Extreme Black Hole Anabasis
EExtreme Black Hole Anabasis
Shahar Hadar, ∗ Alexandru Lupsasca, † and Achilleas P. Porfyriadis
1, 3, ‡ Center for the Fundamental Laws of Nature,Harvard University, Cambridge, MA 02138, USA Princeton Gravity Initiative, Princeton University, Princeton, NJ 08544, USA Black Hole Initiative, Harvard University, Cambridge, MA 02138, USA
Abstract
We study the SL (2) transformation properties of spherically symmetric perturbations of theBertotti-Robinson universe and identify an invariant µ that characterizes the backreaction of theselinear solutions. The only backreaction allowed by Birkhoff’s theorem is one that destroys the AdS × S boundary and builds the exterior of an asymptotically flat Reissner-Nordstr¨om black holewith Q = M (cid:112) − µ/
4. We call such backreaction with boundary condition change an anabasis .We show that the addition of linear anabasis perturbations to Bertotti-Robinson may be thoughtof as a boundary condition that defines a connected
AdS × S . The connected AdS is a nearly- AdS with its SL (2) broken appropriately for it to maintain connection to the asymptotically flatregion of Reissner-Nordstr¨om. We perform a backreaction calculation with matter in the connected AdS × S and show that it correctly captures the dynamics of the asymptotically flat black hole. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - t h ] D ec . INTRODUCTION Birkhoff’s theorem in four dimensions tells us that all spherically symmetric spacetimeswith vanishing Ricci tensor are static and therefore described by the Schwarzschild metric.The theorem extends to the Einstein-Maxwell equations with the Schwarzschild solutionreplaced by the Reissner-Nordstr¨om one. On the other hand, in the Einstein-Maxwell theoryanother spherically symmetric solution of importance is the Bertotti-Robinson universe withmetric given by the direct product of
AdS with a two-sphere. This is consistent withBirkhoff’s theorem because Bertotti-Robinson agrees with the near-horizon of extreme andnear-extreme Reissner-Nordstr¨om, and the theorem is only a local statement. Reissner-Nordstr¨om is asymptotically flat while Bertotti-Robinson is asymptotically
AdS × S .If one considers the linearized Einstein-Maxwell equations around Reissner-Nordstr¨om,then within spherical symmetry one finds a two-parameter family of solutions parametrizedby the change in mass δM and charge δQ relative to the background. In other words,such linearized perturbations are moving towards other Reissner-Nordstr¨om solutions. Onthe other hand, if one considers the linearized Einstein-Maxwell equations around Bertotti-Robinson then one finds a four-parameter family of solutions. Why is that? One may identifytwo of the four parameters with δM ± δQ and derive the corresponding solutions from ap-propriate near-horizon limits of the linearized solutions around Reissner-Nordstr¨om. When δM = δQ this results in a linearized solution around Bertotti-Robinson that respects the SL (2) symmetry associated with the AdS factor of the background. This SL (2)-preservinglinear solution is asymptotically AdS × S and moves towards another Bertotti-Robinson.When δM (cid:54) = δQ the near-horizon limit produces a linear solution around Bertotti-Robinsonthat breaks its SL (2) symmetry. Therefore acting with the background’s SL (2) isometrieswe may obtain two additional linear solutions. The SL (2)-breaking triplet of solutions arenot asymptotically AdS × S and it is well known that the backreaction of these solutionsdestroys the AdS boundary [2]. In the past, this has led to the slogans that “ AdS has nodynamics” or that “ AdS admits no finite energy excitations.” While not incorrect, theseslogans are true only as long as one insists on asymptotically AdS boundary conditions.In this paper, we study the SL (2)-breaking triplet solutions with boundary conditions The precise statement of Birkhoff’s theorem is that a C solution of the Einstein (resp. Einstein-Maxwell)equations which is spherically symmetric in an open set V is locally equivalent to part of the maximallyextended Schwarzschild (resp. Reissner-Nordstr¨om) solution in V (see, e.g., [1]). AdS × S throat to maintain its connection with the exteriorasymptotically flat Reissner-Nordstr¨om. In this context, we first identify an SL (2)-invariantquantity µ associated with any solution in the triplet. Then we show that when µ = 0the corresponding solution may be thought of as beginning to build the asymptotically flatregion of extreme Reissner-Nordstr¨om starting from its AdS × S throat. When µ > Q = M (cid:112) − µ/
4. In other words, when µ ≥ SL (2)-breaking linear solutions makes sense provided we allow the boundary conditionchange that leads to an asymptotically flat nonlinear solution. We call such backreactionwith boundary condition change an anabasis —an adventure of climbing out of the blackhole throat into the weak gravity regime.Next, consider perturbing an extreme Reissner-Nordstr¨om black hole using a sphericallysymmetric infalling matter source with energy-momentum tensor of order (cid:15) (cid:28)
1. Gener-ically, one expects the fully backreacted nonlinear endpoint of this perturbation to be anear-extreme Reissner-Nordstr¨om with Q = M (cid:112) − O ( (cid:15) ) [3]. To leading order in (cid:15) , theinitial and final states only differ in their near-horizon region. More precisely, the near-horizon throat geometry remains locally AdS × S before as well as after the perturbationbut the associated SL (2) symmetry breaking induced by the gluing to the exterior regionis different in the extreme and near-extreme cases. We define the connected AdS × S throat as the geometry obtained by the addition of anabasis perturbations. This may alsobe thought of as a boundary condition for the backreaction calculation. We show that thisleads to a consistent backreaction calculation in AdS × S that captures the dynamics ofthe asymptotically flat black hole.In Section II, we derive the spherically symmetric perturbations of the Bertotti-Robinsonuniverse and study their SL (2) transformation properties. In particular, we identify the in-variant quantity µ associated with each SL (2)-breaking perturbation. Section III singles outstandard Poincar´e and Rindler anabasis perturbations responsible for building the exteriorin extreme and near-extreme Reissner-Nordstr¨om, respectively. Section IV presents the gen-eral transformation from Poincar´e to Rindler AdS × S . In Section V, we do a backreactioncalculation for a pulse of energy (cid:15) in the connected AdS × S throat of a Reissner-Nordstr¨omblack hole. Section VI contains further discussion of our work, especially in relation to theAdS/CFT correspondence and models of two-dimensional dilaton gravity in AdS .3 I. PERTURBATIONS OF BERTOTTI-ROBINSON
The Einstein-Maxwell equations in four dimensions read ( G = c = 1) R µν = 8 πT µν , ∇ µ F µν = 0 , (1)with 4 πT µν = F µρ F ρν − g µν F ρσ F ρσ , and F µν = ∂ µ A ν − ∂ ν A µ . The Bertotti-Robinson universe1 M ds = − r dt + dr r + d Ω , A t = M r , (2)is a spherically symmetric conformally flat exact solution with uniform electromagnetic field F rt = M . Clearly the Bertotti-Robinson metric is a direct product AdS × S . From nowon, we set M = 1 and restore it only when beneficial for clarity.Consider the most general spherically symmetric perturbation h µν = h tt ( t, r ) h tr ( t, r ) 0 0 h rr ( t, r ) 0 0 h θθ ( t, r ) 0 h θθ sin θ ,a µ = (cid:16) a t ( t, r ) a r ( t, r ) 0 0 (cid:17) . (3)The linearized Einstein-Maxwell equations are invariant under the gauge transformations h µν → h µν + L ξ g µν , a µ → a µ + L ξ A µ + ∇ µ Λ , (4)for any vector field ξ and scalar function Λ. Within the spherically symmetric ansatz, wemay use this gauge freedom, with appropriate ξ = ξ t ( t, r ) ∂ t + ξ r ( t, r ) ∂ r , Λ = Λ( t, r ), to set h tt = h rr = a t = 0 , (5)and remove from h tr any addition of the form h tr = c ( r ) + c ( t ) /r for arbitrary c , c . Note,however, that for perturbations around the Bertotti-Robinson solution h θθ is gauge invariant.Therefore, all physical information for perturbations of Bertotti-Robinson is contained in h θθ .We find that the most general solution to the linearized Einstein-Maxwell equationsaround Bertotti-Robinson is given by h θθ = Φ + ar + brt + cr (cid:0) t − /r (cid:1) , (6) h tr = − rt (cid:20) Φ + 2 ar + brt + 23 cr (cid:0) t − /r (cid:1)(cid:21) , (7) f tr = ∂ t a r = h θθ − Φ / . (8)4hus, the spherically symmetric perturbations of Bertotti-Robinson are a four-parameterfamily of solutions parametrized by the constants Φ , a , b , c . A. SL (2) transformations and invariants The background (2) is invariant under the SL (2) transformations associated with the AdS factor: H ( α ) : t → t + α , (9) D ( β ) : t → t/β , r → βr , (10) K ( γ ) : t → t − γ ( t − /r )1 − γt + γ ( t − /r ) , r → r (cid:2) − γt + γ (cid:0) t − /r (cid:1)(cid:3) . (11)Here H ( α ), D ( β ), K ( γ ) are the time translations, dilations, and special conformal trans-formations for real parameters α , β , γ with β >
0. The special conformal coordinatetransformation must be followed by a gauge field transformation A → A + d ln r ( t − /γ )+1 r ( t − /γ ) − .The SL (2) invariance of the background implies that if we act with an SL (2) trans-formation on any of the solutions (6–8) we will obtain another solution to the linearizedEinstein-Maxwell equations around the same background. Note, however, that the SL (2)transformations do not necessarily preserve the gauge (5). Fortunately, as we have pre-viously emphasized, h θθ is gauge invariant and therefore uniquely labels each physicallydistinct solution in every gauge.The four-parameter solution (6) consists of an SL (2)-invariant solution Φ together withthe SL (2)-breaking triplet Φ = ar + brt + cr (cid:0) t − /r (cid:1) . (12)Clearly, the SL (2)-invariant solution Φ corresponds to a rescaling of (2) by M → M + δM with Φ = 2 M δM . In the remainder of the paper, we will focus on the SL (2)-breakingsolutions Φ.The action of the SL (2) transformations on Φ is given by H ( α ) : a → a + bα + cα , b → b + 2 cα , c → c , (13) D ( β ) : a → aβ , b → b , c → c/β , (14) K ( γ ) : a → a , b → b − aγ , c → c − bγ + aγ . (15)5sing the above, we identify the following SL (2) invariant µ = b − ac . (16)Moreover, we note that for µ < a = sgn c (cid:54) = 0 being an additional SL (2)invariant, while for µ = 0 it is sgn( a + c ) that is also SL (2)-invariant.Before we end this section, let us fix two standard choices for the general solution (12).For µ > SL (2) transformation that will setΦ = −√ µ rt , µ > . (17)Similarly, for µ = 0 and sgn( a + c ) = 1 one may setΦ = 2 r , µ = 0 , sgn( a + c ) = 1 . (18)Specifically, when µ > −√ µ rt by acting on (12) with the followingseries of SL (2) transformations: H (cid:18) a √ b − ac (cid:19) ◦ K (cid:18) b + √ b − ac a (cid:19) , for a (cid:54) = 0 , (19) K ( γ ) ◦ H (1 /γ ) ◦ K ( γ ) ◦ K (cid:16) cb (cid:17) , for a = 0 , b > , (20) K (cid:16) cb (cid:17) , for a = 0 , b < . (21)Likewise, when µ = 0 we may get to Φ = sgn( a + c ) 2 r by acting on (12) with the followingseries of SL (2) transformations: D (cid:18) | a | (cid:19) ◦ K (cid:18) b a (cid:19) , for abc (cid:54) = 0 , (22) D (cid:18) γ | c | (cid:19) ◦ K (1 /γ ) ◦ H ( γ ) , for b = a = 0 , (23) D (cid:18) | a | (cid:19) , for b = c = 0 . (24) III. ANABASIS AND THE CONNECTED THROAT
The Bertotti-Robinson solution (2) may be derived from near-horizon near-extremalityscalings of the Reissner-Nordstr¨om black hole solution of mass M and charge Q , withouter/inner horizons at r ± = M ± (cid:112) M − Q , ds = − (cid:18) − M ˆ r + Q ˆ r (cid:19) d ˆ t + (cid:18) − M ˆ r + Q ˆ r (cid:19) − d ˆ r + ˆ r d Ω , ˆ A ˆ t = − Q ˆ r . (25)6his implies that the leading corrections in these scaling limits are, by construction, solu-tions of the linearized Einstein-Maxwell equations around Bertotti-Robinson. There are twoessentially distinct scaling limits of the black hole exterior that yield the Bertotti-Robinsonsolution.The first scaling limit is most simply described by setting Q = M , making the coordinateand gauge transformation, r = ˆ r − MλM , t = λ ˆ tM , A = ˆ A + d ˆ t , (26)to obtain1 M ds = − (cid:18) r λr (cid:19) dt + (cid:18) r λr (cid:19) − dr + (1 + λr ) d Ω , A t = M r λr , (27)and then taking the limit λ →
0. At order O (1) this produces exactly (2). The leadingcorrection is of order O ( λ ) and it is given by h tt = 2 r , h rr = 2 /r , h θθ = 2 r , f rt = − r . (28)By construction, this solves the linearized Einstein-Maxwell equations around (2).Comparing the gauge invariant h θθ in the above with (6) we see that this is the Φ = 2 r solution. Hence the SL (2)-breaking µ = 0 solution Φ = 2 r may be thought of as beginningto build the asymptotically flat region of an extreme Reissner-Nordstr¨om starting from itsnear-horizon Bertotti-Robinson throat. In other words, the nonlinear solution obtained fromthe µ = 0 perturbation of AdS × S , when backreaction is fully taken into account in theEinstein-Maxwell theory, is the extreme Reissner-Nordstr¨om black hole.The second scaling limit is described by setting Q = M √ − λ κ , making the coordinateand gauge transformation, ρ = ˆ r − r + λr + , τ = λ ˆ tM , A = ˆ A + d ˆ t , (29)to obtain1 M ds = − ρ ( ρ + 2 κ + λκρ )(1 + λκ )(1 + λρ ) dτ + (1 + λκ ) (1 + λρ ) ρ ( ρ + 2 κ + λκρ ) dρ + (1 + λκ ) (1 + λρ ) d Ω ,A τ = Mλ (cid:32) − (cid:114) − λκ λκ
11 + λρ (cid:33) , (30) Indeed, one may align (28) with the a = 2, b = c = Φ = 0 solution (6–8) by adjusting the gauge via (4)with ξ = 2 rt∂ t − r ∂ r , Λ = 0. λ →
0. At order O (1) this produces1 M ds = − ρ ( ρ + 2 κ ) dτ + dρ ρ ( ρ + 2 κ ) + d Ω , A τ = M ( ρ + κ ) . (31)The leading correction is of order O ( λ ) and it is given by h ττ = 2 ρ ( ρ + κ ) , h ρρ = 2 ( ρ + 3 κ + 3 κρ ) ρ ( ρ + 2 κ ) , h θθ = 2( ρ + κ ) , f ρτ = − ρ − κ . (32)By construction, this solves the linearized Einstein-Maxwell equations around (31).Locally, the O (1) results of the two scaling limits we have considered [Eqs. (2) and (31)]are diffeomorphic—they are both the Bertotti-Robinson universe. Indeed, the coordinatetransformation [4] τ = − κ ln (cid:0) t − /r (cid:1) , ρ = − κ (1 + rt ) , (33)together with A → A − d Λ , Λ = ln rt − rt +1 = − ln ρρ +2 κ maps (31) to (2). Globally, onthe Penrose diagram of AdS × S , the coordinates in (2) cover a Poincar´e patch whilethe coordinates in (31) cover a Rindler patch. The transformation (33) situates the twopatches relative to each other as shown in Fig. 1. It follows that under this transformationthe leading O ( λ ) correction (32) transforms to a solution of the linearized Einstein-Maxwellequations around (2). Comparing the gauge invariant h θθ = 2( ρ + κ ) = − κrt with (6),we see that this is the Φ = − κrt solution. Hence the SL (2)-breaking √ µ = 2 κ solutionΦ = − κrt may be thought of as beginning to build the asymptotically flat region of anear-extreme Reissner-Nordstr¨om starting from its near-horizon Bertotti-Robinson throat.In other words, the nonlinear solution obtained from the µ > AdS × S ,when backreaction is fully taken into account in the Einstein-Maxwell theory, is the near-extreme Reissner-Nordstr¨om black hole with Q = M (cid:112) − µ/ r and Φ = 2( ρ + κ ) that begin to build the asymptotically flat black hole exteriors are positiveΦ > . (34)Intuitively, this is because Φ measures the increase in the size of the S as one climbs out ofa black hole’s throat towards its asymptotically flat region. In particular, notice that whenthe Rindler anabasis solution Φ = 2( ρ + κ ) is mapped to Φ = − κrt via (33), this leads tothe range rt ≤ − = 0 r = r = r = ∞ ρ = ρ = ρ = ∞ FIG. 1. Penrose diagram of
AdS × S with relative placement of the Poincar´e patch (2) and theRindler patch (31) according to the transformation (33). IV. GENERAL POINCAR´E TO RINDLER TRANSFORMATION
The transformation between the Poincar´e and Rindler backgrounds used in the previoussection [Eq. (33)] maps the Rindler anabasis solution h θθ = 2( ρ + κ ) to the µ > −√ µ rt in Poincar´e coordinates (17) with √ µ = 2 κ . As explained in Section II A, thestandard Φ = −√ µ rt solution is related by SL (2) transformations to any SL (2)-breakingsolution Φ (12) with the same µ >
0. As a result, there is a two-parameter generalizationof the standard Poincar´e to Rindler transformation (33) that can map the h θθ = 2( ρ + κ )Rindler anabasis solution to a general Poincar´e solution Φ (12) with √ µ = 2 κ .On the Penrose diagram of AdS × S , the two-parameter generalization of Fig. 1 allowsfor the Rindler patch to have arbitrary vertical location and size with respect to the Poincar´e9ne. This is shown in Fig. 2. The general coordinate transformation is t = (cid:0) ν (cid:1) e κτ (cid:112) ρ ( ρ + 2 κ ) (cid:16) ρ + κ + ψe κτ (cid:112) ρ ( ρ + 2 κ ) (cid:17)(cid:16) ρ + κ + ψe κτ (cid:112) ρ ( ρ + 2 κ ) (cid:17) − κ − ν ,r = 1(1 + ν ) κ (cid:16) e − κτ (cid:112) ρ ( ρ + 2 κ ) (cid:0) ψ e κτ (cid:1) + 2 ψ ( ρ + κ ) (cid:17) , (35)accompanied by A → A + d Λ,Λ = −
12 ln ρρ + 2 κ + ln ρ + ψe κτ (cid:112) ρ ( ρ + 2 κ ) ρ + 2 κ + ψe κτ (cid:112) ρ ( ρ + 2 κ )= 12 ln [ r ( t + ν ) −
1] [ ψr ( t + ν ) − (1 + ν ) r − ψ ][ r ( t + ν ) + 1] [ ψr ( t + ν ) − (1 + ν ) r + ψ ] , (36)with ψ = ν − χ ≥
0. The derivation of this general transformation is in Appendix A. t = − νt = νχν − χ r = r = r = ∞ ρ = ρ = ρ = ∞ FIG. 2. Penrose diagram of
AdS × S with relative placement of the Poincar´e patch (2) and theRindler patch (31) according to the transformation (35). Using the above general transformation we may ask again: when is it possible to map aPoincar´e solution Φ = ar + brt + cr ( t − /r ) to the Rindler anabasis solution Φ = 2( ρ + κ )?We find that the answer is again: when and only when µ = b − ac >
0. For µ > κ = √ b − ac/ √ µ/ , (37)10nd ν = b − √ b − ac c , χ = a + c √ b − ac , for c < , (38) ν = a/b , χ = a/b , for c = 0 , b > , (39) ν = + ∞ , χ = − a/b , for c = 0 , b < . (40)Notice that the above does not include any solutions with c >
0. The reason is the following.As noted in (34), only solutions with Φ > c >
0, near the boundary r → ∞ , the general Poincar´e solutionis given by Φ ≈ r ( a + bt + ct ), which is positive in a portion of the boundary that has theform ( −∞ , t ) ∪ ( t , + ∞ ). As a result, no µ > c > ρ + κ ) by a single transformation (35–36) everywherenear the boundary. That said, if we allow for topology change when c >
0, it may be possibleto perform anabasis to two separate asymptotically flat regions.
V. BACKREACTION CALCULATION IN THE CONNECTED THROAT
In the previous sections, we have seen that the
AdS backreaction problem for an SL (2)-breaking electrovacuum solution Φ in four-dimensional Einstein-Maxwell theory is consistentas long as one does not insist on maintaining AdS boundary conditions but rather considerssuch linear solutions as black hole anabasis solutions that build the asymptotically flatregions of extreme (for µ = 0) and near-extreme (for µ >
0) Reissner-Nordstr¨om. In thissection, we show that the addition of anabasis perturbations to
AdS may also be thoughtof as a boundary condition for a connected AdS . The connected AdS is a nearly- AdS with its SL (2) broken appropriately for it to maintain connection to the asymptotically flatregion of a Reissner-Nordstr¨om black hole.Consider, for example, throwing a matter pulse of energy (cid:15) > AdS throat of an extreme Reissner-Nordstr¨om. That is, add to the right hand side of the Einsteinequation in (1) the matter energy-momentum tensor8 πT matter vv = (cid:15)δ ( v − v ) , v = t − /r . (41)For (cid:15) (cid:28) O ( (cid:15) ) metric and gauge field perturbation around the Bertotti-11obinson universe that generalizes the solution (6–8) according to h θθ = Φ + ar + brt + cr (cid:0) t − /r (cid:1) − (cid:15) r ( t − v ) − r Θ( v − v ) , (42) h tr = − rt (cid:20) Φ + 2 ar + brt + 23 cr (cid:0) t − /r (cid:1)(cid:21) (43)+ (cid:15) r ( t − v ) − r ( t − v ) + 6 ln( r ( t − v )) + 83 r Θ( v − v ) ,f tr = ∂ t a r = h θθ − Φ / . (44)Before the pulse, for v < v , we impose the causal boundary condition for a connected AdS throat given by the µ = 0 Poincar´e anabasis solution Φ = 2 r , h θθ = 2 r , for v < v . (45)That is to say, we set a = 2 and b = c = Φ = 0. Then after the pulse, for v > v , we get h θθ = 2 r − (cid:15) r ( t − v ) − r , for v > v . (46)This solution after the pulse is a µ = 4 (cid:15) solution which, using the results from Sec. IV, mapsto the Rindler anabasis solution h θθ = 2( ρ + κ ) via (35–36) with κ = √ (cid:15) , ν = 2 √ (cid:15) − v , χ = 1 √ (cid:15) − v √ (cid:15) . (47)We thus see that we have a backreaction calculation in the connected AdS throat that isconsistent with the expectation from the physics of Reissner-Nordstr¨om: Throwing a pulseof energy (cid:15) (cid:28) Q = M “shifts the horizon” and the blackhole becomes near extreme with Q = M √ − (cid:15) . This is shown in Fig. 3. VI. DISCUSSION
In this paper, we have studied backreaction in the context of
AdS × S connected toan asymptotically flat region in four-dimensional Einstein-Maxwell theory. We imposedspherical symmetry but considered both electrovacuum solutions as well as a matter sourcein the form of a null ingoing pulse. We have seen that backreaction with boundary conditionchange, which we call anabasis, is consistent with Reissner-Nordstr¨om physics.In AdS/CFT, anabasis is dual to following the inverse renormalization group flow, fromIR to UV, for an appropriate irrelevant deformation of the boundary field theory that does12 = v t = v + √ (cid:15) v = v FIG. 3. Penrose diagram of
AdS × S . The dashed boundary signifies that this AdS is maintainingconnection with the exterior of Reissner-Nordstr¨om. An ingoing pulse of energy (cid:15) enters the extremethroat (Poincar´e patch) at t = v . The black hole horizon shifts and the throat becomes near-extreme (shaded Rindler patch). not respect AdS boundary conditions. This is not something discussed very often in theAdS/CFT literature for at least two reasons. First, it is a difficult question to study sys-tematically because it is hard to identify appropriate solvable irrelevant deformations ofCFTs. Second, it runs somewhat contrary to the spirit of AdS/CFT which is a completeself-contained theory in itself—a theory in which even when one studies irrelevant deforma-tions, one may wish to restrict oneself to deformations which do not destroy the boundaryof AdS. Historically, of course, AdS/CFT was discovered by a low-energy near-horizon limitfrom string theory in asymptotically flat spacetime. A recent body of work that carries outan anabasis by following a flow for a single-trace irrelevant deformation of a CFT , whichgoes under the name T T and changes
AdS asymptotics to flat with a linear dilaton, maybe found in [5–8]. A different double-trace
T T deformation of CFT has been holographically interpreted as a gravitationaltheory in an AdS that is cut off at a finite interior surface [9] (see also [10]). This is not related toanabasis as it may be obtained from mixed boundary conditions that respect the AdS boundary [11]. AdS anabasis studied in this paper do not rely on the ex-istence of a holographic dual and are expected to be readily generalizable to a wide classof theories with (near-)extreme black holes which universally exhibit AdS -like near-horizongeometries [12]. This includes rotating black holes such as Kerr which near extremality has athroat geometry, the Near-Horizon-Extreme-Kerr (NHEK) solution [13], with backreactionproperties similar to AdS × S [14, 15]. In Appendix B, we give the SL (2)-breaking triplet oflinear perturbations of NHEK that generalizes (6–8). Beyond near-horizon approximations, AdS makes an appearance in other contexts where approximate spacetime decoupling oc-curs, such as the interaction region of colliding shock electromagnetic plane waves [16], ornear certain highly localized matter distributions [17]. The ideas in this paper may also berelevant in such contexts.A model of two-dimensional dilaton gravity in AdS that is solvable with backreaction,as well as with the addition of matter, is the Jackiw-Teitelboim (JT) theory [18, 19]. Thismodel captures many of the universal aspects in the spherically symmetric sector of higherdimensional gravity near extreme black hole horizons, and it has been studied extensivelyfrom the holographic perspective beginning with [20–23]. In JT theory, the geometry isfixed to being locally AdS but the SL (2) is broken by a dilaton Φ JT . Comparing withour gravitational perturbations, we may identify Φ = Φ JT , noting that (12) solves the JTequation of motion ∇ µ ∇ ν Φ JT − g µν ∇ Φ JT + g µν Φ JT = 0 for AdS in Poincar´e coordinates.This is because in the ansatz (3) we have Φ measuring the variation in the size of the S and,in this ansatz, dimensional reduction of higher dimensional gravity down to two dimensionsis known to lead to JT theory with the dilaton Φ JT measuring precisely this variation (see e.g[24–26]). Continuing the comparison, the SL (2)-invariant µ defined in (16) may be identifiedwith the ADM mass of the 2D black holes in JT theory. It follows that the mass of the 2D AdS black hole in JT is the deviation from extremality of the 4D Reissner-Nordstr¨om inEinstein-Maxwell. A comment is in order here. It is often said in the literature that JT isa nearly- AdS theory with the “nearly” part, which is due to the dilaton’s breaking of the SL (2) symmetry of AdS , associated with a departure from extremality. As we have seen inthis paper, however, this is not necessarily so because for µ = 0 the SL (2) may be brokenonly in order to build the exterior of an exactly extreme Reissner-Nordstr¨om.The connected AdS × S , which we defined in Section V in order to perform a consistentReissner-Nordstr¨om backreaction calculation, is nearly- AdS in the sense that its SL (2) has14een broken by the addition of anabasis perturbations that make this AdS an approximateone. We also saw that this SL (2) breaking may be thought of as a choice of boundarycondition for the backreaction calculation. A comprehensive study of various boundaryconditions for the JT theory has been carried out in [27]. However, it appears that none ofthe boundary conditions contained therein would yield an anabasis as they at best correspondto mixed boundary conditions that do not destroy the AdS boundary (of the double-tracetype in AdS/CFT terms). On the other hand, the boundary term used in [28] for what iscalled therein “permeable boundary conditions” appears to be a better candidate for defininga connected AdS in JT theory. Indeed, matching fields across the AdS × S boundary inReissner-Nordstr¨om, as in the calculations of [29, 30], necessitates boundary conditions thatare “leaky” from the AdS point of view.Broadening the pulse used in Section V and replacing it with a finite-width wavepacket,one may arrange to have such a wavepacket enter the AdS × S region by sending in lowenergy waves from past null infinity in Reissner-Nordstr¨om. The calculation may be set upusing matched asymptotic expansions and features leaky boundary conditions [31].Generically, the backreaction of an extreme Reissner-Nordstr¨om black hole results, as inSection V, in a near-extreme one. However, there is a notable exception. In [3] it was foundthat there exist fine-tuned initial data for a massless scalar perturbing extreme Reissner-Nordstr¨om, for which an instability of the scalar field at the event horizon persists forarbitrarily long evolution and leads to a spacetime that may be thought of as a dynamicalextreme black hole. It was observed that, at late times, this dynamical extreme black holehas the same exterior as extreme Reissner-Nordstr¨om but differs from it at the horizon. In[32] the instability of the perturbing massless scalar on extreme Reissner-Nordstr¨om wasanalyzed using the symmetries of its AdS × S throat. It would be interesting to study thedynamical extreme black hole of [3] using a connected AdS × S as defined in this paper. ACKNOWLEDGMENTS
We thank Andrew Strominger for helpful discussions. This work was supported by theBlack Hole Initiative at Harvard University, which is funded by grants from the John Tem-pleton Foundation and the Gordon and Betty Moore Foundation. SH and AL gratefullyacknowledge support from the Jacob Goldfield Foundation. Funding for shared facilities15sed in this research was provided by NSF Grant No. 1707938.
Appendix A: Derivation of the transformation in Sec. IV
In this appendix, we give some details about the derivation of the general Poincar´e toRindler transformation (35–36) in Sec. IV. In particular, we show how this general transfor-mation may be derived by composing a global
AdS time translation with a Poincar´e timetranslation of the basic transformation (33).The global AdS coordinates, with ds = ( − dη + dσ ) / sin σ , are related to the Poincar´ecoordinates via t ± r = tan (cid:0) η ± σ (cid:1) . The translation η → η + η then moves the Poincar´epatch vertically on the Penrose diagram of AdS as shown in the left panel of Fig. 4. In theoverlapping region, with ds = − r dt + dr /r = − ˜ r d ˜ t + d ˜ r / ˜ r , the relation between thetwo different sets of Poincar´e coordinates reads t = − (cid:0) ν (cid:1) ˜ r (cid:0) ˜ t − ν (cid:1) ˜ r (cid:0) ˜ t − ν (cid:1) − − ν ,r = 11 + ν ˜ r (cid:0) ˜ t − ν (cid:1) − r , (A1)with ν = cot( η / A → A + d Λ , Λ = ln r ( t + ν ) − r ( t + ν )+1 .The resizing of the Rindler patch situated as in Fig. 1 is achieved by a translation of thePoincar´e time coordinate in the transformation (33). Specifically, the transformation˜ t = − e − κτ ρ + κ (cid:112) ρ ( ρ + 2 κ ) + χ , ˜ r = 1 κ e κτ (cid:112) ρ ( ρ + 2 κ ) , (A2)together with A → A + d Λ , Λ = ln ˜ r ( ˜ t − χ ) − r ( ˜ t − χ ) +1 implements the mapping shown in the rightpanel of Fig. 4.The transformation (35–36) is a composition of (A1) with (A2). Note that one musthave ν ≥ χ in order for the Rindler patch to be contained entirely inside the Poincar´e onecovered by the ( t, r ) coordinates. 16 = − ν ˜ t = ν r = r = r = ∞ ˜ r = ˜ r = ˜ r = ∞ ˜ t = χ ˜ r = ˜ r = ˜ r = ∞ ρ = ρ = ρ = ∞ FIG. 4. Penrose diagrams of
AdS × S . Left: The two Poincar´e patches are related by a globaltime translation according to the transformation (A1). Right: The Poincar´e and Rindler patchesare situated according to the transformation (A2). Appendix B: SL (2)-breaking NHEK perturbations The Near-Horizon of Extreme-Kerr (NHEK) was obtained in [13] by a scaling limit λ → O (1) NHEK metric is given by ds = 2 M Γ (cid:20) − r dt + dr r + dθ + Λ ( dφ + rdt ) (cid:21) , (B1)Γ( θ ) = 1 + cos θ , Λ( θ ) = 2 sin θ θ . (B2)NHEK has an isometry group given by SL (2) × U (1), where the U (1) is due to axial symmetryand the SL (2) is again associated with the AdS part in the above.An SL (2)-breaking triplet of axially symmetric linear perturbations of NHEK that gen-17ralizes (6–8) is given by h tt = 4Γ ∂ t Φ + r (2Γ − (cid:0) (cid:1) Φ + 4 rA t ΓΛ , h tr = 2 rA r ΓΛ , (B3) h tφ = Φ r (2Γ −
1) Λ + 2 A t ΓΛ , h rr = Φ r , h rφ = 2 A r ΓΛ , (B4) h θθ = Φ , h φφ = Φ (2Γ −
1) Λ , (B5)with A t = − r Γ Λ ∂ r Φ , A r = 1 r (cid:18) − Γ Λ (cid:19) ∂ t Φ , (B6)and Φ as in (12). We derived the above as follows. First, we obtained the anabasis solution,corresponding to Φ = 2 r , from the O ( λ ) term in the expansion of [13]. Then, we applied SL (2) transformations to generate the triplet while adjusting the gauge for clarity. We notethat for NHEK perturbations, even within axial symmetry, h θθ is not gauge-invariant.We expect the anabasis of these three perturbations towards Kerr to proceed in a similarfashion to the analysis carried out above for Reissner-Nordstr¨om. However, it is worthemphasizing that there is no analog of Birkhoff’s theorem for axisymmetric spacetimes andthat there exist axisymmetric propagating gravitational wave perturbations of Kerr andNHEK. These are typically studied in the Newman–Penrose formalism as in [14, 15]. In[33] an attempt was made to find NHEK perturbations using a metric ansatz judiciouslypicked to accommodate the anabasis perturbation to (near-)extreme Kerr. Unfortunately,the solutions found in [33] that go beyond the above triplet are singular at the poles θ = 0 , π . Finally, note that in another gauge, our solution triplet takes the simple form h µν = 2 M Γ r ΦΓ h tr Φ(2 − Γ) r Λ , (B7) h tr = 13 rt (cid:20) ar + brt + 23 cr (cid:0) t + 9 /r (cid:1)(cid:21) . (B8) [1] S. Hawking and G. Ellis, The Large Scale Structure of Space-Time . Cambridge Monographson Mathematical Physics. Cambridge University Press, 2, 2011. In their notation, Ψ diverges at the poles unless χ = Φ, in which case their SL (2)-breaking solution reducesto the above triplet.
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