From the BTZ black hole to JT gravity: geometrizing the island
PPrepared for submission to JHEP
From the BTZ black hole to JT gravity: geometrizingthe island
Evita Verheijden and Erik Verlinde
Institute of Physics & Delta Institute for Theoretical Physics,University of Amsterdam, Science Park 904,1090 GL Amsterdam, The Netherlands
E-mail: [email protected] , [email protected] Abstract:
We study the evaporation of two-dimensional black holes in JT gravity froma three-dimensional point of view. A partial dimensional reduction of AdS in Poincar´ecoordinates leads to an extremal 2D black hole in JT gravity coupled to a ‘bath’: theholographic dual of the remainder of the 3D spacetime. Partially reducing the BTZ blackhole gives us the finite temperature version. We compute the entropy of the radiation usinggeodesics in the three-dimensional spacetime. We then focus on the finite temperature caseand describe the dynamics by introducing time-dependence into the parameter controllingthe reduction. The energy of the black hole decreases linearly as we slowly move thedividing line between black hole and bath. Through a re-scaling of the BTZ parameters wemap this to the more canonical picture of exponential evaporation. Finally, studying theentropy of the radiation over time leads to a geometric representation of the Page curve.The appearance of the island region is explained in a natural and intuitive fashion. a r X i v : . [ h e p - t h ] F e b ontents black holes 4 black hole from AdS black hole from BTZ 72.5 Boundary action 7 black hole 83.2 Finite temperature AdS black hole 9 A.1 Extremal AdS black hole 19A.2 Finite temperature AdS black hole 20 Recent developments have shed new light on the question whether the formation andevaporation of black holes happen in a unitary fashion. This problem, known as the blackhole information problem, boils down to reproducing the Page curve: to save unitarity,the von Neumann entropy of the collected Hawking radiation should initially rise duringevaporation, but start decreasing after reaching a halfway point known as the Page time[1, 2]. However, a semiclassical gravity approach gives an ever-increasing entropy, usuallycalled the Hawking curve. This tension has a formal resolution in holography: replacingthe black hole by its dual representation, it is clear that the process must be unitary.– 1 –ne would like to find the analogous statement from the gravity point of view, e.g. usingholographic entropy tools such as the Ryu-Takayanagi formula and its covariant extension[3, 4]. The recent progress was initiated by two groups [5, 6] that used the minimal QuantumExtremal Surface (QES) [7], the surface that minimizes the generalized entropy, to monitorthe evolution of the entropy of the evaporating black hole.They found that the QES undergoes a transition at the Page time: it jumps frombeing the empty surface to a surface just inside the black hole horizon. This reproducesa Page curve for the black hole in geometric terms, but not for the radiation. Usinga doubly holographic model, i.e. considering a black hole in a gravitational theory withmatter that itself has a holographic dual, the authors of [8] showed that in this set-up onecan simply use the RT/HRT formula in the higher-dimensional dual. This ensures that inthe evaporating black hole picture, the minimal surface of the radiation and the black holecoincide, thus obtaining a Page curve for the radiation as well. The higher-dimensionalgeometry connects the radiation to the black hole interior, such that (at late times) theblack hole interior becomes part of the entanglement wedge of the radiation or ‘bath’. Thisled them to propose a new rule for computing the entropy in gravitational systems, nowknown as the “Island Rule”: S [Rad] QG = min I (cid:110) ext I (cid:104) S [Rad ∪ I ] SCG + Area[ ∂I ]4 G (cid:105)(cid:111) . (1.1)The prescription tells us that to compute the entropy of Hawking radiation in quantumgravity, we should include “quantum extremal islands” in our semi-classical entropy cal-culation. These islands can minimize the entropy, e.g. an island just inside the black holehorizon will include Hawking partners of the radiation. The price to pay is the area of theisland. Finally, one has to extremize and minimize over all possible islands.Most of the recent work takes place in two dimensions. For example, in [6], thespecific model considered was that of two-dimensional JT gravity coupled to a bath. Inthe doubly-holographic model of [8], the JT gravity was coupled to conformal matter (aCFT ), which played the role of the bath. This CFT is then itself the boundary theory ofa dual AdS bulk. The JT gravity lives on a Planck brane, which in the three-dimensionalpicture can be thought of as a Randall-Sundrum or End-of-the-World (EOW) brane. TheEOW brane also appeared in other work on the BHIP by [11]. However, in their set-up,the CFT living on the brane is different from the CFT of the bath. The boundary CFTon its own would describe an equilibrium black hole, but the coupling to the bulk CFTallows the black hole to evaporate. This gives a free parameter c bdy /c bulk , which controls There is no reason to believe the general arguments should fail in higher dimensions, and some explicitextensions to higher-dimensional systems exist, see e.g. [9] for a numerical and [10] for an exact example. – 2 –he Page time. A transition also occurs for the case of thermal equilibrium: even thoughthe black hole does not evaporate, it does “radiate” information away. In a related paper[12], (three of) the authors of [8] also study an AdS black hole in thermal equilibriumwith a bath. They too find a version of the information paradox, and show that the islandrule can resolve it. Interestingly, these islands lie outside of the horizon.In this paper we hope to add to this exciting line of research by combining someof the previous techniques: the doubly-holographic picture, and black holes in thermalequilibrium. We show how to obtain black holes in JT gravity by dimensional reductionfrom Poincar´e AdS and the BTZ geometry, resulting in an extremal and non-extremalJT black hole, respectively. As is common in these constructions, the angular metriccomponent takes on the role of the dilaton. Instead of applying a full dimensional reduction,we will reduce over part of the range of the angular coordinate; thereby, we effectively splitthe geometry into two parts: a JT black hole, and a 2D CFT dual to the remainder of the3D geometry.See figures 1 and 2 for a schematic depiction of the set-up. The 2D CFT willmodel a ‘bath’, and we can now compute the entanglement entropy of an interval in thebath system using RT surfaces.We can introduce dynamics into the finite temperature JT black hole system by givingtime-dependence to the parameter that controls the dimensional reduction. In this way,we can let the black hole ‘geometrically’ evaporate. From the BTZ perspective, we sim-ply move the dividing line between the degrees of freedom that are ‘in’ and ‘out’ of theblack hole. From the JT perspective, the mass decreases linearly and the temperature isfixed. Exploiting a map of BTZ parameters discussed in [13], we can equivalently view thisevaporation as adiabatically decreasing the mass of the BTZ black hole, such that we canconsider it to be in thermal equilibrium at each instant of time. On the JT gravity side,this gives us a more standard exponential evaporation, where the temperature depends ontime. Finally, computing the entropy of the entire bath system for the ‘geometric evapo-ration’, we obtain a Page curve for the radiation entropy.This paper is organized as follows. In section 2 we briefly review black holes in JT gravity,and discuss how to obtain both extremal and non-extremal black holes from AdS bydimensional reduction. In section 3, we compute the generalized entropy for intervals inthe extremal and finite temperature black hole plus bath systems, from a three-dimensionalpoint of view. Then, we will introduce dynamics in section 4, and allow the non-extremalblack hole to slowly evaporate. Finally, we obtain a Page curve for the entropy of theradiation. – 3 – JT gravity from AdS black holes After a brief reminder on extremal and non-extremal black holes in JT gravity, we showhow to obtain these black holes from AdS by dimensional reduction. We end with somecomments on retrieving the boundary action from the dimensional reduction. Jackiw-Teitelboim (JT) gravity is a two-dimensional dilaton gravitational theory (see [14,15] for the original model, and [16, 17] for compact reviews). If we require the classicalbackground metric to be given by AdS with AdS radius (cid:96) , we find the following action: S = 116 πG (cid:34)(cid:90) d x √− g Φ R + (cid:90) d x √− g Φ (cid:18) R + 2 (cid:96) (cid:19)(cid:35) + S matter . (2.1)The first term (with Φ a constant) is topological and determines the extremal entropy;after adding the appropriate boundary term it gives the Euler characteristic of the manifold.The second term is the JT term and the equation of motion for Φ sets R = − (cid:96) , i.e. itforces the spacetime to be asymptotically AdS . S matter is some arbitrary matter systemcoupled to both the metric and the dilaton. The general solution for the metric is ds = − (cid:96) d X + d X − ( X + − X − ) , (2.2)where X + ( u ) and X − ( v ) are general monotonic functions of the lightcone coordinates( u, v ). The AdS boundary is located at X + ( u ) = X − ( v ).The action (2.1) admits black hole solutions, dynamically formed by throwing in matterfrom the boundary. The vacuum equations of motion are solved by the dilaton profileΦ = Φ + 2Φ r − κEX + X − X + − X − , (2.3)where E = M is the mass of the black hole and Φ r = πGκ is an integration constantthat specifies the asymptotic boundary conditions of the dilaton field (notice that Φ isdimensionless, but Φ r has dimensions of length ). In particular, close to the boundary g uu = (cid:96) (cid:15) and Φ − Φ = Φ r (cid:15) , with (cid:15) the UV cutoff. Through the substitution X + ( u ) = 1 √ κE tanh (cid:16) √ κEu (cid:17) , X − ( v ) = 1 √ κE tanh (cid:16) √ κEv (cid:17) , (2.4)we find that the metric and dilaton are periodic in imaginary time with period β = π/ √ κE ⇒ T H = 1 π (cid:114) πGE r . (2.5) In some references, e.g. [18, 19], the action is derived from a four-dimensional parent theory, such thatthe Newton’s constant has dimension (length) . This gives the dilaton Φ dimensions of (length) and theinterpretation of an area. In (2.1), however, we work directly in two dimensions such that G is dimensionless. – 4 –n terms of the lightcone coordinates ( u, v ), the metric and dilaton profile are ds = − π (cid:96) β d u d v sinh πβ ( u − v ) , Φ = Φ + 2 π Φ r β πβ ( u − v ) . (2.6)The future and past horizons are at u = ∞ and v = −∞ . Note that if we take thelimit E → geometry in Poincar´e coordinates and the correspondingdilaton profile: ds = − (cid:96) d X + d X − ( X + − X − ) , Φ = Φ + 2Φ r ( X + − X − ) . (2.7)Since E → T H →
0, we will interpret this solution asthe extremal AdS black hole. In the following sections, we will see how to obtain theseblack holes from three dimensional Anti-de Sitter space. Consider the three-dimensional action S = 116 πG (3) (cid:90) d x √− g ( R (3) − , (2.8)with negative cosmological constant Λ <
0. Solutions are asymptotically AdS with theAdS radius given by Λ = − (cid:96) , and include the BTZ solution. Suppose that we have asolution for which the metric field is independent of one coordinate, which we will call ϕ ,and that it can be written as ds = g µν d x µ d x ν = h ij ( x i ) d x i d x j + φ ( x i ) (cid:96) d ϕ , (2.9)where the indices µ, ν = 0 , , i, j = 0 ,
1. Then the action (2.8) reduces to S = 2 πα(cid:96) πG (3) (cid:90) d x √− hφ ( R (2) − , (2.10)See also [20]. Here, we accounted for a partial reduction controlled by the parameter α ∈ (0 ,
1] for reasons that will become clear later. We see that we retrieved the JT action(2.1) (ignoring the topological piece) for Λ = − (cid:96) , the cosmological constant for AdS gravity. If we identify (cid:96) = (cid:96) and G (3) = (cid:96)G (2) then we are led to conclude that the dilatonin (2.1) is given by Φ = 2 παφ . (2.11)Since the three-dimensional Newton’s constant has dimensions of length, from this reduc-tion we again inherit a dimensionless dilaton (remember that in (2.1), the two-dimensionalNewton’s constant is dimensionless, and therefore the dilaton as well). Note also that the– 5 –onstant Φ that gives the topological part of the JT action can be added by hand, suchthat we get Φ − Φ = 2 πα √ g ϕϕ (cid:96) , (2.12)where we should be careful to satisfy the correct boundary condition (Φ − Φ ) | bdy = Φ r (cid:15) .Taken to the boundary we have α √ g ϕϕ (cid:96) (cid:12)(cid:12)(cid:12) bdy = α(cid:96)(cid:15) , (2.13)such that we should interpret Φ r = 2 π(cid:96)α ≡ Φ r α . (2.14)In what follows we will therefore useΦ = Φ + Φ r √ g ϕϕ (cid:96) = Φ + Φ r φ(cid:96) . (2.15)We will now apply this procedure to Poincar´e AdS and the BTZ black hole. black hole from AdS The metric of AdS in Poincar´e coordinates is given by: ds = (cid:96) z ( − d t + d z + d x ) . (2.16)We would like to reproduce (2.7). First, note that we can use the coordinate transformation z = (cid:96) r and x = (cid:96)ϕ to rewrite the above as ds = − r (cid:96) d t + (cid:96) r d r + r d ϕ . (2.17)This is exactly of the form (2.9), with φ(cid:96) = r . Hence we arrive immediately at theconclusion that AdS in Poincar´e coordinates reduces to a solution of JT gravity. To getexactly (2.7), we can instead use lightcone coordinates X ± = t ± z , in which the metric(2.16) becomes ds = − (cid:96) d X + d X − ( X + − X − ) + 4 (cid:96) d ϕ ( X + − X − ) . (2.18)Therefore, comparing to (2.7) we obtain precisely the AdS Poincar´e metric if we identifythe dilaton [12]: Φ = Φ + Φ r √ g ϕϕ (cid:96) = Φ + 2Φ r X + − X − . (2.19)– 6 – .4 Finite temperature AdS black hole from BTZ We can follow a similar procedure for the finite temperature case. Now, we start from theBTZ geometry, ds = − (cid:32) r − R (cid:96) (cid:33) d t + (cid:32) r − R (cid:96) (cid:33) − d r + r d ϕ , (2.20)where R = 8 GM (cid:96) is the horizon radius and the inverse temperature is β = π(cid:96) R . TheBTZ metric (2.20) is also of the form (2.9) and hence we could immediately identify again φ(cid:96) = r . To make contact with our earlier description of the non-extremal black hole inJT gravity, we change coordinates to u = t + r ∗ , v = t − r ∗ . Here r ∗ is the usual tortoisecoordinate, defined through d r ∗ = (cid:32) r − R (cid:96) (cid:33) − d r . (2.21)Outside the horizon, the metric takes the form ds = − π (cid:96) β d u d v sinh πβ ( u − v ) + 4 π (cid:96) β πβ ( u − v ) d ϕ . (2.22)In the first part we recognize precisely the AdS black hole metric (2.6). Furthermore wecan identify again the dilaton profile [12]Φ = Φ + Φ r √ g ϕϕ (cid:96) = Φ + 2 π Φ r β πβ ( u − v ) . (2.23) In JT gravity, the boundary term in the action famously leads to the Schwarzian action.We wish to reproduce the Schwarzian action from the three-dimensional point of view. TheGibbons-Hawking term is S GH = 18 πG (3) (cid:90) d x √− h (cid:18) K (3) + 2 (cid:96) (cid:19) = 2 πα(cid:96) πG (3) (cid:90) d t (cid:112) − h tt φ b (cid:18) K (3) + 2 (cid:96) (cid:19) , (2.24)where φ b is the boundary value of φ . The trace of the extrinsic curvature splits into twoparts: K (3) = h µν K µν = K (2) + h ϕϕ K ϕϕ . (2.25)We evaluate the two-dimensional K (2) below and first focus on the contribution h ϕϕ K ϕϕ .To perform the dimensional reduction, we initially choose the boundary to be at a fixedvalue of z . We find both in Poincar´e as well as in BTZ h ϕϕ K ϕϕ = − (cid:96) , (2.26)– 7 –hich is expected for the curvature of a circle. Thus, the boundary term in the action is(using Φ = 2 παφ ) S GH = 18 πG (cid:90) d t (cid:112) − h tt Φ b (cid:18) K (2) + 1 (cid:96) (cid:19) . (2.27)We will use this action to describe the dynamics. We will denote the boundary timecoordinate with t , which becomes a parameter for the dynamical boundary trajectory (cid:0) τ ( t ) , z ( t ) (cid:1) [16–18]. Here, τ and z are (fixed) coordinates on the AdS boundary; we willchoose them to be the Poincar´e coordinates. We then require that the boundary of AdS ,i.e. the surface u = v ≡ t , coincides with the general boundary X + ( u ) = X − ( v ). Thisdefines the dynamical boundary time to be X + ( t ) = X − ( t ) ≡ τ ( t ) . (2.28)We demand that the induced metric satisfies g | bdy = h tt = − (cid:96) (cid:15) , which then implies that z = (cid:15) (cid:112) ( τ (cid:48) ) − ( z (cid:48) ) = (cid:15)τ (cid:48) + O ( (cid:15) ). The normal to the boundary z = z ( t ( τ )) is n a = (cid:96)z √ τ (cid:48) − z (cid:48) ( − z (cid:48) , τ (cid:48) ) . (2.29)This gives K (2) ≈ − (cid:96) + (cid:15) (cid:96) { τ, t } , with { τ, t } = τ (cid:48)(cid:48)(cid:48) τ (cid:48) − (cid:0) τ (cid:48)(cid:48) τ (cid:48) (cid:1) the Schwarzian derivative.Thus, the Gibbons-Hawking term evaluates to S GH = 18 πG (cid:90) d t (cid:96)(cid:15) Φ b (cid:18) K (2) + 1 (cid:96) (cid:19) = 18 πG (cid:90) d t Φ r { τ ( t ) , t } , (2.30)where we defined Φ b = Φ r (cid:15) . As before, Φ is dimensionless such that Φ r has dimensions oflength. In this section we compute the generalized entropy for intervals in the extremal and finitetemperature black hole plus bath systems. We do so from the higher-dimensional point ofview discussed in the previous section, i.e. we will use geodesics. black hole In [12], the generalized entropy for an interval in the extremal black hole + bath systemwas computed from the two-dimensional point of view. Here, instead, we want to computethe generalized entropy from the point of view of AdS . The set-up that we have in mind isdepicted in figure 1. Here, we have done a partial dimensional reduction of the ϕ -direction,i.e. instead of integrating the coordinate ϕ in (2.17) over 2 π we integrated over some angle2 πα with α ∈ (0 , JT AdS παb b Figure 1 . AdS -Poincar´e, partially reduced over the angle 2 πα . The purple region is the 2D JTextremal black hole, and the green region is dual to the bath 2D CFT. The blue geodesic computesthe entropy of the region [0 , b ] in the CFT, including the quantum mechanical system. is the JT black hole, the other is dual to the CFT/bath system. Now, if we consideran interval [0 , b ] in the CFT/bath system — which also includes the quantum-mechanicaldegrees of freedom — its entropy will be given by the length of the blue geodesic in figure1. This can be easily computed using embedding coordinates. The details are in appendixA.1. The entropy of an interval of which the endpoints lie at the same radial distance r and separated by an angular interval ∆ ϕ is given by S = 12 G arcsinh r ∆ ϕ (cid:96) . (3.1)Here and in what follows G = G (2) . If we take the endpoints to lie on the boundary, as isthe case for the geodesic in figure 1, one can expand to get S = 14 G (cid:18) Φ + 2 log Φ r + 2 b(cid:96) (cid:19) , (3.2)where we absorbed the UV cutoff in Φ and used Φ r = 2 π(cid:96)α . black hole For the BTZ case, the setup is as in figure 2. Again, we have done a partial dimensionalreduction over the ϕ -coordinate up to 2 πα ∈ (0 , π ]. The corresponding region in the BTZblack hole now reduces to a black hole in JT gravity (purple region). The remainder (greenregion) we view as dual to the bath (2D CFT) degrees of freedom. By decreasing the valueof α from 1 to 0, we can geometrically ‘evaporate’ the black hole. We will comment moreon this in the next section, in which we discuss the dynamics of our model. For now, wewill distinguish two cases: ‘before’ and ‘after’ the Page time or the half-way evaporation– 9 – πα QMCFT JT BTZ (a) t < t
Page πα QMCFT JTBTZ (b) t > t
Page
Figure 2 . We have done a partial reduction over the angle 2 πα . The value of α determines if theblack hole is before (a) or after (b) the Page time. point (figure 2(a) and 2(b), respectively). In both cases, the entropy of an interval [0 , b ] inthe CFT, which also includes the quantum-mechanical degrees of freedom, is given by thelength of a geodesic in BTZ of which the endpoints lie at the boundary (on a fixed timeslice). The details are in appendix A.2. Before taking the endpoints to the boundary, theentropy of such an interval is given by S = 12 G arcsinh 2 π(cid:96) rβ sinh πβ (cid:96) ∆ ϕ , (3.3)where ∆ ϕ is the angular separation of the two points. For the geodesic in figure 3(a), i.e.before the Page time, we then find (expanding for r → ∞ ) S = 14 G (cid:16) Φ + 2 log sinh πβ (cid:0) π(cid:96) (1 − α ) − b (cid:1)(cid:17) , (3.4)where we absorbed the UV cutoff in Φ . After the Page time, the geodesic ‘jumps’ andcrosses the purple region (see figure 3(b)). Thus the entropy is now given by S = 14 G (cid:16) Φ + 2 log sinh πβ (2 π(cid:96)α + 2 b ) (cid:17) . (3.5)Eventually, we are interested in the entropy of the entire bath of radiation, i.e. we wish totake b →
0. We will do so in section 4.5.Notice that we are viewing the full black hole + bath geometry as being in a purestate: we think of the BTZ black hole as the result of some collapsing matter, i.e. thethree-dimensional geometry is in a pure state. Since we are interested in obtaining a Pagecurve, this is also the more natural state to consider. If we instead consider the geometry– 10 – b (a) t < t Page b b (b) t > t
Page
Figure 3 . To find the entropy of the double interval [0 , b ], including the quantum-mechanicaldegrees of freedom, we need to compute the length of the blue geodesics. to be in the thermal state, we have to add a second, disconnected contribution (the BTZblack hole area) to the entropy if the angular interval grows larger than some critical angle[21]. In that case, the transition occurs much later than the halfway evaporation point andwe do not reproduce the Page curve.
The black holes we have discussed so far will not dynamically evaporate, because of thereflecting boundary conditions at infinity. If we allow instead for particles to escape, wecan simulate an evaporation process. This idea lies at the heart of the recent models (seealso e.g. [22, 23], in which Hawking radiation is allowed to escape to the ‘bath’ consistingof a two-dimensional CFT on the half-line). Note that in some other models, such as[12, 24], the black hole is in equilibrium with the bath. In that case, the black hole doesnot evaporate and an island appears outside of the horizon. The model we propose can beargued to lie somewhere in between. For the purposes of the discussion of the evaporationprocess, we will only consider the finite temperature black hole.In the preceding sections, we stated that we can consider different phases of evaporationby means of the parameter α , which controls the dimensional reduction (and is closelyrelated to the dilaton). So far, we did not discuss a dynamical way of changing Φ r ∼ α .In this section, we will add explicit time-dependence to the renormalized dilaton Φ r ( t ) =2 π(cid:96)α ( t ) and see that this indeed results in an evaporating black hole. Then, we will map thisto a more standard evaporation protocol in which the dilaton is fixed, but the temperature(and mass) of the black hole decrease. To do so, we will use a mapping between two BTZ– 11 –eometries with different parameters. We will first briefly discuss this mapping, and thenproceed to compare the two perspectives on evaporation. We consider a map between two BTZ geometries with different parameters introduced in[13]: BTZ(
M λ ; 2 π ) ≡ BTZ( M ; 2 πλ ) . (4.1)On the left hand side we have a BTZ geometry with a mass that can vary; on the righthand side we have a BTZ geometry with a varying conical deficit. We start from the usualBTZ metric ds = − (cid:32) r (cid:96) − π (cid:96) β (cid:33) d t + (cid:32) r (cid:96) − π (cid:96) β (cid:33) − d r + r d ϕ , (4.2)where (cid:96) is the AdS length, the horizon is at R = 8 GM (cid:96) and ϕ ∼ ϕ + 2 π is identified.The inverse temperature is β = π(cid:96) R . Now, consider the transformation r = λ ˜ r, R = λ ˜ R, t = λ − ˜ t, ϕ = λ − ˜ ϕ . (4.3)This keeps the form of the metric invariant as in (4.2), but the periodicity in ˜ ϕ is now 2 πλ .Under this λ -transformation, the entropy S = πR G (3) remains invariant, but the Hawkingtemperature T H = R π(cid:96) gets scaled by λ − . If we pick λ = π(cid:96)β and leave R (or equivalently β ) fixed, we find ds = − (cid:32) ˜ r (cid:96) − (cid:33) d˜ t + (cid:32) ˜ r (cid:96) − (cid:33) − d˜ r + ˜ r d ˜ ϕ . (4.4)In what follows, we will use tilded coordinates to describe the BTZ geometry on which weperformed a partial reduction in section 3.2. Indeed, in this case the temperature is fixedand the periodicity of ˜ ϕ leads us to identify α with λ = π(cid:96)β . We will study the dynamicsof this model in the next section. We will then use the above map in section 4.3 to finduntilded coordinates in which the black hole energy decays exponentially. To study the dynamics of our model, consider again the Schwarzian action S = 18 πG (cid:90) d t Φ r { τ, t } , (4.5)where we will allow for Φ r to be time-dependent. Varying with respect to τ ( t ) gives theequation of motion (in the absence of matter terms)1 τ (cid:48) (cid:0) Φ r { τ, t } (cid:48) + 2Φ (cid:48) r { τ, t } + Φ (cid:48)(cid:48)(cid:48) r (cid:1) = 0 , (4.6)– 12 –here primes denote t -derivatives. If Φ r is constant, this reduces to the more familiar1 τ (cid:48) { τ, t } (cid:48) = 0 , (4.7)i.e. we are looking for non-constant functions τ ( t ) with constant Schwarzian. This leads tothe solution for the non-evaporating black hole, where τ = βπ tanh πβ t , (4.8)with constant inverse temperature β and (constant) ADM energy E = − Φ r πG { τ, t } = 2 π β Φ r πG ≡ E . (4.9)Now we will restore time-dependence and use tilded quantities to distinguish from theabove case. Remember that in the model at hand, we have done a partial reduction of theBTZ geometry, resulting in a JT gravity part with dilaton˜Φ r = 2 π(cid:96)α (˜ t ) ≡ Φ r α (˜ t ) , (4.10)where we defined Φ r ≡ π(cid:96) and α (˜ t ) decreases from 1 to 0. We interpret the remainingpart as holographically dual to a 2D CFT. Now, the decreasing α does not affect thetemperature of the 2D black hole, which is directly inherited from the BTZ black hole.Therefore, we will consider this temperature to be fixed and we will keep τ = βπ tanh πβ ˜ t .From the 2D CFT part of the action we find an extra term in the equation of motion (4.6),which is the incoming minus the outgoing energy flux [6, 17]: − πG (cid:16) ˜Φ r { τ, ˜ t } (cid:48) + 2 ˜Φ (cid:48) r { τ, ˜ t } + ˜Φ (cid:48)(cid:48)(cid:48) r (cid:17) = ˜ T vv (˜ t ) − ˜ T uu (˜ t ) = : ˜ T vv (˜ t ) : − : ˜ T uu (˜ t ) : . (4.11)We assume that the CFT — our ‘bath’ — has perfect absorbing boundary conditions, i.e.there is no flux coming into the AdS spacetime from the boundary/bath: ˜ T vv (˜ t ) = 0. Theoutgoing energy momentum-flux on the other hand is given by the conformal anomaly,: ˜ T uu (˜ t ) : = − c π { τ, ˜ t } . (4.12)One can think of this as simply the effect of moving the dividing line between the JT gravityand CFT part of our BTZ black hole, as in figure 2: we are relabeling which degrees offreedom are ‘in’ and which are ‘out’ of the black hole. Then, since { τ, ˜ t } = − π β = cst,(4.11) gives − πG Φ (cid:48) r = c π , (4.13)which we can solve to find˜Φ r = Φ r α (˜ t ) = Φ r (cid:18) − A t (cid:19) , where A c G Φ r . (4.14)– 13 –e think of A as an evaporation rate (we pick the factor for later convenience). In termsof this evaporation rate, the energy in this coordinate system decreases asd ˜ E d˜ t = T vv − T uu = − c π π β = − E A . (4.15)Here and below, β should be interpreted as a constant, unless explicitly written as β ( t ). In many models of black hole evaporation, the energy decreases exponentially in time (seee.g. [6, 17, 22, 23]). In those models, the two-dimensional JT black hole is put into contactwith an external bath. We will now show how our model relates to this type of evaporation,in which the temperature depends on time and the dilaton is fixed. To do so, we will exploitthe mapping discussed in section 4.1.As a first step, we need to find ˜ t ( t ). A simple solution comes from considering againthe Schwarzian action with constant dilaton and changing t → ˜ t ( t ): S = 18 πG (cid:90) d t Φ r { τ, t } = 18 πG (cid:90) d t Φ r (cid:104)(cid:16) d t d˜ t (cid:17) − { τ, ˜ t } − { ˜ t, t } (cid:105) = 18 πG (cid:90) d˜ t Φ r (cid:16) d˜ t d t (cid:17) { τ, ˜ t } = 18 πG (cid:90) d˜ t ˜Φ r ( t ) { τ, ˜ t } (4.16)where in the second line we assumed that { ˜ t, t } is constant, as we will later confirm. Thenrequiring that the action is invariant leads us to conclude that ˜Φ r ( t ) = Φ r d˜ t d t , i.e.d˜ t d t = α (˜ t ) = 1 − A t . (4.17)Solving (4.17), we find that t and ˜ t are related by˜ t ( t ) = 2 A (1 − e − At/ ) ⇒ d˜ t d t = e − At/ , { ˜ t, t } = − (cid:18) A (cid:19) . (4.18)For a similar discussion on exponential evaporation with a more complicated solution for˜ t ( t ), see [22–24]. We expect that our solution is a good approximation to these more generalmodels in the limit of slow evaporation. We are now ready to connect the linear/geometric evaporation presented in section 4.2 tothe more common exponential decay. As before, we start from the equation of motion,which for constant dilaton reduces to − πG Φ r { τ, t } (cid:48) = : T vv ( t ) : − : T uu ( t ) : . (4.19)– 14 – priori we do not know the energy flux. However, transforming (4.11) gives informationon this. Since the right hand side is manifestly a tensor ( ˜ T ˜ t ˜ t ) both the left and right handside should transform as a tensor. Indeed we find that under ˜ t → ˜ t ( t ) we get − (cid:16) d˜ t d t (cid:17) − (cid:0) Φ r { τ, t } (cid:48) + 2Φ (cid:48) r { τ, t } + Φ (cid:48)(cid:48)(cid:48) r (cid:1) = (cid:16) d˜ t d t (cid:17) − πG ( T vv − T uu ) (4.20)Here, we have not yet assumed that Φ r = cst; we only assumed that it transforms as avector, i.e. ˜Φ r (˜ t ) = d˜ t d t Φ r ( t ). Hence we conclude T vv − T uu = (cid:32) d˜ t d t (cid:33) ( ˜ T vv − ˜ T uu ) = (cid:32) d˜ t d t (cid:33) c π { τ, ˜ t } = − c π (cid:16) { ˜ t, t } − { τ, t } (cid:17) . (4.21)Thus it is clear that where we previously only had a purely outgoing energy-momentumflux, this time there is also an incoming contribution: the bath and black hole exchangeenergy. Using the explicit expression ˜ t ( t ) given in (4.18), we see that the first (incoming)term is a small, constant contribution. Also, the solution (4.18) ensures that (4.21) isconsistent with (4.15). Therefore we conclude that the ADM energy decreases as E = − Φ r πG { τ, t } = E e − At + 12 (cid:16) A (cid:17) Φ r πG , (4.22)and d E d t = − Φ r πG { τ, t } (cid:48) = − c π π β e − At . (4.23)Finally, from the map discussed in section 4.1 we know thatd˜ t d t = 2 π(cid:96)β ( t ) ⇒ β ( t ) = 2 π(cid:96) e A t . (4.24)As also noted in [17], for low evaporation rates, i.e. in the regime A (cid:28) (cid:96) , the evaporationis adiabatic. The length scale Φ r c sets the evaporation time of the black hole. From the Hawking temperature (2.5) we see that the entropy and energy are related via S BH = 2 π (cid:114) E Φ r πG . (4.25)Consider this formula in the tilded coordinate system discussed in section 4.2. We have˜ S BH (˜ t ) = 2 π (cid:115) ˜ E (˜ t ) ˜Φ r (˜ t )4 πG = 2 πβ Φ r G (1 − A t ) , where A cG r . (4.26)– 15 – tS G (Φ + π Φ r β ) Φ G /A S t
Hence this entropy decreases linearly in time ˜ t (remember that β is fixed). In the above, weshould add the extremal entropy S = G Φ . In the untilded coordinate system discussedin 4.4, we find S BH ( t ) = 2 π (cid:114) E ( t )Φ r πG = Φ r G (cid:115) A π β e − At . (4.27)Next, we would like to make contact with our entropy calculations performed in section3. The comparison is most natural in the tilded coordinates. In section 3.2 we found (3.4)and (3.5) before and after the Page time, respectively. We are now interested in the entropyof the entire bath, i.e. we wish to take b →
0. Then before the Page time we find S ˜ t< ˜ t Page = 14 G (cid:18) Φ + 2 log sinh πβ Φ r A t (cid:19) . (4.28)After the Page time we get S ˜ t> ˜ t Page = 14 G (cid:18) Φ + 2 log sinh πβ Φ r (1 − A t ) (cid:19) . (4.29)For high temperatures Φ r (cid:29) β we can approximate this with S ≈ G (cid:16) Φ + Φ r πβ A ˜ t (cid:17) if ˜ t < ˜ t Page14 G (cid:16) Φ + Φ r πβ (1 − A ˜ t ) (cid:17) if ˜ t > ˜ t Page . (4.30)Notice that for ˜ t > ˜ t Page = A − the entropy of the radiation is equal to the entropy of theblack hole in (4.26). The latter is the coarse-grained entropy and follows a Hawking curve.From (4.30) it is clear that we indeed reproduce a Page curve for the radiation entropy;see figure 4. – 16 – ˜ t S ˜ t Figure 5 . The qualitative behavior of the entropy for ∆ ϕ = 2 π A ˜ t (blue) and ∆ ϕ = 2 π (1 − A ˜ t )(orange) shows that it follows the Page curve (red line). To plot we set (cid:96) = G = 1, r ∞ = 10 . ,the evaporation rate A = 5 and R = 2 (cid:96) on the left, R = 10 (cid:96) on the right. The dashed green lineindicates the UV cutoff. If we do not expand for large cutoff, and without zooming in on the regime Φ r (cid:29) β ,we can study the qualitative behavior of the entropy using S = 12 G arcsinh r ∞ R sinh R ϕ , (4.31)where R = π(cid:96) β is the BTZ radius, r ∞ is the cutoff surface, and ∆ ϕ = 2 π A ˜ t before and∆ ϕ = 2 π (cid:16) − A ˜ t (cid:17) after the Page time. This gives figure 5. In this paper, we investigated JT black holes of zero and finite temperature coupled toa bath (a 2D CFT) from a higher-dimensional, geometrical perspective. By performinga partial dimensional reduction from Poincar´e AdS and the BTZ geometry, respectively,we effectively split the three-dimensional spacetime into a two-dimensional black hole anda remainder, of which the holographic dual takes on the role of the bath. The boundaryconditions on the dilaton lead us to identify the renormalized value of the dilaton with theparameter α controlling the dimensional reduction: Φ r = 2 π(cid:96)α . This procedure allowed usto compute the entropy of an interval in the bath/radiation by simply computing geodesiclengths in the three-dimensional spacetime.By making the dimensional reduction parameter α time-dependent, we could modelthe dynamics of the BTZ system and allowed the finite temperature JT black hole toevaporate. From a boundary analysis we showed that the energy decreases linearly intime. The renormalized dilaton is time-dependent and the temperature is fixed. Then,we exploited a mapping of BTZ parameters to connect this linear evaporation to a morecanonical (exponential) evaporation of the energy, in which the renormalized dilaton takes– 17 –n a constant value, and the temperature is time-dependent. Finally, we demonstratedthat the entropy of the radiation/bath system follows a Page curve.We included the extremal black hole mostly as a toy model to study the dimensionalreduction and demonstrate the entropy calculations. We did not include time dependencefor the extremal black hole, because this black hole (which has T H = 0) does not evaporate.A few comments are in place. First, the connection between the dilaton and the black holeentropy naturally arises from our description in a geometrical fashion. This is best seenfor the finite temperature case. From (4.26) we see that the black hole entropy is given bythe value of the dilaton (2.23) at the horizon: S = 14 G Φ (cid:12)(cid:12) hor = 14 G (cid:18) Φ + 2 πβ Φ r (cid:19) . (5.1)Note that the entropy (4.26) is a thermal and coarse-grained entropy; indeed, insertingΦ r (˜ t ) = 2 π(cid:96) (1 − A ˜ t ) does not lead to the Page curve, but instead a linearly decreasing(Hawking) curve. Since we wish to consider the full geometry to be in a pure state, thefine-grained entropy of the JT black hole is equal to that of the radiation (its complement)and thus follows a Page curve as well.As a second comment on our results, note that we did not have to make use of the islandformula to reproduce the Page curve for the radiation. Instead, we naturally find the Pagecurve from the RT prescription, which instructs us to take the minimal length geodesic inthe BTZ geometry. As the interval on the boundary corresponding to the radiation grows,the geodesic ‘jumps’ and its length starts to decrease. In the two-dimensional theory, thisis mirrored by a jump of the quantum extremal surface, thereby including an island inthe generalized entropy calculation. One might like to interpret the region in figure 3(b)bounded by the geodesic and the division between the purple and green regions as theisland. This island lies outside of the horizon, as expected for our adiabatic evaporation.It would be interesting to make this connection more precise.Finally, we would like to point out that the method we have described in this pa-per, i.e. using a partial dimensional reduction to create a black hole and bath withinthe same higher-dimensional system, could in principle be applied to other spacetimes aswell. In particular, it could be worthwhile to apply this procedure to a three-dimensionalSchwarzschild-de Sitter black hole, reducing it to a pure dS spacetime connected to a bath,to see if we can learn more about the (pure) de Sitter horizon and a possible informationparadox. So far, the literature has not been conclusive on the existence of islands in purede Sitter (see e.g. [25–27]), leaving the matter of both the existence as well as the inter-pretation of a Page curve for the de Sitter entropy open for discussion. It could therefore– 18 –rove useful to employ our method — that needs neither quantum extremal surfaces norislands — to add to this discussion. Acknowledgments
It is a pleasure to thank Ben Freivogel, Claire Zukowski, Antonio Rotundo and TheodoraNikolakopoulou for discussions during the initial stage of this project. Our work is sup-ported by the Spinoza grant and the Delta ITP consortium, a program of the NWO thatis funded by the Dutch Ministry of Education, Culture and Science (OCW).
Appendix A Entropy calculations
In this appendix we provide a detailed computation of the entropies (3.2) and (3.4)/(3.5)using the formula for the geodesic distance ∆ s between two points s , s in terms of em-bedding coordinates: − (cid:96) cosh (cid:0) ∆ s/(cid:96) (cid:1) = X µ ( s ) X µ ( s ) . (A.1)The computations are standard [3, 28]; we added them for completeness. A.1 Extremal AdS black hole For the Poincar´e metric, the embedding coordinates are X = z z ( (cid:96) + x − t ) ,X = (cid:96) z t ,X = (cid:96) z x ,X = z − z ( (cid:96) − x + t ) , (A.2)where X and X are timelike, i.e. − ( X ) − ( X ) + ( X ) + ( X ) = − (cid:96) . The geodesicdistance between two points s = ( t , z , x ) and s = ( t , z , x ) is then given bycosh (cid:0) ∆ s/(cid:96) (cid:1) = 12 z z (cid:16) − ( t − t ) + ( x − x ) + z + z (cid:17) . (A.3)For a fixed time slice, the geodesic distance depends only on z , z and ∆ x . If the endpointslie at the same radial distance z = z = z we findcosh (cid:0) ∆ s/(cid:96) (cid:1) = 1 + 12 (cid:18) ∆ xz (cid:19) ⇒ ∆ s = 2 (cid:96) arcsinh ∆ x z . (A.4)For z → (cid:15)(cid:96) with (cid:15) → s = 2 (cid:96) log ∆ x (cid:96) + UV cutoff . (A.5)– 19 –apping the Poincar´e patch to the cylinder using x = (cid:96)ϕ , z = (cid:96) r as in the discussionaround (2.17) we find the entropy of an angular interval ∆ ϕ to be S = 14 G (Φ + 2 log ∆ ϕ ) , (A.6)where we used G (3) = (cid:96)G (2) and absorbed the UV cutoff in Φ . For the angular intervalin figure 1, we have ∆ ϕ = 2 πα + b(cid:96) . Then we can distinguish two regimes b (cid:28) Φ r and b (cid:29) Φ r , leading to S = G (cid:0) Φ + 2 log Φ r /(cid:96) (cid:1) if b (cid:28) Φ r G (Φ + 2 log 2 b/(cid:96) ) if b (cid:29) Φ r , (A.7)where we used Φ r = 2 π(cid:96)α . A.2 Finite temperature AdS black hole For the BTZ metric (2.20) and (2.22) the embedding coordinates are X = (cid:96) R (cid:112) r − R sinh Rt(cid:96) = (cid:96) sinh πβ ( u + v )sinh πβ ( u − v ) ,X = r(cid:96) R cosh Rϕ(cid:96) = cosh πβ (cid:96) ϕ tanh πβ ( u − v ) ,X = r(cid:96) R sinh Rϕ(cid:96) = sinh πβ (cid:96) ϕ tanh πβ ( u − v ) ,X = (cid:96) R (cid:112) r − R cosh Rt(cid:96) = (cid:96) cosh πβ ( u + v )sinh πβ ( u − v ) , (A.8)where again X and X are timelike directions. For two points s = ( t , r , ϕ ) and s = ( t , r , ϕ ) the geodesic distance iscosh (cid:0) ∆ s/(cid:96) (cid:1) = r r R cosh R(cid:96) ( ϕ − ϕ ) − (cid:118)(cid:117)(cid:117)(cid:116)(cid:32) r R − (cid:33) (cid:32) r R − (cid:33) cosh R(cid:96) ( t − t ) , (A.9)such that for a fixed time slice and for two points at the same radius r = r = r one finds1 − cosh (cid:0) ∆ s/(cid:96) (cid:1) = r R (cid:18) − cosh R ∆ ϕ(cid:96) (cid:19) ⇒ ∆ s = 2 (cid:96) arcsinh rR sinh R ∆ ϕ (cid:96) . (A.10)Now, for r → ∞ we can expand to find∆ s = 2 (cid:96) log sinh R ∆ ϕ (cid:96) + UV cutoff , (A.11)leading to an entropy S = 14 G (2) (cid:18) Φ + 2 log sinh πβ (cid:96) ∆ ϕ (cid:19) , (A.12)where we used G (3) = (cid:96)G (2) , R = π(cid:96) β and absorbed the UV cutoff in Φ .– 20 – eferences [1] D.N. Page, Information in black hole radiation , Phys. Rev. Lett. (1993) 3743[ hep-th/9306083 ].[2] D.N. Page, Time Dependence of Hawking Radiation Entropy , JCAP (2013) 028[ ].[3] S. Ryu and T. Takayanagi,
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