Towards Black Hole Evaporation in Jackiw-Teitelboim Gravity

Using a definition of the bulk frame within 2d Jackiw-Teitelboim gravity, we go into the bulk from the Schwarzian boundary. Including the path integral over the Schwarzian degrees of freedom, we discuss the quantum gravitational Unruh effect and the Planckian black-body spectrum of the thermal atmosphere. We analyze matter entanglement entropy and how the entangling surface should be defined in quantum gravity. Finally, we reanalyze a semi-classical model for black hole evaporation studied in JHEP 1607, 139 (2016) and compute the entanglement between early and late Hawking radiation, illustrating information loss in the semi-classical framework.


Introduction
Understanding black hole evaporation and information loss is one of the big unsolved problems in quantum gravity [2]. Ever since the advent of AdS/CFT it has been understood that the evaporation process is unitary since it can be described by a unitary QM theory on the boundary, and hence information cannot be lost. However, this state of affairs is unsatisfactory, as we still have no precise understanding of the bulk mechanism that releases the quantum information during evaporation. One of the hurdles we are faced with here is the difference in language between the two set-ups. Hawking's computation is done in the context of quantum field theory in curved spacetimes, neglecting the matter-gravitational interactions and hence backreaction in the process. Boundary holography on the other hand faces the problem of bulk reconstruction: how does one construct local bulk operators and make contact with local bulk dynamics, as would be constructed by e.g. an infalling observer.
In order to address these problems, it is useful to have interesting toy models where aspects of these computations can be carried out exactly. An interesting model studied primarily in the '90s is the 2d CGHS model [3,4,5,6]. Explicit computations can be performed probing semi-classical Hawking evaporation, but it is asymptotically flat and hence more difficult to embed within a tractable unitary framework. Another model that has attracted a considerable amount of attention during the past few years is Jackiw-Teitelboim (JT) gravity [7,8], a model of 2d asymptotically AdS 2 dilaton gravity with action: defined in terms of the 2d metric g µν and the dilaton field Φ, with cosmological constant Λ = −2/L 2 . The model is topological in the bulk, but it has non-trivial dynamics due to the choice of boundary conditions. This happens in a very similar way as in the Chern-Simons / WZW correspondence.
The only physical degree of freedom in JT gravity is the boundary reparametrization f (t) [9,10,11,1], whose dynamics is governed by the Schwarzian action: which first appeared describing the low-energy dynamics of Sachdev-Ye-Kitaev (SYK) models (see e.g. [12,13,14,15,16,17,18]). This means the quantum gravity path integral only contains inequivalent frames in a fixed AdS 2 ambient space, and is hence far more tractable than expected generically. The quantum gravitational thermal correlation functions are found by computing the thermal Schwarzian path integral with a suitable operator insertion O[f ]. In the semi-classical regime (at large C), this path integral localizes to its classical equation of motion with solution f (τ ) = τ . Such path integrals are always performed in Euclidean signature, with the resulting expressions (carefully) Wick-rotated afterwards to real time [19].
Given any reparametrization map f (t) on the boundary, we can set up a unique bulk frame by shooting in (and extracting) null rays at times t − z and t + z. This, combined with conformal gauge, uniquely defines a bulk point (t, z), and constructs the bulk metric: (1.4) where the conformal factor is hence determined by the construction. This fully fixes the bulk diffeomorphism gauge invariance and relates everything to boundary-intrinsic constructions. We have previously explored this definition of bulk coordinates in [20].
In this note, we present three separate computations that can be done within this model that address aspects of the Hawking evaporation process. The computations are logically distinct and illustrate the power of the JT model to investigate these problems. Section 2 applies the above bulk frame to the bulk matter stress tensor and the matter occupation numbers in the Unruh heat bath surrounding the black hole, i.e. we construct the generalization of Unruh's effect including matter-gravitational interactions. Section 3 discusses the entanglement entropy of matter fields in AdS 2 across a bulk entangling surface, emphasizing an invariant definition of the location of the surface. Section 4 proceeds within the standard semi-classical framework, but includes evaporating boundary conditions. The evaporating black hole in JT gravity was constructed in [1]. Here we extend this study and compute the entanglement of outgoing matter (early radiation) with the remaining interior (late radiation) and obtain an analytic result. The result demonstates information loss within the semi-classical framework quite explicitly.
Of course, ultimately we would want to perform an exact quantum gravitational computation of the evaporating black hole. Initial steps in this direction were taken in [21], but a full understanding is still lacking. We leave this to future work.

Unruh heat bath
We couple the JT model (1.1) to the free boson action: and want to study the effect of particle creation in different matter vacuum states [30], labeled by the reparametrization functions f (u) and f (v) in the bulk geometry (1.4).
Considering the thermal state, we will write down the Unruh energy fluxes in this system, and decompose these to read off how the Planckian black body spectrum and the thermal atmosphere are modified in quantum gravity for the eternal black hole.

Energy flux
Within this matter theory, for a fixed background f , the propagator is well-known: 1 which is the 2d CFT two-point function supplemented with an image charge term such that Dirichlet boundary conditions are specified at the holographic boundary z = 0. The stress tensor components are given by interpreted as the outgoing and ingoing energy densities (Figure 1 left). Bilocal boundary operation insertion at zero temperature, and the bulk injection of energy that it entails. The bulk energy densities T uu and T vv are zero before and after the injections, and non-zero but constant and equal in between the legs of the bilocal.
As composite operators in the quantum theory, these require renormalization. Referring w.r.t. the Poincaré frame and using (2.2) with a point-splitting regularization, we find the renormalized operators: : : (2.5) Series expanding this expression, one rewrites these as 2 where we have introduced the matter central charge c to generalize to a generic matter CFT sector. The boundary time frame is extrapolated into the bulk using these equations, in the light-ray fashion described in the Introduction. 3 In effect, bulk energy densities can be computed by inserting the boundary energy operator T tt (t) = C {F (t), t} (see e.g. (A.9)), up to some prefactors, and reinterpreting the time t as either u or v. 4 This bulk gauge is a choice, but it is one that nicely contains the semi-classical Unruh physics as we illustrate now.
On the saddle f (t) = t, and tanh π β f (u), u = 2π 2 β 2 , leading to the Unruh heat bath [30] : which includes a further quantum thermodynamical correction that is suppressed in the semi-classical regime C → +∞. 6 The measured stress tensor components of observers whose detectors are calibrated to the u, v-vacuum, is hence spacetime-independent. This changes when matter is being injected (or extracted) into the system.

Energy pulses
Classically, energy can be injected through pulses as studied extensively in [9,1]. At the quantum level, this can be done by using bilocal operators of the type in (1.7) ( Figure  1 right). Let us gather some evidence for this interpretation. 2 The CFT expectation value brackets are left implicit from here on. 3 These operators are of the type of (1.3), and should be written as : Tuu[f (u)] : and : Tvv[f (v)] :, which we won't do to avoid cluttering the equations. 4 Note the presence of the factor of C in the boundary energy. This has dimensions of length, and hence indeed the bulk stress tensor has dimension L −2 , and the boundary stress tensor has units of L −1 . 5 The conformal anomaly determines the remaining stress tensor component to be Tuv = c 6L 2 , in terms of the AdS length L. This also holds when doing the full path integral (1.3). This is independent of the temperature and can be viewed as an energy offset E0. 6 By inverse Laplace transforming, for a pure energy quantum state |M , with energy M/2C, we obtain instead :Tuu(u): (2.9) • In Appendix A, we demonstrate the Ward identity for boundary stress tensor insertions T tt (t) in bilocal correlators. Dropping contact terms, and continuing to Lorentzian signature, we write: (2.10) in momentum space interpreted as energy 2C between the legs of the bilocal, and 2C outside. In particular, one finds energy is conserved everywhere except at the bilocal points: interpreted in Fourier space as injecting and extracting an energy at the bilocal points. Since : T uu (u) : and : T vv (v) : are also given by Schwarzian derivatives (2.6), the result (2.11) implies there are no transient phenomena for these bulk stress tensors after crossing energy pulses (Figure 1 right).
• Let us prove that this operator indeed gives the correct semi-classical energy pulse interpretation. In the limit where we take N bilocals of = 1 at the same endpoints and with N ∼ C → +∞ to reach the semi-classical regime, the bulk interpretation is a semi-classical coherent state: a null pulse (m 2 = ( − 1) = 0) of energy E( N, t 12 ) that can be found by solving a transcendental equation [31], followed by a negative energy null pulse with energy −E( N, t 12 ). One finds the bulk energy densities (2.9) in between the null pulses, with M = E( N, t 12 ), and zero outside. The classical time reparametrization profile is and represents the Poincaré frame, going through a thermal phase, and then returning to a (Shapiro time-delayed) Poincaré frame.
• Including this bilocal in the definition of time-evolution operator, in the sense that we define: we can evolve the system back and forward in time, and the energy injection (and extraction) are contained within our H. This is quite analogous to the set-up in [32]. This Hamiltonian H is self-adjoint and explicitly time-dependent and hence energy is indeed not conserved. 7 The configuration generated by H is displayed in Figure 1 right. 8 The classical problem of measuring the ingoing and outgoing energy T vv and T uu within the different regions of Figure 1 right, is translated in the full quantum theory in the operator ordering of the stress tensor w.r.t. the bilocal. For instance, O (τ 1 , τ 2 )T uu (u) β→+∞ always vanishes, irrespective on the value of u w.r.t. τ 1 and τ 2 . The amplitude can be read as taking empty AdS, time-evolving with H to t = u, applying T uu , then evolving back or forward to the first bilocal time and inject energy, evolve to the second bilocal time and extract the energy, to arrive at empty AdS again.

Planckian black body spectrum
Next we perform a spectral decomposition of the Unruh fluxes (2.8). It is well-known, using 2d CFT techniques, that the occupation number of the chiral mode ω in a generic frame f (y) is given by [34]: where u ω (y) = 1 √ 4πω e −iωy is a positive frequency mode in the observer's local frame y. The integral is finite at y 1 = y 2 due to the renormalization w.r.t. the Poincaré state.
Semiclassically, where f (y) = y, the treatment is well-known and leads to the Planckian black body spectrum. Let us briefly write down how this is proven. Firstly, since the integrand only depends on y 1 −y 2 one of the integrals factorizes out and gives a divergent prefactor δ(0). For the remaining integral one can use the Fourier transform formula (2.14) In the limit β → +∞, we obtain the formula: The bilocal operator is in fact the matrix product of two local operators. Each of these is a matrix element in the discrete unitary infinite-dimensional representation of SL(2, R). As such, these operators are unitary operators on the quantum Hilbert space. 8 Note that this is asymmetric in time. Ordinarily one draws shockwaves that impact the boundary and get reflected back into the bulk [33]. That situation is obtained by always evolving with H, even to go back in time after applying the shockwave operator. This is done to obtain the bulk geometry associated to a QM state O |ψ . In our case here, when we evolve back in time, we take away the shockwaves in the process.
which is readily found from the integral definition of the Heaviside function. 9 Hence subtracting out the Poincaré contribution (2.15), one obtains the Planckian spectrum: Beyond semi-classical gravity, we path-integrate over the frame f as in (1.3): Using the fact that the bilocal operator commutes with the Hamiltonian [19]: , H = 0, (2.18) the result again only depends on the difference y 12 . 10 This means the result is independent of y 1 + y 2 , and the integral gives δ(0), the same universal divergence also appearing at the semi-classical level. Using the known expressions for the Schwarzian bilocal, we write: 11 19) the last term coming from (2.15). Performing the τ -integral yields a delta-function δ(ω − k 2 1 + k 2 2 ). The resulting expression only makes sense for ω > 0 and we get: (2.20) The integral, despite starting out Lorentzian, can be done numerically and is plotted in Figure 2. Taking into account both chiral components, and integrating the energy spectral density over ω, one obtains 2 dω ωN ω β = 2 dy :T ±± : = dy √ −gg 00 :T 00 : , (2.21) where in the last equality we used that √ −gg 00 = 1, also off-shell. Indeed, the quantumcorrected Unruh population (2.8) is slightly more energetic than the semi-classical one (2.7), leading to a larger population of the thermal modes as Figure 2 shows. 9 Both of these formulas are regularized separately by moving the pole at t = 0 in the same direction. In most treatments, one first subtracts these terms before performing the Fourier transform, rendering the regularization t → t ± i obsolete. 10 Note that this need not occur for each single path f in the integral. 11 We set C = 1/2 here. The radiation is not precisely thermal. The UV region is dominated by the τ → 0 pole and is the same for the semi-classical Planck spectrum, but deviations due to gravitational interactions are visible at order one energies. This means quantum gravitational effects modify the Unruh process. This also means that there is information stored within the heat bath of the black hole, but it is not visible at the semi-classical level. It is well-known that matter interactions do not influence the thermal character of the Unruh effect [35], basically because one can do perturbation theory and one finds thermal answers at every fixed order. The non-thermality hence fully originates from the matter-gravitational interactions. This is to be expected, and was also observed for the metric tensor itself in [20]. It is remarkable that we are able to obtain an analytic formula describing the exact quantum gravitational Unruh spectrum.

Matter entanglement entropy
Another quantity that can be computed using these methods is the matter CFT entanglement entropy. Divide a spatial slice Σ of AdS 2 in two parts (Figure 3 left). The entanglement entropy between the degrees of freedom left and right of the bulk point u = t + z, v = t − z of the matter CFT is given by the formula: where the UV cut-off δ is measured by the observer in the u, v-coordinate frame. 12 In our language, using the radar construction of bulk points, this corresponds to a boundaryintrinsic choice of UV-cut-off. This equation excludes gravity, and is taken in the matter Poincaré vacuum state, labeled by the coordinate f . We do not need to specify the precise shape of the Cauchy surface Σ of interest due to the foliation independence of S ent and the fact that we insist on the UV-cut-off associated to the u, v-frame. This is illustrated in Figure 3 middle. 13 Classically, using f (u) = tanh π β u, this becomes Setting z → +∞ to obtain the thermal entropy, we write: 14 which is IR-divergent. As a check, the total matter energy on the other hand is obtained by integrating (2.7) over the spatial volume. 15 Using the thermodynamical relation E mat = ∂ β βF , one finds F = −E mat and the thermal entropy S = β(E mat − F ) 13 Different foliations can be obtained by applying conformal transformations F with the fixed point properties F (f (u)) = f (u) and F (f (v)) = f (v). Additionally, insisting on using the same UV-cutoff yields the same formula. Note that this is different than replacing f in (3.1) by F • f in both numerator and denominator, which would change the state to the vacuum in the F -coordinates. 14 We subtracted the zero-temperature entropy S ref = c 6 ln u−v δ from this expression to isolate the thermal piece [39]. This cancels the UV-cutoff dependence and gives an additional contribution c 6 ln β 4πz that is subdominant in the large z-regime. 15 We used T00 = Tuu + Tvv. In principle, one should add to this 2Tuv = c 3L 2 from the conformal anomaly and using R = −2/L 2 for AdS2 JT-gravity, with L the AdS length. This contribution is β-independent and hence does not contribute to the entropy. is indeed given by (3.3). This is the semi-classical result that the thermal entropy of the matter gas surrounding the black hole can be viewed as entanglement entropy of the half-space (Figure 3 right).
The formula (3.1) can also be read as the (analytically continued) geodesic length between two boundary points v and u [40]. Since the information within the matter CFT moves on null rays, the information inside the interval can be mapped into the time interval [t − z, t + z]. One can view this as the boundary observer's ignorance to information prior and after this interval (Figure 4 left). In quantum gravity, we generically expect the entanglement entropy formula (3.1) to be qualitatively influenced in two ways. Firstly, gravitons also contribute to the entanglement entropy and they should be taken into account. Secondly, a conceptually deeper question is how one defines the location of the entangling surface invariantly within quantum gravity. For JT gravity, gravitons are of course absent, but we can deal with the second conceptual issue in a precise way. The location of the entangling surface is at the bulk point (u, v), which is found by the radar definition from the boundary observer's times u and v. Given two boundary times u and v, the entanglement entropy can be viewed as a diff-invariant bulk observable if we construct it as S ent [f (u), f (v)], and it is this operator that we will insert in the gravitational path integral (1.3). The computation can be done by taking the -derivative of the two-point function (1.7) and setting = 0 in the end: (3.5) time-independent as it should be. For small separations z C, we retrieve the semiclassical formula (3.2). Still for a macroscopic black hole but in the very-near horizon regime β C z, the k 2 -integral is dominated by its saddle (leading to the equivalence between microcanonical and canonical ensembles), whereas the large z-regime enforces k 1 ≈ k 2 . Using an integral formula for the Gamma-functions, the computation is identical to that in [29] 16 and, upon subtracting the UV-divergent piece, leads to of the same form as (3.3). Hence the linear increase of the entanglement entropy happens well past the semi-classical regime. This suggests the identification of the total thermal entropy in the thermal atmosphere and the entanglement entropy of the half-space remains true for macroscopic black holes β C, in spite of probing the dangerous deep bulk z C where quantum gravitational effects are expected, see also [20].
For completeness, this computation can be readily extended to the entanglement entropy for a bulk interval ( Figure 5). One uses the following quantum gravity operator S Figure 5: Left: Matter entanglement of a bulk interval between (u 1 , v 1 ) and (u 2 , v 2 )). Right: The Cauchy slice Σ can be deformed into those new surfaces within the blue regions that respect the spacelike nature. insertion: (3.7) As before, this quantity is independent of the precise spatial form of the Cauchy surface Σ on which the entropy is computed. The resulting quantity is a bulk bilocal observable, whose Schwarzian path integral is readily computed by the same trick as above.

Semi-classical entanglement of Hawking particles
Everything up to this point concerned non-evaporating black holes, as black holes in AdS tend to equilibrate instead. In order to allow evaporation, we have to modify the asymptotic boundary conditions from perfect reflection to absorption. This model was studied in [1], and we retake it here. Energy conservation dictates that the total bulk energy can only be modified by in-and outfluxes of matter in the sense that: Besides this pulse, we set the boundary conditions such that nothing reflects back into the bulk: : T vv (t) : = 0 for t > 0 as perfect absorption boundary conditions. Both the total boundary energy T tt (t) = C {f, t} and the ingoing flux :T uu (t): = c 24π {f, t} are given by Schwarzian derivatives. Hence plugging these in (4.1), the energy decays exponentially in the system: where A = c 24πC . This leads to the time reparametrization profile [1]: with α = 24π c √ 2CE 0 . This time reparametrization asymptotes to a fixed value beyond the eternal black hole horizon. The endpoint however does not reach the original Poincaré horizon. The Penrose diagram of the evaporating hole is shown in Figure 6 right.
In any given frame f , the semi-classical two-point correlator is of the form:  Let us now use this solution to compute the entanglement entropy in the matter sector between the early and late Hawking radiation. This computation is similar to that of [6] done for the asymptotically flat CGHS model. Where JT gravity stands out again, is in the full analytic solvability of the problem. The entanglement entropy of the matter fields can be computed using a Cauchy surface Σ that is close to the initial infalling pulse and then reconnects to the Poincaré horizon in the end (Figure 8). Note that we are required to study a Cauchy surface in the entire Poincaré patch as this is our starting geometry. The interval is between (u = , v = ) and (u = t, v = /2) for some infinitesimal whose sole purpose is to make sure the surface is spacelike. As the matter is null and propagates unhindered to the timelike boundary, the entanglement entropy across this interval on Σ measures the entanglement entropy the boundary observer would associate to all radiation he received until some specific time t, measured in his evaporating time frame (see also figure 4 right). The entanglement entropy in the interval in the Poincaré vacuum state, but described using the evaporating frame (4.3) is then found as . information being returned with the Hawking radiation. Hence we do not obtain a Page curve, and information is lost. A similar feature was observed with the analysis of the CGHS model [6]. In both cases, this can be viewed as a quantitative confirmation that the semi-classical Hawking computation is not able to restore information, but this is an artifact of the semi-classical approximation, see also [41].
As a comparison, doing the same computation for the non-evaporating black hole, by plugging f (t) = tanh √ E 0 t into (4.7), one finds an ever-increasing S ren (for large t as S ren ∼ t 2 ). The perfectly thermal Hawking emission does not contain any information whatsoever, and the information contained in the compensating ingoing quanta is lost in the process, see also [36].
We have seen in the previous sections how Schwarzian techniques can be used to go beyond semi-classical gravity. What is required here, is an embedding of this computation within a unitary quantum-mechanical framework. Due to the absorbing boundary conditions however, this is not so simple, and is postponed to future work.

Concluding remarks
The Unruh and Hawking effects are of fundamental importance in understanding quantum black holes, but unfortunately it is very difficult to go beyond the level of matter quantum fields in a curved spacetime. Due to its solvability, Jackiw-Teitelboim gravity is an ideal test-case to attempt to include quantum gravitational effects, which we have studied throughout this work.
Within the set-up of a thermal quantum system, we have studied several bulk diffinvariant operators: the bulk stress tensor components : We defined these objects operationally using only boundary-intrinsic data and studied how quantum gravitational effects modify them from their semi-classical limit.
In the final section, we imposed absorbing boundary conditions at the holographic boundary instead to allow the bulk black hole to evaporate. We computed the entanglement between the early and late Hawking radiation in this model and found information loss within the semi-classical set-up. This was expected and illustrates that unitarity can seemingly be violated as an artifact of the semi-classical perturbative expansion. It would be very interesting to combine these two types of calculations, and reach a quantum understanding of evaporation in this model. This is left to future work. witht(t) as given in (A.2). This generalizes immediately to higher-point functions. Obviously, time-translation invariance t 21 → t 21 +c is lost for any non-constant profile C(t). This simple generalized model also demonstrates that one can compute amplitudes using quasi-statics of the coupling C(t): no dependence on its derivatives is present. This resonates with the exactness of the quasi-static approximation in 2d JT gravity, as we mentioned in Sections 2.2 and 4.
This procedure also has an interpretation in JT gravity [1,9,10,11]. Changing the asymptotic boundary value of the dilaton as with z = ṫ andz = ṫ , requires the time reparametrization
trivial as it can be undone by a conformal transformation. However, when taking the Schwarzian limit, 2d conformal symmetry is taken to 1d reparametrization symmetry, which is explicitly broken in this procedure. As emphasized in [19], the Liouville computation reduces to a minisuperspace Hamiltonian propagation amplitude along the cylinder, where the Schwarzian time coordinate t is identified with the Liouville σ-coordinate, t ≡ σ, and one finds a time-dependent Schwarzian coupling constant C(t) in (A.1). The changing circumference corresponds to a time-dependent Hamiltonian. The boundary ZZ-states and Liouville primary vertex operator insertions happen at a single instant in time t, and our unaffected by this varying radius. The only effect in the minisuperspace Liouville computation is then the replacement: e −tH → e − t 0 dtH(t) , (A. 8) indeed what was found in (A.5).

A.2 Ward identities
We can use the time-dependent coupling C(t) to derive the Ward identity for stress tensor insertions in bilocal correlation functions. A version of the Ward identity for solely stress tensor correlators was explored in [22]. The Ward-like identities for bilocal correlators were derived in [19] using the embedding within Liouville CFT. Finally, another approach is being developed in [42].
At finite temperature, the stress tensor insertion is T tt (t) = −C tan π β f, t . The timedependent thermal generalization of (A.5) is given by the expression: Γ( ± ik 1 ± ik 2 ) (2C(τ 1 )) (2C(τ 2 )) 2π 2 Γ (2 ) , .11) leading to the thermal Ward identity 12) in Fourier space leading to an energy Inverse Laplace transforming this expression, one finds the Ward identity in a fixed energy eigenstate |k with energy k 2 /2C, for which the last term can be rewritten as T tt (t) k O (τ 1 , τ 2 ) k , in agreement with the result of [19].