Semiclassical S-matrix for black holes
IINR-TH/2015-007CERN-PH-TH/2015-063
Prepared for submission to JHEP
Semiclassical S –matrix for black holes Fedor Bezrukov a,b,c
Dmitry Levkov d Sergey Sibiryakov a,e,d a Physics Department, CERN, CH-1211 Geneva 23, Switzerland b Physics Department, University of Connecticut, Storrs, CT 06269-3046, USA c RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA d Institute for Nuclear Research of the Russian Academy of Sciences,60-th October Anniversary Prospect 7a, Moscow 117312, Russia e FSB/ITP/LPPC ´Ecole Polytechnique F´ed´erale de Lausanne, CH-1015 Lausanne, Switzerland
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We propose a semiclassical method to calculate S –matrix elements for two–stage gravitational transitions involving matter collapse into a black hole and evaporationof the latter. The method consistently incorporates back–reaction of the collapsing andemitted quanta on the metric. We illustrate the method in several toy models describingspherical self–gravitating shells in asymptotically flat and AdS space–times. We find thatelectrically neutral shells reflect via the above collapse–evaporation process with probabilityexp( − B ), where B is the Bekenstein–Hawking entropy of the intermediate black hole. Thisis consistent with interpretation of exp( B ) as the number of black hole states. The sameexpression for the probability is obtained in the case of charged shells if one takes intoaccount instability of the Cauchy horizon of the intermediate Reissner–Nordstr¨om blackhole. Our semiclassical method opens a new systematic approach to the gravitational S –matrix in the non–perturbative regime. a r X i v : . [ h e p - t h ] N ov ontents S –matrix for gravitational scattering 52.2 The functional T int [Φ] 8 S –matrix element 153.4 Relation to the Hawking radiation 193.5 Space–time picture 21 L →
35D Gibbons–Hawking terms at the initial– and final–time hypersurfaces 35E Shell self–gravity at order /r Gravitational scattering has been a subject of intensive research over several decades, see [1]and references therein. Besides being of its own value, this study presents an importantstep towards resolution of the information paradox — an apparent clash between unitarityof quantum evolution and black hole thermodynamics [2, 3]. Recently the interest in thisproblem has been spurred by the AMPS (or “firewall”) argument [4, 5] which suggests that– 1 – a) collapsing matter Ψ i (b) black hole (c) Hawking radiation Ψ f Figure 1 . Complete gravitational transition involving formation and evaporation of a black hole. reconciliation of the black hole evaporation with unitarity would require drastic departuresfrom the classical geometry in the vicinity of an old black hole horizon (see [6–8] for relatedworks). Even the minimal versions of such departures appear to be at odds with theequivalence principle. This calls for putting all steps in the logic leading to this result ona firmer footing.Unitarity of quantum gravity is strongly supported by the arguments based onAdS/CFT correspondence [9, 10]. This reasoning is, however, indirect and one wouldlike to develop an explicit framework for testing unitarity of black hole evaporation. Inparticular, one would like to see how the self–consistent quantum evolution leads to thethermal properties of the Hawking radiation and to test the hypothesis [11] that the infor-mation about the initial state producing the black hole is imprinted in subtle correlationsbetween the Hawking quanta.A natural approach is to view the formation of a black hole and its evaporation asa two–stage scattering transition, see Fig. 1. The initial and final states Ψ i and Ψ f ofthis process represent free matter particles and free Hawking quanta in flat space–time.They are the asymptotic states of quantum gravity related by an S –matrix [1, 12, 13].Importantly, the black hole itself, being metastable, does not correspond to an asymtoticstate. The S –matrix is unitary if black hole formation does not lead to information loss.The importance of collapse stage for addressing the information paradox was emphasizedin [14–18].However, calculation of the scattering amplitude for the process in Fig. 1 encounters aformidable obstacle: gravitational interaction becomes strong in the regime of interest andthe standard perturbative methods break down [19, 20]. General considerations [20] sup-ported by the perturbative calculations [19] show that scattering of two trans–Planckianparticles is accompanied by an increasingly intensive emission of soft quanta as the thresh-old of black hole formation is approached. While this is consistent with the qualitativeproperties of the Hawking radiation dominated by many quanta, a detailed comparison isnot available. A new perturbative scheme adapted to processes with many particles in thefinal state has been recently proposed in [21]; however, its domain of applicability is yet tobe understood.In this paper we follow a different route. We propose to focus on exclusive processes– 2 –here both initial and final states contain a large number of soft particles. Specifically, onecan take Ψ i and Ψ f to be coherent states with large occupation numbers corresponding tothe semiclassical wavepackets. We assume that the total energy of the process exceeds thePlanck scale, so that the intermediate black hole has mass well above Planckian. Then theoverall process is expected to be described within the low–energy gravity. Its amplitudecan be evaluated using the semiclassical methods and will yield the black hole S –matrixin the coherent–state basis [22]. A priori, we cannot claim to describe semiclassically thedominant scattering channel with Hawking-like final state which is characterized by lowoccupation numbers . Still, it seems a safe bet to expect that, within its domain of validity ,this approach will provide valuable information on the black hole–mediated amplitudes.A crucial point in the application of semiclassical methods is the correct choice of thesemiclassical solutions. Consider the amplitude (cid:104) Ψ f | ˆ S| Ψ i (cid:105) = (cid:90) D Φ i D Φ f Ψ ∗ f [Φ f ] Ψ i [Φ i ] (cid:90) D Φ e iS [Φ] / (cid:126) (1.1)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 includingmatter 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) mustdescribe collections of free particles in flat space–time. In the semiclassical approach oneevaluates the path integral (1.1) in the saddle–point approximation. The saddle–pointconfiguration must inherit the correct flat–space asymptotics and, in addition, extremize S i.e. solve the classical equations of motion. However, a naive choice of the solution fails tosatisfy the former requirement. Take, for instance, the solution Φ cl describing the classical collapse. It starts from collapsing particles in flat space–time, but arrives to a black holein the asymptotic future. It misses the second stage of the scattering process — the blackhole decay. Thus, it is not admissible as the saddle point of (1.1). One faces the task ofenforcing the correct asymptotics on the saddle–point configurations.Furthermore, the saddle–point solutions describing exclusive transitions are genericallycomplex–valued and should be considered in complexified space–time [22, 23]. Even sub-ject to appropriate boundary conditions, such solutions are typically not unique. Not allof them are relevant: some describe subdominant processes, some other, when substitutedinto the semiclassical expression for the amplitude, give nonsensical results implying expo-nentially large scattering probability. Choosing the dominant physical solution presents anon–trivial challenge. It is conceivable, however, that this dominant amplitude can be obtained with a suitable limitingprocedure. A hint comes from field theory in flat space where the cross section of the process 2 → many isrecovered from the limit of the cross section many → many [23–25]. In line with the common practice, the applicability of the semiclassical method will be verified aposteriori by subjecting the results to various consistency checks. The precise definition of the functional measure is not required in the leading semiclassical approxima-tion, which we will focus on. This property is characteristic of dynamical tunneling phenomena which have been extensively studiedin quantum mechanics with multiple degrees of freedom [26]. – 3 –he method to overcome the two above problems has been developed in Refs. [27–32]in the context of scattering in quantum mechanics with multiple degrees of freedom; itwas applied to field theory in [33, 34]. In this paper we adapt this method to the case ofgravitational scattering.It is worth emphasizing the important difference between our approach and perturba-tive expansion in the classical black hole (or collapsing) geometry, which is often identifiedwith the semiclassical approximation in black hole physics. In the latter case the evapo-ration is accounted for only at the one–loop level. This approach is likely to suffer fromambiguities associated with the separation of the system into a classical background andquantum fluctuations. Instead, in our method the semiclassical solutions by constructionencapsulate black hole decay in the leading order of the semiclassical expansion. Theyconsistently take into account backreaction of the collapsing and emitted matter quantaon the metric. Besides, we will find that the solutions describing the process of Fig. 1 arecomplex–valued. They bypass, via the complexified evolution, the high–curvature regionnear the singularity of the intermediate black hole. Thus, one does not encounter theproblem of resolving the singularity.One the other hand, the complex–valued saddle–point configurations do not admit astraightforward interpretation as classical geometries. In particular, they are meaninglessfor an observer falling into the black hole: the latter measures local correlation functionsgiven by the path integrals in the in–in formalism — with different boundary conditions anddifferent saddle–point configurations as compared to those in Eq. (1.1). This distinctionlies at the heart of the black hole complementarity principle [35].Our approach can be applied to any gravitational system with no symmetry restric-tions. However, the task of solving nonlinear saddle–point equations is rather challenging.In this paper we illustrate the method in several exactly tractable toy models describingspherical gravitating dust shells. We consider neutral and charged shells in asymptoticallyflat and anti–de Sitter (AdS) space–times. Applications to field theory that are of primaryinterest are postponed to future.Although the shell models involve only one collective degree of freedom — the shellradius — they are believed to capture some important features of quantum gravity [36–39]. Indeed, one can crudely regard thin shells as narrow wavepackets of an underlyingfield theory. In Refs. [39–41] emission of Hawking quanta by a black hole is modeled astunneling of spherical shells from under the horizon. The respective emission probabilityincludes back–reaction of the shell on geometry,
P (cid:39) e − ( B i − B f ) , (1.2)where B i and B f are the Bekenstein–Hawking entropies of the black hole before and af-ter the emission. It has been argued in [42] that this formula is consistent with unitaryevolution.In the context of shell models we consider scattering processes similar to those inFig. 1: a classical contracting shell forms a black hole and the latter completely decays dueto quantum fluctuations into an expanding shell. The initial and final states Ψ i and Ψ f of– 4 –he process describe free shells in flat or AdS space–times. Our result for the semiclassicalamplitude (1.1) has the form (cid:104) Ψ f | ˆ S| Ψ i (cid:105) (cid:39) e iS reg / (cid:126) . (1.3)We stress that it includes backreaction effects. The probability of transition is P fi (cid:39) e − S reg / (cid:126) . We show that for neutral shells it reproduces Eq. (1.2) with B i equal to the entropy of theintermediate black hole and B f = 0. This probability is exponentially small at M (cid:29) S –matrixelements which explicitly depend, besides the properties of the intermediate black hole, onthe initial and final states of the process.The paper is organized as follows. In Sec. 2 we introduce general semiclassical methodto compute S –matrix elements for scattering via black hole formation and evaporation. InSec. 3 we apply the method to transitions of a neutral shell in asymptotically flat space–time. We also discuss relation of the scattering processes to the standard thermal radiationof a black hole. This analysis is generalized in Sec. 4 to a neutral shell in asymptoticallyAdS space–time where scattering of the shell admits an AdS/CFT interpretation. A modelwith an electrically charged shell is studied in Sec. 5. Section 6 is devoted to conclusionsand discussion of future directions. Appendices contain technical details. S –matrix for gravitational scattering The S –matrix is defined as (cid:104) Ψ f | ˆ S| Ψ i (cid:105) = lim t i → −∞ t f → + ∞ (cid:104) Ψ f | ˆ U (0 , t f ) ˆ U ( t f , t i ) ˆ U ( t i , | Ψ i (cid:105) , (2.1)where ˆ U is the evolution operator; free evolution operators ˆ U on both sides transformfrom Schr¨odinger to the interaction picture. In our case ˆ U describes quantum transitionin Fig. 1, while ˆ U generates evolution of free matter particles and Hawking quanta in theinitial and final states. The time variable t ∈ [ t i , t f ] is chosen to coincide with the time ofan asymptotic observer at infinity.Using the path integrals for the evolution operators and taking their convolutionswith the wave functionals of the initial and final states, one obtains the path integral– 5 – f → + ∞−∞ ← t i t Ψ i Ψ f S ( t f , + ) S (0 − , t i ) S ( t i , t f ) t f → + ∞−∞ ← t i t Ψ i Ψ f S ( t f , + ) S (0 − , t i ) S ( t i , t f ) 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. representation for the amplitude (2.1), (cid:104) Ψ f | ˆ S| Ψ i (cid:105) = (cid:90) D Φ e iS ( t i , t f )+ iS (0 − , t i )+ iS ( t f , + ) Ψ i [Φ − ] Ψ ∗ f [Φ + ] ≡ (cid:90) D Φ e iS tot [Φ] , (2.2)where Φ = { φ, g µν } collectively denotes matter and gravitational fields along the timecontour in Fig. 2. The quantum measure D Φ should include some ultraviolet regularizationas well as gauge–fixing of the diff–invariance and respective ghosts. A non-perturbativedefinition of this measure presents a well–known challenge. Fortunately, the details of D Φ are irrelevant for our leading–order semiclassical calculations. The interacting andfree actions S and S 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 theendpoints of the contour. In the second equality of Eq. (2.2) we combined all factors in theintegrand into the “total action” S tot [Φ]. Below we mostly focus on nonlinear evolutionfrom t i to t f and take into account contributions from the dashed parts of the contour inFig. 2 at the end of the calculation.To distinguish between different scattering regimes, we introduce a parameter P char-acterizing the initial state [50] — say, its average energy. If P is small, the gravitationalinteraction is weak and the particles scatter trivially without forming a black hole. Inthis 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 andfinal states [22]. However, if P exceeds a certain critical value P ∗ , the classical solutionΦ cl corresponds to formation of a black hole. It therefore fails to interpolate towards theasymptotic out–state Ψ f living in flat space–time. This marks a breakdown of the standardsemiclassical method for the amplitude (2.2).To deal with this obstacle, we introduce a constraint in the path integral which ex-plicitly guarantees that all field configurations Φ from the integration domain have flatspace–time asymptotics [30–32]. Namely, we introduce a functional T int [Φ] with the fol-lowing properties: it is (i) diff–invariant; (ii) positive–definite if Φ is real; (iii) finite if Φapproaches flat space–time at t → ±∞ ; (iv) divergent for any configuration containing ablack hole in the asymptotic future. Roughly speaking, T int [Φ] measures the “lifetime” ofa black hole in the configuration Φ. Possible choices of this functional will be discussed in From now on we work in the Planck units (cid:126) = c = G N = k B = 1. – 6 –he next subsection; for now let us assume that it exists. Then we consider the identity1 = + ∞ (cid:90) dT δ ( T int [Φ] − T ) = + ∞ (cid:90) dT i ∞ (cid:90) − i ∞ d(cid:15) πi e (cid:15) ( T − T int [Φ]) , (2.3)where in the second equality we used the Fourier representation of the δ –function. InsertingEq. (2.3) into the integral (2.2) and changing the order of integration, we obtain, (cid:104) Ψ f | ˆ S| Ψ i (cid:105) = (cid:90) dT d(cid:15) πi e (cid:15)T (cid:90) D Φ e i ( S tot [Φ]+ i(cid:15)T int [Φ]) . (2.4)The inner integral over Φ in Eq. (2.4) has the same form as the original path integral, butwith the modified action S (cid:15) [Φ] ≡ S tot [Φ] + i(cid:15)T int [Φ] . (2.5)By construction, this integral is restricted to configurations Φ with T int [Φ] = T < ∞ , i.e.the ones approaching flat space–time in the asymptotic past and future. In what followswe make this property explicit by deforming the contour of (cid:15) –integration to Re (cid:15) >
0. Thenconfigurations with black holes in the final state i.e. with T int [Φ] = + ∞ , do not contributeinto the integral at all.By now, we have identically rewritten the integral (2.2) in the form (2.4). Its valueclearly does not depend on the form of the regulating functional T int [Φ].Now the amplitude (2.4) is computed by evaluating the integrals over Φ, (cid:15) and T oneafter another in the saddle–point approximation. The saddle–point configuration Φ (cid:15) of theinner integral extremizes the modified action (2.5), while the saddle–point equation for (cid:15) gives the constraint T int [Φ (cid:15) ] = T . (2.6)This implies that Φ (cid:15) has correct flat–space asymptotics. The integral over T is saturatedat (cid:15) = 0. Importantly, we do not substitute (cid:15) = 0 into the saddle–point equations for Φ (cid:15) ,since in that case we would recover the original classical equations together with incorrectasymptotics of the saddle–point solutions. Instead, we understand this equation as thelimit (cid:15) → +0 (2.7)that must be taken at the last stage of the calculation. The condition Re (cid:15) > S reg = lim (cid:15) → +0 S tot [Φ (cid:15) ] , (2.8)where the limit is taken in the end of the calculation.Our modified semiclassical method addresses another problem mentioned in the Intro-duction: it allows to select the relevant saddle–point configurations from the discrete set of Below we consider only the leading semiclassical exponent. The prefactor in the modified semiclassicalapproach was discussed in [30–32]. – 7 –omplex–valued solutions to the semiclassical equations. As discussed in [27–29], at (cid:15) > describing scattering at a small value of the parameter P < P ∗ . By construction, Φ approaches flat space–time at t → ∓∞ and gives the domi-nant contribution to the integral (2.4). Next, we modify the action and gradually increase (cid:15) from (cid:15) = 0 to the positive values constructing a continuous branch of modified solutionsΦ (cid:15) . At (cid:15) → +0 these solutions reduce to Φ and therefore saturate the integral (2.4).We increase the value of P to P > P ∗ assuming that continuously deformed saddle–pointconfigurations Φ (cid:15) remain physical . In this way we obtain the modified solutions and thesemiclassical amplitude at any P . We stress that our continuation procedure cannot beperformed with the original classical solutions which, if continued to P > P ∗ , describeformation of black holes. On the contrary, the modified solutions Φ (cid:15) interpolate betweenthe flat–space asymptotics at any P . They are notably different from the real classical so-lutions at P > P ∗ . At the last step one evaluates the action S tot on the modified solutionsand sends (cid:15) → +0 obtaining the leading semiclassical exponent of the S –matrix element,see Eqs. (1.3), (2.8). T int [Φ]Let us construct the appropriate functional T int [Φ]. This is particularly simple in the caseof reduced models with spherically–symmetric gravitational and matter fields. The generalspherically–symmetric metric has the form ds = g ab ( y ) dy a dy b + r ( y ) d Ω , (2.9)where d Ω is the line element on a unit two–sphere and g ab is the metric in the transversetwo–dimensional space . Importantly, the radius r ( y ) of the sphere transforms as a scalarunder the diffeomorphisms of the y –manifold. Therefore the functional T int = (cid:90) d y √− g w ( r ) F (cid:0) g ab ∂ a r∂ b r (cid:1) , (2.10)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 (cid:46) r w , where r w issome fixed value, and falls off sufficiently fast at large r . An example of functions satisfyingthese conditions is w ( r ) = δ ( r − r w ), F (∆) = (∆ − .To understand the properties of the functional (2.10), we consider the Schwarzschildframe where r is the spatial coordinate and the metric is diagonal. The functional (2.10) In other words, we assume that no Stokes lines [51] are crossed in the course of deformation. Thisconjecture has been verified in multidimensional quantum mechanics by direct comparison of semiclassicaland exact results [27–32, 52]. We use the signature ( − , + , . . . ) for the metrics g µν and g ab . The Greek indices µ, ν, . . . are used for thefour–dimensional tensors, while the Latin ones a, b, . . . = 0 , – 8 –akes the form, T int = (cid:90) dtdr √− g w ( r ) F (cid:0) g (cid:1) . (2.11)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 andfuture. In flat space–time g = 1 and the integrand in Eq. (2.10) vanishes. Thus, theintegral over t is finite if g ab approaches the flat metric at t → ±∞ . Otherwise the integraldiverges. In particular, any classical solution with a black hole in the final state leadsto linear divergence at t → + ∞ because the Schwarzschild metric is static and g (cid:54) = 1.Roughly speaking, T int can be regarded as the Schwarzschild time during which matterfields efficiently interact with gravity inside the region r < r w . If matter leaves this regionin finite time, T int takes finite values. It diverges otherwise. Since the functional (2.10) isdiff–invariant, these properties do not depend on the particular choice of the coordinatesystem.The above construction will be sufficient for the purposes of the present paper. Beyondthe spherical symmetry one can use the functionals T int [Φ] that involve, e.g., an integralof the square of the Riemann tensor, or the Arnowitt–Deser–Misner (ADM) mass inside alarge volume. Recall the final result for the S –matrix does not depend on the precise choiceof the functional T int . Of course, this functional should satisfy the conditions (i)-(iv) listedbefore Eq. (2.3). We illustrate the method of Sec. 2 in the spherically symmetric model of gravity with thindust shell for matter. The latter is parameterized by a single collective coordinate — theshell radius r ( τ ) — depending on the proper time along the shell τ . This is a dramaticsimplification 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 wavepacketsin field theory. In particular, one expects that the shell model captures essential featuresof gravitational transition between such wavepackets. The minimal action for a spherical dust shell is S shell = − m (cid:90) dτ (3.1)where m is the shell mass. However, such a shell always collapses into a black hole andhence is not sufficient for our purposes. Indeed, as explained in Sec. 2.1, in order to selectthe physically relevant semiclassical solutions we need a parameter P such that an initiallycontracting shell reflects classically at P < P ∗ and forms a black hole at P > P ∗ . Wetherefore generalize the model (3.1). To this end we assume that the shell is assembledfrom particles with nonzero angular momenta. At each point on the shell the velocities of Note that our approach does not require complete solution of the quantum shell model which may beambiguous. Rather, we look for complex solutions of the classical equations saturating the path integral. – 9 –he constituent particles are uniformly distributed in the tangential directions, so that theoverall configuration is spherically–symmetric . The corresponding shell action is [54] S shell = − (cid:90) m eff dτ , m = m + L /r ( τ ) , (3.2)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. Inwhat follows we switch between the scattering regimes by changing the parameter L ≡ P − .For completeness we derive the action (3.2) in Appendix A.Gravitational sector of the model is described by the Einstein–Hilbert action with theGibbons–Hawking term, S EH = 116 π (cid:90) V d x √− g R , (3.3) S GH = 18 π (cid:90) ∂ V κ d σ (cid:112) | h | ( K − K ) . (3.4)Here the metric g µν and curvature scalar R are defined inside the space–time volume V with the boundary ∂ 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 → ∓∞ . InEq. (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 gravitationalaction in flat space–time, we subtract the regulator K which is equal to the flat–spaceextrinsic curvature of the boundary [55]. For the sphere at infinity K = 2 /r ∞ , while theinitial– and final–time hypersurfaces have K = 0. The Gibbons–Hawking term (3.4) willplay an important role in our analysis.Let us first discuss the classical dynamics of the system. Equations of motion followfrom variation of the total action S = S shell + S EH + S GH (3.5)with respect to the metric g µν and the shell trajectory y a ( τ ). In the regions inside andoutside the shell the metric satisfies vacuum Einstein equations and therefore, due toBirkhoff 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 ds ± = − f ± ( r ) dt ± + dr f ± ( r ) + r d Ω , (3.6)where f − = 1 , f + = 1 − M/r . (3.7) Similar models are used in astrophysics to describe structure formation [53]. We impose the Dirichlet boundary conditions on the variations of g µν at ∂ V . – 10 – a) r ( τ )Flat( t − , r )Schwarzschild, M ( t + , r ) (b) r ( τ ) r w Flat M = M + i ˜ (cid:15) M Figure 3 . Gravitational field of the spherical dust shell: (a) in the original model, (b) in the modelwith modification at r ≈ r w . The parameter M is the ADM mass which coincides with the total energy of the shell. Inwhat follows we will also use the Schwarzschild radius r h ≡ M . For the validity of thesemiclassical approach we assume that the energy is higher than Planckian, M (cid:29) r + V eff ( r ) = 0 , (3.8) V eff = f − − r m (cid:18) f + − f − − m r (cid:19) (3.9)= 1 − (cid:0) L + m r + 2 M r (cid:1) r ( L + m r ) . This equation incorporates gravitational effects as well as the backreaction of the shell onthe spacetime metric. The potential V eff ( r ) goes to −∞ at r → − M /m at r = + ∞ , see Fig. 4. At large enough L the potential crosseszero at the points A and A (cid:48) — the turning points of classical motion. A shell coming frominfinity reflects from the point A back to r = + ∞ . When L decreases, the turning pointsapproach each other and coalesce at a certain critical value L = L ∗ . At even smaller L the turning points migrate into the complex plane, see Fig. 5 (upper left panel), and thepotential 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 Recall that the shell energy M is always larger than its rest mass m . For a massless shell ( m = 0) the critical value is L ∗ = 27 M / – 11 – eff r r h A A L > L ∗ L < L ∗ Figure 4 . Effective potential for the shell motion. solutions r = r ( τ ) are implicitly defined as r ( τ ) (cid:90) dr (cid:112) − V eff ( r ) = τ , (3.10)where the square root is positive at r → + ∞ + i
0. The indefinite integral is performedalong the contour C running from r = + ∞ − i r = + ∞ + i A — the branch point of the integrand (see the upper left panel of Fig. 5). As L islowered, the branch point moves and the integration contour stays attached to it. However,at L = L ∗ when the branch points A and A (cid:48) coalesce, the contour C is undefined. It istherefore impossible to obtain reflected semiclassical solutions at L < L ∗ from the classicalsolutions at L > L ∗ . To find physically relevant reflected trajectories at
L < L ∗ , we use the method of Sec. 2 andadd an imaginary term i(cid:15)T int to the action. We consider T int of the form (2.10), where thefunction w ( r ) is concentrated in the vicinity of r = r w . The radius r w is chosen to be largeenough, in particular, larger than the Schwarzschild radius r h and the position r A of theright turning point A . Then the Einstein equations are modified only at r ≈ r w , whereasthe geometries inside and outside of this layer are given by the Schwarzschild solutionswith masses M (cid:48) and M , see Fig. 3b. To connect these masses, we solve the modifiedEinstein equations in the vicinity of r w . Inserting general spherically symmetric metric inthe Schwarzschild frame, ds = − f ( t, r ) dt + dr ˜ f ( t, r ) + r d Ω , (3.11)into the ( tt ) component of Einstein equations, we obtain, ∂ r ˜ fr − − ˜ fr = 2 i(cid:15)r w ( r ) F ( ˜ f ) . (3.12)– 12 – = 0: A Ar h r C (cid:15) > A Ar h rAA r h r C Figure 5 . Motion of the turning points in the complex r –plane. The points move along the arrowsas L decreases. The cases of classical shell (upper left panel) and shell with the modified action(right panels) are shown. The solution reads ,˜ f = 1 − M ( r ) r , where ˜ M ( r ) = M + i(cid:15) (cid:90) + ∞ r dr (cid:48) w ( r (cid:48) ) F ( ˜ f ) . (3.13)This gives the relation M (cid:48) = M + i ˜ (cid:15) , ˜ (cid:15) = (cid:15) (cid:90) r ≈ r w dr (cid:48) wF . (3.14)Here ˜ (cid:15) > M of thesystem is conserved in the course of the evolution. It coincides with the initial and finalenergies of the shell which are, in turn, equal, as will be shown in Sec. 3.3, to the initial– andfinal–state energies in the quantum scattering problem. Thus, M is real, while the mass M (cid:48) of the Schwarzschild space–time surrounding the shell acquires a positive imaginarypart . The shell dynamics in this case is still described by Eq. (3.8), where M is replacedby M (cid:48) in the potential (3.9). Below we find semiclassical solutions for small ˜ (cid:15) >
0. In theend ˜ (cid:15) 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 reflectedtrajectory is obtained with the same contour C as before, see the upper left panel of Fig. 5. The function ˜ f is time–independent due to the ( tr ) equation. In this setup the method of Sec. 2 is equivalent to analytic continuation of the scattering amplitudeinto the upper half–plane of complex ADM energy, cf. [28]. – 13 –he modification of V eff becomes important when L gets close to L ∗ and the two turningpoints A and A (cid:48) approach each other. Expanding the potential in the neighborhood of themaximum, we write, V eff ( r ) ≈ V max − µ ( r − r max ) , (3.15)where V max , µ and r max depend on L and M (cid:48) . For real M (cid:48) = M the extremal value V max is real and crosses zero when L crosses L ∗ , whereas the parameters µ > r max remain approximately constant. The shift of M (cid:48) into the upper complex half–plane givesa negative imaginary part to V max ,Im V max = ˜ (cid:15) ∂V max ∂M (cid:48) + O (˜ (cid:15) ) < , (3.16)where the last inequality follows from the explicit form (3.9). Now, it is straightforward totrack the motion of the turning points using Eq. (3.15) as L decreases below L ∗ . Namely, A and A (cid:48) are shifted into the lower and upper half–planes as shown in Fig. 5 (upper rightpanel). 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) whilekeeping it attached to the same turning point. As we anticipated in Sec. 2, a smooth branchof reflected semiclassical solutions parameterized by L exists in the modified system.If L is slightly smaller than L ∗ , the relevant saddle–point trajectories reflect atRe r A > r h and hence never cross the horizon. A natural interpretation of the correspond-ing 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 semi-classical trajectories in this limit describe complete gravitational transitions proceeding viaformation and decay of a black hole.After establishing the correspondence between the solutions at
L > L ∗ and L < L ∗ , wetake ˜ (cid:15) = 0. This leaves us with the original semiclassical equations which do not includethe imaginary regularization terms.We numerically traced the motion of the turning point A as L decreases from large tosmall values, see Fig. 5 (lower panel). It approaches the singularity r = 0 at L →
0. Thisbehavior is confirmed analytically in Appendix C. Thus, at small L the contour C goesessentially along the real axis making only a tiny excursion into the complex plane nearthe singularity. It encircles the horizon r = r h from below.One remark is in order. For the validity of the low–energy gravity the trajectory shouldstay in the region of sub-Planckian curvature, R µνλρ R µνλρ ∼ M /r (cid:28)
1. This translatesinto the requirement for the turning point | r A | (cid:29) M / . (3.17)On the other hand, we will see shortly that the dependence of the semiclassical amplitudeon L drops off at L (cid:28) L ∗ ∼ M ; we always understand the limit L → r A from Appendix C one verifies that the condition(3.17) can be satisfied simultaneously with L (cid:28) L ∗ in the semiclassical regime M (cid:29) This may be impossible in more complicated systems [27, 28, 30, 31] where the relevant saddle–pointtrajectories do not exist at ˜ (cid:15) = 0. In that case one works at nonzero ˜ (cid:15) till the end of the calculation. – 14 – + C t t i t f t f A πiM πiM Figure 6 . The time contour corresponding to the semiclassical solution at small L . Solid anddashed 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) iscomputed 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 theSchwarzschild time t + ( r ) along this contour. We write, t + ( r ) = (cid:90) r dr dt + dr = (cid:90) r dr (cid:112) − V eff ( r ) (cid:112) f + ( r ) − V eff ( r ) f + ( r ) , (3.18)where the indefinite integral runs along C . In Eq. (3.18) we used the the definition of theproper time implying f + (cid:18) dt + dr (cid:19) = 1˙ r + 1 f + , and expressed ˙ r from Eq. (3.8). The integrand in Eq. (3.18) has a pole at the horizon r = r h , f + ( r h ) = 0, which is encircled from below, see Fig. 5, lower panel. The half–residue at this pole contributes iπr h to t + each time the contour C passes close to it. Thecontributions have the same sign: although the contour C passes the horizon in the oppositedirections, 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 theturning point A ; this contribution vanishes at L → C t of the contour C is shown in Fig. 6, solid line. Adding free evolutionfrom t + = 0 − to t + = t i and from t + = t f to t + = 0 + (dashed lines), we obtain the contouranalogous to the one in Fig. 2. One should not worry about the complex value of t f inFig. 6: the limit t f → + ∞ in the definition of S –matrix implies that S reg does not dependon t f . Besides, the semiclassical solution r = r ( t + ) is an analytic function of t + and thecontour C t can be deformed in complex plane as long as it does not cross the singularities of r ( t + ). Below we calculate the action along C t because the shell position and the metricsare real in the initial and final parts of this contour. This simplifies the calculation of theGibbons–Hawking terms at t + = t i and t + = t f . In fact, C t is separated from the real time axis by a singularity where r ( t + ) = 0. This is the usualsituation for tunneling solutions in quantum mechanics and field theory [27, 28]. Thus, S reg cannot becomputed along the contour in Fig. 2; rather, C t or an equivalent contour should be used. – 15 – nteracting action. Now, we evaluate the action of the interacting system S ( t i , t f )entering S reg . We rewrite the shell action as S shell = − (cid:90) C dr √− V eff m eff , (3.19)where Eq. (3.10) was taken into account. The Einstein–Hilbert action (3.3) is simplifiedusing the trace of the Einstein equations, R = − πT µshell µ , and the energy–momentumtensor of the shell computed in Appendix B, Eq. (B.5), S EH = (cid:90) τ f τ i dτ m m eff = (cid:90) C dr √− V eff m m eff . (3.20)An important contribution comes from the Gibbons–Hawking term at spatial infinity r = r ∞ → + ∞ . The extrinsic curvature reads, K (cid:12)(cid:12)(cid:12) r ∞ = rf (cid:48) + + 4 f + r (cid:112) f + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r ∞ = 2 r ∞ − Mr ∞ + O ( r − ∞ ) . (3.21)The first term here is canceled by the regulator K in Eq. (3.4). The remaining expressionis finite at r ∞ → + ∞ , S GH (cid:12)(cid:12)(cid:12) r ∞ = − M (cid:90) dt + = − M (cid:90) C dr √− V eff (cid:112) f + − V eff f + , (3.22)where we transformed to integral running along the contour C using Eq. (3.18). Note thatthis contribution contains an imaginary partIm S GH (cid:12)(cid:12)(cid:12) r ∞ = − M t f − t i ) . (3.23)Finally, in Appendix D we evaluate the Gibbons–Hawking terms at the initial– and final–time hypersurfaces. The result is S GH (cid:12)(cid:12)(cid:12) t i,f = √ M − m r i,f + M (2 M − m )4 √ M − m , (3.24)where r i,f are the radii of the shell at the endpoints of the contour C . The latter radii arereal, and so are the terms (3.24).Summing up the above contributions, one obtains, S ( t i , t f ) = (cid:90) C dr √− V eff (cid:20) m − m m eff − M (cid:112) f + − V eff f + (cid:21) + √ M − m r i + r f ) + M (2 M − m )2 √ M − m . (3.25)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 onlythe phase of the reflection amplitude but not its absolute value.– 16 – nitial and final–state contributions. The linear divergence in Eq. (3.25) is relatedto free motion of the shell in the asymptotic region r → + ∞ , whereas the logarithmic oneis due to the 1 /r tails of the gravitational interaction in this region. Though the 1 /r termsin the Lagrangian represent vanishingly small gravitational forces in the initial and finalstates, they produce logarithmic divergences in S ( t i , t f ) when integrated over the shelltrajectory. To obtain a finite matrix element, we include these terms in the definition ofthe free action S . In Appendix E the latter action is computed for the shell with energy M , S (0 − , t i ) = (cid:90) r i r p i ( r, M ) dr − M t i , S ( t f , + ) = (cid:90) r r f p f ( r, M ) dr + M t f , (3.26)where r , are the positions of the shell at t + = 0 ∓ and p i,f ( r, M ) = ∓ (cid:20)(cid:112) M − m + M (2 M − m )2 r √ M − m (cid:21) (3.27)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 ) and Ψ f ( r )of the initial and final states. We consider the semiclassical wavefunctions of the shell withfixed energy E ,Ψ i ( r ) (cid:39) exp (cid:26) i (cid:90) r r p i ( r (cid:48) , E ) dr (cid:48) (cid:27) , Ψ f ( r ) (cid:39) exp (cid:26) i (cid:90) r r p f ( r (cid:48) , E ) dr (cid:48) (cid:27) , (3.28)where p i,f are the same as in Eq. (3.27). In fact, the energy E is equal to the energy of thesemiclassical solution, E = M . Indeed, the path integral (2.2) includes integration overthe initial and final configurations of the system, i.e. over r and r in the shell model. Thecondition for the stationary value of r reads, ∂∂r log Ψ i + i ∂∂r S (0 − , t i ) = 0 ⇒ p i ( r , E ) = p i ( r , M ) , (3.29)and similarly for r . This implies equality of E and M . Note that the parameter r (cid:29) M in Eq. (3.28) fixes the phases of the initial and final wavefunctions. Namely, the phasesvanish at r = r . The result.
Combining the contributions (3.26), (3.28) with Eq. (3.25), we obtain theexponent of the S –matrix element (2.2), S reg = (cid:90) C dr √− V eff (cid:20) m − m m eff + M (cid:112) f + − V eff f + (cid:21) − (cid:112) M − m (cid:18) r i + r f − r (cid:19) + M (2 M − m )2 √ M − m (cid:16) − log( r i r f /r ) (cid:17) . (3.30)Notice that this result does not depend on any regularization parameters. It is straightfor-ward to check that Eq. (3.30) is finite in the limit r i,f → + ∞ . In Fig. 7 we plot its real and This relies on the freedom in splitting the total Lagrangian into “free” and “interacting” parts. – 17 – L ∗ /M L/M R e S r e g / M . π L ∗ /M L/M I m S r e g / M (a) (b) Figure 7 . Real (a) and imaginary (b) parts of S reg at m = 0 as functions of L . For the real partwe take r = 10 M . imaginary parts as functions of L for the case of massless shell ( m = 0). The imaginarypart vanishes for the values L ≥ L ∗ corresponding to classical reflection. At smaller L theimaginary part is positive implying that the reflection probability P fi (cid:39) e − S reg is exponentially suppressed. Importantly, S reg does not receive large contributions fromthe small– r region near the spacetime singularity. It is therefore not sensitive to the effectsof trans–Planckian physics.In the most interesting case of vanishing centrifugal barrier L → S reg comes from the residue at the horizon r h = 2 M in Eq. (3.30), recallthe contour C in Fig. 5. The respective value of the suppression exponent is2Im S reg (cid:12)(cid:12)(cid:12) L =0 = 2 πM Res r = r h (cid:112) f + − V eff √− V eff f + = πr h . (3.31)This result has important physical implications. First, Eq. (3.31) depends only on the totalenergy M of the shell but not on its rest mass m . Second, the suppression coincides withthe Bekenstein–Hawking entropy of a black hole with mass M . The same suppression wasobtained in [39, 40] for the probability of emitting the total black hole mass in the form of asingle shell. We conclude that Eq. (3.31) admits physical interpretation as the probabilityof the two–stage reflection process where the black hole is formed in classical collapsewith probability of order 1, and decays afterwards into a single shell with exponentiallysuppressed probability.One may be puzzled by the fact that, according to Eq. (3.30), the suppression receivesequal contributions from the two parts of the shell trajectory crossing the horizon in theinward and outward directions. Note, however, that the respective parts of the integral(3.30) do not have individual physical meaning. Indeed, we reduced the original two–dimensional integral for the action to the form (3.30) by integrating over sections of constantSchwarzschild time. Another choice of the sections would lead to an expression with a– 18 –ifferent integrand. In particular, using constant–time slices in Painlev´e or Finkelsteincoordinates one obtains no imaginary contribution to S reg from the inward motion of theshell, whereas the contribution from the outward motion is doubled. The net result for theprobability is, of course, the same. The above result unambiguously shows that the shell model, if taken seriously as afull quantum theory, suffers from the information paradox. Indeed, transition betweenthe only two asymptotic states in this theory — contracting and expanding shell — isexponentially suppressed. Either the theory is intrinsically non–unitary or one has to takeinto consideration an additional asymptotic state of non–evaporating eternal black holeformed in the scattering process with probability 1 − P fi .On the other hand, the origin of the exponential suppression is clear if one adoptsa modest interpretation of the shell model as describing scattering between the narrowwavepackets in field theory. Hawking effect implies that the black hole decays predomi-nantly into configurations with high multiplicity of soft quanta. Its decay into a single hardwavepacket is entropically suppressed. One can therefore argue [42] that the suppression(3.31) is compatible with unitarity of field theory. However, the analysis of this section isclearly insufficient to make any conclusive statements in the field theoretic context.As a final remark, let us emphasize that besides the reflection probability our methodallows one to calculate the phase of the scattering amplitude Re S reg . At L = m = 0 it canbe found analytically, Re S reg = 2 M r + 2 M log( r / M ) + M . (3.32)Notice that apart from the trivial first term, the phase shift is proportional to the entropy B ∝ M ; this is compatible with the dependence conjectured in [20]. The phase (3.32)explicitly depends on the parameter r of the initial– and final–state wavefunctions. In this section we deviate from the main line of the paper which studies transitions betweenfree–particle initial and final states, and consider scattering of a shell off an eternal pre–existing black hole. This will allow us to establish a closer relation of our approach to theresults of [39, 40] and the Hawking radiation. We will focus on the scattering probabilityand 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 themetric functions f − = 1 − M BH /r , f + = 1 − M BH + M ) /r , (3.33)where M BH is the eternal black hole mass and M denotes, as before, the energy of theshell. The inner and outer metrics possess horizons at r − h = 2 M BH and r + h = 2( M BH + M ),respectively. The shell motion is still described by Eq. (3.8), where the effective potential isobtained by substituting expressions (3.33) into the first line of Eq. (3.9). Next, the global Note that our semiclassical method is free of uncertainties [58–60] appearing in the approach of [39]. – 19 – h e ll r ∞ r Figure 8 . Penrose diagram for scattering of a classical shell from an eternal black hole. space–time has an additional boundary r = r (cid:48)∞ → + ∞ at the second spatial infinity ofthe eternal black hole, see Fig. 8. We have to include the corresponding Gibbons–Hawkingterm, cf. Eq. (3.22), S GH (cid:12)(cid:12)(cid:12) r (cid:48)∞ = − M BH (cid:90) dt − . (3.34)Finally, the eternal black hole in the initial and final states contributes into the free action S . We use the Hamiltonian action of an isolated featureless black hole in empty space–time [61], S ,BH = − M BH (cid:90) dt + , (3.35)where, as usual, the time variable coincides with the asymptotic time . Since we do notequip the black hole with any degrees of freedom , its initial– and final–state wavefunctionsare Ψ BH = 1.Adding new terms (3.34), (3.35) to the action (3.30), one obtains ,Im S reg = Im (cid:90) C dr √− V eff (cid:34) m − m m eff + M BH + M (cid:112) f + − V eff f + − M BH (cid:112) f − − V eff f − (cid:35) , (3.36)where the integration contour C is similar to that in Fig. 5 (lower panel), it bypasses thetwo horizons r − h and r + h in the lower half of complex r –plane. In the interesting limit ofvanishing centrifugal barrier L → S reg (cid:12)(cid:12)(cid:12) L =0 = 2 π ( M BH + M ) Res r = r + h (cid:112) f + − V eff √− V eff f + − πM BH Res r = r − h (cid:112) f − − V eff √− V eff f − = π (cid:2) ( r + h ) − ( r − h ) (cid:3) . (3.37) To compute the phase of the scattering amplitude, one should also take into account long–range inter-action of the black hole with the shell in the initial and final states. The choice of a model for an isolated black hole is the main source of uncertainties in scattering problemswith eternal black holes. Recall that the free action enters this formula with the negative sign, S ,BH = M BH ( t f − t i ). Note alsothat the original Gibbons–Hawking term (3.22) is proportional to the total ADM mass M BH + M . – 20 –nterpretation of this result is similar to that in the previous section. At the first stage oftransition the black hole swallows the shell with probability of order one and grows to themass M BH + M . Subsequent emission of the shell with mass M involves suppression P fi (cid:39) e − B + + B − , (3.38)where B ± = π ( r ± h ) are the entropies of the intermediate and final black holes. Thissuppression coincides with the results of [39, 40].At M BH = 0 the process of this section reduces to reflection of a single self-gravitatingshell and expression (3.37) coincides with Eq. (3.31). In the other limiting case M (cid:28) M BH the shell moves in external black hole metric without back–reaction. Reflection probabilityin this case reduces to the Boltzmann exponent P fi (cid:39) e − M/T H , where we introduced the Hawking temperature T H = 1 / (8 πM BH ). One concludes thatreflection of low–energy shells proceeds via infall into the black hole and Hawking evapo-ration, whereas at larger M the probability (3.38) includes back–reaction effects. Let us return to the model with a single shell considered in Secs. 3.1–3.3. In the previousanalysis we integrated out the non–dynamical metric degrees of freedom and worked withthe semiclassical shell trajectory ( t + ( τ ) , r ( τ )). It is instructive to visualize this trajectory inregular coordinates of the outer space–time. Below we consider the case of ultrarelativisticshell with small angular momentum: L → M (cid:29) m . One introduces Kruskalcoordinates for the outer metric, U = − ( r/ M (cid:48) − / e ( r − t + ) / M (cid:48) , V = ( r/ M (cid:48) − / e ( r + t + ) / M (cid:48) . (3.39)We choose the branch of the square root in these expressions by recalling that M (cid:48) differsfrom the physical energy M by an infinitesimal imaginary shift, see Eq. (3.14). The initialpart of the shell trajectory from t + = t i to the turning point A (Figs. 5, 6) is approximatelymapped to a light ray V = V > L → A is close to the singularity r = 0, but does not coincide with it. At theturning 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, and the direction of τ is opposite to that of the Kruskal time U + V . The corresponding evolution is represented bythe interval ( A, t f ) in Fig. 9. We conclude that at t + = t f the shell emerges in the oppositeasymptotic region in the Kruskal extension of the black hole geometry. This conclusion mayseem puzzling. However, the puzzle is resolved by the observation that the two asymptoticregions are related by analytic continuation in time. Indeed it is clear from Eqs. (3.39) thatthe shift t + (cid:55)→ t + − πM 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 thereal time axis (point t (cid:48) f in Fig. 6). At t + = t (cid:48) f the shell emerges in the right asymptoticregion with future–directed proper time τ . The process in Fig. 9 can be viewed as a Although the shell coordinate r ( t (cid:48) f ) is complex, we identify the asymptotic region using Re r ( t (cid:48) f ) whichis much larger than the imaginary part. – 21 – i t f At f U V
Figure 9 . Trajectory of the shell in Kruskal coordinates of the outer metric. Black dashed linesshow the singularities
U V = 1. shell–antishell annihilation which is turned by the analytic continuation into the transitionof a single shell from t i to t (cid:48) f .Now, we write down the space–time metric for the saddle–point solution at m = 0and L →
0. Recall that in this case the shell moves along the real r –axis. We thereforeintroduce 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), ds = − (cid:16) − M (cid:48) r shell ( t + ) (cid:17) dt + dr + r d Ω , r < r shell ( t + ) − (cid:16) − M (cid:48) r (cid:17) dt + dr − M (cid:48) r + r d Ω , r > r shell ( t + ) , (3.40)where we changed the inner time t − to t + by matching them at the shell worldsheet r = r shell ( t + ). Importantly, the metric (3.40) is regular at the origin r = 0 which is neverreached by the shell. It is also well defined at r h = 2 M due to the imaginary part of M (cid:48) ; in the vicinity of the Schwarzschild horizon r h the metric components are essentiallycomplex. Discontinuity of Eq. (3.40) at r = r shell ( t + ) is a consequence of the δ –functionsingularity in the shell energy–momentum tensor. This makes the analytic continuation ofthe metric ill–defined in the vicinity of the shell trajectory. We expect that this drawbackdisappears in the realistic field–theory setup where the saddle–point metric will be smooth(and complex–valued) in Schwarzschild coordinates. In this and subsequent sections we subject our method to further tests in more complicatedshell models. Here we consider a massless shell in 4-dimensional AdS space-time. Theanalysis 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–Hilbertaction is supplemented by the cosmological constant term, S EH = 116 π (cid:90) V d x √− g ( R − . (4.1)– 22 –ere Λ ≡ − /l , l is the AdS radius. The Gibbons–Hawking term has the form (3.4),where now the regulator at the distant sphere K (cid:12)(cid:12) r ∞ = 2 l + lr ∞ (4.2)is chosen to cancel the gravitational action of an empty AdS . The metric inside andoutside the shell is AdS and AdS–Schwarzschild, respectively, f − = 1 + r l , f + = 1 − Mr + r l , (4.3)where M is the shell energy. The trajectory of the shell obeys Eq. (3.8) with the effectivepotential given by the first line of Eq. (3.9), V eff = 1 + r l − ( L + 2 M r ) r L . (4.4)The (cid:15) –modification again promotes M in this expression to M (cid:48) = M + i ˜ (cid:15) . Repeating theprocedure of Sec. 3.2, we start from the reflected trajectory at large L . Keeping ˜ (cid:15) >
0, wetrace the motion of the turning point as L decreases . The result is a family of contours C spanned by the trajectory in the complex r –plane. These are similar to the contours inFig. 5. In particular, at L → C mostly runs along the real axis encircling theAdS–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, thespace–time curvature is now non-zero everywhere. Trace of the Einstein’s equations gives R = 4Λ. The Einstein–Hilbert action takes the form, S EH = Λ2 (cid:20) (cid:90) dt − r shell (cid:90) r dr + (cid:90) dt + r ∞ (cid:90) r shell r dr (cid:21) = (cid:90) shell r l ( dt + − dt − ) − r ∞ l (cid:90) dt + . (4.5)The last term diverging at r ∞ → ∞ is canceled by the similar contribution in the Gibbons–Hawking term at spatial infinity, S GH (cid:12)(cid:12) r ∞ = (cid:18) r ∞ l − M (cid:19) (cid:90) dt + . (4.6)Second, unlike the case of asymptotically flat space–time, Gibbons–Hawking terms at theinitial– and final–time hypersurfaces t + = t i,f vanish, see Appendix D. Finally, the canon-ical momenta of the free shell in AdS, p i,f ( r, M ) = M (1 + r /l ) − L r (1 + r /l ) , (4.7)are negligible in the asymptotic region r → + ∞ . Thus, the terms involving p i,f in the freeaction (3.26) and in the initial and final wavefunctions (3.28) are vanishingly small if the Alternatively, one can start from the flat–space trajectory and continuously deform it by introducingthe AdS radius l . In the massless case the trace of the shell energy–momentum tensor vanishes, T µshell µ = 0, see Eqs. (B.3). They follow from the dispersion relation m = − g M − g rr p . – 23 –ormalization point r is large enough. This leaves only the temporal contributions in thefree actions, S (0 − , t i ) + S ( t f , + ) = M ( t f − t i ) . (4.8)Summing up Eqs. (4.5), (4.6), (4.8) and the shell action (3.2), we obtain, S reg = (cid:90) C dr √− V eff (cid:20) r l (cid:18) (cid:112) f + − V eff f + − (cid:112) f − − V eff f − (cid:19) + M (cid:112) f + − V eff f + − Lr (cid:21) −→ (cid:90) ∞ dr (cid:20) r l (cid:18) f + − f − (cid:19) + M f + (cid:21) at L → , (4.9)where the integration contour in the last expression goes below the pole at r = r h . Theintegral (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 theinitial 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 isrelated to the residue of the integrand at r h ,2Im S reg = 2 π (cid:18) r h l + M (cid:19) Res r = r h f + ( r ) = πr h . (4.10)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 hascombined into r h which is a complicated function of the AdS–Schwarzschild parameters M and l . Exponential suppression of the shell reflection has a natural interpretation within theAdS/CFT correspondence [9, 62, 63]. The latter establishes relationship between gravityin AdS and strongly interacting conformal field theory (CFT). Consider three–dimensionalCFT on a manifold with topology R × S parameterized by time t and spherical angles θ .This is the topology of the AdS boundary, so one can think of the CFT as living on thisboundary. Let us build the CFT dual for transitions of a gravitating shell in AdS . Assumethe CFT has a marginal scalar operator ˆ O ( t, θ ); its conformal dimension is ∆ = 3. Thisoperator is dual to a massless scalar field φ in AdS .Consider now the composite operatorˆ O M ( t ) = exp (cid:26)(cid:90) dt d θ G M ( t − t ) ˆ O ( t, θ ) (cid:27) , (4.11)where G M ( t ) is a top–hat function of width ∆ t (cid:29) /M . This operator is dual to a sphericalwavepacket (coherent state) of the φ –field emitted at time t from the boundary towardsthe center of AdS [64, 65]. With an appropriate normalization of G M ( t ), the energy of thewavepacket is M . Similarly, the operator ˆ O + M ( t ) is dual to the wavepacket absorbed onthe boundary at time t . Then the correlator G M = (cid:104) ˆ O + M ( πl ) ˆ O M (0) (cid:105) (4.12) The time πl is needed for the wavepacket to reach the center of AdS and come back if it movesclassically [65]. – 24 – FT O M (0)ˆ O + M ( πl ) AdS Figure 10 . Conformal diagram for scattering of a massless shell in AdS . Creation and annihilationof the shell at the AdS boundary correspond to insertions of composite operators in the CFT dual. is proportional to the amplitude for reflection of the contracting wavepacket back to theboundary. If the width of the wavepacket is small enough, ∆ t (cid:28) l , the φ –field can be treatedin the eikonal approximation and the wavepacket follows a sharply defined trajectory. Inthis way we arrive to the transition of a massless spherical shell in AdS , see Fig. 10.Exponential suppression of the transition probability implies respective suppression ofthe correlator (4.12). However, the latter suppression is natural in CFT because the statecreated by the composite operator ˆ O M (0) is very special. Submitted to time evolution,it evolves into a thermal equilibrium which poorly correlates with the state destroyed byˆ O + M ( πl ). Restriction of the full quantum theory in AdS to a single shell is equivalentto a brute–force amputation of states with many soft quanta in unitary CFT . Since thelatter are mainly produced during thermalization, the amputation procedure leaves us withexponentially suppressed S –matrix elements. Another interesting extension of the shell model is obtained by endowing the shell withelectric charge. The corresponding action is the sum of Eq. (3.5) and the electromagneticcontribution S EM = − π (cid:90) d x √− g F µν − Q (cid:90) shell A a dy a , (5.1)where A µ is the electromagnetic field with stress tensor F µν = ∂ µ A ν − ∂ ν A µ and Q is theshell charge. This leads to Reissner–Nordstr¨om (RN) metric outside the shell and emptyflat space–time inside, f + = 1 − Mr + Q r , A = Qr ; f − = 1 , A − = 0 . (5.2)Other components of A µ are zero everywhere. Importantly, the outside metric has twohorizons r ( ± ) h = M ± (cid:112) M − Q (5.3)– 25 – r (+) h r ( − ) h AA C rr h A A C (a) (b) Figure 11 . Motion of the turning points and the contour C defining the trajectory for (a) themodel 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 reflectsclassically. Since the latter classical reflections proceed without any centrifugal barrier, weset L = 0 henceforth. The semiclassical trajectories will be obtained by continuous changeof the shell charge Q .The evolution of the shell is still described by Eq. (3.8) with the effective potentialconstructed from the metric functions (5.2), V eff = 1 − ( m − Q + 2 M r ) m r . (5.4)This potential always has two turning points on the real axis, r A,A (cid:48) = Q − m M ∓ m ) . (5.5)The shell reflects classically from the rightmost turning point r A at Q > M . In the oppositecase
Q < M the turning points are covered by the horizons, and the real classical solutionsdescribe black hole formation.We find the relevant semiclassical solutions at
Q < M using (cid:15) –modification. Sincethe modification term (2.10) does not involve the electromagnetic field, it does not affectthe charge Q giving, as before, an imaginary shift to the mass, M (cid:55)→ M + i ˜ (cid:15) . A notabledifference 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 for any Q . On theother hand, the horizons (5.3) approach the real axis with Im r (+) h > r ( − ) h < Q decreases. Thus, the saddle–point trajectories are defined along the contour C in Fig. 11abypassing r (+) h and r ( − ) h 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 startsin the asymptotic region I , crosses the outer and inner horizons r (+) h and r ( − ) h , repels fromthe time–like singularity due to electromagnetic interaction, and finally re–emerges in theasymptotic region I (cid:48) . At first glance, this trajectory has different topology as comparedto the classical reflected solutions at Q > M : the latter stay in the region I at the final The overall trajectory is nevertheless complex because t + ∈ C , see below. – 26 – (+) h r ( − ) h II O (+) O ( − ) Figure 12 . Conformal diagram for the extended Reissner–Nordstr¨om space–time. Semiclassicaltrajectory running along the contour C in Fig. 11a is shown by the red line. The grey region doesnot exist in theories with dynamical charged fields. time t + = t f . However, following Sec. 3.5 we recall that the Schwarzschild time t + of thesemiclassical trajectory is complex in the region I (cid:48) ,Im ( t f − t i ) = 2 πf (cid:48) + ( r (+) h ) − πf (cid:48) + ( r ( − ) h ) , (5.6)where we used Eq. (3.18) and denoted by t i and t f the values of t + at the initial andfinal endpoints of the contour C in Fig.11a. Continuing t f to real values, we obtain thesemiclassical trajectory arriving to the region I in the infinite future , cf. Sec. 3.5. Thisis what one expects since the asymptotic behavior of the semiclassical trajectories is notchanged in the course of continuous deformations.Let us now evaluate the reflection probability. Although the contour C is real, itreceives imaginary contributions from the residues at the horizons. Imaginary part of thetotal action comes from Eq. (3.30) and the electromagnetic term (5.1). The latter takesthe form, S EM = − (cid:90) d x √− g (cid:18) F µν π + A µ j µ (cid:19) = 116 π (cid:90) d x √− g F µν , (5.7)where we introduced the shell current j µ , used Maxwell equations ∇ µ F µν = 4 πj ν andintegrated by parts. From Eq. (5.2) we find, S EM = 14 (cid:90) dt + ∞ (cid:90) r shell r dr (cid:18) − Q r (cid:19) = − Q (cid:90) shell dt + r . (5.8) Indeed, the coordinate systems that are regular at the horizons r (+) h and r ( − ) h , are periodic in theimaginary part of t + with periods 4 πi/f (cid:48) + ( r (+) h ) and 4 πi/f (cid:48) + ( r ( − ) h ), respectively. Analytic continuation toreal t f implies shifts by half–periods in both systems, see Eq. (5.6). This corresponds to reflections of thetrajectory final point in Fig. 12 with respect to points O (+) and O ( − ) . Gibbons–Hawking terms at t = t i,f are different in the case of charged shell from those in Eq. (3.30).However, they are real and do not contribute into Im S tot . – 27 –ombining this with Eq. (3.30), we obtain,2Im S reg =Im (cid:90) C dr √− V eff (cid:20) − m + (cid:18) M − Q r (cid:19) (cid:112) f + − V eff f + (cid:21) =2 π (cid:2) Res r = r (+) h − Res r = r ( − ) h (cid:3)(cid:18) M − Q r (cid:19) f + ( r ) = π (cid:0) r (+) h (cid:1) − π (cid:0) r ( − ) h (cid:1) . (5.9)After non–trivial cancellation we again arrive to a rather simple expression. However, thistime 2Im S tot is not equal to the entropy of the RN black hole, B RN = π (cid:0) r (+) h (cid:1) .The physical interpretation of this result is unclear. We believe that it is an artifactof viewing the charged shell as an elementary object. Indeed, in quantum mechanics of anelementary shell the reflection probability should vanish at the brink Q = M of classicallyallowed transitions. It cannot be equal to B RN which does not have this property unlikethe expression (5.9). We now explain how the result is altered in a more realistic setup. Recall that the inner structure of charged black holes in theories with dynamical fields isdifferent 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 [44–47].Besides, pair creation of charged particles forces the singularity to discharge [43, 48, 49].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–timeincluding the Cauchy horizon and beyond (the grey region in Fig. 12) is never formed inclassical collapse.Let us mimic the above discharge phenomenon in the model of a single shell. Althoughgauge invariance forbids non–conservation of the shell charge Q , we can achieve essentiallythe same effect on the space–time geometry by switching off the electromagnetic interactionat r →
0. To this end we assume spherical symmetry and introduce a dependence of theelectromagnetic coupling on the radius . This leads to the action S (cid:48) EM = − π (cid:90) d x √− g F µν e ( r/Q ) − Q (cid:90) shell A a dy a , (5.10)where e ( x ) is a positive form–factor starting from e = 0 at x = 0 and approaching e → x → + ∞ . We further assume e ( x ) < x , (5.11)the meaning of this assumption will become clear shortly. Note that the action (5.10) isinvariant under gauge transformations, as well as diffeomorphisms preserving the sphericalsymmetry. The width of the form–factor e ( r/Q ) in Eq. (5.10) scales linearly with Q tomimic larger discharge regions at larger Q .The new action (5.10) leads to the following solution outside the shell, f + = 1 − Mr + Qr a ( r/Q ) , A = a ( r/Q ) , where a ( x ) = ∞ (cid:90) x e ( x (cid:48) ) x (cid:48) dx (cid:48) . (5.12) Alternatively, the discharge can be modeled by introducing nonlinear dielectric permittivity [66]. – 28 –he 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 = 2 M at Q = 0, monotonically decreases with Q and reaches zero at Q ∗ = 2 M/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 effectivepotential for the shell motion, cf. Eq. (5.4), V eff = 1 − (cid:0) M r − Qr a ( r/Q ) + m (cid:1) m r , (5.13)introduces (cid:15) –regularization, M (cid:55)→ M (cid:48) = M + i ˜ (cid:15) , and studies motion of the turning pointsof the shell trajectory as Q decreases. If M, Q (cid:29) m , this analysis can be performed forgeneral e ( x ). In this case the relevant turning point r A is real and positive for Q > Q ∗ .At Q ≈ Q ∗ it comes to the vicinity of the origin r = 0 where the function a ( r/Q ) can beexpanded up to the linear term. The position of the turning point is r A = 1 b (cid:20) − M + m + a (0) Q (cid:115)(cid:18) M − m − a (0) Q (cid:19) − m b (cid:21) , (5.14)where b ≡ − da/dx (cid:12)(cid:12) x =0 is positive according to Eq. (5.12). As Q decreases within theinterval Q ∗ − m (1 − b ) a (0) > Q > Q ∗ − m (1 + b ) a (0) (5.15)the turning point makes an excursion into the lower half of the r –plane, goes below theorigin 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 isshown 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 thatin the case of neutral shell.Finally, we evaluate the imaginary part of the action. The electromagnetic contributionis similar to Eq. (5.8), S (cid:48) EM = − Q (cid:90) shell e dt + r . (5.16)However, in contrast to Sec. 5.1, the trace of the gauge field energy–momentum tensor doesnot vanish due to explicit dependence of the gauge coupling on r (cf. Eq. (B.3b)), T (cid:48) µEM µ = F µν r πe dedr . (5.17)This produces non–zero scalar curvature R = − πT (cid:48) µEM µ in the outer region of the shell,and the Einstein–Hilbert action receives an additional contribution,∆ S EH = − (cid:90) dt + (cid:90) ∞ r shell r dr (cid:18) − Q e r (cid:19) re dedr = (cid:90) shell dt + (cid:18) Q e r − Qa (cid:19) , (5.18)– 29 –here in the second equality we integrated by parts. Combining everything together, weobtain (cf. Eq. (5.9)),2Im S reg = Im (cid:90) C dr √− V eff (cid:20) − m + (cid:18) M − Q e r − Qa (cid:19) (cid:112) f + − V eff f + (cid:21) , = 2 π Res r = r h (cid:18) M − Q e r − Qa (cid:19) f + = πr h , (5.19)where non–trivial cancellation happens in the last equality for any e ( x ). To sum up, weaccounted 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 . In this paper we proposed a semiclassical method to calculate the S –matrix elements for thetwo–stage transitions involving collapse of multiparticle states into a black hole and decay ofthe latter into free particles. Our semiclassical approach does not require full quantizationof gravity. Nevertheless, it consistently incorporates backreaction of the collapsing andemitted quanta on the geometry. It reduces evaluation of the S –matrix elements to findingcomplex–valued solutions of the coupled classical Einstein and matter field equations withcertain boundary conditions.An important technical ingredient of the method is the regularization enforcing thesemiclassical solutions to interpolate between the in- and out- asymptotic states consist-ing of free particles in flat space–time. As a consequence, one works with the completesemiclassical solutions describing formation and decay of the intermediate black hole. Thisdistinguishes our approach from the standard semiclassical expansion in the black holebackground. In addition, the same regularization allows us to select the relevant semi-classical configurations by continuous deformation of the real solutions describing classicalscattering at lower energies. The final result for the S –matrix elements does not dependenton the details of the regularization.We illustrated the method in a number of toy models with matter in the form of thinshells. We have found that the relevant semiclassical solutions are complex–valued anddefined in the complexified space–time in the case of black hole–mediated processes. Theyavoid the high–curvature region near the black hole singularity thus justifying our use ofthe semiclassical low–energy gravity. In particular, the Planck–scale physics near the blackhole singularity is irrelevant for the processes considered in this paper.The method has yielded sensible results for transition amplitudes in the shell mod-els. Namely, we have found that the probabilities of the two–stage shell transitions areexponentially suppressed by the Bekenstein–Hawking entropies of the intermediate blackholes. If the shell model is taken seriously as a full quantum theory, this result implies thatits S –matrix is non-unitary. However, the same result is natural and consistent with uni-tarity if the shells are interpreted as describing scatterings of narrow wavepackets in field We do not discuss the phase of the scattering amplitude as it essentially depends on our choice of thedischarge model. – 30 –heory. The exponential suppression appears because we consider a very special exclusiveprocess: formation of a black hole by a sharp wavepacket followed by its decay into thesame packet. Our result coincides with the probability of black hole decay into a single shellfound within the tunneling approach to Hawking radiation [39, 40] and is consistent withinterpretation of the Bekenstein–Hawking entropy as the number of black hole microstates[42]. Considering the shell in AdS space–time we discussed the result from the AdS/CFTviewpoint.In the case of charged shells our method reproduces the entropy suppression only ifinstability of the Reissner–Nordstr¨om Cauchy horizon with respect to pair–production ofcharged particles is taken into account. This suggests that the latter process is crucial forunitarity of transitions with charged black holes at the intermediate stages.Besides the overall probability, our method yields the phase of the transition amplitude.The latter carries important information about the scattering process, in particular, aboutits initial and final states. In the case of a neutral shell in asymptotically flat space–time thephase contains a logarithmically divergent term due to long–range Newtonian interactionsand terms proportional to the black hole entropy. This is consistent with the behaviorconjectured in [20].We consider the above successes as an encouraging confirmation of the viability of ourapproach.The shell models are too simple to address many interesting questions about the blackhole S –matrix. These include the expected growth of the transition probability when thefinal state approaches the Hawking–like state with many quanta, and the recent conjectureabout sensitivity of the amplitudes to small changes in the initial and final states [67]. Astudy of these issues will require application of our method to a genuinely field–theoreticsetup. Let us anticipate the scheme of such analysis. As an example, consider a scalar field φ minimally coupled to gravity. For simplicity, one can restrict to transitions between initialand final states with particles in the s –wave. These states are invariant under rotations.Then the respective semiclassical solutions are also expected to possess this symmetry.They satisfy the complexified wave– and Einstein equations in the spherically symmetricmodel of gravity plus a scalar field . One can use the simplest Schwarzschild coordinates( t, r ) which are well–defined for complex r and t , though other coordinate systems may beconvenient for practical reasons. One starts from wavepackets with small amplitudes φ 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, one increases φ and obtains saddle–point solutions for the black hole–mediated transitions. The space–time manifold, if needed, should be deformed to complex values of coordinates — awayfrom the singularities of the solutions. We argued in Sec. 2 that the modified solutionsare 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 limitof vanishing modification (cid:15) → +0. Evaluation of S –matrix elements is thus reduced to Another interesting arena for application of the semiclassical method is two–dimensional dilaton grav-ity [68]. – 31 –olution of two–dimensional complexified field equations, which can be performed on thepresent–day computers.One may be sceptical about the restriction to the spherically symmetric sector whichleaves out a large portion of the original Hilbert space. In particular, all states contain-ing gravitons are dropped off because a massless spin-2 particle cannot be in an s –wave.Nevertheless, the S –matrix in this sector is likely to be rich enough to provide valuableinformation about the properties of black hole–mediated scattering. In particular, it issufficient for addressing the questions mentioned in the previous paragraph.Furthermore, one can envisage tests of the S –matrix unitarity purely within the semi-classical approach. Indeed, consider the matrix element of the operator S † S between twocoherent states with the mode amplitudes a k and b k ( k is the mode wavenumber), (cid:104) a |S † S| b (cid:105) = (cid:90) D c k D c ∗ k e − (cid:82) dk c ∗ k c k (cid:104) a |S † | c (cid:105)(cid:104) c |S| b (cid:105) , (6.1)where on the r.h.s. we inserted the sum over the (over-)complete set of intermediate coherentstates. For sifficiently distinct semiclassical states | a (cid:105) and | b (cid:105) the integrand in Eq. (6.1) is arapidly oscillating function. Then, it is natural to assume that the integral will be saturatedby a unique saddle–point state | c (cid:105) which does not coincide with the dominant final statesof the transitions starting from | a (cid:105) and | b (cid:105) . This suggests that the amplitudes (cid:104) a |S † | c (cid:105) and (cid:104) c |S| b (cid:105) correspond to rare exclusive processes and can be evaluated semiclassically.Substituting them in Eq. (6.1) and comparing the result with the matrix element of theunity operator, (cid:104) a | | b (cid:105) = e (cid:82) dk a ∗ k b k , (6.2)one will perform a strong unitarity test for the gravitational S –matrix. Of course, thisdiscussion relies on several speculative assumptions that must be verified. We plan toreturn to this subject in the future.Finally, in this paper we have focused on the scattering processes with fixed initialand final states which are described in the in-out formalism. In principle, the semiclassicalapproach can be also applied to other quantities, e.g. the results of measurements performedby an observer infalling with the collapsing matter. The latter quantitites are naturallydefined in the in-in formalism with the corresponding modification of the path integral.One can evaluate the new path integral using the saddle–point technique. Importantly, thenew semiclassical solutions need not coincide with those appearing in the calculation of the S –matrix. If they turn out to be different, it would imply that the infalling and outsideobservers describe the collapse/evaporation process with different semiclassical geometries.This will be an interesting test of the the black hole complementarity principle. We leavethe study of this exciting topic for future. Acknowledgments
We are grateful to D. Blas, V. Berezin, R. Brustein, S. Dubovsky, G. Dvali, M. Fitkevich,V. Frolov, J. Garriga, V. Mukhanov, V. Rubakov, A. Smirnov, I. Tkachev and T. Vachas-pati for useful discussions. We thank A. Barvinsky, A. Boyarsky and A. Vikman for– 32 –ncouraging interest. This work was supported by the RFBR grant 12-02-01203-a (DL)and the Swiss National Science Foundation (SS).
A A shell of rotating dust particles
Consider a collection of dust particles uniformly distributed on a sphere. Each partice hasmass δm and absolute value δL of angular momentum. We assume no preferred directionin particle velocities, so that their angular momenta sum up to zero. This configurationis spherically–symmetric, as well as the collective gravitational field. Since the sphericalsymmetry is preserved in the course of classical evolution, the particles remain distributedon the sphere of radius r ( τ ) at any time τ forming an infinitely thin shell.Each particle is described by the action δS = − δm (cid:90) | ds | = − δm (cid:90) dτ (cid:113) − g ab ˙ y a ˙ y b − r ( τ ) ˙ ϕ , (A.1)where in the second equality we substituted the spherically symmetric metric (2.9) andintroduced the time parameter τ . To construct the action for r ( τ ), we integrate out themotion of the particle along the angular variable ϕ using conservation of angular momentum δL = δmr ˙ ϕ (cid:112) − g ab ˙ y a ˙ y b − r ˙ ϕ . (A.2)It would be incorrect to express ˙ ϕ from this formula and substitute it into Eq. (A.1). Topreserve the equations of motion, we perform the substitution in the Hamiltonian δH = p a ˙ y a + δL ˙ ϕ − δ L , (A.3)where p a and δL are the canonical momenta for y a and ϕ , whereas δ L is the Lagrangianin Eq. (A.1). Expressing ˙ ϕ from Eq. (A.2), we obtain, δH = p a ˙ y a + (cid:112) − g ab ˙ y a ˙ y b (cid:112) δm + δL /r . (A.4)From this expression one reads off the action for r ( τ ), δ ˜ S = − (cid:90) dτ (cid:112) δm + δL /r , (A.5)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 S shell = N δ ˜ S = − (cid:90) dτ (cid:112) m + L /r ( τ ) , (A.6)where N is the number of particles, m = N δm is their total mass and L = N δL is thesum of absolute values of the particles’ angular momenta. We stress that L is not the totalangular momentum of the shell. The latter is zero because the particles rotate in differentdirections. – 33 – Equation of motion for the shell
In this appendix we derive equation of motion for the model with the action (3.5). Westart by obtaining expression for the shell energy–momentum tensor. Let us introducecoordinates ( y a , θ α ) such that the metric (2.9) is continuous across the shell. Here θ α , α = 2 , (cid:90) d y d Ω4 π δ (2) ( y − y ( τ )) = 1 , (B.1)we recast the shell action (3.2) as an integral over the four–dimensional space–time, S shell = − (cid:90) d y d Ω4 π (cid:90) dτ m eff (cid:112) − g ab ˙ y a ˙ y b δ (2) ( y − y ( τ )) . (B.2)Here τ is regarded as a general time parameter. The energy–momentum tensor of the shellis obtained by varying Eq. (B.2) with respect to g ab and r ( y ), T abshell = 2 √− g δS shell δ g ab = ˙ y a ˙ y b m eff πr (cid:90) dτ δ (2) ( y − y ( τ )) √− g , (B.3a) T αshell α = 2 r √− g δS shell δr = L πr m eff (cid:90) dτ δ (2) ( y − y ( τ )) √− g , (B.3b)where in the final expressions we again set τ equal to the proper time. It is straightforwardto see that the τ –integrals in Eqs. (B.3) produce δ –functions of the geodesic distance n from the shell, δ ( n ) = (cid:90) dτ δ (2) ( y − y ( τ )) √− g . (B.4)We finally arrive at T µνshell = t µνshell δ ( n ) , t abshell = m eff ˙ y a ˙ y b πr , t αshell β = δ αβ L πm eff r , (B.5)where T αshell β ∝ δ αβ due to spherical symmetry.Equation of motion for the shell is the consequence of Israel junction conditions whichfollow from the Einstein equations. The latter conditions relate t µνshell to the jump in theextrinsic curvature across the shell [56, 57]( K µν ) + − ( K µν ) − = − π (cid:18) t µshell ν − h µν t λshell λ (cid:19) . (B.6)Here h µν 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 theoutward–pointing normal, n µ ∂ r x µ >
0. Transforming the metric (3.6) into the continuouscoordinate system, we obtain,( K ab ) ± = − ˙ y a ˙ y b ¨ r + f (cid:48)± / (cid:112) ˙ r + f ± , ( K αβ ) ± = δ αβ (cid:112) ˙ r + f ± r , (B.7) Schwarzschild coordinates in Eq. (3.6) are discontinuous at the shell worldsheet. – 34 –here dot means derivative with respect to τ . From Eq. (B.6) we derive the equations, (cid:112) ˙ r + f + − (cid:112) ˙ r + f − = − m eff r , (B.8)¨ r + f (cid:48) + / (cid:112) ˙ r + f + − ¨ r + f (cid:48)− / (cid:112) ˙ r + f − = L m eff r + m eff r . (B.9)Only the first equation is independent, since the second is proportional to its time deriva-tive. We conclude that Einstein equations are fulfilled in the entire space–time providedthe metrics inside and outside the shell are given by Eqs. (3.6), (3.7) and Eq. (B.8) holdsat the shell worldsheet. The latter equation is equivalent to Eqs. (3.8), (3.9) from the maintext.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 describedby a single coordinate r ( τ ), and its equations of motion are equivalent to conservation ofthe energy–momentum tensor. The latter conservation, however, is ensured by the Einsteinequations. C Turning points at L → Turning points are zeros of the effective potential V eff , Eq. (3.9). The latter has six zeros r i , i = 1 . . .
6, which can be expressed analytically at L →
0. We distinguish the cases ofmassive and massless shell.a) m (cid:54) = 0: r , = − m M ∓ m ) , r , = iLm + M L m ∓ √ M L / m / e − iπ/ ,r , = − iLm + M L m ∓ √ M L / m / e iπ/ . b) m = 0: r , = − L / (2 M ) / ∓ L M , r , = L / (2 M ) / e iπ/ ∓ L M , r , = L / (2 M ) / e − iπ/ ∓ L M .
All turning points approach zero at L → r , in the massive case. Numericallytracing their motion as L decreases from L ∗ , we find that the physical turning point A ofthe reflected trajectory is r 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 arevanishingly small. However, this expectation is incorrect. Indeed, it is natural to definethe initial and final hypersurfaces as t + = const outside of the shell and t − = const insideit. Since the metric is discontinuous in the Schwarzschild coordinates, the inner and outer– 35 – τ ξ + ξ − t + = c o n s t t − = c o n s t s h e ll Figure 13 . Final–time hypersurface in the Gaussian normal coordinates associated with the shell. parts of the surfaces meet at an angle which gives rise to non–zero extrinsic curvature, seeFig. 13.For concreteness we focus on the final–time hypersurface. In the Schwarzschild coor-dinates the normal vectors to its inner and outer parts are ξ µ − = (1 / (cid:112) f − , , , , ξ µ + = (1 / (cid:112) f + , , , . (D.1)It is easy to see that the extrinsic curvature K = ∇ µ ξ µ is zero everywhere except for thetwo–dimensional sphere at the intersection the hypersurface with the shell worldsheet. Letus 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, arethe spherical angles. In this frame the metric in the neighborhood of the shell is essentiallyflat; 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 themon τ µ and n µ — tangent and normal vectors of the shell. The latter in the inner and outerSchwarzschild coordinates have the form, τ µ = (cid:18) (cid:112) ˙ r + f ± f ± , ˙ r, , (cid:19) , n µ = (cid:18) ˙ rf ± , (cid:112) ˙ r + f ± , , (cid:19) . (D.2)Evaluating the scalar products of (D.1) and (D.2), we find, ξ µ ± = ch ψ ± τ µ − sh ψ ± n µ , sh ψ ± ≡ ˙ r (cid:112) f ± . (D.3)As expected, the normals ξ µ ± do not coincide at the position of the shell. To compute thesurface 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, S GH = − π (cid:90) r d θ (cid:90) dn dsdn K = r ψ + − ψ − ) , (D.5)where in the second equality we used ds = dn/ ch ψ for the proper length along the final–time hypersurface and K = ∂ µ ξ µ = − ch ψ ψ (cid:48) for its extrinsic curvature. Next, we express– 36 – ± ( r ) from the shell equation of motion (3.8) and expand Eq. (D.5) at large r . Keepingonly non–vanishing terms at r = r f → + ∞ , we obtain Eq. (3.24) for the final–timeGibbons–Hawking term.For the initial–time hypersurface the derivation is the same, the only difference is inthe sign of ξ µ which is now past–directed. However, this is compensated by the change ofsign of ˙ r . One concludes that the Gibbons–Hawking term at t + = t i is obtained from theone 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 shellin AdS. It is straightforward to see that in the latter case the Gibbons–Hawking termsvanish at r i,f → ∞ due to growth of the metric functions (4.3) at large r . E Shell self–gravity at order /r Let us construct the action for a neutral shell in asymptotically flat space–time taking intoaccount its self–gravity at order 1 /r . To this end we recall that the shell is assembled fromparticles of mass δm , see Appendix A. Every particle moves in the mean field of otherparticles. Thus, a new particle added to the shell changes the action of the system by δS = − (cid:90) δm dτ = (cid:90) dt + (cid:18) − δm (cid:112) − v + δm (1 + v ) √ − v ¯ Mr (cid:19) , (E.1)where v = dr/dt + is the shell velocity in the asymptotic coordinates, ¯ M is its energy, andwe expanded the proper time dτ up to the first order in 1 /r in the second equality. At theleading order in 1 /r , ¯ M = ¯ m √ − v , (E.2)where ¯ m 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, S = (cid:90) dt + (cid:18) − m (cid:112) − v + m (1 + v )2 r (1 − v ) (cid:19) . (E.3)From this expression one reads off the canonical momentum and energy of the shell, p = mv √ − v + 2 m vr (1 − v ) , (E.4) M = m √ − v + m ( − v + v )2 r (1 − v ) . (E.5)Expressing the shell velocity from Eq. (E.5) and substituting it into Eq. (E.4), we obtainEq. (3.27) from the main text. Angular motion of the particle gives 1 /r contributions to the Lagrangian which are irrelevant in ourapproximation. In this calculation the 1 /r terms are treated as corrections. – 37 – eferences [1] S. B. Giddings, The gravitational S-matrix: Erice lectures , arXiv:1105.2036 .[2] S. W. Hawking, Particle Creation by Black Holes , Commun. Math. Phys. (1975) 199[ Erratum-ibid. (1976) 206].[3] S. W. Hawking, Breakdown of Predictability in Gravitational Collapse , Phys. Rev. D (1976) 2460.[4] A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity orFirewalls? , JHEP (2013) 062, [ arXiv:1207.3123 ].[5] A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully,
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