Scaling and universality in extremal black hole perturbations

Journal of High Energy Physics, Jun 2018

Abstract We show that the emergent near-horizon conformal symmetry of extremal black holes gives rise to universal behavior in perturbing fields, both near and far from the black hole horizon. The scale-invariance of the near-horizon region entails power law time-dependence with three universal features: (1) the decay off the horizon is always precisely twice as fast as the decay on the horizon; (2) the special rates of 1/t off the horizon and \( 1/\sqrt{v} \) on the horizon commonly occur; and (3) sufficiently high-order transverse derivatives grow on the horizon (Aretakis instability). The results are simply understood in terms of near-horizon (AdS2) holography. We first show how the general features follow from symmetry alone and then go on to present the detailed universal behavior of scalar, electromagnetic, and gravitational perturbations of d-dimensional electrovacuum black holes.

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Scaling and universality in extremal black hole perturbations

Received: May Scaling and universality in extremal black hole Samuel E. Gralla 0 1 Peter Zimmerman 0 1 0 Department of Physics, University of Arizona , USA 1 1118 E. Fourth Street, Tucson, Arizona, 85721 , U.S.A We show that the emergent near-horizon conformal symmetry of extremal black holes gives rise to universal behavior in perturbing elds, both near and far from the black hole horizon. The scale-invariance of the near-horizon region entails power law timedependence with three universal features: (1) the decay o the horizon is always precisely twice as fast as the decay on the horizon; (2) the special rates of 1=t o the horizon and 1=pv on the horizon commonly occur; and (3) su ciently high-order transverse derivatives grow on the horizon (Aretakis instability). The results are simply understood in terms of nearhorizon (AdS2) holography. We rst show how the general features follow from symmetry alone and then go on to present the detailed universal behavior of scalar, electromagnetic, and gravitational perturbations of d-dimensional electrovacuum black holes. Black Holes; Conformal and W Symmetry; AdS-CFT Correspondence; Black - HJEP06(218) 3.3.1 3.3.2 3.3.3 3.3.4 Principal case I Principal case II Principal case III Supplementary case 4 SO(2; 1) near-horizon geometries 5 Full geometries Charged scalar elds Gravitational and electromagnetic perturbations Separable case Critical frequency Critical tail 5.3.1 5.3.2 Far region and o -horizon tail Near region and on-horizon tail Non-separable equations Electromagnetic and gravitational perturbations 5.6 Summary 6 Example: extremal Kerr-Newman-AdS Near-horizon geometry Elliptic equation and exponents Matching to the full geometry Critical tail 2.1 2.2 2.3 2.4 3.1 3.2 4.1 4.2 5.1 5.2 5.3 5.4 5.5 6.1 6.2 6.3 6.4 A Extremal planar Reissner-Nordstrom AdS (RN-AdS) B Discrete modes { i { 1 Introduction and summary 2 Argument from symmetry Boundary and horizon coordinates and gauge Symmetries Power laws from symmetry Aretakis instability from symmetry 2.5 Scaling dimension 3 Charged scalars in AdS2 with a uniform electric eld Dirichlet/Neumann conditions and holography Mixed boundary conditions 3.3 Frequency-independent mixed boundary conditions 1 Introduction and summary In the greater enterprise of black hole physics, extremal black holes | those at the edge of the allowed parameter space | play a special role. Their distinct mathematical properties generally demand separate analysis, while their privileged physical status gives them special interest. In astrophysical or condensed-matter applications, extremality corresponds to the interesting limits of high spin and low temperature, respectively. In quantum gravity, the lack of Hawking radiation makes extremal black holes thermodynamically stable and hence easier to study. The extremal limit has also seen a recent surge of interest from mathematicians interested in stability. All of these various fora for black holes | astrophysics, condensed matter physics, quantum gravity, and mathematics | involve in an essential way the study of their perturbations. One perturbs the spacetime and/or other elds involved, or for simplicity considers a test eld propagating on the geometry. From the behavior of these perturbing elds results many interesting quantities, such as the Hawking radiation spectrum, the propagator of a holographically dual theory, the gravitational-wave emission from some process, or the stability (or instability!) of the black hole itself. While four-dimensional, asymptotically at, electrovacuum black holes are remarkably constrained (they \have no hair"), modern problems of interest increasingly demand higher and lower dimensions, additional elds like scalars and spinors, and/or the presence of vacuum energy that modi es the boundary behavior. The complexity of this \black hole zoo" motivates the search for universal features, independent of the details of any speci c case. In physics quite generally, universal behavior emerges near special, \critical" points exhibiting emergent conformal symmetry. In black hole physics, the near-horizon region of an extremal black hole functions as such a point, as it sees the emergence of the twodimensional (global) conformal group SO(2; 1) as a spacetime symmetry [1{4].1 In this paper we nd associated universality in perturbing elds both near and far from the horizon. In particular, we show that (under certain conditions) the near-horizon region gives rise to power law time-dependence2 in each angular mode, with three universal features: 1. The decay o the horizon is always precisely twice as fast as the decay on the horizon. 2. The special rates of 1=t o the horizon and 1=pv on the horizon commonly occur (i.e. over nite regions of parameter space, without ne tuning). 3. Su ciently high-order transverse derivatives grow on the horizon (Aretakis instability). 1For extensions to local conformal symmetries, see [5]. 2We assume a stationary black hole with N commuting axisymmetries, and by \time-dependence" we mean Killing time along the orbits of a timelike linear combination of these Killing elds. (The o -horizon decay rate does not depend on which Killing eld one chooses; for an example see eq. (6.23b) and discussion rates will be visible at late times, as in the known 1=t [6] and 1=pv [7] tails of massless perturbations of extremal Kerr. If not, they should still be identi able at intermediate times, as in the transient 1=t decay occurring for charged perturbations of extremal KerrNewman [8]. At the very least, they can be identi ed with spectral analysis, as they are associated with a calculable special frequency in each example, often the superradiant bound. The special rates of 1=t and 1=pv occur over large swaths of parameter space in our analysis, and we therefore expect that these rates will appear much more generally than the known examples, functioning as \calling cards" for an extremal black hole. The universality can be traced to the shared AdS2 factor in extremal near-horizon geometries [3] and is simply understood in holographic terms. Each angular mode features a special frequency near which the dynamics are governed by a eld in AdS2 with some scaling dimension h. Elementary symmetry considerations force the bulk-boundary and boundary-boundary propagators to scale with exponents h and 2h, respectively, where D is the action of an in nitesimal dilation. The key observation is that these propagators encode the e ects of AdS2 on externally sourced perturbations (initial data of compact support away from the horizon), since the AdS2 boundary functions as the gateway of the near-horizon region, corresponding to o -horizon properties of externally sourced on-horizon properties. The relative factor of 2 in eq. (1.1) accounts for the rst result above, while the bound Re[h] 1=2 on AdS2 scaling dimensions accounts for the second. The third result, the Aretakis instability, is also a direct consequence of the symmetry (1.1) [9]. We esh out these arguments in section 2 below. While the symmetry argument captures the essence of our results, it is far from the whole story. In particular, the argument assumes that holographic propagators can be de ned for some exponent h, which is possible only for certain choices of AdS2 boundary conditions (typically Dirichlet). In black hole perturbation problems, the AdS2 boundary conditions are determined by the physics of the far region, and we are not free to adjust them to satisfy our holographic urges. In fact, in many important cases, such as for Kerr black holes, these conditions are such that dynamics in pure AdS2 would not even be wellposed! After presenting the symmetry argument in section 2, we go on to tell the full story in sections 3{5 in terms of the range of boundary conditions that can arise in practice. We delineate the parameter space where the basic results 1{3 survive, giving conditions that can be checked for any particular perturbation problem of interest. We give an example of applying the formalism in section 6. Some of the main features and results of our analysis have been noticed before in the holographic condensed matter literature. In particular, beginning with ref. [10], it was recognized that AdS2 scaling behavior emerges at frequencies near the chemical potential, giving rise to power laws in the dual theory. However, this body of literature has not, to our knowledge, considered decay on the horizon or discussed the growth of derivatives (Aretakis { 2 { e, e^, ^ ^ P P ~ P ; ; ; ; : : : I; J; K; : : : i; j; k; : : : a; b; c; d; charge and mass of complex scalar eld perturbation charge and mass of AdS2 perturbation a quantity P de ned on the AdS2 base space a quantity P de ned on the ber space a quantity P de ned on the full geometry indices on the d 2-dimensional ber space indices running over the N azimuthal angles I on the ber space indices running over the d N 2 coordinates yi on the ber space indices for tensor elds on the d-dimensional near-horizon spacetime L; L0 multi-index for multipoles may say that the 1=t gravitational-wave tail of extremal Kerr [6, 11, 12] is an observational signature of AdS2. The remainder of this paper is organized as follows. In section 2 we show how the so(2; 1) symmetries of AdS2 with a uniform electric eld dictate the main results. In section 3 we give a detailed study of charged, massive scalar elds in AdS2 with a uniform electric eld. In section 4 we study general near-horizon geometries, in which AdS2 appears as a \base space" dictating the dynamics of each angular mode. In section 5 we discuss full extremal geometries. In section 6 we apply the formalism to perturbations of four dimensional Kerr-Newman-AdS. Finally, appendix A discusses non-compact horizons using the example of planar RN-AdS, while appendix B discusses modes which require special care, which we call discrete. Our notation is summarized in table 1. 2 Argument from symmetry Before diving into the full calculations, we show how the main features can be derived from symmetry alone, subject to the presence of holography-friendly boundary conditions for perturbations of AdS2. In the full calculation, these conditions arise for the modes (called \supplementary" in our terminology) which respect the near-horizon BF bound. The analysis of the BF-violating modes has many similar features, but does not lend itself as neatly to the language of holography, and we defer to the detailed calculations of section 3. 2.1 Boundary and horizon coordinates and gauge The eld equation for a charged eld in AdS2 accompanied by a uniform electric eld arises in the near-zone dimensional reduction of the full perturbation problem near a special { 3 { and de ne the associated Lie derivative by the rules Charged elds will have similar derivative laws as appropriate. For example, for a complex scalar ^ of charge e^ we de ne (2.1) (2.2) (2.3) (2.4) H^ . (2.5) (2.6a) (2.6b) (2.7) where v = t 1=x; A^0 = A^ + d(ln x) Horizon-adapted: ds^2 = x2dv2 + 2dvdx; A^0 = xdv: The future horizon is described by x = 0 in these coordinates. 2.2 Symmetries A spacetime symmetry of a metric g and electromagnetic gauge eld A is generated by a vector X such that $X g = 0 and $X A = df for some function f . The spacetime symmetry generators of (2.1) are given by ^ H which satisfy the so(2; 1) commutation relations [H^+; H^ ] = 2H0 and [H^ ; H^0] = With our gauge choice, the gauge eld is Lie-derived by H^0 and H^+ but not by H^ . In fact there is no gauge where A^ is invariant under all three generators. This makes it convenient to introduce a generalized Lie derivative corresponding to a simultaneous in nitesimal change of coordinates and gauge [13, 14]. We use an overbar to denote pairs of a vector eld and a scalar, frequency. The AdS2 future Poincare horizon is identi ed with the event horizon of the black hole, while the AdS2 boundary is identi ed with an overlap region | colloquially the entrance to the throat region | where near and far expansions are matched. It will be convenient to use separate coordinates and gauge when considering the horizon and the boundary. In boundary-adapted coordinates and gauge, we have Boundary-adapted: ds^2 = x2dt2 + A^ = xdt; dx2 x2 ; which cover the Poincare patch x > 0. (We use hats to distinguish AdS2 quantities | see $X A^ = $X A^ + d : $X ^ = $X ^ + ie^ ^: { 4 { The commutator of two pairs acts only on the spacetime part: [X1; X2] := [X1; X2], where Xi = (Xi; i). In this language the spacetime symmetries may be written in boundaryadapted coordinates and gauge as Boundary-adapted: H0 = (H^0; 0); H+ = (H^+; 0); H = (H^ ; 2=x): (2.8) These satisfy the so(2; 1) commutation relations while leaving the metric and gauge eld invariant as $X g^ = 0 and $X A^ = 0. Under a nite U( 1 ) gauge transformation A ! A+d , the generator X = (X; ) changes by gauge (2.2) we have (using = ln x) ! $X . Thus for ingoing coordinates and Horizon-adapted: H0 = (H^0; 1); H+ = (H^+; 0); H = (H^ ; 2v); HJEP06(218) where the ingoing-coordinate representation of the Killing vectors is ^ H (2.9) (2.10) (2.11) (2.12) (2.13) (2.14) 2.3 Power laws from symmetry Having introduced the relevant notion of symmetry, we now give a simple argument showing the existence of power law tails xed by the scaling dimension set by the boundary asymptotics of AdS2 bulk elds. The two-point function GBB(v; x; v0; x0) of an AdS2 eld (BB for \bulk-bulk") must respect all the spacetime symmetries. Respecting the dilation symmetry means $H0 GBB = 0; where the generalized Lie derivative $ acts on both spacetime points of the two-point function. In holography one de nes a bulk-boundary propagator GB@ by selecting the exponent h such that the following limit exists, x0!1 GB@ (t; x; t0) := lim (x0)hGBB(t; x; t0; x0): This limit is to be taken in boundary-adapted coordinates and gauge (2.1). The symmetry (2.11) of the bulk-bulk propagator immediately implies That is, going to the boundary (or equivalently matching to the far-zone) breaks the full symmetry down to a scaling self-similarity. We may also take the bulk point to the boundary or to the horizon, de ning boundary-boundary and horizon-boundary propagators by x!1 x!0 limit is taken in horizon-adapted coordinates and gauge.3 The symmetry (2.13) [or (2.11)] 3The horizon-boundary propagator relates points on the horizon to points on the boundary and hence appears naturally in \mixed" coordinates v and t0. Here v is an a ne parameter along a generator of the (degenerate) horizon, while t is the time coordinate of the boundary theory. Note, however, that we could equivalently use v on the boundary, since v = t 1=x ! t as x ! 1. To de ne GH@ in a single coordinate system and gauge, we could take the boundary limit in the ingoing coordinates and gauge by suitably modifying the conformal factor xh 7! x h ie^ in (2.12) to account for the gauge transformation. { 5 { now implies (2.15) That is, an additional h appears for each point taken to the boundary. The key observation now is that each of these objects is intrinsically one-dimensional, so the self-similarity only allows power laws. Setting t0 = 0 without loss of generality,4 from eqs. (2.8) and (2.9) we (2.16) HJEP06(218) The boundary-boundary propagator governs perturbations that propagate in and out of the near-horizon region, giving rise to a t 2h tail in the external region. On the other hand, the horizon-boundary propagator governs perturbations that propagate only in, giving rise to a v h ie^ tail on the horizon. The decay is set by the real part of h,5 which always di ers by precisely a factor of 2 between the horizon and boundary correlators. This is the rst universal result mentioned in the introduction. The second universal result requires a modicum of AdS2 physics, which is the scaling dimension h for Dirichlet boundary conditions of a charged scalar eld, h+ := 1=2 + p1=4 + ^2 e^2: (2.17) The real part is at most 1=2, giving rise to 1=t and 1=pv decay when this bound is saturated. The scaling dimension is related to symmetry in that h+(h+ 1) is the Casimir of so(2; 1) on the boundary (section 2.5 below), but the precise formula (2.17) requires the eld equations. 2.4 Aretakis instability from symmetry To see the Aretakis instability, we return to the self-similarity of the full bulk-boundary propagator (2.15), generalizing arguments presented in [9] in the context of the Kerr spacetime. We again set t0 = 0 without loss of generality. The equation can be solved in boundary- or horizon-adapated coordinates and gauge, where the generalized Lie derivative acts on AdS2 scalars as Boundary-adapted: Horizon-adapted: In the horizon-adapted gauge the general solution to (2.15) is 4The boundary-boundary and horizon-boundary propagators also inherit the time-translation symmetry and v t0, respectively. 5Here we assume that the charge e^ is a real number. In fact, for electromagnetic and gravitational perturbations the e ective charge has an imaginary part. However, the decay of invariants is still set by the real part of h | see discussion in section 5.5 below. (2.18a) (2.18b) (2.19) { 6 { for some function f . By assumption the propagator is smooth on the horizon, so f is smooth at xv = 0. It follows immediately that (2.20) where f (n) is the nth ordinary derivative. That is, taking a transverse derivative adds a power of v, such that su ciently high-order derivatives grow along the horizon | the Aretakis instability. This implies that infalling observers experience large gradients [15]. However, scalars constructed from the eld remain small, since all such quantities will inherit self-similarity from (2.13) with some exponent h0 = nh where n is a positive integer that counts the number of times the eld appears in the formula for the scalar invariant. The invariant then takes the general form (2.19) with h ! nh, i.e. it decays on the horizon at the rate v nRe[h].5 Tensor elds and their decay can be treated in a similar way [9]. See also refs. [16, 17] for complimentary discussions. 2.5 Scaling dimension The precise scaling dimension h requires the eld equations and cannot follow from symmetry alone. However, we can illustrate the role of the symmetries for charged scalars by noting that the wave operator can be written in terms of the generators as D^ 2 = + e^2; := $H0 ($H0 1) $H $H+ : (2.21) The quadratic operator is the Casimir of so(2; 1). In de ning holographic propagators by limits involving multiplication by xh, we have assumed boundary conditions such that the eld goes as x h at large x in boundary-adapted coordinates and gauge. The scaling self-similarity (2.13) of the bulk-boundary propagator means that GB@ t 2hx h at large x. Using eq. (2.21) one computes [t 2hx h] = h(h 1)t 2hx h 1 + O(1=x2) ; which is interpreted as \ = h(h 1) on the boundary." In fact, the Casimir is proportional to the identity on any irreducible representation, and h(h 1) is a standard name. Imposing the charged, massive scalar wave equation and using this relationship gives The solutions are low. These x the large-x behavior x h for bulk elds, as we recover explicitly in section 3 be0 = 2 ^ 1) + e^2 ^ 2 ^ : h = 1=2 Charged scalars in AdS2 with a uniform electric eld We now give a detailed study of charged, massive scalar elds ^ in AdS2 with a uniform electric eld. The eld equation is ^ 2 ^ = 0; D^ := r^ ie^A^; where r^ is the metric-compatible derivative on AdS2 and e^ is the scalar charge of the eld.6 We work in boundary-adapted coordinates and gauge (2.1). The equation separates under the ansatz giving the radial equation ^ = e i!tR(x); d dx x 2 d dx + denotes asymptotic equality in the sense that a(x) b(x) as 6Here we allow the charge e^ to be any complex number. In the applications to full geometries that we consider below, e^ will be real for scalar perturbations and have integer imaginary part for electromagnetic and gravitational perturbations | see eqs. (4.9) and (4.14) below. { 8 { x ! 1. The R are ^ R ( 2i!) h < (h ie^) 8 (2h ) :( 2i!=x)h ; A convenient set of solutions are represented as Whittaker functions [26], which we denote by ^ Rin = Wie^;h+ 1=2( 2i!=x); ^ R = ( 2i!) h Mie^;h 1=2( 2i!=x); (3.4) where h was given previously in eq. (2.24). Only two solutions are needed to span the general solution to (3.3), but we nd it convenient to introduce all three. Note, however, that when h+ = 1=2 the R^ solutions are the same, and when h+ = 1; 3=2; 2; : : : the R^ solution does not exist. Henceforth we will assume that 2h+ is not a positive integer (or equivalently that 2h is not zero or a negative integer). However, all expressions that are well-de ned in the limit that 2h+ becomes an integer do in fact hold in that case. We refer to h as the weight of the bulk eld or the scaling dimension of the corresponding CFT operator (see discussion in section 3.1 below). The character of the weights (real, complex, or integer) determine many important features of the dynamics. We summarize the important cases for our work in table 2. Generic solutions behave as a linear combination of x h+ and x h at the boundary solutions are distinguished by behaving solely as x h . Their asymptotics (2h ) (h +ie^) e i!=x( 2i!=x) ie^+ ei!=x+ (h ie^) i( 2i!=x)ie^; x ! 0; Name principal discrete De nition B < 0; not discrete depending on the values of the mass ^ and charge e^. B := 1=4 + ^ 2 2 e^ . This terminology originates in the representation theory of the near-horizon symmetry group SO(2; 1) [18, 19], although our de nition of discrete di ers when e^ 6= 0. The condition B > 0 is also called the Breitenlohner-Freedman (BF) bound [20{25]. Properties of h We have de ned h+ := 1=2 + p B with follow from the properties of h+ using the relation h+ + h = 1. x ! x0 is equivalent to a(x)=b(x) ! 1 as x ! x0. Notice that each of the R has waves moving in both directions e i!=x at the horizon x ! 0. The Rin solution is solutions distinguished by having only waves traveling into the black hole. Its asymptotics are where where 2 = (t t0) (x frequencies by ^ Rin (ei!=x( 2i!=x)ie^; x0) is the invariant delta-function of AdS2. We decompose G^ into We will nd it convenient to consider Green functions, which satisfy where c is a real number chosen to put the integration contour in the region where g^ is analytic. (This is just the inverse Laplace transform with s = i!, which agrees with the Fourier transform for causal propagation.7) In the mode decomposition for G^ we have introduced an AdS2 \transfer function" g^(x; x0), which satis es eq. (3.3) with (x x0) on the right-hand side instead of zero, d dx x 2 d dx + 7The inverse Laplace transform does not extend before t = t0, but for causal propagation G^ will vanish for these times anyway. When we perform inverse Laplace transforms, below, we will regard G^ as vanishing for t < t0. Our preference for Laplace over Fourier arises from the convenience of the former for initial value problems. { 9 { (3.6) (3.7) (3.8) (3.9) Specifying the dynamics requires a choice of boundary conditions. We will do so by making a choice of Green function G^. The most common choices are to solve eq. (3.10) by R^in(x<)R^ (x>) ; W (3.11) (3.12) where x< and x> are (respectively) the lesser and greater of the points x and x0 and W = x2(R^inR^0 R^i0nR^ ) is a constant given by (12.14.26 of ref. [26])8 W = ( 2i!)1 h (2h )= (h ie^): The presence of R^in enforces causal propagation. The use of g^+ forces boundary behavior of x h+ , normally called \Dirichlet" conditions, while the use of g^ forces boundary behavior of x h normally called \Neumann" conditions [25].9 These conditions allow the de nition of a boundary eld (\dual operator") O^b; (t) by ^ Ob; = lim x h ^ ; x!1 mal transformation transforms as O^b; dua operator. ! o the leading behavior, which can be regarded as living on the boundary x ! 1 with metric limx!1 x 2ds2 = dt2. The dilation symmetry (x ! 1x; t ! t) of the bulk induces a global confordt2 ! 2 ( dt2) on the boundary, under which the operator O^b; h O^b; . In this sense h is the conformal scaling dimension of the The two-point function for the dual eld is inherited from the bulk dynamics at large x. We rst de ne a bulk-boundary transfer function by taking one point to in nity and peeling x0!1 (h ie^) (2h ) ( 2i!)h 1R^in(x): In this limit the inverse transform (3.9) may be computed exactly from eq. (9) in section 5.20 of ref. [28].10 Provided that h+ + ie^ 2= Z>0; the time-domain bulk-boundary Green function G^B@ (2.12) is given by ^ h (tx 1) 2 ie^ h (tx 1) ie^ h 1 + 2 (tx 1); (3.16) 8If h ie^ = 0 then R^ becomes proportional to R^in and all mode solutions satisfy both boundary conditions. We assume h + ie^ 6= 0 to avoid this physical pathology. 9The Neumann condition may only be imposed when 1=2 < h+ < 1 (0 < h > 1=2). The Neumann version of all formulae below should be ignored when this is not satis ed. Note that in some references ^ R+ is called the Neumann mode because it is the one whose coe cient would be speci ed in Neumann boundary conditions. 10The original Bateman manuscript [27] gives a restricted range Re(h) < 1 for this transform. However, by using the integral representation of the Whittaker W function given in eq. (3.5.16) of [29], we have found that integral transform still applies when h+ equivalent to h+ + ie^ 6= 1; 2; 3; : : :, since h+ ie^ = 1; 2; 3; : : : cannot occur [see eq. (2.17)]. ie^ 6= 1; 2; 3; : : :. In the present context this condition is with C := (h computation is also possible.11 In eq. (3.16) we have set t0 = 0 without loss of generality; this quantity may be restored by sending t ! t t0. Notice that the function enforces causality, since t = 1=x is the (onesided) light cone of the boundary point t0 = 0. The coe cient C vanishes in the special case h+ + ie^ 2 Z>0 that was excluded in eq. (3.15); the meaning of this case in the full geometry is discussed in appendix B. Notice, however, that the bulk-boundary propagator does remain well-de ned in the case that h+ is a half-integer that was excluded earlier. We have therefore computed the result for half-integer h+ by analytic continuation; a direct We may now compute the boundary-boundary correlator (2.14) by taking x to in nity v 2 ie^ h (3.17) (3.18) (3.19) (3.20) (3.21) x!1 ^ (t): In the holographic interpretation, this de nes the dynamics of the dual operator. Eq. (3.18) also follows from the Son/Starinets [30] prescription, where the frequency-domain boundary-boundary correlator is de ned by ratios of A+ and A as A A = (1 2h ) (h (2h ) (1 h ie^) ie^) ( 2i!)2h 1 ; where we have xed the normalization N = 1=(2h 1) to obtain exact agreement with eq. (3.18) after inverse Laplace (or Fourier) transform. Below we will nd use for an abbreviated notation, A = (2 2h ) (1 h ie^) ; G := A A+ = (2h+ 1) (1 2h+) (h+ (2h+) (1 h+ ie^) ie^) ; in which the Dirichlet two-point function is G 2h+ 1 ( 2i!)2h+ 1: The expression (3.16) for the bulk-boundary propagator is not regular on the horizon. Changing the bulk point to horizon-regular coordinates and gauge (2.3) gives Horizon-adapted gauge: ^ ie^ h 1 + vx ie^ h 2 (v): (3.22) Again, causality is manifest as v = 0 is the one-sided light cone of the boundary point t0 = 0. Finally we may take x ! 0 to produce the boundary-horizon propagator (2.14), ^ v 2 (v): (3.23) 11If h+ is a half-integer, the character of the Whittaker W function changes by picking up a logarithmic dependence (see for example DLMF eq. (13.14.8) [26]). The inverse Laplace transform in this case may be performed using eq. (6) in [7]. where C is given by eq. (3.17). These take the general form required by conformal symmetry (2.15) and causality, with the coe cient C and the exponent h+ arising from the details of AdS2. We now consider a more general scenario where the boundary condition set at in nity is a mixture of Neumann and Dirichlet, while preserving the ingoing boundary condition at x = 0. For this, we introduce a new function R^mix as the linear combination R^mix := B+R+ + B R^ ; ^ B+ x h+ + B x h ; For e ect, we now collect the results (3.18) and (3.23) and restore the t0 coordinate, 12 (v t0) t0) 2h h (t ie^ t0); (v t0); (3.24a) (3.24b) (3.25a) (3.25b) (3.26) (3.27) (3.28) may in general depend on ! (but not x). The physics is contained in the ratio of B+ and B , which we denote by N, We are mainly interested in the case where N is independent of !, but for this section we leave it arbitrary. The Green function for perturbations respecting this ratio of fallo s is N := B+ B : g^mix = R^in(x<)R^mix(x>) ; Wmix where Wmix is a constant given by x2 times the Wronskian of the in and mix solutions. For mixed boundary conditions, we assume that 2h+ is not an integer. In this case we may use eq. (13.14.33) of [26] to express R^in in terms of the barred coe cients introduced in (3.20), ^ Rin = A G( 2i!)h+ R^+ + ( 2i!)h R^ 2h+ 1 : The mixed transfer function may then be written g^mix = where again x< and x> are respectively the lesser and greater of x and x0. Observe that with these mixed boundary conditions, the denominator (i.e. W) is now a di erence of terms. Consequently, the analytic structure of the transfer function has changed nontrivially, with new poles and additional branch structure having emerged. For Dirichlet or Neumann conditions, we proceeded to de ne bulk-boundary and boundary-boundary correlators by peeling o leading behavior at large values of x. In 3.3 Frequency-independent mixed boundary conditions The mixed boundary conditions are characterized by prescribing the ratio N(!) of the two boundary behaviors. We are mainly interested in the case where this ratio is independent of !, and we henceforth assume The important quantity to consider is := N G( 2i!)2h+ 1; which appears in the demoninator of the transfer function (3.29) (and (3.30) and (3.31)). At this stage it is useful to treat the principal and supplementary cases (table 2) separately. For principal elds where h+ = 1=2 + ir with r > 0, we need three further sub-cases: principal case I: principal case II: principal case III: e r jG=Nj < e jG=Nj e3 r r jG=Nj > e3 r: The supplementary case further requires a fourth independent analysis. We now give each In case I, the rst term in dominates (its modulus is always larger than that of the second term), and we may expand 1= in a geometric series in ( 2i!)2h+ 1G=N. This allows us to invert the large-x0 transfer function term-by-term via eq. (9) in section 5.20 of in turn. 3.3.1 Principal case I ref. [28],12 giving the mixed case this is not possible because both behaviors x h appear. However, we will still need the large-x asymptotics, which [using (3.25b)] are given by g^mix(x; x0 ! 1) ( 2i!)2h+ 1GR^+ + R^ Nx0 h+ + x0 h g^mix(x ! 1; x0 ! 1) ( 2i!)2h+ 1Gx h+ + x h Nx0 h+ + x0 h Nx0 h+ +x0 h x ie^ t 1=x 2 1=2 ir ie^ (t 1=x) (3.35) 2F~1 (h+ ie^; 1 h+ ie^; 1 h+ ie^+(1 2h+)n; (tx 1)=2) ; 12As before, the integral transform does not hold in the case h+ + ie^ 2 Z>0. However, this special case cannot arise for a principal mode for which the imaginary part of the charge is not half-integer, so there is no corresponding restriction on (3.35). where 2F~1(a; b; c; z) = 2F1(a; b; c; z)= (c). Applying eq. (15.8.2) of [26], we nd that the large-x behavior of (3.35) is G^mix(t; x ! 1; x0 ! 1) after substituting for h = 1=2 3.3.2 Principal case II ir. In the limit N ! 1 we recover the Dirichlet result are the same order of magnitude, and there are in nitely many zeros, corresponding to quasinormal mode frequencies !n. We are unable to invert the Laplace transform exactly in this case, but we can give the spectrum of quasinormal modes as13 !n = 1 2 e 21r (arg(G=N)+2 n) sin " lnjG=Nj 2r i cos lnjG=Nj 2r # ; n 2 Z: (3.37) Upon inspecting the imaginary part of (3.37), we see that a mode instability occurs when e r < jG=Nj < e r; (instability criterion, principal case): (3.38) 3.3.3 Principal case III In case III, the second term in dominates and we may expand in N=(G( 2i!)2h+ 1)). This turns out to be simply related to case I by r.h.s. of (3.35) with h+ ! h and N ! 1=N. (3.39) (The same substitution can also be made in (3.36) to obtain the large-x limit.) Note that under h+ ! h , we have r ! r and G ! 1=G, and A+ ! A , and A ! A+. 3.3.4 Supplementary case In the supplementary case there are always a nite number of zeros of , and again we do not invert exactly. Assuming h+ > 1=2, the resonances are at !n = exp 1 2 lnjG=Nj 1 2h+ " sin 2 n arg(G=N) 2h+ 1 where n n n+; where = oor i cos arg(G=N) 2 n arg(G=N) 2h+ (2h+ 1 1) 2 # ; (3.40) (3.41) 13To demonstrate this claim let z 2i!. Then the pole condition reads z 2ir = exp 21r (arg(G=N) + 2 n) . Now, as ! = 12 iz, one nds ! = 2i jzj (cos arg z + i sin arg z). G=N, or e2r arg ze 2ir lnjzj = jG=Njei arg G=N. It follows that arg z = 21r lnjG=Nj and jzj = n = is imposed by our restriction to =2 < arg ! < 3 =2. Evidently, a mode instability arises when 2 n 2 (2h+ (instability criterion, supplementary case): 1) < arg (G=N) < 2 n + (2h+ 1) ; 2 (3.42) We note, however, that supplementary modes in the full geometry will have their late-time behavior dictated by the Dirichlet case (see dicsussion in section 5.3 below), and will not see this spectrum of modes. 4 SO(2; 1) near-horizon geometries A near-horizon limit can be de ned for any degenerate Killing horizon, giving rise to a universal form for all near-horizon geometries [3, 31, 32]. When eld equations are imposed, the form simpli es further [33]. In particular, we can encompass all known EinsteinMaxwell solutions, as well as all known restrictions on solutions, with the general ansatz HJEP06(218) ds2 = L2(y)ds^2 + IJ (y) d I + kI A^ d J + kJ A^ + ij (y)dyidyj ; A = QI (y) d I + kI A^ ; where ds^2 and A^ are given in eq. (2.1).14 The I form azimuthal angles ( I in N orthogonal planes. We use capital roman letters I; J; K; : : : for these coordinates. The remaining d N 2 coordinates are denoted yi, with indices i; j; k; : : : . The kI are constants, while L, IJ , ij , and QI are functions of yi that are determined by the eld equations. This ansatz trivially generalizes those of [4, 33{35]. We will use the language of ber bundles. The total space is the d-dimensional manifold, with base space (t; x) and ber (yi; I ). We regard g^ and A^ as the metric and gauge eld on the base space. For the ber, we assign a metric g and gauge eld A given by pullback to constant x and t surfaces, ds2 = IJ d I d J + ij dyidyj ; A = QI d I : When necessary, we will use Greek mid-alphabet indices ; ; ; ; : : : to index tensors on the ber. We denote the covariant derivative compatible with the ber metric by r . The ansatz (4.1) has symmetry group SO(2; 1) U( 1 )N , with the factors associated with the base space and ber, respectively. The generators are given by (4.1a) (4.1b) I + 2 ) (4.2) (4.3) H0 = H^0; These satisfy the commutation relations [H+; H ] = 2H0 and [H ; H0] = H of SO(2; 1), with each U( 1 ) generator WI commuting with everything. 14Our conventions for A^ and kI relate to those in Durkee and Reall [34] by A^ = A^DR and kI = kDIR. The coordinates (t; x; yi; I ) do not extend to the horizon x = 0. The metric and gauge eld are regular, however, in \ingoing" coordinates v = t 1=x; 'I = I kI ln x: (4.4) This transformation leaves the ber (4.2) invariant while inducing the coordinate and gauge change (2.3) on the base space (2.1). That is, after making the transformation (4.4), the metric takes the same form (4.1) with I ! 'I and A^ ! A^0 and the hatted quantities now expressed in ingoing coordinates (2.3). The Killing elds (4.3) transform as H0 = H^0 + kI WI ; where eq. (2.10) gives the AdS2 Killing elds in ingoing coordinates. Notice how the relationships between hatted and unhatted Killing elds in eqs. (4.3) and (4.5) correspond to the generalized Killing eld pairings in eqs. (2.8) and (2.9), respectively. Consider now a charged massive (complex) scalar eld in the near-horizon geometry The wave equation separates under the mode ansatz [34] D2 2 = 0; Da = ra ieAa: = ^(t; x)Y ( I ; yi); Y = eimI I P (yi): D L2D Y + E L 2 2 Y = 0; (4.6) (4.7) (4.8) The Y functions satisfy a self-adjoint elliptic equation on the ber, where D = r ieA is the ber covariant derivative. For a compact ber, the operator appearing in (4.8) is self-adjoint with respect to the natural L2 inner product on the ber. This guarantees a complete, orthogonal set of eigenfunctions labeled by a discrete set of eigenvalues mI and E. For a non-compact ber there will generally be a continuous spectrum of allowed values for E. In what follows we will assume the ber is compact, but the analysis of individual modes applies more generally, and results can typically be converted to non-compact cases by exchanging sums for integrals in the usual way. An example is given in appendix A. The equation for the eld ^(t; x) is D^ 2 ^ 2 ^ = 0 with ie^A^ is the gauge covariant derivative on the AdS2 base space. That is, (t; x) obeys the equation for a charged scalar on AdS2, with the eigenvalues mI and E setting the e ective mass and charge.15 The solutions of this equation were studied in 15For the bene t of the reader, we note that our de nition for the AdS2 e ective scalar eld mass in this section di ers from that originally used by Durkee and Reall [34]: ^2 = ^DR 2 2 e^ . section 3. From eq. (2.24), the exponents h are given by h = 1=2 q 1=4 + E (kI mI )2: Expressing a mode (4.7) in regular coordinates (4.4) gives = ^0(v; x)Y ('I ; yi); ^0 = eimI kI ln x ^; showing how the coordinate change (4.4) properly induces the gauge change (2.3) on the AdS2 eld ^ with charge e^ = kI mI . Gravitational and electromagnetic perturbations Gravitational and electromagnetic perturbations of vacuum black holes also satisfy decoupled equations in the near-horizon geometry (4.1) [34]. One introduces a \tensor Hertz potential" b = 1::: jbj with jbj indices living on the ber, where b = magnetic and gravitational perturbations, respectively. The Hertz potential is constructed 1; 2 for electrofrom a null basis for spacetime, whose real null directions ` and n are chosen to be n = 1 p 1 p (4.12a) The eld equation for the Hertz potential separates under the ansatz 1::: jbj = ^b(t; x)Y 1::: jbj ( I ; yi); Y 1::: jbj = eimI I P 1::: jbj (yi): (4.13) The angular eigentensors Y 1::: jbj satisfy self-adjoint elliptic equations (eqs. (2.20) and (2.29) of [34]; see also eqs. (72) and (100) of [36]) and are suitably orthogonal and complete [34], with real eigenvalues Eb. As in the case of scalar perturbations, the ^b satisfy the AdS2 charged, massive scalar wave equation [34],16 2 ^b = 0 with ^2 = Eb + e^2; e^ = kI mI ib; (4.14) which may be compared with eq. (4.9) for the charged scalar. From eq. (2.24), the exponents h are given by h = 1=2 = 1=2 Notice that while the e ective mass and charge can each be complex, the combination appearing under the square root is real. This ensures that the weight h has the same general properties of the scalar case [table 2]. The null basis (4.12) is not regular on the future horizon, since the vector n vanishes there, while the vector ` blows up. This can be xed by rescaling the vectors as ` ! `0 = x`; n ! n0 = x 1n: 16Our convention for the e ective electric charge e^ relates to that of Durkee and Reall [34] by complex conjugation: e^ = e^DR. This results from our choice of elds with boost weight b < 0 for the Hertz potentials. (4.15) (4.16) (4.10) (4.11) HJEP06(218) These rescaled vectors are given in ingoing coordinates (4.4) by n0 = 1 `0 = 1 p (4.17) revealing that, on the horizon, `0 is tangent to the generators, while n0 is transverse. The precise form of (4.16) is chosen so that the simultaneous change of coordinates (4.4) and null basis (4.16) corresponds in AdS2 to the change (2.3) to horizon-adapted coordinates and U( 1 ) gauge. In particular, the Hertz potential has boost-weight b [meaning that b ! 0b = xb b under (4.16)], so a mode (4.13) of b becomes b ! 0b = ^b0(v; x)Y ('I ; yi); ^b0 = ei(mI kI ib) ln x ^b; where we now suppress ber indices on Y . That is, the AdS2 eld ^b properly transforms as a complex scalar eld of charge e^ = kI mI ib under the simultaneous change of coordinates (4.4) and null basis (4.16) for the Hertz potential. Eq. (4.18) may be compared with eq. (4.11), to which it reduces when b = 0. Yet another change of null basis is useful for understanding the decay of electromagnetic and gravitational perturbations. The original basis (4.12) possessed the convenient property that each of its members is invariant under dilations by H0. This symmetry was destroyed by the change (4.16), but can be restored by a further time-dependent rescaling as `00 = v`0 = xv`; n00 = v 1n0 = (xv) 1n: The new legs `00 and n00 are dilation invariant (Lie-derived by H0) as well as regular on the future horizon. The new 0b0 is given by 0b0 = vb 0b: Note that 0b0 does not correspond to some ^b00 on AdS2 (i.e. there is no analog of (4.18)), since we rescale the null basis without the corresponding change of coordinates needed to ensure the proper U( 1 ) gauge transformation of ^b. However, we may still discuss the symmetry properties of 0b0 as a eld in the SO(2; 1) near-horizon geometry, and these will play a key role in our discussion of the electromagnetic and gravitational perturbations constructed from 0b0 in section 5.5 below. 5 Full geometries Now consider a spacetime and electromagnetic eld geometry whose near-horizon limit takes the form (4.1). That is, we consider a stationary, axisymmetric17 Einstein-Maxwell solution, denoted g~; A~, and suppose that there exist coordinates (t; x; yi; I ) together with a U( 1 ) gauge choice such that limit (4.18) HJEP06(218) (4.19) (4.20) x ! 0 xing xt (near-horizon limit) (5.1) 17That is, we suppose there are N commuting Killing elds with closed orbits. Every stationary black hole has at least one such Killing eld [37]. recovers the geometry (4.1).18 This xes the free functions and parameters in the ansatz; see section 6 below for an example. We rst present the case of scalar perturbations and then discuss electromagnetic and gravitational perturbations in section 5.5. A charged, massive scalar eld in the full geometry satis es D~ 2 2 = 0; where r~ is the metric-compatible derivative on the full geometry, making D~ the gaugecovariant derivative. We will require that mode solutions of this equation suitably match onto those of the near-horizon geometry. We will rst consider the case where the equation separates on the full geometry, by which we mean that the mode ansatz (5.2) (5.3) (5.5) (5.6) (5.7) ! 0. HJEP06(218) = e i!tR~(x)Y~ ( I ; yi); gives rise to an ordinary di erential equation for R~(x) and an elliptic (self-adjoint) PDE for the angular functions Y~ . These equations will in general depend on ! and mI as well as an additional separation constant E~. The radial functions R~ are assumed to satisfy a linear second-order ODE. In a given problem, boundary conditions will be imposed at x = 0 (the horizon) as well as at larger x (typically at asymptotic in nity). We denote the corresponding solutions by \in" and \1" R~in(x) : satis es ingoing boundary conditions as x ! 0; R~1(x) : satis es boundary conditions at some x > 0 (typically x ! 1): (5.4a) (5.4b) Below we will make the notion of ingoing boundary conditions precise by demanding a match to the AdS2 ingoing solution [eq. (5.13)]. On the other hand, we leave R~1(x) arbitrary to cover a general choice of boundary conditions. To consider solutions arising from initial data, we introduce a Green function G~, which satis es D~ 2 2 G~ = d ; where d is the invariant delta-function in d-dimensions. We mode decompose G~ as G~ = 2 X e i!(t t0)Y~L( I ; yi)Y~L ( 0I ; y0i)g~L(x; x0) d!: Here we introduce the notation L for the collection fE~; mI g indexing the eigenvalues. Each g~L serves as the transfer function for its mode. In what follows we will usually suppress the index L. The transfer functions satisfy equations of the form x0): 18More formally, the limit is given by changing coordinates to T = t and X = x= and letting Importantly, the function f (x0) is independent of ! since it follows from the principal (highest-derivative) part of D~ 2 together with metric factors in the coordinate expression of d. The homogeneous solutions are just the R~ as already de ned. To properly give causal dynamics, the transfer function must agree with the in solution for x < x0 and with the 1 solution for x > x0. Matching at the support of the delta function gives g~(x; x0) = f (x0) R~in(x<)R~1(x>) ; W [R~in; R~1] where W [R~in; R~1] denotes the Wronskian of R~in and R~1, evaluated at x0. The near-horizon limit was given above as x ! 0 xing xt. Working in frequency space, the relevant notion of near-horizon limit is x ! 0 xing x=!. This choice preserves the e i!t factor in the mode ansatz (5.3) as x ! 0 xing tx, ensuring that the limiting eigenfunctions satisfy the appropriate equations in the near-horizon geometry. We may thus equivalently phrase the near-horizon limit as ! ! 0 xing x=!, showing an association between the near-horizon geometry and the critical frequency ! = 0.19 To analyze the full behavior near the critical frequency, we will also need a second, \far" limit of ! ! 0 xing x. That is, we use the terminology: near limit: ! ! 0 far limit: ! ! 0 xing x=! xing x: !!0 lim Y~ ( I ; yi) = Y ( I ; yi): The distinction between near and far is irrelevant for the angular eigenfunctions Y~ , which do not depend on x. Since these will limit to solutions of the near-horizon elliptic equation (4.8), we can make them match for each index by requiring For the radial functions we must consider both near and far limits, using the method of matched asymptotic expansions. We will assume the following properties:20 R~far satisfying (i) The far limit of the radial equation has linearly independent solutions R~+far and R~+far R~far x h+ ; x h ; ln(x)x h+ in the assumptions below. 19The critical frequency is always zero in the coordinates we have chosen. In speci c metrics, it may be non-zero in the common coordinate systems. For example, in Boyer-Lindquist coordinates for Kerr, ! = m H where m is the azimuthal number of the mode and H is the horizon frequency. This is more commonly known as the superradiant bound frequency. See section 6 for a detailed analysis of extremal 20In the special case where 2h+ is an integer, the overlap region behavior x h+ should be replaced with (5.8) HJEP06(218) (5.9a) (5.9b) (5.10) (5.11a) (5.11b) (5.12) (5.13) (5.14a) (5.14b) (5.14c) (5.14d) (ii) The far limit of the 1 function solves the far limit of the radial equation, (iii) The near limit of the in function is equal to the AdS2 in function, !!0 x x R~1far := lim R~1 = B+R~+far + B R~far: R~innear := lim Rin = R^in: ~ !x!x=0! R~innear R~ifnar R~near 1 R~far 1 A+x h+ + A x h ; A+x h+ + A x h ; B+x h+ + B x h ; B+x h+ + B x h ; In these assumptions, h+; e^, and ^ are to be computed from the near-horizon eigenvalues E and mI using eqs. (4.9) and (4.10). The e ective AdS2 boundary conditions (3.26) can be determined from the small-x behavior of the far 1 function, The above assumptions are not all independent, including signi cant redundancy in order to establish notation. Assumption (i) guarantees that the far limit can match on to the near limit by demanding the appropriate asymptotic behaviors of x h+ and x h . Assumption (ii) ensures that the 1 function properly limits to a solution of the limiting radial equation. Assumption (iii) ensures that the in function properly limits to the in solution in the near-horizon geometry. Finally, assumption (iv) ensures that each of the in and 1 solutions properly matches according to the method of matched asymptotic expansions. In particular, we can write the far in function as R~ifnar = A+R~+far + A R~far; (5.15) where A+ and A are given in eq. (3.7). 5.3 Critical tail By following a standard sequence of steps (done, e.g., for the Kerr spacetime in [9]), the solution arising from initial data supported away from the horizon can be reduced to a sum over convolutions of initial data modes with the transfer function for each mode. Thus transfer functions capture the generic behavior of the eld in the sense that features in the transfer functions will generically manifest in the eld in a calculable way. We will study the behavior of the transfer functions near the critical frequency ! = 0. If the full transfer function has no other singular points in the complex plane at equal or larger (real part of) frequency, the results will correspond to the late-time behavior of the eld. More generally, we can still expect the behavior near ! = 0 to show up in some way during the evolution, much as simple poles (quasinormal modes) are typically seen as identi able damped oscillations at intermediate times. Here we will focus on the ! ! 0 limit of the transfer functions, whose associated behavior we call the \critical tail" in light of the association with critical phenomena [9, 10, 38]. More precisely, we de ne the critical tail ~ Gtail(t) of each mode to be the inverse Laplace transform of the leading non-analytic term in the ! ! 0 expansion of the transfer function g~.21 There are di erent tails for the far and near regions, which we treat separately below. As in the case of pure AdS2, for computation purposes we assume that 2h+ is not an integer; however, the results remain valid in that limit, so we e ectively treat 2h+ 2 Z by analytic continuation (see footnotes 11 and 20). We further assume that the mode is not discrete [see table 2]; this case is discussed in appendix B. Far region and o -horizon tail We begin by considering the eld o of the horizon (x > 0), whose small-! features correspond to the \far" limit ! ! 0 xing x. The initial data is assumed to be supported away from the horizon, and hence we similarly consider x0 to be in the far region. The relevant limit of the transfer function (5.8) is thus stands for \far far" and where the Wronskian W is computed at the point x0. The !-independent function f (x0) was introduced in eq. (5.7). Using eqs. (5.15), (5.12), (3.20), we may express g~ as g~ (x; x0) = ( 2i!)2h+ 1GR~+far(x<) + R~far(x<) In eq. (5.17), the !-dependence is completely explicit: f , G, N and R~far are all independent of !. Alternatively, we can repackage the !-dependence in terms of the Dirichlet =1 boundary-boundary correlator (3.21) of near-horizon holography, g~ (x; x0) = take the form Supplementary modes. In the case of supplementary modes (see table 2), h+ > 1=2 !2h+ 1 is small as ! ! 0. This makes the far-far transfer function (5.18) g~ 21If there are in nitely many poles as ! ! 0 [case II of the classi cation (3.34)], we avoid the term \tail" as the behavior will not be a simple power law. (5.16) : : (5.17) (5.18) (5.19) where the functions a and b are independent of !. The rst term does not contribute at late times (it is a \contact term"). The second term is the leading non-analytic contribution for small !, which de nes the critical tail. The inverse Laplace transform of this term is just the boundary-boundary retarded correlator (3.18), giving the tail as (5.20) Here and below, / means equality up to multiplication by a function of the coordinates not displayed (in this case x and x0). The factor b(x; x0), which may be straightforwardly determined from eq. (5.18), encodes the non-universal physics of the far region. The time Principal modes. For principal modes where h+ = 1=2 + ir with r > 0 (see table 2), !2h+ 1 = !2ir is a logarithmic phase and the transfer function (5.17) does not simplify as ! ! 0. However, it has the same analytic structure as the boundaryboundary mixed transfer function given previously in eq. (3.31), so we may take advantage of previously derived results in AdS2 (section 3.3) to obtain the inverse Laplace transform. The results depend upon the ratio G=N and the imaginary part of h+, denoted r. The unstable range of eq. (3.38) corresponds in the present small-! context to an in nite number of poles along a line as ! ! 0, which has been seen in previous work [10, 39] to correspond to a condensate-type (\superconducting") instability of the spacetime, Range for condensate instability: e r < jG=Nj < e r: Outside of this range, the dynamics associated with ! ! 0 will be stable. In the range e r < jG=Nj < e3 r there are stable poles with spectrum given in eq. (3.37). It is presently unclear what the associated dynamics are in this case. In the remaining ranges jG=Nj < e jG=Nj > e3 r (cases I and III, respectively), the inverse Laplace transform of (5.17) may r and be determined from eq. (3.36) as 1 n=0 G~tail(t; x; x0) = t 1 X cn (x; x0) + dn (x; x0)t 2ir t 2inr; where the lower and upper signs correspond to cases I and III, respectively. The nonuniversal coe cients cn (x; x0) and dn (x; x0), whose precise form is irrelevant for our arguments, may be determined from eq. (3.36). The 1=t tail is universal. 5.3.2 Near region and on-horizon tail We now consider the eld on the horizon (x = 0), whose small-! features are visible in the \near" limit ! ! 0 xing x=!. Again, the initial data is assumed to be supported away from the horizon, and we thus consider x0 to be in the far region. The relevant limit of the transfer function (5.8) in this case is given by g~nf (x; x0) = f (x0) R~innear(x)R~far(x0) 1 ; (5.21) (5.22) (5.23) where nf stands for \near far". As usual, W is computed at the point x0, explaining the appearance of the far functions. Crucially, this means that W is independent of frequency. Using eqs. (3.28) and (5.12), we may write g~nf (x; x0) = (1 This is in fact proportional to the large-x0 mixed AdS2 transfer function (3.30). Unlike in the far-far case (5.17), the !-dependence in (5.24) is not all explicit, since the AdS2 Supplementary modes. Recall that !2h+ 1 is a small quantity for supplementary modes. As ! ! 0 we may the drop the !-dependent term in the denominator of (5.24), giving g~nf (x; x0) (5.24) (5.25) (5.26) (5.27) (5.28) (5.29) coordaintes,22 G~tnafil(v; x; x0) = u(x0)G^B+@ (x; v) / 2 ie^ h+ 1 + vx ie^ h+ 2 ; where the precise form of the function u(x0) is unnecessary. That is, the near-far transfer function is proportional to the AdS2 Dirichlet boundary-bulk propagator as ! ! 0. In the time domain this is just G^B@ as given in (3.16) or (3.22). We present the result in ingoing which may be compared to eq. (5.20). The details of the far region appear in the nonuniversal function u(x0), but the complete near-region behavior (small x and large v) is universal. In particular, the Aretakis instability is present and the on-horizon tail is v ie^ h+ . We can emphasize the latter by evaluating on the horizon and expressing in terms of the horizon-boundary correlator (3.23), G~tnafiljx=0 = u(x0)G^H+@ (v) / v ie^ h+ : Recall that !2h+ 1 is a logarithmic phase for principal modes. We can no longer drop any terms in eq. (5.24), but we can make use of the fact that it is proportional to the large-x0 mixed AdS2 transfer function (3.30), g~nf = w(x0)g^mix(x; x0 ! 1); where by g^mix(x; x0 ! 1) we mean the right-hand side of eq. (3.30). The inverse Laplace transform was computed in eq. (3.35), and we have coordinate v such that v = t eq. (4.4)], with modes de ned relative to eimI'I . 1=x as x ! 0 22Strictly speaking, we have not de ned an ingoing coordinate v in the full geometry; here we mean any xing xt (or equivalently xv) and ' I = I kI ln x [see G~tnafil(v; x; x0) = p 1 v y (x0) X(v=2) 2irn ir ie^z n(vx); 1 n=0 where zn(vx) := (N=G)n2F~1 (h+ ie^; 1 ie^; 1 (1 (5.30) and where the sign is for jG=Nje r < 1 (case I) and + sign is chosen for jG=Nje 3 r > 1 (case III). Again, the non-universal coe cients y may be determined from relations given previously. Although the eld (5.29) is not precisely self-similar, it is given by a sum of terms, each of which is self-similar with the same real part of scaling exponent (in this case 1=2). This property was called \weakly self-similar" in ref. [9], where it was shown to still entail the Aretakis instability. The properties of the hypergeometric function guarantee that eq. (5.29) interpolates between 1=pv decay on the horizon x = 0 and 1=v decay as x ! 1 (reproducing the far-far result of 1=t). On the horizon the hypergeometric function is equal to unity, so we have Gtnafiljx=0 = p 1 y (x0) X 1 n=0 N G n (v=2) 2irn ir ie^; (5.31) HJEP06(218) (case III). In particular, the on-horizon decay is universally like 1=pv. where again the sign is for jG=Nje r < 1 (case I) and + sign is chosen for jG=Nje 3 r > 1 Non-separable equations Above we assumed that the wave equation separates in the full geometry, which is rather restrictive. However, the analysis generalizes straightforwardly to non-separable geometries, as we are mainly interested in the e ects of the near-horizon region, where the equations still separate. Considering a non-separable full geometry merely complicates the analysis of the non-universal factors, as we now explain. functions R~L, which takes the form The important point is that the mode decomposition remains well-de ned even when the eld equations do not separate. In fact, since we no longer care about separating the equations, we can simplify by choosing Y~ = Y , i.e., we use the near-horizon angular eigenfunctions to decompose the full perturbation. Adopting the ansatz (5.3) for each angular index L = fE; m1; : : : ; mN g now results in a coupled set of equations for the radial ALL0 (x)@x2R~L0 + BLL0 (x; !)@xR~L0 + CLL0 (x; !)R~L0 = 0; for some matrix-valued functions A, B, and C, with implied summation over repeated indices L0. We can again de ne the 1 and in solutions by appropriate boundary conditions, as in eqs. (5.4). Instead of the decomposition (5.6) for the Green function, we now adopt G~ = 2 1+ic L;L0 X e i!(t t0)Y~L( I ; yi)Y~L0 ( 0I ; y0i)g~LL0 (x; x0) d!; with transfer functions g~LL0 satisfying ALL00 (x) @x2 g~L00L0 (x; x0) + BLL00 (x; !) @xg~L00L0 (x; x0) + CLL00 (x; !) g~L00L0 (x; x0) = LL0 (x0) (x (5.32) (5.33) x0) (5.34) for some !-independent functions LL0 (x0) coming from the metric determinant. The solution satisfying the boundary conditions is then g~LL0 (x; x0) = fLL0 (x0) R~Lin(x<)R~L1(x>) W [R~in(x0); R~L1(x0)] L (5.35) where W is the Wronskian and fLL0 = (A 1)LL00 L00L0 , assuming A is invertible. As de ned, there is no sum over the L indices on the radial functions. This is the same formula as eq. (5.8) above, except that now f (x0) depends on the indices L and L0. We now make the identical assumptions (i){(iv) of section 5.2 to ensure proper matching of near and far expansions near the critical frequency ! ! 0. The remaining calculations of the critical tail follow without change. Electromagnetic and gravitational perturbations We treat electromagnetic and gravitational perturbations using the Hertz potential formalism [36, 40{44]. We choose any stationary, axisymmetric null basis for spacetime that reduces to eqs. (4.12) in the near-horizon limit. Although the equations for the Hertz potentials b ( ber indices suppressed as in section 4.2) will not separate in the full spacetime, by the same arguments of section 5.4 we can still use the assumptions and calculations of this section for the purposes of calculating the critical tail. There is a minor subtlety in translating the results in that di erent null bases are used on and o the horizon. In particular, scalar results presented in (t; x) coordinates are promoted to Hertz potential results in the original null basis with near-horizon limit (4.12), while scalar results presented in (v; x) coordinates refer to Hertz potentials in the horizon-regular null basis (4.16).23 The qualitatively new feature of the electromagnetic and gravitational cases is the need to construct24 the perturbation from the Hertz potential b. O the horizon, the critical tail of the Hertz potential is a well-behaved function of (t; x) and there is no subtlety: in a suitable gauge, the metric (or electromagnetic) perturbation and all its derivatives will share the same power law decay (or at least decay no slower). On the horizon, it is convenient to use the near-horizon dilation symmetry to organize the calculation and results [9]. We rst consider the case of a supplementary mode in the primed null basis (4.16), for which symmetry considerations (section 2, eqs. (2.13) and (2.18b)) or direct calculations (section 5.3, eq. (5.26)) show that $H0 0b = ( h b) 0b; (5.36) eq. (4.18). true more generally. where H0 is given in ingoing coordinates in eq. (4.5).25 Here we refer to the critical tail of an individual supplementary mode. When a tensor T satis es $H0 T = pT we say that it is dilation self-similar with weight p. Thus the weight of 0b is h 23This follows from the fact that the passage to regular coordinates and gauge in AdS2 corresponds to a simultaneous change of coordinates and null basis in the full geometry; see discussion surrounding 24For gravitational perturbations of Kerr, it has been shown that the metric may be constructed from the Hertz potentials up to gauge and non-dynamical degrees of freedom [45]. We expect the same to be { 26 { To leverage the dilation symmetry we change to the dilation-invariant (and still horizon-regular) null basis (4.19). This sets 0b ! 0b0 = v b 0b [eq. (4.20)], so that $H0 0b0 = h 0b0: (5.37) $H0 g = h g; (5.38) HJEP06(218) Importantly, the dilation-weight h of 0b0 is independent of the boost weight b. Since 0b0 is de ned relative to a dilation-invariant null basis, the perturbation construction procedure preserves the dilation weight [9]. That is, there exists a gauge where the metric perturbation g corresponding to the critical tail of a mode 0b0 satis es and similarly for the gauge potential A of a mode of an electromagnetic perturbation. The above analysis was for a single supplementary mode. Although principal modes are not precisely self-similar, they decompose into a sum [eq. (5.29)] of self-similar terms of dilation-weight pi, all with the same real part Re[pi] = 1=2. This property was called weak self-similarity in ref. [9]. The nal metric perturbation (or electromagnetic gauge potential) inherits the weak self-similarity of its constituent modes. That is, the critical tail of the metric perturbation (in a suitable gauge) is a sum of terms, each of which satis es $H0 g = p g; Re[p] 1=2: (5.39) As explained in more detail in [9] (and barring subtleties in the convergence of the sum), the p with the smallest real part sets the decay of any component of any tensor constructed geometrically from the perturbation. In particular, we may conclude that, despite the growth of transverse derivatives along the event horizon, all scalar invariants decay. 5.6 Summary In this section we have demonstrated the detailed relationship between full-spacetime perturbations near the critical frequency (the \critical tail") and corresponding perturbations of AdS2 with a constant electric For scalars or Hertz potentials eld. We now summarize for the reader's convenience. in a stationary null basis:26 For supplementary modes o the horizon, the critical tail is given by the Dirichlet For supplementary modes on the horizon, the critical tail is given by the Dirichlet horizon-boundary correlator G^H+@ (v) / v ie^ h+ [eq. (5.27) above]. For supplementary modes asymptotically near the horizon (x 0 xing xv), the critical tail is given by the Dirichlet bulk-boundary propagator G^B+@ (x; v) / v ie^ h+ 1 + v2x ie^ h+ [eq. (5.26) above], whose self-similarity gives rise to the Aretakis instability. 26Results on the horizon refer to a null basis reducing to eq. (4.16) in the near-horizon limit, while results o the horizon refer to a static null basis reducing to eq. (4.12) in the near-horizon limit. For principal modes, the critical tail is determined by the mixed AdS2 two-point function. Provided there are no poles, the critical tail universally decays like 1=t [eq. (5.22)] and 1=pv [eq. (5.31)] o and on the horizon, respectively, and the Aretakis instability is present [eq. (5.29)]. Hertz potentials in the dynamical null basis (4.19) always decay (section 5.5 above), implying in particular that scalar invariants decay. 6 Example: extremal Kerr-Newman-AdS HJEP06(218) Although our main concern has been with identifying universal features, the results of this paper can also be applied to particular perturbation problems. We now summarize the recipe and give an example of its use. The rst step in determining the critical tail of an extremal spacetime is to take the near-horizon limit by nding coordinates and gauge such that the limit (5.1) exists and gives a metric of the form (4.1). These coordinates de ne the \critical frequency" by ! = 0, where ! is conjugate to the t coordinate of the limit. (In general the critical frequency will be non-zero in some original coordinate system in which the metric was written down.) The limiting metric xes the free functions in the general near-horizon metric (4.1), which de nes an elliptic equation for angular eigenfunctions (eq. (4.8) for scalars and eqs. (2.20) and (2.29) of [34] for electromagnetic or gravitational perturbations), whose spectrum of eigenvalues must be computed. The e ective mass and charge are then given by simple formulas [eqs. (4.9) and (4.14)], from which the exponents h+ of each mode may be computed using (2.17). Real exponents (supplementary modes) give rise to a critical tail with power law decay of v h+ and t 2h+ on and o the horizon, respectively. (See section 5.6 for a summary of the details.) If any of the exponents is complex (a principal mode), then a more detailed analysis of the far region is required to determine the critical tail. One must solve the far equations to determine the e ective AdS2 boundary conditions N of each mode, as described below eq. (5.14d). Then, for each principal mode one must check certain conditions. For modes satisfying eq. (5.21), there will be a condensatetype instability. For modes in case II of (3.34) [but not satisfying (5.21)], we have not analyzed the behavior. For modes in cases I or III of (3.34), the critical tail will have the universal 1=pv and 1=t behavior on and o the horizon, respectively, whose properties are summarized in section 5.6. We now illustrate this procedure in the example of four-dimensional extremal KerrNewman-AdS (KN-AdS), which may be written in coordinates (t~; r~; ; ~) as [46] a sin2 d ~ 2 a sin2 d ~ ; dr~2 + d 2 + sin2 adt~ r~2 + a2 d ~ 2 ; (6.1a) (6.1b) where = (r~2 + a2) 1 + 2M r~ + q2; = 1 = 1 = r~2 + a2 cos2 : The spacetime characterizes a rotating black hole when 0 < a < `. The event horizon is located at the outermost real root of , denoted r+. The complete family depends on four parameters M; a; q; `, which are interpreted in [47]. The extremal family, which is our primary interest here, is the parameter set of solutions for which 2 1 + 3r+2 q 2 2 M = 1 1 + 2 2 q2 ! `2 : 1 2 In the extremal case, may be expanded near the horizon as where, following the notation in [48], we have introduced (r+2 + a2)r+2 (r~=r+ 2 1)2 + O(r~=r+ 1)3; r02 = (r+2 + a2)(1 r+2=`2) q2=`2 + 3r+2=`2(2 r+2=`2) : Near-horizon geometry gauge by Following [48, 49], we obtain the near horizon spacetime by changing coordinates and t = ~ t x = ~ H t; A = A~ + H dt~; where the horizon frequency H and electrical potential H are de ned by Letting x ! 0 xing tx (and and ) gives the near-horizon geometry as H = r+2 + a2 H = qr+ : ds2 = L2( )ds^2 + ( )d 2 + ( ) d + kA^ 2 where L2( ) = Q( ) = q + r 2 a2 cos2 2 +r+ H ( ) = k = 2 H r+ ; ( ) = sin2 + a 2 2 ; H (6.2) (6.3) (6.4) (6.5) (6.6) (6.7) (6.8a) (6.8b) (6.9) with + := r+2 + a2 cos2 ; := 0 : (6.10) In eq. (6.8), ds^2 and A^ are the boundary-adapted AdS2 metric and gauge (2.1). Comparing with the general form (4.1), we identify the ber as the (topological) sphere covered by coordinates x = (yi; I ) = ( ; ). (In this example of a two-dimensional ber with a single U( 1 ) symmetry , we do not need explicit indices I or i, and we denote y = .) Eq. (6.9) xes all of the free functions in the general ansatz (4.1). The critical frequency is HJEP06(218) by de nition ! = 0 for the notion of frequency ! conjugate to the near-horizon coordinate t in the near-horizon gauge A. In terms of the original frequency !~ conjugate to the original coordinate t~ in the original gauge A~, this becomes critical frequency: !~ = m H + e H : (6.11) In the original coordinates, the critical tail will be associated with time-dependence of this characteristic frequency [eq. (6.23b) below]. Elliptic equation and exponents We consider scalar perturbations for simplicity. Using eq. (6.9), the elliptic equation (4.8) for the near-horizon angular eigenfunctions Y is given by 2H 2+(m eQ)2 csc2 a2 E Y = 0: (6.12) The spectrum E can be determined by solving this equation. In the limit of a massless perturbation to Kerr, E SpheroidalEigenvalue[l; m; im=2]. m2 = K, where K is given in Mathematica by The e ective mass and charge of the AdS2 perturbations are given by eq. (4.9), D~ 2 2 = 0; D~ = r~ ieA~; and the scaling dimension h+ is [from (4.10) or (2.17)] e^ = km = 2 H r+m= ; + p1=4 + E (2 H r+m= )2: In the Kerr-Newman limit, where the AdS4 length scale ` is taken to in nity, = 1 and H is given by (6.7) with = 1. To further take the RN limit, where ` ! 1 and a ! 0, an additional change of gauge is needed [48]. 6.3 Matching to the full geometry In the full KN-AdS geometry, the massive charged scalar equation (6.13) (6.14) (6.15) the poles. The radial equation is where 2 2 r~ K R~ = 0; separates under the harmonic decomposition = e i!t~R~(r~)Y~ ( ; ); Y~ := eim ~S( ); P 2 Z2 where we have introduced = a cos following [50]. The !-dependent functions S satisfy the massive spheroidal equation of the Heun type with eigenvalue K S = 0; where P = (! + m H + e H ) (a2 2 a2=`2) and = (1 2=`2)(a2 2). This equation may be solved using Sturm-Liouville methods by imposing regularity at (6.16) (6.17) (6.18) (6.19) (6.20) (6.21) (6.22) Z = (! + m H + e H ) (r~2 + a2) eQ r~: This equation may be used to determine the e ective AdS2 boundary conditions N, given a choice of true boundary conditions at the AdS4 boundary. Unfortunately, the equation is of Heun type on an in nite domain, meaning this analysis would have to proceed numerically. However, we can still check analyticaly that the far solutions properly match to AdS2 by taking the far limit ! ! 0 and then examining the small-x behavior. Setting ! = 0 in (6.18) and then expanding near x = 0 (r~ = r+) using the Frobenius method, the Frobenius indices are found to be 1 2 s 1 4 2r+2 + Kj!=0 2mr+ H + e H (r+2 2 2 : This entails behavior of x h at small x in the far limit, and must match the h of (6.14), determined in the near limit. Comparing the two expressions xes the near-horizon eigenvalue E in terms of the ! ! 0 limit of the general eigenvalue K to be E = Kj!=0 + (r+ ) 2 e H (r+2 r +2 2 4mr+2 H + e H (r+2 a2) : With this identi cation, and the previous identi cations (6.13), the far limit of a generic solution satis es R~far for some constants D . This means that far solutions will properly match to near region solutions, as assumed in section 5.2. Having veri ed that a near-far match can be achieved, the next logical step is to determine the full details of the critical tail by computing the eigenvalues E and the e ective nearhorizon boundary conditions N. This was done for the Kerr spacetime in ref. [9]. In this more general setting we do not attempt this calculation and instead present the range of possible critical tails. This entails simply utilizing the formulas (6.14) or (6.20) for h+ in the general results summarized in section 5.6. For example, for a supplementary mode at xed and x > 0, the critical tails are / t 2h+ eim ; / t~ 2h+ e i(m H +e H )t~eim ~; (near-horizon coordinates and gauge (6.8)); (original coordinates and gauge (6.1)); (6.23b) HJEP06(218) where k was given previously in (6.9). Notice the appearance of the phase e im H t~ when we re-express in the original coordinates and gauge. Including also the on-horizon result, we may summarize the situation for supplementary modes as v h+ ikm; on a horizon generator in the regular gauge (6.8) t~ 2h+ e i(m H +e H )t~; xed (r~; ; ~) o the horizon in the original gauge (6.1); (6.24) where v is an a ne parameter on the horizon generators. For principal modes (of case I or case III in (3.34)), the analysis is similar, using the more complicated phase structure summarized in section 5.6. The critical decay is like 1=t~ and 1=pv o and on the horizon, respectively. Since the formula (6.20) for h+ clearly indicates the potential for principal modes, we can expect to see these universal rates at least in some region of parameter space. Indeed, 1=t~ and 1=pv decay is known for massless perturbations of extremal Kerr [9], while 1=t~ intermediate-time behavior was seen for certain massive, charged perturbations of extremal Kerr-Newman [8]. Acknowledgments We thank Sean Hartnoll for helpful conversations. This work was supported by NSF grant 1506027 to the University of Arizona. A Extremal planar Reissner-Nordstrom AdS (RN-AdS) RN-AdS black holes have an important place in the AdS/CFT correspondence, as they are solutions to higher-dimensional supergravity truncations with appropriate compactications [51, 52]. Here we consider the planar limit of the extremal RN-AdS solution, which plays a role in holographic models for certain condensed matter systems near their quantum critical point [39]. Although our framework has assumed a compact horizon, most of the calculations remain relevant to the non-compact horizon case. Here we brie y review perturbations of extremal planar RN-AdS black holes to expedite comparison with the holographic condensed matter literature. In planar static coordinates ( ; r; ~y), the RN-AdS solution is given in d spacetime dimensions by [52] N d 2 + N 1dr2 + (r=`)2d~y2; N = A~ = Q 1 r rd 3 d ; r 2 m rd 3 + q 2 r2d 6 x = ; t = ; ds2 = `22ds^2 + r 2 `+2 d~y2; A = QA^; Q = 3)Q`22 D~ := r~ ( ; r; ~y) = d d 1 k (2 )d 1 R~k(r) ei~k ~y i! ; Z { 33 { where ds^2 is given in (2.1). The near-horizon gauge eld is related to the AdS2 potential A^ = xdt by Consider now a charged eld with mass We utilize the planar and time-translation symmetries by adopting the mode-decomposition where the momentum vector k is given by k = (!; ~k) and ~k ~y denotes the Euclidean scalar product on the d 2-dimensional transverse space. This gives rise to the radial equation N R~k00 + ! + eA~ !2 p N ` ~k !2 r 2 R~k = 0; where r+ is the location of the outer horizon and Q := q r d 3 3) `2 = (d 2)(d 2 1) : Here is the cosmological constant. Using the standard relation for the Hawking temperature T = N 0(r+)=(4 ) we nd that, at extremality (T = 0), r 2 N = +2 (r=r+ 1)2 + O (r=r+ 1)3 ; where `22 = `2=((d coordinates 2)(d 1)) is the square of the scalar curvature of AdS2. The near-horizon geometry of the extremal solution may be obtained by introducing and taking x ! 0 xing tx as usual. The resulting metric is R d 2 AdS2 (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.9) where prime denotes the radial derivative. For the zero-temperature (extremal) background, setting ! = 0 in (A.9) results in a radial equation which is di cult to work with. However, by taking the limit r ! r+ at ! = 0 using (A.3) we see that the solutions must have the asymptotics R~far (for some coe cients D , di erent for each solution), where h 1 2 r 1 ^2 = `22 2 + `22(`~k=r+)2; e^2 = e2Q2: (A.10) (A.11) (A.12) From eq. (A.11) we can read o the e ective AdS2 mass and charge as The analysis of the critical tail then proceeds identically, giving the behaviors summarized in section 5.6 with the formula (A.11) for h+. The key di erence from the compact case is that the modes are labeled by a continuous parameter ~k instead of a discrete list E. Correspondingly, each mode does not have compact spatial support. This makes it more di cult to generalize conclusions about individual modes to generic perturbations. Work in this direction is underway [53]. B Discrete modes Modes with weight h+ + ie^ 2 Z>0 are called discrete in our classi cation [table 2]. We have excluded these modes from consideration above, and we cannot treat them as limiting cases, since the coe cient C+ (3.17) appearing in the front of the power law tails vanishes in the discrete limit. Direct treatment of discrete modes in AdS2 reveals that the mode functions are analytic at ! = 0 and hence do not give any power law tail. The present framework does not make a universal prediction for discrete modes. However, discrete modes do in general possess power law tails (and the Aretakis instability), at least in the well-studied cases of four-dimensional, asymptotically at black holes. Examples of discrete modes include axisymmetric perturbations of the extremal Kerr spacetime [7], neutral (massless) scalar mode perturbations of extremal ReissnerNordstrom [14] and axisymmetric gravitational perturbations of higher-dimensional extremal rotating black holes and black rings [54]. In the four-dimensional cases we have studied in detail, the matched asymptotic expansion for the discrete modes di ers from the one we assume here (section 5.2) in two important ways. First, the far expansion must be de ned as ! ! 0 xing x! (instead of ! ! 0 xing x) in order to satisfy the outgoing conditions that de ne the 1 solution. Second, in both near and far expansions one must keep subleading terms (in !) in the equations of motion in order to satisfy all boundary and matching conditions. The correct near functions are Whittaker functions with an e ective frequency-dependent charge e^ = ! ib, which do not satisfy the wave equation in AdS2. 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Samuel E. Gralla, Peter Zimmerman. Scaling and universality in extremal black hole perturbations, Journal of High Energy Physics, 2018, 61, DOI: 10.1007/JHEP06(2018)061