3.8) with the effective potential constructed from the metric functions (5.2), This potential always has two turning points on the real axis, The shell reflects classically from the rightmost turning point r A at Q > M . In the opposite case Q < M the turning points are covered by the horizons, and the real classical solutions describe black hole formation. We find the relevant semiclassical solutions at Q < M using -modification. Since the modification term (2.10) does not involve the electromagnetic field, it does not affect the charge Q giving, as before, an imaginary shift to the mass, M → M + i˜ . A notable difference from the case of Sec. 3 is that the turning points (5.5) are almost real at Q < M . The semiclassical trajectories therefore run close to the real r-axis 27 for any Q. On the other hand, the horizons (5.3) approach the real axis with Im r from below and from above, respectively. Since the semiclassical motion of the shell at Q < M proceeds with almost real r(τ ), we can visualize its trajectory in the extended RN geometry, see Fig. 12. The shell starts in the asymptotic region I, crosses the outer and inner horizons r h , repels from the time-like singularity due to electromagnetic interaction, and finally re-emerges in the asymptotic region I . At first glance, this trajectory has different topology as compared to the classical reflected solutions at Q > M : the latter stay in the region I at the final 27 The overall trajectory is nevertheless complex because t+ ∈ C, see below. time t + = t f . However, following Sec. 3.5 we recall that the Schwarzschild time t + of the semiclassical trajectory is complex in the region I , where we used Eq. (3.17) and denoted by t i and t f the values of t + at the initial and final endpoints of the contour C in Fig.11a. Continuing t f to real values, we obtain the semiclassical trajectory arriving to the region I in the infinite future 28 , cf. Sec. 3.5. This is what one expects since the asymptotic behavior of the semiclassical trajectories is not changed in the course of continuous deformations. Let us now evaluate the reflection probability. Although the contour C is real, it receives imaginary contributions from the residues at the horizons. Imaginary part of the total action comes 29 from Eq. (3.29) and the electromagnetic term (5.1). The latter takes the form, where we introduced the shell current j µ , used Maxwell equations ∇ µ F µν = 4πj ν and integrated by parts. From Eq. (5.2) we find, (5.8) 28 Indeed, the coordinate systems that are regular at the horizons r However, they are real and do not contribute into Im Stot.
Combining this with Eq. (3.29), we obtain, After non-trivial cancellation we again arrive to a rather simple expression. However, this time 2Im S tot is not equal to the entropy of the RN black hole, B RN = π r The physical interpretation of this result is unclear. We believe that it is an artifact of viewing the charged shell as an elementary object. Indeed, in quantum mechanics of an elementary shell the reflection probability should vanish at the brink Q = M of classically allowed transitions. It cannot be equal to B RN which does not have this property unlike the expression (5.9). We now explain how the result is altered in a more realistic setup.

Model with discharge
Recall that the inner structure of charged black holes in theories with dynamical fields is different from the maximal extension of the RN metric. Namely, the RN Cauchy horizon r (−) h suffers from instability due to mass inflation and turns into a singularity [38][39][40]. Besides, pair creation of charged particles forces the singularity to discharge [37,41,42]. As a result, the geometry near the singularity resembles that of a Schwarzschild black hole, and the singularity itself is space-like. The part of the maximally extended RN space-time including the Cauchy horizon and beyond (the grey region in Fig. 12) is never formed in classical collapse.
Let us mimic the above discharge phenomenon in the model of a single shell. Although gauge invariance forbids non-conservation of the shell charge Q, we can achieve essentially the same effect on the space-time geometry by switching off the electromagnetic interaction at r → 0. To this end we assume spherical symmetry and introduce a dependence of the electromagnetic coupling on the radius. This leads to the action where e(x) is a positive form-factor starting from e = 0 at x = 0 and approaching e → 1 at x → +∞. We further assume e(x) < x , (5.11) the meaning of this assumption will become clear shortly. Note that the action (5.10) is invariant under gauge transformations, as well as diffeomorphisms preserving the spherical symmetry. The width of the form-factor e(r/Q) in Eq. (5.10) scales linearly with Q to mimic larger discharge regions at larger Q.
The new action (5.10) leads to the following solution outside the shell, The space-time inside the shell is still empty and flat. As expected, the function f + corresponds to the RN metric at large r and the Schwarzschild one at r → 0. Moreover, the horizon r h satisfying f + (r h ) = 0 is unique due to the condition (5.11). It starts from r h = 2M at Q = 0, monotonically decreases with Q and reaches zero at Q * = 2M/a(0). At Q > Q * the horizon is absent and the shell reflects classically. The subsequent analysis proceeds along the lines of Secs. 3, 4. One introduces effective potential for the shell motion, cf. Eq. (5.4), where b 2 ≡ −da/dx x=0 is positive according to Eq. (5.12). As Q decreases within the interval the turning point makes an excursion into the lower half of the r-plane, goes below the origin and returns to the real axis on the negative side, see Fig 11b. For smaller charges r A is small and stays on the negative real axis. The contour C defining the trajectory is shown in Fig. 11b. It bypasses the horizon r h from below, goes close to the singularity, encircles the turning point and returns back to infinity. This behavior is analogous to that in the case of neutral shell. Finally, we evaluate the imaginary part of the action. The electromagnetic contribution is similar to Eq. (5.8), However, in contrast to Sec. 5.1, the trace of the gauge field energy-momentum tensor does not vanish due to explicit dependence of the gauge coupling on r (cf. Eq. (B.3b)), This produces non-zero scalar curvature R = −8πT µ EM µ in the outer region of the shell, and the Einstein-Hilbert action receives an additional contribution, where in the second equality we integrated by parts. Combining everything together, we obtain (cf. Eq. (5.9)), where non-trivial cancellation happens in the last equality for any e(x). To sum up, we accounted for the discharge of the black hole singularity and recovered the intuitive result: the reflection probability is suppressed by the entropy of the intermediate black hole 30 .

Conclusions and outlook
In this paper we developed a consistent semiclassical method to calculate the S-matrix elements for the two-stage transitions involving collapse of matter into a black hole and decay of the latter into free particles. We applied the method to a number of models with matter in the form of thin shells and obtained sensible results for transition amplitudes. We discussed the respective semiclassical solutions and their interpretation. We demonstrated that the probabilities of the two-stage shell transitions are exponentially suppressed by the Bekenstein-Hawking entropies of the intermediate black holes. If the shell model is taken seriously as a full quantum theory, this result implies that its S-matrix is non-unitary. However, the same result is natural and consistent with unitarity if the shells are interpreted as describing scatterings of narrow wavepackets in field theory. It coincides with the probability of black hole decay into a single shell found within the tunneling approach to Hawking radiation [33,34] and is consistent with interpretation of the Bekenstein-Hawking entropy as the number of black hole microstates [36]. Considering the shell in AdS 4 space-time we discussed the result from the AdS/CFT viewpoint. We consider these successes as an encouraging confirmation of the viability of our approach.
In the case of charged shells our method reproduces the entropy suppression only if instability of the Reissner-Nordström Cauchy horizon with respect to pair-production of charged particles is taken into account. This suggests that the latter process is crucial for unitarity of transitions with charged black holes at the intermediate stages.
It will be interesting to apply our method to field theory. Let us anticipate the scheme of such analysis. As an example, consider a spherically-symmetric scalar field φ minimally coupled to gravity 31 . Its classical evolution is described by the wave equation, while Einstein-Hilbert equations reduce to constraints. One can use the simplest Schwarzschild coordinates (t, r) which are well-defined for complex r and t, though other coordinate systems may be convenient for practical reasons. One starts from wavepackets with small amplitudes φ 0 which scatter trivially in flat space-time. Then one adds the complex term (2.5), (2.10) to the classical action and finds the modified saddle-point solutions. Finally, 30 We do not discuss the phase of the scattering amplitude as it essentially depends on our choice of the discharge model. 31 Another interesting arena for application of the method is two-dimensional dilaton gravity [61].
one increases φ 0 and obtains saddle-point solutions for the black hole-mediated transitions. The space-time manifold, if needed, should be deformed to complex values of coordinatesaway from the singularities of the solutions. We argued in Sec. 2 that the modified solutions are guaranteed to approach flat space-time at t → +∞ and as such, describe scattering. The S-matrix element (1.3) is then related to the saddle-point action S reg in the limit of vanishing modification → +0. The above procedure reduces evaluation of S-matrix elements to solution of two-dimensional complexified field equations, which can be performed on the present-day computers. At this point one may wonder whether the leading-order semiclassical results will be useful for addressing the unitarity of the S-matrix. At first sight, the unity operator S † S = 1 does not appear to be "semiclassical." However, its matrix elements in the coherent-state representation a|1|b = e dk a * k b k (6.1) have perfect exponential form, where |a and |b are eigenstates of the annihilation operators with eigenvalues a k and b k [44]. Comparison of Eq. (6.1) with the leading semiclassical exponent of a|S † S|b will provide a strong unitarity test for the gravitational S-matrix.

A A shell of rotating dust particles
Consider a collection of dust particles uniformly distributed on a sphere. Each partice has mass δm and absolute value δL of angular momentum. We assume no preferred direction in particle velocities, so that their angular momenta sum up to zero. This configuration is spherically-symmetric, as well as the collective gravitational field. Since the spherical symmetry is preserved in the course of classical evolution, the particles remain distributed on the sphere of radius r(τ ) at any time τ forming an infinitely thin shell. Each particle is described by the action where in the second equality we substituted the spherically symmetric metric (2.9) and introduced the time parameter τ . To construct the action for r(τ ), we integrate out the motion of the particle along the angular variable ϕ using conservation of angular momentum δL = δmr 2φ −g abẏ aẏb − r 2φ2 .
It would be incorrect to expressφ from this formula and substitute it into Eq. (A.1). To preserve the equations of motion, we perform the substitution in the Hamiltonian where p a and δL are the canonical momenta for y a and ϕ, whereas δL is the Lagrangian in Eq. (A.1). Expressingφ from Eq. (A.2), we obtain, δH = p aẏ a + −g abẏ aẏb δm 2 + δL 2 /r 2 .
From this expression one reads off the action for r(τ ), where we fixed τ to be the proper time along the shell. We finally sum up the actions (A.5) of individual particles into the shell action where N is the number of particles, m = N δm is their total mass and L = N δL is the sum of absolute values of the particles' angular momenta. We stress that L is not the total angular momentum of the shell. The latter is zero because the particles rotate in different directions.

B Equation of motion for the shell
In this appendix we derive equation of motion for the model with the action (3.5). We start by obtaining expression for the shell energy-momentum tensor. Let us introduce coordinates (y a , θ α ) such that the metric (2.9) is continuous 32 across the shell. Here θ α , α = 2, 3 are the spherical angles. Using the identity we recast the shell action (3.2) as an integral over the four-dimensional space-time, Here τ is regarded as a general time parameter. The energy-momentum tensor of the shell is obtained by varying Eq. (B.2) with respect to g ab and r 2 (y), 32 Schwarzschild coordinates in Eq. (3.6) are discontinuous at the shell worldsheet.
where in the final expressions we again set τ equal to the proper time. It is straightforward to see that the τ -integrals in Eqs. (B.3) produce δ-functions of the geodesic distance n from the shell, We finally arrive at where T α shell β ∝ δ α β due to spherical symmetry. Equation of motion for the shell is the consequence of Israel junction conditions which follow from the Einstein equations. The latter conditions relate t µν shell to the jump in the extrinsic curvature across the shell [51,52] Here h µ ν is the induced metric on the shell, K µν is its extrinsic curvature, the subscripts ± denote quantities outside (+) and inside (−) the shell. We define both (K µν ) ± using the outward-pointing normal, n µ ∂ r x µ > 0. Transforming the metric (3.6) into the continuous coordinate system, we obtain, where dot means derivative with respect to τ . From Eq. (B.6) we derive the equations, Only the first equation is independent, since the second is proportional to its time derivative. We conclude that Einstein equations are fulfilled in the entire space-time provided the metrics inside and outside the shell are given by Eqs. (3.6), (3.7) and Eq. (B.8) holds at the shell worldsheet. The latter equation is equivalent to Eqs. (3.8), (3.9) from the main text. The action (3.5) must be also extremized with respect to the shell trajectory y a (τ ). However, the resulting equation is a consequence of Eq. (B.8). Indeed, the shell is described by a single coordinate r(τ ), and its equations of motion are equivalent to conservation of the energy-momentum tensor. The latter conservation, however, is ensured by the Einstein equations.
All turning points approach zero at L → 0 except for r 1,2 in the massive case. Numerically tracing their motion as L decreases from L * , we find that the physical turning point A of the reflected trajectory is r 6 in both cases.

D Gibbons-Hawking terms at the initial-and final-time hypersurfaces
Since the space-time is almost flat in the beginning and end of the scattering process, one might naively expect that the Gibbons-Hawking terms at t + = t i and t + = t f are vanishingly small. However, this expectation is incorrect. Indeed, it is natural to define the initial and final hypersurfaces as t + = const outside of the shell and t − = const inside it. Since the metric is discontinuous in the Schwarzschild coordinates, the inner and outer parts of the surfaces meet at an angle which gives rise to non-zero extrinsic curvature, see Fig. 13. For concreteness we focus on the final-time hypersurface. In the Schwarzschild coordinates the normal vectors to its inner and outer parts are It is easy to see that the extrinsic curvature K = ∇ µ ξ µ is zero everywhere except for the two-dimensional sphere at the intersection the hypersurface with the shell worldsheet. Let us introduce a Gaussian normal frame (τ, n, θ α ) in the vicinity of the shell, see Fig. 13. Here τ is the proper time on the shell, n is the geodesic distance from it, and θ α , α = 2, 3, are the spherical angles. In this frame the metric in the neighborhood of the shell is essentially flat; corrections due to nonzero curvature are irrelevant for our discussion.
To find the components of ξ µ + and ξ µ − in Gaussian normal coordinates, we project them on τ µ and n µ -tangent and normal vectors of the shell. The latter in the inner and outer Schwarzschild coordinates have the form, Evaluating the scalar products of (D.1) and (D.2), we find, As expected, the normals ξ µ ± do not coincide at the position of the shell. To compute the surface integral in the Gibbons-Hawking term, we regularize the jump by replacing (D.3) with ξ µ = ch ψ(n) τ µ − sh ψ(n) n µ , (D. 4) where ψ(n) is a smooth function interpolating between ψ − and ψ + . The expression (3.4) takes the form, where in the second equality we used ds = dn/ ch ψ for the proper length along the finaltime hypersurface and K = ∂ µ ξ µ = − ch ψ ψ for its extrinsic curvature. Next, we express ψ ± (r) from the shell equation of motion (3.8) and expand Eq. (D.5) at large r. Keeping only non-vanishing terms at r = r f → +∞, we obtain Eq. (3.23) for the final-time Gibbons-Hawking term. For the initial-time hypersurface the derivation is the same, the only difference is in the sign of ξ µ which is now past-directed. However, this is compensated by the change of sign ofṙ. One concludes that the Gibbons-Hawking term at t + = t i is obtained from the one at t + = t f by the substitution r f → r i .
Note that expression (D.5) is valid also in the model of Sec. 4 describing massless shell in AdS. It is straightforward to see that in the latter case the Gibbons-Hawking terms vanish at r i,f → ∞ due to growth of the metric functions (4.3) at large r. E Shell self-gravity at order 1/r Let us construct the action for a neutral shell in asymptotically flat space-time taking into account its self-gravity at order 1/r. To this end we recall that the shell is assembled from particles of mass δm, see Appendix A. Every particle moves in the mean field of other particles. Thus, a new particle added to the shell changes the action of the system 33 by 33 Angular motion of the particle gives 1/r 2 contributions to the Lagrangian which are irrelevant in our approximation.
where v = dr/dt + is the shell velocity in the asymptotic coordinates,M is its energy, and we expanded the proper time dτ up to the first order in 1/r in the second equality. At the leading order in 1/r,M wherem is the shell mass before adding the particle. Now, we integrate Eq. (E.1) from m = 0 to the actual shell mass m and obtain the desired action, From this expression one reads off the canonical momentum and energy of the shell, Expressing the shell velocity from Eq. (E.5) and substituting 34 it into Eq. (E.4), we obtain Eq. (3.26) from the main text.