Extreme Black Hole Anabasis

We study the $\mathsf{SL}(2)$ transformation properties of spherically symmetric perturbations of the Bertotti-Robinson universe and identify an invariant $\mu$ that characterizes the backreaction of these linear solutions. The only backreaction allowed by Birkhoff's theorem is one that destroys the $AdS_2\times S^2$ boundary and builds the exterior of an asymptotically flat Reissner-Nordstr\"om black hole with $Q=M\sqrt{1-\mu/4}$. We call such backreaction with boundary condition change an anabasis. We show that the addition of linear anabasis perturbations to Bertotti-Robinson may be thought of as a boundary condition that defines a connected $AdS_2\times S^2$. The connected $AdS_2$ is a nearly-$AdS_2$ with its $\mathsf{SL}(2)$ broken appropriately for it to maintain connection to the asymptotically flat region of Reissner-Nordstr\"om. We perform a backreaction calculation with matter in the connected $AdS_2\times S^2$ and show that it correctly captures the dynamics of the asymptotically flat black hole.


I. INTRODUCTION
Birkhoff's theorem in four dimensions tells us that all spherically symmetric spacetimes with vanishing Ricci tensor are static and therefore described by the Schwarzschild metric.
The theorem extends to the Einstein-Maxwell equations with the Schwarzschild solution replaced by the Reissner-Nordström one. On the other hand, in the Einstein-Maxwell theory another spherically symmetric solution of importance is the Bertotti-Robinson universe with metric given by the direct product of AdS 2 with a two-sphere. This is consistent with Birkhoff's theorem because Bertotti-Robinson agrees with the near-horizon of extreme and near-extreme Reissner-Nordström, and the theorem is only a local statement. When δM = δQ the near-horizon limit produces a linear solution around Bertotti-Robinson that breaks its SL(2) symmetry. Therefore acting with the background's SL(2) isometries we may obtain two additional linear solutions. The SL(2)-breaking triplet of solutions are not asymptotically AdS 2 × S 2 and it is well known that the backreaction of these solutions destroys the AdS 2 boundary [2]. In the past, this has led to the slogans that "AdS 2 has no dynamics" or that "AdS 2 admits no finite energy excitations." While not incorrect, these slogans are true only as long as one insists on asymptotically AdS 2 boundary conditions.
In this paper, we study the SL(2)-breaking triplet solutions with boundary conditions 1 The precise statement of Birkhoff's theorem is that a C 2 solution of the Einstein (resp. Einstein-Maxwell) equations which is spherically symmetric in an open set V is locally equivalent to part of the maximally extended Schwarzschild (resp. Reissner-Nordström) solution in V (see, e.g., [1]). that allow the near-horizon AdS 2 × S 2 throat to maintain its connection with the exterior asymptotically flat Reissner-Nordström. In this context, we first identify an SL(2)-invariant quantity µ associated with any solution in the triplet. Then we show that when µ = 0 the corresponding solution may be thought of as beginning to build the asymptotically flat region of extreme Reissner-Nordström starting from its AdS 2 × S 2 throat. When µ > 0 the corresponding solution is beginning to build the asymptotically flat region of near-extreme Reissner-Nordström with Q = M 1 − µ/4. In other words, when µ ≥ 0 the backreaction of these SL(2)-breaking linear solutions makes sense provided we allow the boundary condition change that leads to an asymptotically flat nonlinear solution. We call such backreaction with boundary condition change an anabasis-an adventure of climbing out of the black hole throat into the weak gravity regime.
Next, consider perturbing an extreme Reissner-Nordström black hole using a spherically symmetric infalling matter source with energy-momentum tensor of order 1. Generically, one expects the fully backreacted nonlinear endpoint of this perturbation to be a near-extreme Reissner-Nordström with Q = M 1 − O( ) [3]. To leading order in , the initial and final states only differ in their near-horizon region. More precisely, the nearhorizon throat geometry remains locally AdS 2 × S 2 before as well as after the perturbation but the associated SL(2) symmetry breaking induced by the gluing to the exterior region is different in the extreme and near-extreme cases. We define the connected AdS 2 × S 2 throat as the geometry obtained by the addition of anabasis perturbations. This may also be thought of as a boundary condition for the backreaction calculation. We show that this leads to a consistent backreaction calculation in AdS 2 × S 2 that captures the dynamics of the asymptotically flat black hole.
In Section II, we derive the spherically symmetric perturbations of the Bertotti-Robinson universe and study their SL(2) transformation properties. In particular, we identify the invariant quantity µ associated with each SL(2)-breaking perturbation. Section III singles out standard Poincaré and Rindler anabasis perturbations responsible for building the exterior in extreme and near-extreme Reissner-Nordström, respectively. Section IV presents the general transformation from Poincaré to Rindler AdS 2 × S 2 . In Section V, we do a backreaction calculation for a pulse of energy in the connected AdS 2 ×S 2 throat of a Reissner-Nordström black hole. Section VI contains further discussion of our work, especially in relation to the AdS/CFT correspondence and models of two-dimensional dilaton gravity in AdS 2 .

II. PERTURBATIONS OF BERTOTTI-ROBINSON
The Einstein-Maxwell equations in four dimensions read (G = c = 1) 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 2 × S 2 . From now on, we set M = 1 and restore it only when beneficial for clarity.
Consider the most general spherically symmetric perturbation The linearized Einstein-Maxwell equations are invariant under the gauge transformations for any vector field ξ and scalar function Λ. Within the spherically symmetric ansatz, we may use this gauge freedom, with appropriate ξ = ξ t (t, r)∂ t + ξ r (t, r)∂ r , Λ = Λ(t, r), to set and remove from h tr any addition of the form h tr = c 1 (r) + c 2 (t)/r for arbitrary c 1 , c 2 . 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 equations around Bertotti-Robinson is given by Thus, the spherically symmetric perturbations of Bertotti-Robinson are a four-parameter family of solutions parametrized by the constants Φ 0 , a, b, c.

A. SL(2) transformations and invariants
The background (2) is invariant under the SL(2) transformations associated with the AdS 2 factor: Here H(α), D(β), K(γ) are the time translations, dilations, and special conformal transformations for real parameters α, β, γ with β > 0. The special conformal coordinate transformation must be followed by a gauge field transformation A → A + d ln r(t−1/γ)+1 r(t−1/γ)−1 . The SL(2) invariance of the background implies that if we act with an SL(2) transformation on any of the solutions (6-8) we will obtain another solution to the linearized Einstein-Maxwell equations around the same background. Note, however, that the SL (2) transformations do not necessarily preserve the gauge (5). Fortunately, as we have previously emphasized, h θθ is gauge invariant and therefore uniquely labels each physically distinct solution in every gauge.
The four-parameter solution (6) consists of an SL(2)-invariant solution Φ 0 together with the SL(2)-breaking triplet Clearly, the SL(2)-invariant solution Φ 0 corresponds to a rescaling of (2) by M → M + δM In the remainder of the paper, we will focus on the SL(2)-breaking The action of the SL(2) transformations on Φ is given by Using the above, we identify the following SL(2) invariant Moreover, we note that for µ < 0 we have sgn a = sgn c = 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 > 0 one may always find an SL(2) transformation that will set Similarly, for µ = 0 and sgn(a + c) = 1 one may set Specifically, when µ > 0 we may get to Φ = − √ µ rt by acting on (12) with the following series of SL(2) transformations: Likewise, when µ = 0 we may get to Φ = sgn(a + c) 2r by acting on (12) with the following series of SL(2) transformations:

III. ANABASIS AND THE CONNECTED THROAT
The Bertotti-Robinson solution (2) may be derived from near-horizon near-extremality scalings of the Reissner-Nordström black hole solution of mass M and charge Q, with outer/inner horizons at r ± = M ± M 2 − Q 2 , tions of the linearized Einstein-Maxwell equations around Bertotti-Robinson. There are two essentially distinct scaling limits of the black hole exterior that yield the Bertotti-Robinson solution.
The first scaling limit is most simply described by setting Q = M , making the coordinate and gauge transformation, to obtain and then taking the limit λ → 0. At order O(1) this produces exactly (2). The leading correction is of order O(λ) and it is given by 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 = 2r solution. 2 Hence the SL(2)-breaking µ = 0 solution Φ = 2r may be thought of as beginning to build the asymptotically flat region of an extreme Reissner-Nordström starting from its near-horizon Bertotti-Robinson throat. In other words, the nonlinear solution obtained from the µ = 0 perturbation of AdS 2 × S 2 , when backreaction is fully taken into account in the Einstein-Maxwell theory, is the extreme Reissner-Nordström black hole.
The second scaling limit is described by setting Q = M √ 1 − λ 2 κ 2 , making the coordinate and gauge transformation, to obtain and then taking the limit λ → 0. At order O(1) this produces The leading correction is of order O(λ) and it is given by 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 coordinate together with A → A − dΛ , Λ = 1 2 ln rt−1 rt+1 = − 1 2 ln ρ ρ+2κ maps (31) to (2). Globally, on the Penrose diagram of AdS 2 × S 2 , the coordinates in (2)   and Φ = 2(ρ + κ) that begin to build the asymptotically flat black hole exteriors are positive Intuitively, this is because Φ measures the increase in the size of the S 2 as one climbs out of a black hole's throat towards its asymptotically flat region. In particular, notice that when the Rindler anabasis solution Φ = 2(ρ + κ) is mapped to Φ = −2κrt via (33), this leads to the range rt ≤ −1 shown in Fig. 1. Rindler anabasis solution to a general Poincaré solution Φ (12) with √ µ = 2κ.
On the Penrose diagram of AdS 2 × S 2 , the two-parameter generalization of Fig. 1 allows for the Rindler patch to have arbitrary vertical location and size with respect to the Poincaré one. This is shown in Fig. 2. The general coordinate transformation is t = 1 + ν 2 e κτ ρ(ρ + 2κ) ρ + κ + ψe κτ ρ(ρ + 2κ) accompanied by A → A + dΛ, with ψ = ν − χ ≥ 0. The derivation of this general transformation is in Appendix A. Using the above general transformation we may ask again: when is it possible to map a Poincaré solution Φ = ar +brt+cr (t 2 − 1/r 2 ) to the Rindler anabasis solution Φ = 2(ρ+κ)?
We find that the answer is again: when and only when µ = b 2 − 4ac > 0. For µ > 0 we find that the parameter identification is Notice that the above does not include any solutions with c > 0. The reason is the following.
As noted in (34) For 1 we may find an O( ) metric and gauge field perturbation around the Bertotti-Robinson universe that generalizes the solution (6-8) according to Before the pulse, for v < v 0 , we impose the causal boundary condition for a connected AdS 2 throat given by the µ = 0 Poincaré anabasis solution Φ = 2r, That is to say, we set a = 2 and b = c = Φ 0 = 0. Then after the pulse, for v > v 0 , we get This solution after the pulse is a µ = 4 solution which, using the results from Sec. IV, maps to the Rindler anabasis solution h θθ = 2(ρ + κ) via (35-36) with We thus see that we have a backreaction calculation in the connected AdS 2 throat that is consistent with the expectation from the physics of Reissner-Nordström: Throwing a pulse of energy 1 into the extreme black hole with Q = M "shifts the horizon" and the black hole becomes near extreme with Q = M √ 1 − . This is shown in Fig. 3.

VI. DISCUSSION
In this paper, we have studied backreaction in the context of AdS 2 × S 2 connected to an asymptotically flat region in four-dimensional Einstein-Maxwell theory. We imposed spherical symmetry but considered both electrovacuum solutions as well as a matter source in the form of a null ingoing pulse. We have seen that backreaction with boundary condition change, which we call anabasis, is consistent with Reissner-Nordström physics.
In AdS/CFT, anabasis is dual to following the inverse renormalization group flow, from IR to UV, for an appropriate irrelevant deformation of the boundary field theory that does not respect AdS boundary conditions. This is not something discussed very often in the AdS/CFT literature for at least two reasons. First, it is a difficult question to study systematically because it is hard to identify appropriate solvable irrelevant deformations of CFTs. Second, it runs somewhat contrary to the spirit of AdS/CFT which is a complete self-contained theory in itself-a theory in which even when one studies irrelevant deformations, one may wish to restrict oneself to deformations which do not destroy the boundary of AdS. Historically, of course, AdS/CFT was discovered by a low-energy near-horizon limit from string theory in asymptotically flat spacetime. A recent body of work that carries out an anabasis by following a flow for a single-trace irrelevant deformation of a CFT 2 , which goes under the name T T and changes AdS 3 asymptotics to flat with a linear dilaton, may be found in [5][6][7][8]. 3 3 A different double-trace T T deformation of CFT 2 has been holographically interpreted as a gravitational theory in an AdS 3 that is cut off at a finite interior surface [9] (see also [10]). This is not related to anabasis as it may be obtained from mixed boundary conditions that respect the AdS 3 boundary [11].
The gravitational aspects of AdS 2 anabasis studied in this paper do not rely on the existence of a holographic dual and are expected to be readily generalizable to a wide class of theories with (near-)extreme black holes which universally exhibit AdS 2 -like near-horizon geometries [12]. This includes rotating black holes such as Kerr which near extremality has a throat geometry, the Near-Horizon-Extreme-Kerr (NHEK) solution [13], with backreaction properties similar to AdS 2 ×S 2 [14,15]. In Appendix B, we give the SL(2)-breaking triplet of linear perturbations of NHEK that generalizes (6)(7)(8). Beyond near-horizon approximations, AdS 2 makes an appearance in other contexts where approximate spacetime decoupling occurs, such as the interaction region of colliding shock electromagnetic plane waves [16], or near certain highly localized matter distributions [17]. The ideas in this paper may also be relevant in such contexts.
A model of two-dimensional dilaton gravity in AdS 2 that is solvable with backreaction, as well as with the addition of matter, is the Jackiw-Teitelboim (JT) theory [18,19]. This model captures many of the universal aspects in the spherically symmetric sector of higher dimensional gravity near extreme black hole horizons, and it has been studied extensively from the holographic perspective beginning with [20][21][22][23]. In JT theory, the geometry is fixed to being locally AdS 2 but the SL(2) is broken by a dilaton Φ JT . Comparing with our gravitational perturbations, we may identify Φ = Φ JT , noting that (12) solves the JT equation of motion ∇ µ ∇ ν Φ JT − g µν ∇ 2 Φ JT + g µν Φ JT = 0 for AdS 2 in Poincaré coordinates. This is because in the ansatz (3) we have Φ measuring the variation in the size of the S 2 and, in this ansatz, dimensional reduction of higher dimensional gravity down to two dimensions is known to lead to JT theory with the dilaton Φ JT measuring precisely this variation (see e.g [24][25][26]). Continuing the comparison, the SL(2)-invariant µ defined in (16) may be identified with the ADM mass of the 2D black holes in JT theory. It follows that the mass of the 2D AdS 2 black hole in JT is the deviation from extremality of the 4D Reissner-Nordström in Einstein-Maxwell. A comment is in order here. It is often said in the literature that JT is a nearly-AdS 2 theory with the "nearly" part, which is due to the dilaton's breaking of the SL(2) symmetry of AdS 2 , associated with a departure from extremality. As we have seen in this paper, however, this is not necessarily so because for µ = 0 the SL(2) may be broken only in order to build the exterior of an exactly extreme Reissner-Nordström.
The connected AdS 2 × S 2 , which we defined in Section V in order to perform a consistent Reissner-Nordström backreaction calculation, is nearly-AdS 2 in the sense that its SL(2) has been broken by the addition of anabasis perturbations that make this AdS 2 an approximate one. We also saw that this SL(2) breaking may be thought of as a choice of boundary condition for the backreaction calculation. A comprehensive study of various boundary conditions for the JT theory has been carried out in [27]. However, it appears that none of the boundary conditions contained therein would yield an anabasis as they at best correspond to mixed boundary conditions that do not destroy the AdS 2 boundary (of the double-trace type in AdS/CFT terms). On the other hand, the boundary term used in [28] for what is called therein "permeable boundary conditions" appears to be a better candidate for defining a connected AdS 2 in JT theory. Indeed, matching fields across the AdS 2 × S 2 boundary in Reissner-Nordström, as in the calculations of [29,30], necessitates boundary conditions that are "leaky" from the AdS 2 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 2 × S 2 region by sending in low energy waves from past null infinity in Reissner-Nordström. The calculation may be set up using matched asymptotic expansions and features leaky boundary conditions [31].
Generically, the backreaction of an extreme Reissner-Nordström black hole results, as in Section V, in a near-extreme one. However, there is a notable exception. In [3] it was found that there exist fine-tuned initial data for a massless scalar perturbing extreme Reissner-Nordström, for which an instability of the scalar field at the event horizon persists for arbitrarily long evolution and leads to a spacetime that may be thought of as a dynamical extreme black hole. It was observed that, at late times, this dynamical extreme black hole has the same exterior as extreme Reissner-Nordström but differs from it at the horizon. In [32] the instability of the perturbing massless scalar on extreme Reissner-Nordström was analyzed using the symmetries of its AdS 2 × S 2 throat. It would be interesting to study the dynamical extreme black hole of [3] using a connected AdS 2 × S 2 as defined in this paper.
We expect the anabasis of these three perturbations towards Kerr to proceed in a similar fashion to the analysis carried out above for Reissner-Nordström. However, it is worth emphasizing that there is no analog of Birkhoff's theorem for axisymmetric spacetimes and that there exist axisymmetric propagating gravitational wave perturbations of Kerr and NHEK. 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 judiciously picked 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. 4 Finally, note that in another gauge, our solution triplet takes the simple form h tr = 1 3 rt 2ar + brt + 2 3 cr t 2 + 9/r 2 .