Gravitational perturbations from NHEK to Kerr
Alejandra Castro, Victor Godet, Joan Simón, Wei Song, Boyang Yu
GGravitational perturbations from NHEK to Kerr
Alejandra Castro a , Victor Godet a,b , Joan Simón c , Wei Song d, e , and Boyang Yu d, e a Institute for Theoretical Physics, University of Amsterdam, Science Park 904, Postbus 94485,1090 GL Amsterdam, The Netherlands b International Centre for Theoretical Sciences (ICTS-TIFR), Tata Institute of Fundamental Research,Shivakote, Hesaraghatta, Bangalore 560089, India c School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh,Edinburgh EH9 3FD, United Kingdom d Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China e Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China [email protected], [email protected], [email protected], [email protected],[email protected]
ABSTRACT
We revisit the spectrum of linear gravitational perturbations of the (near-)extreme Kerrblack hole. Our aim is to characterise those perturbations that are responsible for the devia-tions away from extremality, and to contrast them with the linearized perturbations treatedin the Newman-Penrose formalism. For the near horizon region of the (near-)extreme Kerrsolution, i.e. the (near-)NHEK background, we provide a complete characterisation of ax-isymmetric modes. This involves an infinite tower of propagating modes together with themuch subtler low-lying mode sectors that contain the deformations driving the black holeaway from extremality. Our analysis includes their effects on the line element, their contri-butions to Iyer-Wald charges around the NHEK geometry, and how to reconstitute them asgravitational perturbations on Kerr. We present in detail how regularity conditions along theangular variables modify the dynamical properties of the low-lying sector, and in particulartheir role in the new developments of nearly-AdS holography. a r X i v : . [ h e p - t h ] F e b ontents (cid:96) = 0 and (cid:96) = 1 sl (2) charges and angular momentum 29 (cid:96) = 0 and (cid:96) = 1 A Aspects of Teukolsky formalism 52
A.1 Overview 52A.2 Newman-Penrose formalism & master equations 54A.3 Kerr & NHEK specifics 56A.4 Wald’s theorem [30]. 59A.5 Further identities and proofs 60
B Nearly-AdS holography 62C Plebański–Demiański type D solutions 64 C.1 Changing extremal mass 65C.2 Adding NUT charge 65C.3 Accelerated NHEK 66
D Isometries of (near-)NHEK 66 Introduction
Gravitational perturbations of a black hole illustrate the invaluable interplay between theoryand experiment (or numerical simulation) in general relativity. For example, any progress inthe analytic calculations controlling the physics of extremal mass ratio inspirals (EMRI) canbe of experimental relevance since it can give rise to more accurate waveforms used in thedata analysis algorithms searching for gravitational wave signals. This synergy between theory, experiment, and numerical simulations, has been furthercrossed in recent years when the primary black hole in the binary system is near-extremal.The enhancement of symmetry from R × u (1) to sl (2 , R ) × u (1) in the near horizon region andthe use of asymptotic matching techniques allows the computation of some observables eitherexactly, or with high analytic accuracy; see [9–17] and references therein. What this bodyof work stresses is that the gravitational radiation from near-extremal primaries has ratherunique features and can be used as a smoking gun for identifying these objects in the sky. A further, and more recent, development in the theory side has been the identificationof the relevant degrees of freedom describing the low energy physics driving a black holeaway from extremality. Based on ideas from nearly-AdS holography [22, 23], these degreesof freedom arise from breaking the reparametrization symmetries of the AdS throat thatappear in the near horizon region of the extremal black hole. This mechanism includes aspontaneous plus an explicit symmetry breaking pattern, leading to the construction of aneffective field theory description for the resulting pseudo-Goldstone modes. This low energysector determines important aspects of the gravitational backreaction, and several propertiesthat are key to our microscopic (quantum) understanding of black hole physics.However, whereas gravitational perturbations of Kerr black holes are typically formulatedusing the Teukolsky formalism [24–26], the description of nearly-AdS holography physicsis typically done in the context of Jackiw-Teitelboim (JT) gravity [27, 28], or similar two-dimensional gravitational theories. The former is a gauge invariant description based on theNewman-Penrose formalism whose relation to measurable quantities in gravitational wavephysics is known. The latter is based on some specific choice of gauge and is typically tied tothe near horizon geometry, from which its universality comes from. It is natural to ask howthe features of JT gravity appear in the Teukolsky formulation and how they are glued to theasymptotically flat physics that we observe. We will refer to the gravitational perturbations More precisely, EMRIs are one of the most exciting sources of gravitational radiation for the space-baseddetector LISA [1]. However, they are also very challenging to model and to extract data [2–4]. This is becauseEMRIs will be observable for a large number of cycles before plunge, involving eccentric and inclined orbitsup to a few cycles before the latter. This introduces a huge amount of complexity to encode and extract suchinformation in models, but at the same time makes them suitable to test GR [5–7]. See [8] for a broaderperspective on the relevance of LISA for theoretical physics. These features also leave a trace in the dynamics of the transition from inspiral to plunge in a circularequatorial orbit. In [18, 19], new potential terms responsible for different scaling behaviours were identified inthe near-extremal regime, extending the original analysis by Ori and Thorne [20]. In fact, if near-extremal Kerrblack holes exist and are observed, they are predicted to have much higher parameter estimation sensitivity,using gravitational wave probes, than regular rotating Kerr black holes and the origin for such increase can,once more, be traced to the existence of a throat in the near horizon geometry [21]. igure 1 . The near-extremal Kerr geometry, highlighting its NHEK portion. Gravitational pertur-bations of the near region, described by the NHEK geometry, are glued to perturbations in the farregion, which includes the asymptotically flat region of the black hole. The red circle represents theasymptotic boundary of the AdS throat, which lies in the matching region for the perturbations. that encode these features as the JT sector, and fields that obey the same dynamics as thescalar field in JT gravity will be called JT modes.Following these motivations and observations, the purpose of this work is twofold. First,we generalize the original results in [29] and relate axisymmetric gravitational perturbationsaround the near horizon geometry of the extremal Kerr black hole (NHEK) to the gaugeinvariant Weyl scalars appearing in the Teukolsky formulation. This involves an infinite towerof (near-)AdS modes together with the much subtler low-lying mode sectors that containmarginal extremal deformations and a JT sector responsible for driving the system away fromextremality. Second, we discuss how to glue the previous near horizon relations to the fullasymptotically flat (near-)extremal Kerr geometry. Fig. 1 depicts the various regions in thegeometry that are used in this gluing procedure. For the low-lying modes this is an intricatetask as we will discuss in detail: for smooth perturbations, diffeomorphisms enter in thisprocess which become physical states on NHEK, and there are also cases when the singularnature of some of these modes adds novel features to the matching procedure.Before presenting our general strategy and main results, and in an attempt to make thiswork minimally self-contained while taking into account the expertise of different readers, wehave included several appendices at the end of this manuscript providing brief reviews ondifferent topics. App. A discusses some relevant aspects of the standard Teukolsky formalismof gauge invariant gravitational perturbations, applied to both the Kerr black hole and theNHEK geometry, and the content of Wald’s theorem regarding gravitational perturbations onKerr [30]. App, B reviews some of the ideas in nearly-AdS holography.3 .1 Summary of our strategy and results The (near-)extremal Kerr black hole is a particular example of (near)-extremal black holeswhere the ideas of nearly-AdS holography, reviewed in App. B, should apply. However, theexplicit breaking of the spherical symmetry makes the identification of the JT sector subtle,and as we will show, adds new intricacies to the low energy description. Prior work thatincorporates aspects of rotation in nearly-AdS holography includes [31–36] and see [37–42]for extensive work on the three-dimensional BTZ black hole. Here we follow, and generalize,the approach presented in [29].To start our summary, we first focus on our results regarding the gravitational perturba-tions on NHEK. We describe axisymmetric gravitational perturbations of NHEK as follows ds = J (cid:0) θ + (cid:15)χ ( x , θ ) (cid:1) (cid:104) g ab d x a d x b + d θ (cid:105) +4 J sin θ θ + (cid:15)χ ( x , θ ) (d φ + A a d x a + (cid:15) A a d x a ) + O ( (cid:15) ) . (1.1)Here J = M is the maximal amount of angular momentum allowed by requiring the absence ofnaked singularities. Note that we are setting G = 1 . At linear order in (cid:15) , these perturbationsare determined by a single scalar χ ( x , θ ) satisfying (cid:3) χ + sin θ cos θ ∂ θ (cid:18) cos θ sin θ ∂ θ (cid:16) χ cos θ (cid:17)(cid:19) = 0 , (1.2)where (cid:3) = ∇ a ∇ a is the Laplacian on AdS with coordinates denoted by x . After separationof variables, χ ( x , θ ) = sin θ (cid:88) (cid:96) S (cid:96) ( θ ) χ (cid:96) ( x ) , (1.3)the (cid:96) -modes on the sphere correspond to associated Legendre polynomials with (cid:96) ∈ Z and (cid:96) ≥ , while the function χ (cid:96) ( x ) satisfies the wave equation (cid:3) χ (cid:96) = (cid:96) ( (cid:96) + 1) χ (cid:96) , (1.4)on AdS . These perturbations describe a tower of AdS modes with conformal dimension ∆ = (cid:96) + 1 ≥ .Since this description is not gauge invariant, its relation to the gauge invariant quantitiesappearing at linear order in the Teukolsky formalism is not apparent. By computing theappropriate Weyl scalars, Ψ and Ψ , we show that modes with (cid:96) ≥ have a one-to-onecorrespondence with outgoing and incoming modes in the Teukolsky formalism. Hence, theseare physical and it is in accordance to prior work on gravitational perturbations in NHEK[43–45]. We will refer to them as propagating modes.There are no associated Legendre polynomials with (cid:96) = 0 , . However, these modes areallowed by the AdS Breitenlohner-Freedman bound [46]. We will refer to these as low-lying Furthermore, both Ψ and Ψ diverge at the location of such singularities. We point out that requiring the absence ofsuch divergences, i.e. Ψ = Ψ = 0 , which is a much milder condition that setting χ (cid:96) to zero,gives rise to two constraints. This is interesting for three reasons:• For Ψ = Ψ = 0 , the resulting perturbation due to χ is not a diffeomorphism plus achange of mass and/or angular momentum. This does not contradict Wald’s theorem[30] since these modes produce conical singularities on the geometry.• For (cid:96) = 1 , combining these two constraints with the AdS wave equation, coming fromthe linearized Einstein’s equations, can be shown to be equivalent to the JT equationsof motion ∇ a ∇ b Φ JT − g ab (cid:3) Φ JT + g ab Φ JT = 0 . (1.5)This equation is the key feature of nearly-AdS holography. As described in App. B,from the dynamics of (1.5) one can infer the low energy sector that arises due to thesymmetry breaking pattern.• For (cid:96) = 0 , the same constraint gives rise to a constant zero mode.Thus, including the low-lying modes in the AdS tower gives rise to an extra irrelevant per-turbation with ∆ = 2 ( (cid:96) = 1 ) and a marginal perturbation with ∆ = 1 ( (cid:96) = 0 ). Whenrequiring Ψ = Ψ = 0 , the former satisfies the JT equations of motion while the latter is aconstant zero mode. However, both perturbations remain singular since the perturbed metrichas conical singularities at either the north or south poles.To have a complete characterization of the JT sector, we balance the conical singularityof the (cid:96) = 1 low-lying mode using the following mechanism. First, we show that the Killingvectors ζ of AdS backgrounds of the form ds = Λ( θ )( g AdS + d θ ) + Γ( θ )(d φ + A a d x a ) , (1.6)which include NHEK as a particular case, are in one–to–one correspondence with a scalar field Φ( x ) solving the JT equations of motion (1.5) and a constant zero mode c ( φ ) . More explicitly, ζ = ε ba ∇ b Φ ζ ∂ a + (Φ ζ + ε ab A a ∇ b Φ ζ ) ∂ φ , with Φ ζ ≡ c ( φ ) + Φ JT . (1.7)It is important to note that these Killing vectors are determined by the same differentialequations that govern the AdS low-lying modes with vanishing Weyl scalars. Hence, Killingvectors of NHEK naturally encode a second copy of the previously identified (cid:96) = 1 and (cid:96) = 0 This observation was recently made in a similar context to ours in an appendix in [47]. Since these Weyl scalars are computed on near-NHEK, it is natural to interpret these conditions as theabsence of ingoing and outgoing energy flux at the horizon [24–26, 48]. modes. However, these are non-dynamical . To make them dynamical, we observe thata non-single valued diffeomorphism ξ µ ( x , θ, φ ) = (cid:15) φ ζ µ ( x , θ ) , (1.8)with ζ µ ( x , θ ) a Killing vector on NHEK, acting on g NHEK , gives rise to an axisymmetricperturbation, i.e. satisfying ∂ φ ( L ξ g NHEK ) = 0 . Even though the latter is locally pure gauge, itis a physical singular perturbation. Singular, because it gives rise to a conical defect, as onemay have expected from being generated by a non-single valued diffeomorphism. Physical,because it gives rise to non-trivial Iyer-Wald charges, as we explicitly compute in Sec. 4.2.The resulting perturbation takes the form ds = J (cid:0) θ + (cid:15)χ ( x , θ ) (cid:1) (cid:104) ( g ab + (cid:15)h ab ) d x a d x b + d θ (cid:105) (1.9) +4 J sin θ (1 + (cid:15) Φ( x ))1 + cos θ + (cid:15)χ ( x , θ ) (d φ + A a d x a + (cid:15) A a d x a ) + O ( (cid:15) ) , with the dynamical constraints satisfied by h ab and A a given in Sec. 3.2.1. We show that thephysics of the JT sector is driven by the NHEK perturbation with Φ JT = χ . This is the choicebalancing the conical singularities associated with each (cid:96) = 1 mode. We confirm the physicalinterpretation of this near horizon perturbation as a change of mass (plus a local diffeomor-phism), in agreement with Wald’s theorem, by gluing this near horizon perturbation with afull Kerr perturbation in Sec. 5.2. A similar mechanism to balance conical singularities appliesto the (cid:96) = 0 mode, albeit singularities in this sector have interesting physical interpretationsdiscussed in Sec. 3.2.2.Our analysis of axisymmetric low-lying modes in NHEK identifies all the possibilitiesallowed by Wald’s theorem. In the same gauge as in (1.1), these perturbations are characterisedby χ ( x , θ ) = Φ JT ( x ) + 12 (1 + cos θ ) c ( φ ) , (1.10)where we stressed the nature of the JT mode and we included the zero mode c ( φ ) . In Sec. 4.2,we compute the Iyer-Wald charges carried by these near horizon perturbations. There arethree sl (2) charges δ Q ζ − = − (cid:15)J c + , δ Q ζ = (cid:15) J c , δ Q ζ + = − (cid:15) J c − , (1.11)corresponding to the three independent solutions Φ JT = c i Φ ζ i for the JT modes, one pergenerator of sl (2) , and the u (1) charge δ Q ∂ φ = − (cid:15) J c ( φ ) . (1.12)Having identified the dynamical mechanism that is characteristic of nearly-AdS holog-raphy in NHEK, we undertake the second main goal in this work in Sec 5: how to describe6hese near horizon perturbations as full Kerr perturbations. The matching procedure of theperturbations is as follows. The starting point is the decoupling limit that relates the Kerrand NHEK backgrounds, a singular coordinate transformation on near-extreme Kerr of theform ˜ r = √ J + λ (cid:18) r + τ r (cid:19) , ˜ t = 2 J tλ , ˜ φ = φ + √ J tλ , λ → . (1.13)Starting from a perturbation on Kerr, we implement this limit. Our requirement is that themetric perturbations, and associated Iyer-Wald charges, do not diverge as we take λ → .This allows us to match our analysis of perturbations on NHEK with those on Kerr. Ourdiscussion here follows the same organization as above: we distinguish the propagating andlow-lying modes. A summary of our results is presented in Table 1.Our reconstruction strategy for propagating modes is standard. Since the Hertz potentialdetermines the metric perturbation in the ingoing radiation gauge (IRG), we use asymptoticmatching techniques for near-extremal Kerr [26, 45] to solve the master equation (5.3) satisfiedby the Hertz potential. Specifically, we follow a three-step algorithm: we first glue the KerrHertz potential ˜Ψ H to the NHEK Hertz potential Ψ H , reconstruct the NHEK perturbationin IRG using the latter, and finally use a small diffeomorphism to bring the perturbation tothe gauge (1.1). Our main technical result is the relation between our scalar perturbation χ ( x , θ ) and Ψ H : χ ( x , θ ) = − sin θ l a l b ∇ a ∇ b Ψ H ( x , θ ) . (1.14)Furthermore, decomposing Ψ H ( x , θ ) = (cid:80) (cid:96) ≥ U (cid:96) ( x ) S (cid:96) ( θ ) , we also obtain the inverse relation U (cid:96) ( x ) = − (cid:96) − (cid:96) ( (cid:96) + 1)( (cid:96) + 2) n a n b ∇ a ∇ b χ (cid:96) ( x ) . (1.15)Here l a and n a are tetrads for AdS , given in App. A.5. These explicit maps relate, for (cid:96) ≥ ,our specific gauge with the more common radiation gauge used in gravitational wave physics.In the reconstruction of low-lying mode perturbations, we consider two different cases:smooth and singular perturbations. Smooth perturbations have vanishing Weyl scalars andtheir would-be conical singularity is compensated by the transformation (1.8). Thus, theirdescription in the full Kerr geometry is constrained by Wald’s theorem, and we give a detailedanalysis in Sec. 5.2. Singular perturbations can have both vanishing or non-vanishing Weylscalars. We treat these using similar techniques as the ones outlined for the propagatingmodes, and it is the focus of Sec. 6.According to Wald’s theorem, the reconstruction of smooth low-lying perturbations inKerr, with metric ˜ g , must be a linear combination of mass ( δ M ˜ g ) , angular momentum ( δ J ˜ g ) perturbations and a diffeomorphism ( L ˜ ξ ˜ g ) , i.e. , δ ˜ g = δ M ˜ g + δ J ˜ g + (cid:15) L ˜ ξ ˜ g , (1.16)7 odes ∆ Weyl scalars Properties on Kerr propagating (cid:96) ≥ (cid:96) ( (cid:96) + 1) non-zero Well-behaved axisymmetric perturbationssmooth (cid:96) = 1 M (or J ) deformation with fixed J (or M ),plus diffeomorphismssmooth (cid:96) = 0 M and J deformation with fixed J = M ,plus diffeomorphisms.singular (cid:96) = 0 , (cid:96) = 0 C -metric deformationsingular (cid:96) = 1 zero (NHEK)non-zero (Kerr) (cid:63) Separable Hertz potential
Table 1 . Summary of the different classes of perturbations considered in our work. Here (cid:96) is anonnegative integer that controls the angular θ dependence. ∆ = (cid:96) + 1 is the conformal dimension ofthe perturbations as viewed in NHEK. The third column describes the value of the Weyl scalars ˜Ψ and ˜Ψ on Kerr. The last column briefly describes properties of the perturbations on Kerr. (cid:63) Depending on the specific configuration with vanishing Weyl scalars on NHEK, the resulting Weylscalar on Kerr can be either zero or non-zero. with δM ∼ (cid:15) and δJ ∼ (cid:15) . We show that all finite perturbations of this type as λ → ,correspond to smooth low-lying perturbations characterised by (1.10). We also quantify theirnear horizon charges using (1.11)-(1.12). These results give extra support to the physicalinterpretation of these near horizon perturbations. We refer the readers to the discussion inSec. 5.2 for details. The key features we would like to highlight are:1. In the absence of any explicit perturbation of Kerr, we can obtain a perturbation ofNHEK as the first correction to the near horizon decoupling limit (1.13). This corre-sponds to the choice (cid:15) ∼ λ with (cid:15)χ ( x , θ ) = (cid:15)χ ( x ) = (cid:15) Φ JT ( x ) = 2 √ J λ (cid:18) r + τ r (cid:19) . (1.17)The action of SL(2 , R ) can generate the full multiplet as in (1.11).2. There is also a sector characterised by diffeomorphisms on Kerr with δ M ˜ g = δ J ˜ g = 0 .Restricting ourselves to single-valued diffeomorphisms with support on the sphere, weshow the diffeomorphisms ˜ ξ that are well defined in the decoupling limit have a nearhorizon expansion ξ [ λ ] = (cid:15) λ − ( ξ (-1) + λξ (0) + · · · ) , (1.18)with ξ (-1) = a i ζ i + a φ ζ ( φ ) . (1.19)8ere ξ [ λ ] is the pullback of ˜ ξ from Kerr to NHEK under the decoupling limit; ζ i and ζ ( φ ) are Killing vectors of NHEK. We identify this perturbation with a near horizonperturbation (1.9) with χ = Φ JT . The Killing vector part carries sl (2) × u (1) chargeswith c = 0 and c ± (cid:54) = 0 , i.e. , these transformations carry neither energy nor angularmomentum.It is very important to remark that although the procedure starts from a diffeomorphismon the Kerr geometry, as one takes the decoupling limit, the resulting perturbation is not a diffeomorphism on NHEK. Also, in our discussion, ξ (0) is constructed such thatthe resulting perturbation matches with (1.9): this is a choice of boundary conditionson NHEK. And for this choice, ξ (0) does not contribute to the Iyer-Wald charges.3. Mass and angular momentum perturbations are described as follows. The first observa-tion is that δ M g [ λ ] ∼ δM λ − . If δM ∼ λ (cid:15) , the perturbation is finite and correspondsto a nearby near-NHEK with Hawking temperature τ (cid:48) H = τ H (cid:18) √ J λ − δM π τ H (cid:19) . (1.20)However the resulting perturbation carries no sl (2) charges. When δM ∼ λ(cid:15) , we cancombine this transformation with (1.18) to again obtain a NHEK perturbation with χ = Φ JT in (1.9). The result is (cid:15) χ ( x , θ ) = (cid:15) Φ( x ) = 4 λ − δMτ (cid:18) r + τ r (cid:19) . (1.21)The difference with (1.17) is that we don’t need to identify (cid:15) with λ , and hence theIyer-Wald charges (1.11) are finite even when λ = 0 .For an angular momentum perturbation, δ J g [ λ ] , the discussion and outcome is similaras that of the mass perturbation.4. One interesting class of marginal perturbations corresponds to δJ = 2 √ J δM , i.e. de-formations of the mass and angular momentum while keeping the black hole extremal.In this case, ( δ M + δ J ) g [ λ ] = δM ( λ − h (-1) + h (0) + · · · ) . (1.22)The most interesting scaling is when δM ∼ λ (cid:15) , which can be combined with a diffeo-morphism to give one of our modes χ ( x , θ ) = Φ( x ) = c ( φ ) , δM = c ( φ ) √ J . (1.23)Finally, we consider the reconstruction in Kerr of the singular low-lying modes, i.e. , modeswith (cid:96) = 0 , for which we impose no regularity conditions on the sphere. The details are inSec. 6. A very interesting feature in this case is that the Hertz potential Ψ H on NHEK does9 ot allow for a separation of variables ansatz, but has to be written as a sum of two terms.More explicitly, we have for (cid:96) = 1Ψ H = S ( θ ) U ( x ) − S inhom ( θ ) n a n b ∇ a ∇ b χ ( x ) . (1.24)Here S inhom ( θ ) is given by (6.6), and we have l a ∇ a U ( x ) = n a ∇ a χ ( x ) . (1.25)A similar construction also holds for (cid:96) = 0 . Our analysis includes a discussion on how to applya matching procedure to ˜Ψ H , and the challenges that a non-separable solution pose. Anotherstriking feature concerns the special case when Ψ = Ψ = 0 on NHEK: we show that thecorresponding Weyl scalars on Kerr are in general non-trivial, and only become zero in thedecoupling limit.This paper is organized as follows. In Sec. 2, after reviewing some aspects of NHEK andnear-NHEK, including how it is obtained as a near horizon limit of (near-)extremal Kerr, wedescribe in Sec. 2.1 an explicit one–to–one correspondence between Killing vectors of NHEK-like geometries and a scalar field solving the JT equations of motion (plus a constant zeromode). In Sec. 3, we start our study of axisymmetric perturbations in NHEK. Propagatingmodes are discussed in Sec. 3.1, while low-lying modes are presented in Sec. 3.2. The balancingmechanism giving rise to the smooth JT mode is discussed in Sec. 3.2.1, while the discussionof marginal deformations of NHEK corresponding to the ones allowed by Wald’s theorem isgiven in Sec. 3.2.2, though the technical derivations are left to App. C. Iyer-Wald chargesof our NHEK perturbations are computed in Sec. 4. In Sec. 5, we explain our techniquesto match/glue our NHEK perturbations with Kerr perturbations using Wald’s theorem andthe reconstruction of gauge invariant perturbations based on the Hertz potential. Finally, wediscuss in Sec. 6 some properties of singular low-lying perturbations. Our appendices includevarious complementary material related to the main sections. The Kerr black hole metric in Boyer-Lindquist coordinates is ds = − Σ ∆(˜ r + a ) − ∆ a sin θ d˜ t + Σ (cid:18) d˜ r ∆ + d θ (cid:19) + sin θ Σ ((˜ r + a ) − ∆ a sin θ ) (cid:18) d ˜ φ − aM ˜ r (˜ r + a ) − ∆ a sin θ d˜ t (cid:19) , (2.1)with ∆ = (˜ r − r − )(˜ r − r + ) , Σ = ˜ r + a cos θ . (2.2)The outer ( r + ) and inner ( r − ) horizons are r ± = M ± √ M − a . We set Newton’s constant G = 1 , so that M is the mass and J = aM is the angular momentum of the black hole.10ilde coordinates (˜ t, ˜ r, ˜ φ ) refer to the asymptotically flat black hole to distinguish them from ( t, r, φ ) in the near horizon geometry below; θ is unchanged.We are interested in extreme Kerr corresponding to J = M , i.e. a = M , when bothhorizons coalesce ( r + = r − ) . These black holes develop an AdS throat in the region close tothe horizon that can be decoupled from the asymptotically flat description in (2.1) by takingthe limit λ → in the change of coordinates ˜ r = r + + λr ˜ t = 2 r tλ , ˜ φ = φ + r + tλ , (2.3)while keeping all other parameters fixed. This near horizon limit leads to the line element ds = g NHEK µν d x µ d x ν = J (1 + cos θ ) (cid:20) − r d t + d r r + d θ (cid:21) + J θ θ [d φ + r d t ] . (2.4)This is the Near Horizon geometry of Extreme Kerr (NHEK) [49, 50].The isometries for the full Kerr geometry (2.1), given by R × u (1) , are enhanced to sl (2 , R ) × u (1) in NHEK (2.4). The four Killing vectors generating the latter are ζ − = ∂ t , ζ = t∂ t − r∂ r , ζ + = (cid:18) r + t (cid:19) ∂ t − rt∂ r − r ∂ φ , (2.5)and ζ ( φ ) = ∂ φ . (2.6)The decoupling limit (2.3) introduces some arbitrariness on how we relate the AdS time t withthe asymptotically flat time ˜ t . This freedom gives rise to a set of diffeomorphisms preservingthe asymptotic structure of the NHEK metric. More explicitly, we take [51, 52] t −→ f ( t ) + 2 f (cid:48)(cid:48) ( t ) f (cid:48) ( t ) r f (cid:48) ( t ) − f (cid:48)(cid:48) ( t ) ,r −→ r f (cid:48) ( t ) − f (cid:48)(cid:48) ( t ) r f (cid:48) ( t ) ,φ −→ φ + log (cid:18) rf (cid:48) ( t ) − f (cid:48)(cid:48) ( t )2 rf (cid:48) ( t ) + f (cid:48)(cid:48) ( t ) (cid:19) . (2.7)The arbitrariness is reflected on the arbitrary function f ( t ) that redefines the time in the nearhorizon region. Acting on (2.4), this diffeomorphism gives ds = J (1 + cos θ ) (cid:34) − r (cid:18) { f ( t ) , t } r (cid:19) d t + d r r + d θ (cid:35) (2.8) + 4 J sin θ θ (cid:20) d φ + r (cid:18) − { f ( t ) , t } r (cid:19) d t (cid:21) , { f ( t ) , t } = (cid:18) f (cid:48)(cid:48) f (cid:48) (cid:19) (cid:48) − (cid:18) f (cid:48)(cid:48) f (cid:48) (cid:19) . (2.9)We can see here that leading terms as r → ∞ in (2.8) approach (2.4), i.e. the diffeomorphism(2.7) only affects subleading components of the line elements in an expansion in r . It isimportant to note that these are not the same boundary conditions used in Kerr/CFT [50]:the set of allowed diffeomorphisms there does not overlap with those here (with the exemptionof the u (1) Killing vector).In the analysis of the subsequent sections, a particular choice of f ( t ) selects a backgroundfor which we will quantify the gravitational perturbations. For example, f ( t ) = t returns usto (2.4). Another choice which we will use frequently is f ( t ) = e τt , with τ constant. It followsthat { f ( t ) , t } = − τ , leading to the background ds = J (1 + cos θ ) (cid:34) − r (cid:18) − τ r (cid:19) d t + d r r + d θ (cid:35) (2.10) + 4 J sin θ θ (cid:20) d φ + r (cid:18) τ r (cid:19) d t (cid:21) . This corresponds to the so-called near-NHEK geometry [44]. It can be also obtained froma near-extremal Kerr black hole (2.1), where the decoupling limit (2.3) allows for a smallincrease of the mass while keeping the angular momentum of the black hole fixed to J . Moreconcretely, we introduce a small deviation away from extremality of the form r ± = √ J ± λτ + τ λ √ J + O ( λ ) , (2.11)with τ finite and positive, and the decoupling limit λ → becomes ˜ r = √ J + λ (cid:18) r + τ r (cid:19) , ˜ t = 2 J tλ , ˜ φ = φ + √ J tλ , (2.12)where the choice of coordinate r is a choice of gauge to keep the radial metric componentindependent of τ . These steps lead to the near-NHEK geometry (2.10). In this subsection we revisit the isometries of (near-)NHEK using a more covariant formalism.We will extend the original discussion in [53] for AdS to backgrounds of the form ds = Λ( θ )( g ab d x a d x b + d θ ) + Γ( θ )(d φ + A a d x a ) . (2.13)12 ( θ ) and Γ( θ ) are two functions of θ . Here x a are coordinates in 2D, and the metric g ab corresponds to a locally AdS spacetime. Tensors and other covariant objects below are definedrelative to this spacetime, e.g., covariant derivatives ( ∇ a ) or the Laplacian ( (cid:3) = ∇ a ∇ a ) . A a is a gauge field supported on this 2D spacetime, and the field strength associated to it is d A = − ε ab d x a ∧ d x b , (2.14)with ε ab the Levi-Civita tensor. Our expressions will turn out to the covariant with respectto the AdS metric, and hence hold as well for the nearly-AdS geometries such as the one in(2.8). Still, it will be convenient to write some expressions explicitly; a choice of backgroundwe will commonly use for the AdS metric and gauge field are g ab d x a d x b = − r d t + d r r , A a d x a = r d t , (2.15)with Levi-Civita tensor ε tr = 1 .In the following we will build a one-to-one map between the isometries of (2.13) and ascalar field satisfying some suitable equations of motion. We will show this scalar field corre-sponds to the JT field in parallel with [53], with an additional term due to the axisymmetryof the background (2.13).Let ζ be a Killing vector field of (2.13). It follows from ∇ θ ζ θ = 0 that ζ θ = 0 . We canthen split the Killing vector ζ into a 2D vector field ζ a ∂ a and the φ -component ζ φ accordingto ζ = ζ a ( x a , φ ) ∂ a + ζ φ ( x a , φ ) ∂ φ . (2.16)The variation of the line element under the diffeomorphism generated by (2.16) equals δ ζ ( ds ) = 2Λ( θ )( ∇ ( a ζ b ) d x a d x b + g ab ∂ φ ζ b d x a d φ )+2Γ( θ )(( ∂ φ ζ φ + A a ∂ φ ζ a )d φ + ( ∂ a ζ φ + L ζ A a )d x a )(d φ + A a d x a ) , (2.17)where all indices are raised and lowered by g ab . Since Λ( θ ) and Γ( θ ) are two independentfunctions, and ζ, g ab , A a are independent of θ , the Killing equations guaranteeing the vanishingof δ ζ ( ds ) reduce to ∂ φ ζ a = ∂ φ ζ φ = 0 , (2.18) ∇ ( a ζ b ) = 0 , (2.19) ∂ a ζ φ + L ζ A a = 0 . (2.20)The first implies that ζ only depends on 2D coordinates, i.e. ζ a = ζ a ( x a ) , ζ φ = ζ φ ( x a ) ,whereas (2.19) implies the 2D vector ζ a ∂ a satisfies a 2D Killing equation. Contracting (2.19) The subsequent discussion does not depend on the explicit form of Λ( θ ) and Γ( θ ) . For NHEK and near-NHEK, they can be read from (2.4) and (2.10), respectively. g ab , we get ζ a ∂ a is divergence free, i.e. ∇ · ζ = 0 , enabling us to write it as the curl ofan scalar Φ ζ ζ a = ε ba ∇ b Φ ζ . (2.21)Integrating (2.20) determines the full Killing vector ζ to be ζ = ε ba ∇ b Φ ζ ∂ a + (Φ ζ + ε ab A a ∇ b Φ ζ ) ∂ φ , (2.22)where we used (2.14), (2.19)-(2.21) and absorbed the integral constant into Φ ζ . Notice thatgiven a Killing vector ζ , the scalar Φ ζ can be reconstructed by Φ ζ = ζ φ + A a ζ a . (2.23)Finally, substituting (2.21) into the 2D Killing equations (2.19), we get ∇ ( a ζ b ) = ε a c ( ∇ c ∇ b Φ ζ − g cb (cid:3) Φ ζ ) = 0 . (2.24)This is equivalent to a set of differential equations T ab [Φ ζ ] ≡ ∇ a ∇ b Φ ζ − g ab (cid:3) Φ ζ = 0 . (2.25)Thus, the existence of Killing vectors ζ solving (2.18)-(2.20) is equivalent to (2.25) evaluatedon the background (2.13).We now show that the solutions to (2.25) are equivalent to the JT modes (1.5) plus theaddition of a zero mode. First, we note that (2.25) is the traceless portion of (1.5), andhence any solution to the JT equation will comply with T ab [Φ ζ ] = 0 . However (2.25) has oneadditional solution. To see this, evaluate the divergence of T ab [Φ ζ ] : this gives ∇ a T ab [Φ ζ ] = 12 ∇ b ( (cid:3) − ζ = 0 . (2.26)Its general solution is a linear combination of a constant mode, which we will denote as c ( φ ) ,and the solution to ( (cid:3) − ζ = 0 . (2.27)It is then clear that (2.27) together with the Killing equation (2.25) is equivalent to the JTequations (1.5). Hence, the general solution of (2.25) consists of JT modes and a zero modewhich we cast as Φ ζ ≡ c ( φ ) + Φ JT . (2.28)To sum up, given a Killing vector ζ , we can construct an scalar field Φ ζ via (2.23) satisfyingthe equations of motion (2.25). Conversely, given a scalar Φ ζ satisfying (2.25), the vector field(2.22) is an isometry. This establishes the sought equivalence between the isometries of thebackground (2.13) and a linear combination of JT modes and a zero mode c ( φ ) as reflected in142.28).We close this general discussion by connecting the above conclusion with the explicitKilling vectors in (2.5) for the NHEK geometry (2.4). Since ζ Φ is a Killing vector, it is alinear combination of the sl (2 , R ) × u (1) generators ζ Φ = c − ζ − + c ζ + c + ζ + + c ( φ ) ∂ φ , (2.29)with c ± , constants. The corresponding scalars, dual to the u (1) and the sl (2 , R ) isometries,are the zero mode c ( φ ) and the JT modes Φ JT , respectively Φ ζ = c ( φ ) + Φ JT , Φ JT = c i Φ ζ i , i = − , , + . (2.30)where the components of the JT field for (2.15) read c i Φ ζ i = c − r + c r t + c + (cid:18) r t − r (cid:19) . (2.31)See App. D for a construction of the Killing vectors and Φ JT for near-NHEK.Given the one-to-one map between ζ i and Φ ζ i , the set Φ ζ i forms a representation of sl (2 , R ) . Let us define the bilinear η ij ≡ η (Φ ζ i , Φ ζ j ) = − (cid:16) ∇ a Φ ζ i ∇ a Φ ζ j − Φ ζ i Φ ζ j (cid:17) . (2.32)This is invariant under the adjoint action, i.e. η ([ ζ i , ζ j ] , ζ k ) + η ( ζ j , [ ζ i , ζ k ]) = 0 . Since the sl (2) algebra is simple, the invariant bilinear form is unique up to a constant factor. By explicitcomputation, one can verify that η ij is just the Killing form of the sl (2) algebra, whose nonzeroentries are given by η − + ≡ η (Φ ζ − , Φ ζ + ) = η (Φ ζ + , Φ ζ − ) ≡ η + − = − , η ≡ η (Φ ζ , Φ ζ ) = 2 . (2.33)This bilinear form can be used to write the coefficients c i in (2.30) in terms of Φ ζ i and Φ JT using the inverse matrix η ij to η ij c i = η ij η (Φ ζ j , Φ JT ) . (2.34)More explicitly, c − = − η (Φ ζ + , Φ JT ) , c = 12 η (Φ ζ , Φ JT ) , c + = − η (Φ ζ − , Φ JT ) . (2.35)It is also useful to record the identity Φ [ ζ i ,ζ j ] = ε ba ∇ b Φ ζ i ∇ a Φ ζ j = ζ ai ∇ a Φ ζ j , (2.36)for any ζ i , ζ j ∈ sl (2) , which simply follows from the algebra.15 Gravitational perturbations on NHEK
In this section we will characterise axisymmetric ( φ -independent) gravitational perturbationsaround the NHEK background (2.4), generalizing the results in [29]. In particular, we willrelate our description of these perturbations to the more familiar Teukolsky formalism forgravitational perturbations [24, 54]: this will allow us to distinguish among excitations thatcorrespond to normalizable propagating degrees of freedom and modes that affect the globalproperties of the black hole.Let us cast the gravitational fluctuations around the generalized NHEK background as ds = J (cid:0) θ + (cid:15)χ ( x , θ ) (cid:1) (cid:104) g ab d x a d x b + d θ (cid:105) +4 J sin θ θ + (cid:15)χ ( x , θ ) (d φ + A a d x a + (cid:15) A ) + O ( (cid:15) ) . (3.1)The background, corresponding to (cid:15) = 0 , is described in (2.13). The axisymmetric deforma-tions from NHEK we have introduced here involve an scalar field χ ( x , θ ) and a one-form A supported in the x a = ( t, r ) subspace A = A a ( x , θ )d x a . (3.2)Consider the metric (3.1) at linear order in (cid:15) . To study the dynamics of the perturbations,we impose that (3.1) satisfies the linearized vacuum Einstein equations, i.e. R µν = R (0) µν + (cid:15)R (1) µν + O ( (cid:15) ) ! = 0 . (3.3)Setting R (1) µν = 0 , gives ∂ θ (cid:16) ∇ b A b (cid:17) = 0 , sin θ ∂ θ (cid:16) ε ab A b (cid:17) + cot θ ∇ a χ = 0 ,ε ab ∇ a A b − cos θ sin θ ∂ θ (cid:16) χ cos θ (cid:17) = 0 , (3.4)and (cid:3) χ + sin θ cos θ ∂ θ (cid:18) cos θ sin θ ∂ θ (cid:16) χ cos θ (cid:17)(cid:19) = 0 , (3.5)where (cid:3) is the Laplacian on AdS . Note that once χ is specified, it is straightforward to solvefor A a from (3.4). For this reason, from now on we will treat χ as the independent variablefor the metric perturbation, whose equation of motion is given by (3.5).It is not common to cast gravitational perturbations for the Kerr black hole, or its nearhorizon NHEK geometry, as explicitly as in (3.1). The drawbacks of starting from such an Here and in subsequent expressions we are using the shorthand notation f ( x ) := f ( x a ) = f ( t, r ) . In the following, we explainhow to systematically overcome both of them.The perturbations of the Kerr metric are commonly characterised by the Weyl scalarsin the Teukolsky formalism, since these are gauge invariant quantities at linear order. InApp. A we review the general strategy of this approach, and provide the relevant definitions.In particular we will focus on Ψ = C µναβ l µ m ν l α m β , Ψ = C µναβ n µ ¯ m ν n α ¯ m β , (3.6)where C µναβ is the Weyl tensor and the vectors l µ , n µ and m µ are introduced in (A.9) andidentified in (A.26) for NHEK. Ψ and Ψ are the Weyl scalars that characterise in a dif-feomorphism invariant way a massless spin s = ± perturbation. To relate the Teukolskyformalism with our ansatz, we evaluate (3.6) using our perturbations (3.1). To linear order in (cid:15) , this gives Ψ = − (cid:15) θ l a l b ∇ a ∇ b χ + O ( (cid:15) ) , Ψ = − (cid:15) J sin θ (1 − i cos θ ) n a n b ∇ a ∇ b χ + O ( (cid:15) ) , (3.7)where n and l are the AdS counterparts of (A.26), defined in (A.40). Note that in deriving(3.7) we have not used the equations of motion for χ , (3.5), but we did use (3.4).The direct relation between χ ( x , θ ) and Ψ , captured by (3.7) establishes the physicalcontent of our scalar perturbations χ ( x , θ ) . In the next subsection we will make this connectionmore explicit by analyzing the solution to (3.5) and placing it in the context of the solutionsto the Teukolsky equation (A.36). In this subsection we describe the physical content encoded in the scalar perturbation χ ( x , θ ) in (3.1). Its equation of motion (3.5) allows to use separation of variables χ ( x , θ ) = sin θ S ( θ ) χ ( x ) . (3.8)It follows S ( θ ) satisfies S (cid:48)(cid:48) + cot θ S (cid:48) + (cid:18) K − θ (cid:19) S = 0 . (3.9) To emphasize, the advantages of using (3.1) are that it was simple to solve the linearized Einstein equations,and we can easily quantify their effect in the spacetime as we will see in subsequent analysis. χ ( x ) satisfies (cid:3) χ ( x ) = Kχ ( x ) . (3.10)At this stage K is a real eigenvalue relating the angular equation (3.9) to the Laplacian onAdS in (3.10). Using the terminology of the AdS/CFT correspondence, the AdS conformaldimension equals ∆ ± = 12 ± (cid:114)
14 +
K . (3.11)We observe that for K > − , the field χ ( x ) is above the Breitenlohner-Freedman stabilitybound in AdS [55].The allowed values of K can be assessed from properties of the solutions to (3.9). Changingits independent variable to x = cos θ , the latter becomes ∂ x (cid:0) (1 − x ) ∂ x S (cid:1) + (cid:18) K − − x (cid:19) S = 0 . (3.12)We recognise this as a particular case of the spin-weighted spheroidal harmonics [24] ∂ x (cid:0) (1 − x ) ∂ x S (cid:1) + (cid:18) λ + s + c x − csx − ( m + sx ) − x (cid:19) S = 0 , (3.13)corresponding to the specific values m = c = 0 , λ = K − s ( s + 1) , s = ± . (3.14)To identify the space of normalizable solutions, with respect to the inner product inheritedfrom the Sturm-Liouville theory, notice that (3.12), or (3.9), is mathematically equivalent tothe standard spherical harmonics equation. Hence, the general solution to (3.12) is S ( θ ) = c P (2) (cid:96) (cos θ ) + c Q (2) (cid:96) (cos θ ) , (cid:96) = 12 ( − √ K ) , (3.15)where P ( m ) (cid:96) and Q ( m ) (cid:96) are the associated Legendre functions. Requiring the solutions to be smooth and normalizable functions of x = cos θ restricts (cid:96) to be (cid:96) ∈ Z , (cid:96) ≥ , (3.16)and discards the Q -branch of solutions in (3.15). To sum up, regularity of the solutions to(3.5) in the angular θ -direction gives rise to the mode expansion χ ( x , θ ) = sin θ (cid:88) (cid:96) ≥ S (cid:96) ( θ ) χ (cid:96) ( x ) . (3.17)18he spin-weighted spherical harmonic S (cid:96) ( θ ) equals an associated Legendre polynomial S (cid:96) ( θ ) = P (2) (cid:96) (cos θ ) , (cid:96) = 2 , , . . . , (3.18)and χ (cid:96) ( x ) satisfies the AdS wave equation (cid:3) χ (cid:96) = (cid:96) ( (cid:96) + 1) χ (cid:96) , (3.19)where we used that the separation constant is K = (cid:96) ( (cid:96) +1) in (3.10) . In the AdS terminology,these regular modes are interpreted as fields of conformal dimension ∆ = (cid:96) + 1 ≥ . Hence,they correspond to irrelevant operators in the context of AdS/CFT.Requiring smoothness and normalizability is common in the discussion of gravitationalperturbations when determining a basis of angular eigenfunctions. To relate this discussionfurther to the traditional literature, we return to the Weyl scalars in the Teukolsky formalism:for each single mode (cid:96) in (3.17) inserted in (3.7), we find Ψ = − (cid:15) S (cid:96) ( θ ) l a l b ∇ a ∇ b χ (cid:96) ( x ) + O ( (cid:15) ) , Ψ = − (cid:15) J S (cid:96) ( θ )(1 − i cos θ ) n a n b ∇ a ∇ b χ (cid:96) ( x ) + O ( (cid:15) ) . (3.20)It is evident that the same special function S (cid:96) ( θ ) controls both χ ( x , θ ) and Ψ , . In particular,they satisfy the same ODE (A.38). Furthermore, it is also straightforward to verify that l a l b ∇ a ∇ b χ (cid:96) ( x ) , n a n b ∇ a ∇ b χ (cid:96) ( x ) , (3.21)correspond to U s ( x ) in (A.37) with s = ± respectively, and the radial equation (A.39) iscompatible with the wave equation (3.19). All these features identify χ ( x , θ ) in terms ofTeukolsky modes and show that our NHEK ansatz captures all the gravitational modes in the m = 0 sector. As we will further discuss in Sec. 5, this is correct for (cid:96) ≥ because (3.21) isnon-zero for these propagating modes. The discussion is subtler for the (cid:96) = 0 , sectors, as weshall start discussing in Sec. 3.2.The two derivative combinations in (3.21) have a natural interpretation which is manifestwhen working in Eddington-Finkelstein coordinates, either ( u, r ) or ( v, r ) , u = t + 1 r , v = t − r . (3.22)These are smooth coordinates across the horizons allowing to write the AdS Poincaré metric(2.15) as ds = − r d v + 2d v d r = − r d u − u d r . (3.23) Here m is the Fourier mode for the azimuthal direction as defined in (A.30). Ψ = − (cid:15) S (cid:96) ( θ ) ∂ r χ (cid:96) ( u, r ) + O ( (cid:15) ) : outgoing mode , Ψ = − (cid:15)r J S (cid:96) ( θ )(1 − i cos θ ) ∂ r χ (cid:96) ( v, r ) + O ( (cid:15) ) : ingoing mode . (3.24)This matches the physical interpretation of Ψ and Ψ as describing outgoing and ingoing fluxfor the AdS scalar perturbations, as customary in the Teukolsky formalism [24]. (cid:96) = 0 and (cid:96) = 1 Based on the regularity conditions around (3.18) satisfied by the functions S (cid:96) ( θ ) , it wouldseem natural to end the discussion of the spectrum of axisymmetric perturbations there.However, it is worth exploring whether there is any physics in the solutions that are notregular on the sphere. We will see that these modes tamper with the global properties of thegeometry. Furthermore, they do it in an interesting way that will allow us to identify the JTmode responsible for making the extremal Kerr black hole non-extremal, as we will discuss insubsequent sections.Let us relax the smoothness and normalizable restrictions on S (cid:96) ( θ ) in (3.15), by allowingmeromorphic solutions on the sphere while respecting the Breitenlohner-Freedman bound in(3.10). This permits two more values of K : K = 0 : ∆ = 1 , (cid:96) = 0 ,K = 2 : ∆ = 2 , (cid:96) = 1 . (3.25)We have the decomposition χ ( x , θ ) = sin θ (cid:88) (cid:96) =0 , S (cid:96) ( θ ) χ (cid:96) ( x ) , (3.26)where S (cid:96) ( θ ) should solve (3.9). For (cid:96) = 0 and (cid:96) = 1 , there is no associated Legendre polyno-mials of the first kind but we find in each case the two linearly independent solutions (cid:96) = 0 : S ( θ ) = 1sin θ (cid:0) s +0 (1 + cos θ ) + s − cos θ (cid:1) , (3.27) (cid:96) = 1 : S ( θ ) = 1sin θ (cid:0) s +1 + s − cos θ (cid:0) cos θ − (cid:1)(cid:1) , (3.28)with s ± and s ± constants, differing in their parity properties under θ → π − θ . Both S ( θ ) an S ( θ ) are singular at the north and/or the south pole. A suitable choice of s ± (cid:96) can cancelone of the two singularities, but never both. As a consequence, they are non-normalizablewith the inner product inherited from the Sturm-Liouville theory associated with the linear Indeed we have P (2) (cid:96) (cos θ ) = 0 for (cid:96) = 0 , . Note that one of the two solutions with (cid:96) = 0 , is an associatedLegendre function of the second kind: Q (2)0 (cos θ ) = 2 cos θ/ sin θ and Q (2)1 (cos θ ) = 2 / sin θ . (cid:96) = 1 : the JT mode. Instead of disregarding the (cid:96) = 1 sector by setting χ ( x ) = 0 , we canimpose a softer condition on χ ( x ) by requiring the Weyl scalars (3.20) to vanish: Ψ = Ψ = 0 . (3.29)This leads to two constraints l a l b ∇ a ∇ b χ = 0 , n a n b ∇ a ∇ b χ = 0 , (3.30)on top of the equation of motion (3.19) which reads (cid:3) χ = 2 χ . (3.31)A simple computation shows these three conditions are equivalent to ∇ a ∇ b χ − g ab (cid:3) χ + g ab χ = 0 , (3.32)which we recognize as the equation of motion (1.5) for the dilaton field in Jackiw-Teitelboimgravity [27, 28]. Hence χ behaves like a JT mode.At this stage there is an important remark about the properties of χ . Based on Wald’stheorem for the Kerr geometry [30], it is tempting to conclude that imposing (3.29) leadsto a trivial perturbation, i.e. a diffeomorphism possibly combined with a change of massand/or angular momentum. However, this is the wrong conclusion since one can explicitlyverify that it’s impossible to cast the line element (3.1), under the restriction (3.30), as adiffeomorphism. Hence, χ carries additional information besides its potential interpretationas a change of the constant parameters in NHEK. We will return to this point in Sec. 6 as wediscuss the matching conditions of perturbations that have vanishing Weyl scalar contributionsand its interplay with Wald’s theorem. (cid:96) = 0 : marginal deformations. The (cid:96) = 0 mode corresponds to a marginal operatorwith conformal dimension ∆ = 1 . Hence, this should correspond to perturbations preservingextremality. As above, despite the singularities of S ( θ ) , we will not set χ ( x ) = 0 but, This requirement can be interpreted as demanding finiteness of the ingoing and outgoing energy fluxesassociated with the perturbation, as measured by integrating Ψ and Ψ on the sphere [24–26, 48]. Wald’s theorem also implies that for Kerr perturbations, imposing Ψ = 0 is equivalent to imposing Ψ = 0 . Our analysis shows that this conclusion is incorrect on NHEK since the first and second lines of (3.30)are independent. Ψ = Ψ = 0 . This leads to l a l b ∇ a ∇ b χ = 0 , n a n b ∇ a ∇ b χ = 0 , (3.33)which together with (cid:3) χ = 0 is equivalent to the equation ∇ a ∇ b χ = 0 . (3.34)Its unique solution is χ = const . (3.35) Conical singularities.
The pathologies associated to the poles in (3.28) leave an imprinton the geometry, even after imposing (3.32). Indeed, the metric of the sphere at fixed 2Dcoordinates x a in (3.1) is given by ds (cid:12)(cid:12)(cid:12) x = J (cid:0) θ + (cid:15)χ ( x , θ ) (cid:1) d θ + 4 J sin θ θ (cid:18) − (cid:15)χ ( x , θ )1 + cos θ (cid:19) d φ + O ( (cid:15) ) . (3.36)The troublesome points are the poles θ = 0 , π where the one-form d φ is ill-defined. Near thesepoints, the term linear in (cid:15) takes the form ds (cid:12)(cid:12)(cid:12) x ∼ θ → ,π J (cid:0) d θ + sin θ d φ (cid:1) + (cid:15)J χ ( x , θ ) (cid:0) d θ − sin θ d φ (cid:1) + O (sin θ ) + O ( (cid:15) ) . (3.37)This makes manifest the presence of conical singularities at both poles. We will show nextthat when the conditions (3.32) and (3.34) are satisfied, and for the even modes s +0 and s +1 ,these conical singularities can be cancelled by an appropriate diffeomorphism. It is interesting to identify a perturbation within the propagating sector leading to the JTmode. However, it is disappointing the latter has a conical singularity. In this section we willshow that this singularity can be removed by acting with a non-single valued diffeomorphism.In order to potentially remove the conical singularity in (3.37) we will tamper with thetopology of the sphere as follows. Consider a non-single valued diffeomorphism of the form ξ µ ( x , θ, φ ) = (cid:15) φ ζ µ ( x , θ ) . (3.38) Note that for (cid:96) ≥ , the modes are described by associated Legendre Polynomial which vanish at θ = 0 , π .And hence, these perturbation are well supported on the sphere as expected. g φφ , among other effects. It is also not welldefined on the sphere for an arbitrary ζ µ ( x , θ ) . However, we will tolerate this provided theresulting Lie derivative is single-valued on the sphere, so we impose ∂ φ ( L ξ g µν ) = 0 . (3.39)where g is the NHEK metric (2.4). For (3.38) we have ∂ φ ( L ξ g µν ) = (cid:15) L ζ g µν , and hence we can comply with (3.39) provided that ζ is one of the NHEK Killing vectors. Asin (2.22), we will write it in the basis ζ = ε ba ∇ b Φ ∂ a + (Φ + ε ab A a ∇ b Φ) ∂ φ , (3.40)where Φ( x ) = c ( φ ) + Φ JT , Φ JT = c i Φ ζ i ( x ) . (3.41)Recall that c ( φ ) parametrizes the u (1) isometries, and c i the sl (2) symmetry of NHEK; Φ JT obeys the JT equations (1.5). The transformation now has the desired properties: For instance,applying (3.38) with (3.40) to the background (2.13), one finds that the fiber changes as d φ + A a d x a → (cid:16) (cid:15) x ) (cid:17) d φ + A a d x a , (3.42)which has the effect of modifying the size of the sphere. Using the jargon of AdS/CFT, thistransformation can be interpreted as turning on an irrelevant deformation with ∆ = 2 due to Φ JT , and a marginal deformation, with ∆ = 1 , due to c ( φ ) .Applying (3.38), with (3.40), to the perturbation (3.1) leads to the metric ds = J (cid:0) θ + (cid:15)χ ( x , θ ) (cid:1) (cid:104) ( g ab + (cid:15)h ab ) d x a d x b + d θ (cid:105) (3.44) +4 J sin θ (1 + (cid:15) Φ( x ))1 + cos θ + (cid:15)χ ( x , θ ) (d φ + A a d x a + (cid:15) A ) + O ( (cid:15) ) . The fields h ab ( x ) and A are determined by Φ( x ) and χ ( x , θ ) . The contribution of χ ( x , θ ) tothese modes is given by (3.4) and h ab = 0 ; the dependence on Φ( x ) is given by A = −
12 Φ( x ) A a d x a − (cid:0) θ (cid:1) θ ε ab ∇ b Φ d x a , (3.45) To place the result in the same gauge as in [29], we add a correction term which doesn’t affect the sphere,with the total diffeomorphism being ξ = 12 φ ζ + ξ corr , ξ corr ≡ ε ab ( ∇ c A b ) ∂ c Φ ∂ a . (3.43) h ab = A ( b ε a ) c ∇ c Φ . (3.46)Next, we will show how Φ( x ) can be used to cancel the conical singularity described in(3.37). The metric of the sphere, obtained by considering a slice at fixed x in (3.44), takesthe form ds (cid:12)(cid:12)(cid:12) x = J (cid:18) (1 + cos θ ) d θ + 4 sin θ θ d φ (cid:19) (3.47) + (cid:15) (cid:20) χ ( x , θ ) d θ + 4 sin θ θ (cid:18) Φ( x ) − χ ( x , θ )1 + cos θ (cid:19) d φ (cid:21) + O ( (cid:15) ) . Expanding the above metric near the poles θ = 0 , π , we obtain the condition for the absenceof conical singularity Φ( x ) = χ ( x , ⇐⇒ regular at θ = 0 , (3.48) Φ( x ) = χ ( x , π ) ⇐⇒ regular at θ = π . (3.49)This condition selects the parity even solutions in (3.27), i.e. we need to set s − = 0 = s − .For the (cid:96) = 1 ( ∆ = 2 ) mode, we have to equate the JT component of Φ( x ) in (3.41) to χ , i.e. Φ JT ( x ) = χ ( x ) , (3.50)to obtain a regular perturbation. For (cid:96) = 0 ( ∆ = 1 ), demanding regularity gives c ( φ ) = 12 χ . (3.51)Still for (cid:96) = 0 one could allow singular behaviours, which we will explain in the next subsection.Let us summarize our findings on the (cid:96) = 1 sector of NHEK perturbations. There are twofields with conformal dimension ∆ = 2 whose origin and main features are the following :1. χ ( x ) arises as part of the tower of AdS modes contained within the Weyl scalars Ψ , . Due to the poles in the angular eigenfunction S ( θ ) (3.28), the Weyl scalars wouldtypically diverge at both poles (3.20). Demanding the vanishing of the Weyl scalars, theperturbation χ ( x ) becomes equivalent to a JT mode solving the JT equations (1.5).However, even after imposing the latter, this mode remains physical and the perturbedgeometry contains conical singularities.2. Φ JT is generated by a non-single valued diffeomorphism modifying the size and shapeof the 2-sphere, while adding a further conical singularity. Preservation of the axialsymmetry again leads to the JT equations (1.5).The combination of both modes, together with (3.50), gives rise to a smooth perturbationdriving the extreme Kerr black hole away from extremality. It is this combination that we24ill colloquially refer to as the JT sector. We will confirm this interpretation by computingthe contribution of these modes to the Iyer-Wald charges in Sec. 4. Furthermore, in Sec. 5,we will show these smooth modes can be glued to asymptotically flat modes corresponding toa change in the mass of the extreme Kerr black hole, in agreement with Wald’s theorem [30]. The complete family of type D spacetimes in 4D Einstein gravity with
Λ = 0 is contained inthe Plebański–Demiański family of solutions. In addition to the mass and angular momentumof the Kerr geometry, the metric also has a NUT parameter n and an acceleration parameter α .The accelerating Kerr black hole is also known as the spinning C -metric [56, 57]. The upshotis that the (cid:96) = 0 mode described above captures deformations of the NHEK corresponding tochanging these parameters while preserving extremality. We only present the results here andrefer to App. C for the derivations. Change of extremal entropy.
We consider the perturbation of the NHEK metric (2.4)corresponding to a change of extremal mass J → J + (cid:15) δJ + O ( (cid:15) ) . (3.52)After taking the decoupling limit, this leads to the perturbation (3.44) with χ ( x , θ ) = δJJ (1 + cos θ ) , Φ( x ) = 2 δJJ . (3.53)We recognize this as the even ( s − = 0 ) ∆ = 1 mode together with the Φ = c ( φ ) modein (3.51), which cancels the canonical singularities at θ = 0 and θ = π . According to theprevious section, Φ = c ( φ ) is simply generated by a rescaling of the angle φ : φ → (cid:16) (cid:15) c ( φ ) (cid:17) φ . (3.54)We will see this perturbation again in Sec. 4.2 as a contribution of the marginal deformationto angular momentum. Towards the C -metric. The perturbation towards the spinning C -metric is obtained bytaking the NUT parameter n = 0 and the acceleration parameter α = (cid:15) δα + O ( (cid:15) ) . (3.55)In the extremal case J = M , the decoupling limit leads to the perturbation (3.44) with χ ( x , θ ) = 4 J δα cos θ , Φ( x ) = 0 . (3.56) The parameters n and α in this subsection and App. C to describe the NUT parameter and accelerationshould not be confused with the Newman-Penrose variables defined in App. A. The context of the discussionshould make clear the distinction.
25e recognize the odd ( s +0 = 0 ) ∆ = 1 mode. Since Φ( x ) = 0 , we have conical singularities atboth the south and north poles of the sphere. These singularities follow from the correspondingsingularities of the C -metric. We note that one of the two conical singularities can be cancelledby rescaling the angle φ according to (3.54). A possible physical interpretation of thesesingularities in terms of meromorphic superrotations was proposed in [58]. Towards Kerr-NUT.
The addition of NUT charge corresponds to the perturbation withNUT parameter n = (cid:15) δn + O ( (cid:15) ) . (3.57)and with acceleration parameter α = 0 . In the extremal case M = a − n ≡ M , thedecoupling limit leads to the perturbation (3.44) with χ ( x , θ ) = 2 δnM cos θ , Φ( x ) = 2 δnM . (3.58)We see here that the Φ( x ) = c ( φ ) mode cancels the conical singularity at θ = 0 but not at θ = π . This leads to a conical singularity at the south pole which is interpreted as comingfrom the corresponding singularity in the Kerr-NUT geometry. The constant value of Φ( x ) can be changed by rescaling the angle φ according to (3.54). For example, this can be usedto move the conical singularity to the north pole by changing the sign of Φ( x ) . After decoding the spectrum of axisymmetric perturbations on NHEK, in this section wequantify the Iyer-Wald Noether charges associated to them. The emphasis will be mostlyplaced on the low-lying modes with (cid:96) = 1 , ( ∆ = 2 , ) which affect the global properties ofthe NHEK background. As we will see, conservation and finiteness of the gravitational chargesof these modes is tied to the global regularity requirements discussed in the prior section. We would like to collect a pair of facts of the covariant formalism for gravitational charges àla Iyer-Wald [59] for the Einstein-Hilbert action in four dimensions : the existence of Noethercharges and their relevance to reproduce the first law of black hole thermodynamics. Werecommend [60] for a more extensive and pedagogical review. Our main goal is to highlightthat the presence of singularities and sources, such as the conical singularities carried by someof the (near-)NHEK perturbations can still lead to finite quantities affecting the conservationproperties of the ought-to-be (near-)NHEK charges.Let L [ g ] be the Lagrangian 4-form, its variation defines the 3-form pre-symplectic potential26 [ δg, g ] δ L [ g ] = E [ g ] δg + dΘ[ δg, g ] , (4.1)where E [ g ] are the Einstein’s equations, g is the on-shell background metric and δg is anarbitrary variation near g . Given a pair δ i g ( i = 1 , of such variations, the Lee-Waldsymplectic current is defined as ω [ δ g, δ g ] = δ Θ[ δ g, g ] − δ Θ[ δ g, g ] . (4.2)Notice this is a 2-form in phase space, i.e. with arguments δ i g , and a 3-form in spacetime.When one of the perturbations is a diffeomorphism generated by a vector field ξ , i.e. δ g = δ ξ g ,the integral of the symplectic current over a co-dimension one spatial region Σ equals theinfinitesimal variation of the Hamiltonian δH ξ [ g ; Σ] = (cid:90) Σ ω [ δ ξ g, δg ] . (4.3)Furthermore, since ω [ δ ξ g, δg ] is closed, up to the linearised equations of motion δE gµν , it canlocally be written as an exact form, i.e. there exists a 2-form k ξ satisfying ω [ δ ξ g, δg ] − δE gµν ξ µ ε ν = d k ξ [ δg, g ] , (4.4)where ε ν is defined as the particular case of the volume form of a ( d + 1 − k ) -dimensionalsurface in a ( d + 1) -dimensional spacetime ε µ ··· µ k = 1 k !( d + 1 − k )! √− g ε µ ··· µ k µ k +1 ··· µ d +1 d x µ k +1 ∧ · · · ∧ d x µ d +1 . (4.5)Using Stokes’ theorem, the Hamiltonian (4.3) can be written as a surface integral δH ξ [ g ; Σ] − (cid:90) Σ δE gµν ξ µ ε ν = (cid:90) ∂ Σ k ξ [ δg, g ] . (4.6)Notice that whenever ξ is a Killing vector of the background g and δg satisfies the linearisedEinstein’s equations everywhere in the region Σ , the left hand side of (4.6) vanishes. Thisallows to define the Noether charge for a spatial region bounded by a closed co-dimension twosurface C as δ Q ξ [ g ; C ] ≡ (cid:90) C k ξ [ δg, g ] . (4.7)The latter is invariant if the surface C is deformed to another surface C (cid:48) which is homologousto C . This conclusion does not hold if there are sources or singularities between both surfaces Note that Θ[ δg, g ] is ambiguous up to terms of the form, Θ → Θ + δµ + d Y , which are important fordefining a classical phase space in the presence of boundaries [61, 62]. In the following will be ignoring thosecontributions since they seem to not affect our final results; still it might be worth to investigate them moreclosely. and C (cid:48) . This fact will play an important role in our specific discussions for linearisedperturbations of near-NHEK.Next, we review how this formalism captures the first law of black hole thermodynamics.Consider an stationary background solution containing a bifurcate Killing horizon H . For anappropriate choice of constant angular velocity Ω H , the generator of this null surface is theKilling vector ξ = ∂ t + Ω H ∂ φ , (4.8)with the defining properties that it vanishes on this surface, ξ | H = 0 , and the Hawkingtemperature ( T H ) is given by ∇ [ µ ξ ν ] (cid:12)(cid:12)(cid:12) H = 2 πT H ε µν , (4.9)where ε µν is the binormal vector on the bifurcation surface, i.e. the entries of (4.5) with k = 2 at H . The infinitesimal entropy can be defined as the Noether charge associated with thehorizon generator [63] δS = 1 T H δ Q ξ [ g, H ] . (4.10)For the Einstein-Hilbert action the expression for k ξ [ δg, g ] in (4.7) is k ξ [ δg, g ] = δ (cid:16) π ∇ µ ξ ν ε µν (cid:17) − ξ · Θ[ δg, g ] . (4.11)Since the horizon generator vanishes at the bifurcation surface H , the second term of (4.11)vanishes, and the first term is a total variation. This allows to integrate the infinitesimalentropy in (4.10) in the solution (tangent) space, recovering the well known result that theentropy is given by the horizon area S = Area[ H ]4 . (4.12)On the other hand, the surface charge could have also been evaluated at infinity, δ Q ξ [ g ; C ∞ ] ≡ δM − Ω H δJ , (4.13)where we used δM = δQ ∂ t [ g ; C ∞ ] , δJ = − δQ ∂ φ [ g ; C ∞ ] . (4.14)On the space of all Kerr black hole solutions parameterised by the mass and angular mo-mentum, we can choose Σ to be the spatial region between the event horizon and asymptoticinfinity, where the variation δg satisfies the linearised Einstein’s equations with no mattersources. Charge conservation gives rise to the first law of black hole thermodynamics T H δS = δ Q ξ [ g ; H ] = δ Q ξ [ g ; C ∞ ] = δM − Ω H δJ . (4.15)As stressed earlier, this argument would fail if the perturbation δg encodes a singularity. This28s the case we will discuss in the next subsection for near-NHEK perturbations. sl (2) charges and angular momentum Let us compute the Iyer-Wald charge differences for the gravitational perturbations of Sec. 3associated to the isometries of near-NHEK. As discussed in Sec. 2.1, the latter consist of three sl (2) generators ζ ± , and an additional u (1) generator ∂ φ . These preserve the near-NHEK(background) metric g , whereas δg will be one of the axisymmetric near-NHEK perturbations,including both the propagating and the low-lying modes.Consider the near-NHEK background metric (2.10). The explicit expression for k ξ in(4.11) is given by k ξ [ δg, g ] = 18 π ε µν (cid:16) ξ ν ∇ µ δg σσ − ξ ν ∇ σ δg µσ + ξ σ ∇ ν δg µσ + δg σσ ∇ ν ξ µ − δg σν ∇ σ ξ µ (cid:17) . (4.16)Since near-NHEK has a bifurcation horizon, the discussion leading to (4.6) and (4.7) applies,modulo the presence of singularities and/or sources. In particular, it is natural to define thetotal gravitational charge of near-NHEK perturbations as the surface charge evaluated on aclosed co-dimensional two sphere C ∞ at the asymptotic boundary δ Q ξ [ C ∞ ] , while the chargeof the black hole δ Q ξ [ H ] can be evaluated at the event horizon. However, explicit calculationsshow these charges depend on the choice of surface whenever the perturbations δg of near-NHEK have conical singularities at the poles located at θ = 0 , π . This is because at theselocations, Einstein’s equations are not satisfied. Notice the left hand side of (4.6) would stillvanish if we chose a source free region Σ (cid:48) . For instance, we can choose Σ (cid:48) to be the constanttime slice t = t between the horizon and asymptotic infinity, excluding the strings from thenorth and south poles. The boundary ∂ Σ (cid:48) now contains H , C ∞ , and the strings from the northand south poles. This means the following relation must be satisfied δ Q ξ [ C ∞ ] = δ Q ξ [ H ] + δ Q ξ [ N ] + δ Q ξ [ S ] , (4.17)where δ Q ξ [ N ] and δ Q ξ [ S ] stand for the extra boundary contribution at each pole. An equiv-alent interpretation of the above equation is to choose Σ as the constant time slice t = t between the horizon and asymptotic infinity. The conical singularities at the poles can beunderstood as adding a source localised along the spin axis. Using Einstein’s equations inthe presence of matter, the additional term on the left hand side of (4.6) is the matter stresstensor.We can write a more general charge conservation equation by considering 2-sphere shells C r at a constant radius r , rather than at infinity and/or at the event horizon. The relation29etween the charge at radius r and r is given by δ Q ξ [ C r ] ≡ (cid:73) C r ( k ξ ) θφ = δ Q ξ [ C r ] + (cid:90) rr d r (cid:90) π d φ (cid:16) ( k ξ ) rφ (cid:12)(cid:12)(cid:12) θ = π − ( k ξ ) rφ (cid:12)(cid:12)(cid:12) θ =0 (cid:17) . (4.18)The latter can be derived by choosing Σ (cid:48) as a region bounded by the shells at r , r and thestrings between both shells. What we learn is that if k ξ is a regular 2-form, the last two termsvanish. However, both the low-lying χ modes with (cid:96) = 0 , and the Φ perturbation, produceconical singularities. Hence, k ξ is no longer regular and we must keep the second and thirdterms to comply with (4.4). In the following, we present the values of these charges for themodes discussed in Sec. 3. sl (2) × u (1) charges for axisymmetric modes. Consider the metric variation δg µν due tothe terms linear in (cid:15) in (3.1). These are metric perturbations induced by the axisymmetricmodes χ . We will compute the charges (4.18) associated to the Killing vector ζ , the sl (2) × u (1) generators in (2.22), evaluated on the surface at x = ( t = t , r = r ) . The result can be expressed in terms of the dual scalars Φ ζ in (2.30) as δ χ Q ζ [ C r ] = (cid:73) C r ( k ζ ) θφ = − (cid:15) J Φ ζ ( x ) (cid:18) cos θ (1 + cos θ ) χ ( x , θ ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ = πθ =0 . (4.19)For modes with (cid:96) ≥ , χ ( x , θ ) vanishes at the poles. Hence their contribution to thecharges (4.19) is zero. However, for the low-lying modes the contribution is non-trivial. For (cid:96) = 1 ( ∆ = 2 ) we select s +1 = 1 and s − = 0 in (3.28), i.e. χ ( x , θ ) = χ ( x ) ; the corresponding sl (2) charges are δ χ Q ζ [ C r ] = (cid:15) J Φ ζ ( x ) χ ( x ) . (4.20)For the (cid:96) = 0 ( ∆ = 1 ) mode, we will set χ ( x , θ ) = (1 + cos θ ) χ ( x ) for simplicity. Thecorresponding charges are δ χ Q ζ [ C r ] = (cid:15) J Φ ζ ( x ) χ ( x ) . (4.21)Both (cid:96) = 1 and (cid:96) = 0 charges depend on both t and r for generic solutions. The dependenceon t implies that these charges are not conserved and the dependence on r implies theexistence of sources between the two sphere shells with different radius. This is due to the30act that the χ ( x , θ ) perturbation will create a conical singularity unless compensated by a Φ mode, as discussed in Sec. 3. sl (2) × u (1) charges for the global mode Φ . Let us compute the same charges evaluatedon the same surfaces due to metric variations induced by the global mode Φ , as described bythe metric perturbations in (3.44), where Φ( x ) = Φ JT + c ( φ ) = c i Φ ζ i + c ( φ ) , (4.22)and we set χ ( x , θ ) = 0 . The resulting expression at C r reads δ Φ Q ζ [ C r ] = (cid:15) J (cid:16) Φ (cid:3) Φ ζ + Φ ζ (cid:3) Φ − ∇ a Φ ζ ∇ a Φ − ζ Φ (cid:17)(cid:12)(cid:12)(cid:12) x = x . (4.23)For the sl (2) charges, we can use the relation (2.27) to reduce both Φ and Φ ζ i , leading to δ Φ Q ζ i [ C r ] = (cid:15) J (cid:16) − ∇ a Φ ζ i ∇ a Φ − ζ Φ (cid:17)(cid:12)(cid:12)(cid:12) x = x = (cid:15) J (cid:16) η ij c j − ζ i Φ (cid:17)(cid:12)(cid:12)(cid:12) x = x . (4.24)In the last equality we used the definition of the sl (2) Killing form η ij defined in (2.33). Forthe u (1) charge we simply obtain δ Φ Q ∂ φ [ r ] = (cid:15) J ( (cid:3) Φ( x ) − x )) . (4.25)As for the low-lying axisymmetric modes, the charges are again neither conserved nor positionindependent due to the presence of conical singularities. sl (2) × u (1) charges for smooth low-lying perturbations. As discussed in Sec. 3, regularperturbations require the low-lying modes of χ ( x , θ ) to be accompanied by a Φ mode satisfying χ ( x , θ ) = Φ JT ( x ) + 12 (1 + cos θ ) c ( φ ) . (4.26)Adding the charge formula (4.19) for the χ mode and (4.23) for the Φ mode, we find fourcharges for the regular perturbation: the three sl (2) charges δ Q ζ − = − (cid:15) J c + , δ Q ζ = (cid:15) J c , δ Q ζ + = − (cid:15) J c − , (4.27)and the u (1) charge δ Q ∂ φ = − (cid:15) J c ( φ ) . (4.28)All these charges are both conserved and independent of the radius due to the regularity ofthese perturbations, in agreement with the general discussion based on the covariant formal-ism. Note that in (4.27) we wrote Φ JT ( x ) = c i Φ ζ i , with c i constant, and used (2.33). The31alue obtained in (4.28) is in accordance to the change in δJ discussed around (3.52)-(3.54). Thermodynamics for near-NHEK.
Now, we aim to place the sl (2) × u (1) charges ina thermodynamic context, for which the appropriate geometry is the near-NHEK solution(2.10). This background is by itself a black hole, with a horizon generator ξ = ∂ t + Ω H ∂ φ , Ω H = − τ , (4.29)for which we can read the horizon and the effective temperature, r h = τ , τ H = τ π . (4.30)The three local sl (2) generators, up to automorphism, are given by (2.22), and the explicitprofile for the three dual scalars is Φ ζ = 1 τ (cid:18) r + τ r (cid:19) , Φ ζ ± = 1 τ (cid:18) r − τ r (cid:19) e ± τt , (4.31)which corresponds to the Killing vectors ζ = 1 τ ∂ t , ζ ± = (cid:18) r + τ τ (4 r − τ ) ∂ t ∓ r∂ r − τ r r − τ ∂ φ (cid:19) e ± τt . (4.32)The smooth perturbation for the low-lying modes, with (cid:96) = 1 , , is described by (4.26) wherein particular Φ JT = τ (cid:0) c + Φ ζ + + c Φ ζ + c − Φ ζ − (cid:1) , (4.33)with Φ ζ i as in (4.31) and c i are constants. The overall factor of τ is introduced for the large r behavior of Φ JT to depend only on the constants c i .The sl (2) × u (1) charges for near-NHEK are given by (4.27)-(4.28). And in particular,the energy associated to ∂ t and the angular momentum are δE ≡ δ Q ∂ t = (cid:15) J τ c ,δJ ≡ − δ Q ∂ φ = (cid:15) J c ( φ ) . (4.34)Note that the increase of energy depends on τ , i.e. a quadratic response on temperature τ H modulated by the JT field as expected from the 2D gravity arguments in [22, 23].As was argued before, the variation of the entropy due to the perturbation is given bythe Iyer-Wald charge associated with horizon generator δS ≡ τ H δ Q ξ = δE − Ω H δJτ H = π(cid:15) J ( c τ + c ( φ ) ) . (4.35)32ne can easily verify that this is just the variation of the area of the bifurcation surface H δS = δ Area ( H )4 = π(cid:15) J Φ (cid:12)(cid:12)(cid:12) r = r h , (4.36)since Φ ζ ± | r = r h = 0 and Φ ζ | r = r h = 1 . Therefore, the variation of the entropy satisfies the arealaw as it should be. In this section, we match the near and far region perturbations in Kerr. Following the nearhorizon perturbations presented in Sec. 3, we will divide the discussion into propagating modesfor which the Weyl scalars are non-vanishing, and regular (smooth) low-lying modes.Since this section focuses heavily on interpolating between Kerr and NHEK, let us re-inforce the notation used in Sec. 2: variables with a tilde, such as, ˜ g µν or ˜ x µ , correspond toquantities in the Kerr geometry (2.1); and variables without a tilde correspond to quantitiesdefined on near-NHEK (2.10). They are related by the decoupling limit (2.10) with thedeformation (2.11). Fig. 1 depicts the relevant regions of the near-extremal Kerr geometry.
Our aim is to extend the propagating χ -modes with (cid:96) ≥ discussed in Sec. 3.1 into the farregion of Kerr. Since our treatment of gravitational perturbations, via the introduction of χ ( x , θ ) , is unconventional, we will briefly discuss how to carry out the matching procedureand place them in the context of well known results in the literature, in particular [45]. Thiswill also serve to contrast against the subtleties that arise in the low-lying sector.The χ -modes constitute a complete set of normalizable modes with non-vanishing Weylscalars on NHEK. There are multiple ways to perform the matching. Here, we use the Hertzmap because it gives a concise relation between the perturbed metric h µν and a scalar potential Ψ H , called the Hertz potential. On a vacuum type-D spacetime, which encompasses Kerr,this map is given by h IRG µν = (cid:15) (cid:110) l ( µ m ν ) [( D − ρ + ¯ ρ )( δ + 4 β + 3ˆ τ ) + ( δ + 3 β − ¯ α − ˆ τ − ¯ π )( D + 3 ρ )] − l µ l ν ( δ + 3 β + ¯ α − ˆ τ )( δ + 4 β + 3ˆ τ ) − m µ m ν ( D − ρ )( D + 3 ρ ) (cid:111) Ψ H + c.c. (5.1)where h IRG µν denotes the perturbation in the ingoing radiation gauge l µ h IRG µν = g µν h IRG µν = 0 . (5.2) Several of our results are valid more generally for (2.13), i.e. for any locally AdS background. But forconcreteness we will write our results for the near-NHEK geometry.
33n these definitions l µ and m µ are the complex tetrads defined in (A.9). The Einstein equationfor the perturbation h IRG µν translates into the Teukolsky equation for the Hertz potential Ψ H : (cid:104) ( ˆ∆ + 3 γ − ¯ γ + ¯ µ )( D + 3 ρ ) − (¯ δ + ¯ β + 3 α − ¯ τ )( δ + 4 β + 3ˆ τ ) − ψ (cid:105) Ψ H = 0 . (5.3)The operators and quantities that enter in (5.1) and (5.3) are defined in App. A.In the following we describe a three-step algorithm to relate the Hertz potential in Kerrto the χ -modes. These steps involve first solving for the Hertz potential in the far and nearregion (which we will define below), then reconstructing the metric in terms of Ψ H , and finallyestablishing the relation to χ ( x , θ ) . Step 1: ˜Ψ H → Ψ H . Let ˜Ψ H and Ψ H denote the Hertz potentials in Kerr and NHEK,respectively. The first task is to solve the Hertz potential on the Kerr geometry in a lowfrequency regime by a matching procedure to the Hertz potential in the near horizon region.Consider axisymmetric Hertz potentials ˜Ψ H (˜ x , θ ) on the Kerr background, where we use ˜ x to collectively denote the 2D coordinates (˜ t, ˜ r ) . The master equation (5.3) gives (cid:16) L ˜ x − (cid:17) ˜Ψ H (˜ x , θ ) = L θ ˜Ψ H (˜ x , θ ) + (cid:16) i a cos θ ∂ ˜ t − a cos θ ∂ t (cid:17) ˜Ψ H (˜ x , θ ) , (5.4)where the differential operator L θ acts on the angular coordinate θ , and L ˜ x acts on the 2Dcoordinates ˜ x = (˜ t, ˜ r ) , L θ ≡ ∂ θ + cot θ ∂ θ − θ , (5.5) L ˜ x ≡ (˜ r + a ) − a ∆∆ ∂ t − ∆ ∂ ˜ r (cid:16) ∆ − ∂ ˜ r (cid:17) + 4 (cid:16) M (˜ r − a )∆ − ˜ r (cid:17) ∂ ˜ t . (5.6)Due to the terms in the parenthesis on the right hand side of (5.4), it is in general not possibleto have separation of variables in the form ˜Ψ H (˜ x , θ ) = ˜ U (˜ x ) S ( θ ) . One possible exceptionis to consider a regime of parameters where the contribution from the non-separable part isnegligible compared to the other terms. While this possibility is potentially interesting, wewill not explore it in this paper. A more obvious choice is to perform a Fourier expansion: ˜Ψ H (˜ x , θ ) = (cid:90) d ˜ ω (cid:88) (cid:96) e − i ˜ ω ˜ t ˜ R (cid:96) ˜ ω (˜ r ) ˜ S (cid:96) ˜ ω ( θ ) . (5.7)Plugging it into (5.4), we get two decoupled differential equations: one for the angular depen-dence (cid:16) L θ + K (cid:96) ˜ ω (cid:17) ˜ S (cid:96) ˜ ω + (cid:16) a ˜ ω cos θ + a ˜ ω cos θ (cid:17) ˜ S (cid:96) ˜ ω = 0 , (5.8)and one for the radial dependence (cid:16) L ˜ ω, ˜ r − − K (cid:96) ˜ ω (cid:17) ˜ R (cid:96) ˜ ω = 0 , (5.9)34here L ˜ ω, ˜ r is the operator L ˜ x in frequency basis, obtained from (5.6) with the replacement ∂ ˜ t → − i ˜ ω . The eigenvalue K (cid:96) ˜ ω in (5.8) also serves as a separation of variable constant withthe radial equation (5.9).Both the angular and radial equations are within the class of Heun’s differential equa-tions, for which it is not known how to construct an explicit solution for generic parameters.However, there are simplifications for the low energy excitations a ˜ ω (cid:28) , and modes nearthe superradiant bound in the near-extremal Kerr background [25, 49, 64]. Since the latterpermits a well defined decoupling limit, it can be analyzed in NHEK, and it is relevant forour purposes. In addition, requiring axisymmetric modes to be near the superradiant boundalso implies that they have low frequencies. Putting all this together, we will solve for ˜Ψ H ina near extremal Kerr background in the regime a ˜ ω (cid:28) , ˜ τ H ≡ r + − r − r + (cid:28) , with n ≡ M ˜ τ H ˜ ω fixed . (5.10)First, the solutions to the angular equation (5.8) greatly simplify. Imposing regularity of S (cid:96) ˜ ω at θ = 0 , π determines K (cid:96) ˜ ω [25, 65]. The corresponding solutions S (cid:96) ˜ ω are called spin-weightedspheroidal harmonics which form an orthonormal basis of functions of θ ∈ [0 , π ] with the innerproduct (cid:90) π S (cid:96) ˜ ω ( θ ) S ∗ (cid:96) (cid:48) ˜ ω ( θ ) sin θ dθ = 2 πδ (cid:96)(cid:96) (cid:48) . (5.11)For small ˜ ω , we have K (cid:96) ˜ ω ∼ (cid:96) ( (cid:96) + 1) + O (˜ ω ) , (5.12)with (cid:96) ≥ a positive integer. Therefore, in the low frequency regime a ˜ ω → , we have ˜ S (cid:96) ˜ ω ( θ ) → S (cid:96) ( θ ) , i.e. the same spin-2 spherical harmonics in (A.38) and (3.18) describing theangular dependence of the gravitational perturbations in NHEK given in Sec. 3.1.Second, the radial equation (5.9) can be solved in both the near region ˜ r − r + r + (cid:28) andthe far region ˜ τ H (cid:28) ˜ r − r + r + . These solutions can then be glued together by matching theirasymptotic expansions in the region of overlap where ˜ τ H (cid:28) ˜ r − r + r + (cid:28) . This procedure hasbeen systematically studied in the literature; see for example [45]. This matching conditiongives us an expression for ˜Ψ H everywhere in the regime (5.10), and reconstructs the metric ˜ h IRG µν through the map (5.1).Our last and most important portion of this first step is to relate the Hertz potential inKerr ˜Ψ H to the Hertz potential in NHEK Ψ H . The corresponding NHEK perturbation h IRG µν in IRG gauge is also related to Ψ H by (5.1) and is expected to match the Kerr perturbationin the decoupling limit (2.3), i.e. lim λ → ˜ h IRG µν d˜ x µ d˜ x ν = h IRG µν d x µ d x ν . (5.13)In order to get a well defined limit for the line element as λ → , we require the Hertz potential35o satisfy the matching condition lim λ → λ − ˜Ψ H (˜ x , θ ) = Ψ H ( x , θ ) , (5.14)where ˜ x is given in terms of x and λ by the decoupling limit (2.12).The master equation (5.3) for the Hertz potential in the NHEK can be written as ( L x − L θ − H ( x , θ ) = 0 , (5.15)where L θ was defined in (5.5) and the differential operator acting on the 2D part is given by L x ≡ n a ∇ a )( l b ∇ b ) + 4 n b A b ( l a ∇ a ) , (5.16)One can explicitly check that the 2D differential operator in Kerr (5.6) directly reduces to(5.16) in NHEK lim λ → L ˜ x = L x . (5.17)This means that the master equation (5.4) reduces to the master equation (5.15) in NHEKin the decoupling limit, provided the terms in the parenthesis on the right hand of (5.4) aresubleading. Notice the latter is indeed satisfied in our low frequency regime (5.10).To make the matching more explicit, let us decompose the Hertz potential Ψ H ( x , θ ) inFourier modes as Ψ H ( x , θ ) = (cid:90) dω (cid:88) (cid:96) e − iωt R (cid:96)ω ( r ) S (cid:96) ( θ ) , L θ S (cid:96) = − (cid:96) ( (cid:96) + 1) S (cid:96) . (5.18)Notice that n = M ˜ ω ˜ τ H = ωτ , where ω is the frequency used in the near horizon variables (5.18).This makes the implementation of this limit compatible with (5.10). Therefore solutions tothe master equation (5.4) and (5.15) will comply with the matching condition (5.14) if werequire R (cid:96)ω = lim λ → λ − ˜ R (cid:96) ˜ ω , S (cid:96) = lim λ → S (cid:96) ˜ ω , (5.19)in the Fourier basis.Finally, let us comment on the Weyl scalars in this context. The relation between the Weylscalars and the Hertz potential is given by (A.22). In NHEK, (A.22) simplifies significantly,and for axisymmetric modes we have Ψ ( x , θ ) = (cid:15) l a ∇ a ) Ψ H ( x , θ ) , Ψ ( x , θ ) = (cid:15) J (1 − i cos θ ) L θ ( L θ + 2)Ψ H ( x , θ ) . (5.20)36n Kerr, the Weyl scalars can be expanded in the decoupling limit (2.3) as ˜Ψ = (cid:15) λ − l a ∇ a ) Ψ H + O ( λ )) = (cid:15)λ − (Ψ + O ( λ )) , ˜Ψ = (cid:15) λ J (1 − i cos θ ) ( L θ ( L θ + 2)Ψ H + O ( λ )) = (cid:15)λ (Ψ + O ( λ )) , (5.21)which is indeed consistent with (5.14).To recapitulate, the Hertz potential ˜Ψ H in Kerr and the Hertz potential Ψ H in NHEK canbe related by the gluing condition (5.14) in the regime of parameters (5.10). More explicitly,in the Fourier basis, solutions in Kerr and in NHEK are related through (5.19). Step 2: Reconstruction in NHEK, Ψ H → h IRG . The decoupling limit of ˜Ψ H in (5.14)leads to the Hertz potential in NHEK, which takes the form (5.18). In this limit, the angulardependence reduces to S (cid:96) , which is independent of ω . Hence it is also consistent to cast (5.18)into the form Ψ H ( x , θ ) = (cid:88) (cid:96) ≥ U (cid:96) ( x ) S (cid:96) ( θ ) , (5.22)with no Fourier decomposition in time. This will allow our expressions in NHEK to becovariant with respect to the 2D coordinates x = ( t, r ) .Assuming Ψ H ( x , θ ) to be real, we use (5.1) to reconstruct h IRG µν . In the tetrad basis, wehave h IRG µν = 12 h IRG nn (cid:96) µ (cid:96) ν + h IRG mm ¯ m µ ¯ m ν − h IRG nm ( (cid:96) µ ¯ m ν + (cid:96) ν ¯ m µ ) + h.c. , (5.23)with components h IRG nn = − (cid:15) sin θJ (1 + cos θ ) ( L θ + 2)Ψ H ,h IRG mm = h IRG ¯ m ¯ m = − (cid:15) l a l b ∇ a ∇ b Ψ H ,h IRG nm = − (cid:15) (1 + i cos θ ) √ J sin θ l a ∇ a ∂ θ (cid:18) sin θ θ Ψ H (cid:19) ,h IRG n ¯ m = − (cid:15) (1 − i cos θ ) √ J sin θ l a ∇ a ∂ θ (cid:18) sin θ θ Ψ H (cid:19) . (5.24)We are particularly interested in the angular components of the metric perturbation h IRG θθ = − (1 + cos θ ) θ h IRG φφ = J sin θ h IRG mm = − (cid:15)J sin θ l a l b ∇ a ∇ b Ψ H , (5.25)since they will suffice to illustrate how to relate Ψ H to the χ -modes.37 tep 3: Gauge transformation from h IRG to h χ . To relate Ψ H to the χ -modes, we needto bring h IRG µν to the same gauge as in (3.1) via a diffeomorphism ˆ ξh χ = h IRG + (cid:15) L ˆ ξ g NHEK . (5.26)Here h χµν denotes the metric perturbation that we used in (3.1). To construct ˆ ξ , notice thatboth (5.25) and (3.1) preserve the determinant of the sphere, i.e. h θθ and h φφ are relatedaccording to the first line in (5.25). To preserve this feature, we set ˆ ξ θ = 0 . This ensures h IRG θθ = h χθθ , h IRG φφ = h χφφ . (5.27)Choosing the diffeomorphism ˆ ξ = − sin θ θ l a ∇ a Ψ H l µ ∂ µ + (cid:90) d θ cot θ l a l b ∇ a ∇ b Ψ H ∂ φ , (5.28)the resulting components h χµν from (5.26) satisfy all the gauge conditions in (3.1). Matchingcomponents, we find χ ( x , θ ) = − sin θ l a l b ∇ a ∇ b Ψ H ( x , θ ) . (5.29)Thus, given a Hertz potential Ψ H characterising a NHEK perturbation, this relation deter-mines the corresponding χ ( x , θ ) propagating modes. Note that (3.7), (5.20), and (5.29) arecompatible relations among the χ -modes, Weyl scalars, and Hertz potential.Following the steps ˜Ψ H → Ψ H → h IRG → h χ , we established a map from the UV to theIR. Namely, given a Kerr perturbation in the ingoing radiation gauge and reconstructed froma Hertz potential ˜Ψ H , we can take the decoupling limit and get a Hertz potential Ψ H in thenear horizon region by (5.14), from which we can read the χ mode using (5.29).Conversely, given a χ mode in the NHEK region, the Hertz potential Ψ H can be deter-mined by solving (5.29) together with the master equation (5.15). Then Ψ H can further beglued to a Hertz potential ˜Ψ H in Kerr. This process can be regarded as being from the IR tothe UV. In fact, (5.29) can be inverted for (cid:96) ≥ allowing us to express Ψ H in terms of χ ina compact form. Indeed, acting with the operator n a n b ∇ a ∇ b on both sides of (5.29), we get n a n b ∇ a ∇ b χ = −
14 sin θ L θ ( L θ + 2)Ψ H , (5.30)where on the right hand side we have first used the relation (A.43) and then the masterequation (5.15). Using the separation of variables as in (3.17) and (5.22), the relation (5.30)leads to the compact relation between the 2D parts of χ and Ψ H , U (cid:96) ( x ) = − (cid:96) − (cid:96) ( (cid:96) + 1)( (cid:96) + 2) n a n b ∇ a ∇ b χ (cid:96) ( x ) . (5.31)Therefore, given a solution of χ ( x , θ ) , we can use (5.31) to obtain the corresponding Hertz38otential. Altogether, relations (5.29) and (5.31) give a one-to-one correspondence betweenthe solution space of χ and Ψ H .This conclusion is the central result in our reconstruction of propagating modes sinceit provides an explicit relation between the χ ( x , θ ) modes and the Hertz potential Ψ H inNHEK, allowing us to extend the χ ( x , θ ) modes to the full Kerr geometry using the asymptoticmatching of the Hertz potentials in (5.14). (cid:96) = 0 and (cid:96) = 1 Having understood the reconstruction of the regular perturbations with (cid:96) ≥ , we would liketo extend this result for the low-lying modes, including the smooth JT modes driving thesystem out of extremality. While specific parts of the propagating modes analysis are stillapplicable to (cid:96) = 0 , , such as the gauge fixing diffeomorphism (5.28) and the relation (5.29),it is also evident that (5.31) breaks down for the low-lying modes. All of this boils down tothe delicate nature of the modes we have discussed in Sec. 3 and Sec. 4. They are intrinsicallyproblematic since their angular dependence is supported by meromorphic functions on thesphere. It is only after balancing these singularities that we can discuss them as well-behavedperturbations.In this subsection we focus on how to reconstruct smooth low-lying modes, i.e. , thosecomplying with the regularity conditions in Sec. 3.2.1. These modes have therefore vanishingWeyl scalars and are well-behaved on the sphere. As a result, they fall into the class of Kerrperturbations considered in [30]. Our task is then to match them: starting from a smoothperturbation in Kerr with vanishing Weyl scalars, we will identify those that are smooth andfinite as we take λ → in the decoupling limit (2.10). This will allow us to interpret them asperturbations of the NHEK geometry, and relate them to our analysis in Sec. 3.2.1.In [30], it was proved that smooth perturbations on Kerr black holes with vanishing Weylscalar can only be a linear combination of changing the mass, angular momentum, and adiffeomorphism, namely δ ˜ g = δ M ˜ g + δ J ˜ g + (cid:15) L ˜ ξ ˜ g . (5.32)Here δM = O ( (cid:15) ) and δJ = O ( (cid:15) ) . Since the NHEK perturbations we consider are axisymmetric,we also focus on axisymmetric perturbations in Kerr. This requires ∂ ˜ φ δ ˜ g = ∂ ˜ φ L ˜ ξ ˜ g = L ∂ ˜ φ ˜ ξ ˜ g = 0 . (5.33)This implies the vector field ∂ ˜ φ ˜ ξ must be a Kerr isometry, i.e. a linear combination of ∂ ˜ t and ∂ ˜ φ . However, when integrating for the vector field ˜ ξ , the non-vanishing part of ∂ ˜ φ ˜ ξ leads to alinear piece in ˜ φ . As discussed in Sec. 3, such non-single valued diffeomorphisms create conicalsingularities and we will not be considered in this section. Hence, from now on, we restrictto the case ∂ ˜ φ ˜ ξ = 0 . We will now show how the different perturbations allowed by Wald’stheorem in (5.32) are related to perturbations in the near horizon limit as presented in (3.44).39efore starting this analysis, let us introduce some convenient notation. Given a tensor ˜ T in Kerr, the near horizon expansion is given by performing the coordinate transformation(2.12) and expanding in λ . This procedure defines a tensor T [ λ ] in NHEK given by T [ λ ] µ ··· µ n ν ··· ν m = ∂x µ ∂ ˜ x α · · · ∂x µ n ∂ ˜ x α n ∂ ˜ x β ∂x ν · · · ∂ ˜ x β m ∂x ν m ˜ T α ··· α n β ··· β m . (5.34)Note that T [ λ ] is just the pullback of ˜ T from Kerr to NHEK under the map x (cid:55)→ ˜ x given in(2.12). We then study the tensor T [ λ ] in an expansion in λ . Now we discuss Kerr perturbationsgiven by (5.32) surviving the near horizon limit (2.12). Decoupling limit with no additional perturbations.
Before turning on any perturba-tion, let us consider the λ expansion of the near extremal Kerr metric ˜ g itself g [ λ ] = g NHEK + λ g (1) + · · · . (5.35)The leading order term g NHEK gives the near-NHEK metric (2.10). The O ( λ ) term g (1) satisfiesthe near-NHEK linearized Einstein’s equations and can be viewed as a near-NHEK pertur-bation with perturbative parameter (cid:15) ∼ λ . It is this agreement that motivated the ansatz in(3.44) as originally introduced in [29].Let us be more precise. We can show that λg (1) , together with the radial redefinition r → r + λ τ r √ J (4 r − τ ) , (5.36)gives a time independent perturbation of the type described in (3.44) with (cid:15) χ ( x , θ ) = (cid:15) χ ( x ) = (cid:15) Φ JT ( x ) = 2 λ √ J (cid:18) r + τ r (cid:19) . (5.37)This means the coefficients in (4.33) are given by c = 2 λ(cid:15) √ J , c ± = 0 . (5.38)We learn the leading near horizon expansion mode can be interpreted as a particular smoothperturbation in near-NHEK λ g (1) = (cid:0) δg (cid:1) χ =Φ JT , (5.39)where δg refers to (3.44). In particular, we note that the angular components of g (1) can bewritten as ( g (1) ) θθ = 2 √ J τ Φ ∂ t , ( g (1) ) φφ = 4 √ J sin θ cos θ (1 + cos θ ) τ Φ ∂ t . (5.40)This is the anabasis discussed in [47], and also discussed in [32] for five-dimensional blackholes. Notice that by performing a finite SL(2 , R ) coordinate transformation on (5.35), the40ear-NHEK background metric g NHEK remains invariant, but one can get more general JTmodes (4.33) than the stationary one in (5.37). Of course, this is consistent with the sym-metry breaking discussion reviewed in App. B. g NHEK is invariant under the full
SL(2 , R ) , butNHEK perturbations captured by JT modes (4.33) break this isometry group. The specificperturbation coming from Kerr is time translationally invariant, hence it must correspond to(5.38), i.e. it must be associated with Φ ∂ t . However, from a purely IR perspective, the actionof SL(2 , R ) on the latter can still generate the full multiplet of NHEK perturbations.Since the interpretation of the near-NHEK perturbation requires (cid:15) ∼ λ , this mode disap-pears when λ = 0 . In the following, we will consider near-NHEK perturbations surviving the λ → , i.e. keeping (cid:15) fixed as λ → . Diffeomorphism sector.
Let us start our analysis of Kerr perturbations consistent withvanishing Weyl scalars, see (5.32), by focusing on those generated by a diffeomorphism pa-rameterized by the vector ˜ ξ , i.e. we set δ M ˜ g = δ J ˜ g = 0 in (5.32). This vector can be expandedin λ as ξ [ λ ] = λ n ( ξ (n) + λξ (n+1) + · · · ) , (5.41)where n is an integer which can be negative. The corresponding diffeomorphism acting onKerr admits a λ expansion according to λ n (cid:16) L ξ (n) g NHEK + λ ( L ξ (n+1) g NHEK + L ξ (n) g (1) ) + · · · (cid:17) . (5.42)Our goal is to restrict the form of ξ [ λ ] by demanding regularity of (5.42). We claim n ≥ − in order to have a finite perturbation as λ → . To prove this, suppose n ≤ − . Regularityrequires the first two terms in the expansion to vanish L ξ (n) g NHEK = 0 , L ξ (n+1) g NHEK = −L ξ (n) g (1) . (5.43)The first equation implies that ξ (n) has to be a NHEK isometry, so that ξ (n) = a − ζ − + a ζ + a + ζ + + a φ ζ ( φ ) . (5.44)We already showed g (1) is a particular mode with χ = Φ JT . Hence, g (1) is not pure gauge.Applying the isometry (5.44) to g (1) simply rewrites such modes in a different coordinatesystem keeping the same metric. Thus L ξ (n) g (1) can still not be written as a diffeomorphismacting on NHEK, and the second equation in (5.42) has no solution for ξ (n+1) . Therefore, weconclude that n ≥ − .Let us consider n = − . The expansion (5.41) starts from λ − ξ (-1) , where ξ (-1) is anisometry of NHEK as in (5.44) in order to make (5.42) finite. It follows that the most generaldiffeomorphism surviving the decoupling limit is given by ˆ h ≡ L ξ (-1) g (1) + L ξ (0) g NHEK . (5.45)41ext we adjust the subleading term ξ (0) such that ˆ h takes the form (3.44) of our NHEKperturbations. This gives the following conditions ∂ θ (cid:16) δ log det g S (cid:17) = ∂ θ (cid:16) g − θθ ˆ h θθ + g − φφ ˆ h φφ (cid:17) = 0 , (5.46) g − θθ ˆ h θθ + g − φφ ˆ h φφ ≡ Φ( x ) , (5.47) ˆ h θθ ≡ J χ ( x , θ ) , (5.48) ˆ h tθ = ˆ h rθ = ˆ h φθ = 0 . (5.49)In addition we have to adjust the AdS components of (3.44), on which we will commentbriefly. Using (5.40), and the isometry-scalar relation (2.22), we get (cid:104) L ξ (-1) g (1) (cid:105) θθ = 2 τ √ J ξ a (-1) ∇ a Φ ∂ t = 2 τ √ J Φ [ ξ (-1) ,∂ t ] , (5.50) (cid:104) L ξ (-1) g (1) (cid:105) φφ = 2 τ √ J sin (2 θ )(1 + cos θ ) Φ [ ξ (-1) ,∂ t ] , (5.51)where (2.36) was used to write the right hand side as the dual scalar of a commutator. While L ξ (-1) g (1) already satisfies the condition (5.46), requiring the latter for L ξ (0) g NHEK amounts to ∂ θ ξ θ (0) + cot θ ∂ θ ξ θ (0) − θ ξ θ (0) = 0 . (5.52)This is the spin-weighted spherical harmonics equation (A.38), with K = 0 and spin 1, whosenon-vanishing solutions are always divergent at the poles. Therefore for smooth, axisymmetricperturbations, we conclude that ξ θ (0) = 0 and ξ (0) will not contribute to ˆ h θθ and ˆ h φφ . Theremaining gauge conditions (5.47)-(5.49), when combined with (5.50), lead to a consistentmatching of the ˆ h µν components provided we identify χ ( x ) = χ ( x ) , Φ = Φ JT , (5.53)together with the constraint [ ξ (-1) ,∂ t ] = √ J χ = √ J Φ JT = √ J Φ c i ζ i . (5.54)The latter reduces to a relation between Killing vectors ξ (-1) , ∂ t ] = √ J c i ζ i . (5.55)Hence, given a diffeomorphism with near horizon expansion (5.41) starting at n = − , theisometry ξ (-1) with arbitrary parameters as in (5.44), determines a near-NHEK perturbationof the form (3.44) that is fully determined by (5.55) using the sl (2) commutation relations.42olving for the constants c i gives c = 0 , c ± = ∓ τ √ J a ± . (5.56)We can read the corresponding sl (2) × u (1) charges of the near-NHEK perturbation in (4.27)-(4.28). However, we see from (4.34) that both the energy and the angular momentum arezero in the near horizon region. This is expected since these perturbations originate from thedecoupling limit of a diffeomorphism in the full Kerr geometry.We have determined ξ (-1) up to the isometries of Kerr ∂ ˜ t and ∂ ˜ φ which act trivially. Nowwe need to solve the remaining gauge fixing conditions that ˆ h µν should satisfy to determine ξ (0) . This is straightforward, and the details just depend on the choice of AdS coordinates in(3.44) and residual transformations that affect h ab and A a in (3.45)-(3.46). In general it takesthe form ξ (0) = ξ res + ξ sub (5.57)where ξ sub is determined by requiring h ab to satisfy (3.46) and ξ res is the residual gaugetransformation originating from the ambiguity of A in solving (3.4), ξ sub = 4 τ r ( r − τ r + 16 τ )( e τt a + + e − τt a − ) √ J (4 r − τ ) ∂ t + τ r (4 r + τ )( e τt a + − e − τt a − ) √ J (4 r − τ ) ∂ r ξ res = f ( t, r ) ∂ φ (5.58)where f ( t, r ) is an arbitrary function. One can explicitly check that the perturbation generatedby ξ (0) in (5.45) carries no sl (2) × u (1) charges.Finally if the transformation in (5.41) starts from the zeroth order in λ , then the dif-feomorphism surviving the decoupling limit and preserving the gauge choices (5.46)-(5.49) isalso given by ξ res , which is a pure diffeomorphism in NHEK that is smooth and has trivialIyer-Wald charges associated to it.To summarize, a general diffeomorphism in Kerr that reduces to a finite metric perturba-tion in NHEK has an expansion (5.41) starting from n = − , whose first two leading ordersare ξ [ λ ] = λ − ( a i ζ i + a φ ζ ( φ ) + λξ (0) + · · · ) . (5.59)The near horizon limit of the perturbation takes the form (5.45), and it can be identified witha near horizon perturbation (3.44) with χ = Φ JT carrying sl (2) × u (1) charges (4.27)-(4.28)evaluated for (5.56). Mass perturbation.
Let us continue with the analysis of the individual Kerr perturbationsin (5.32) by turning on a mass perturbation while keeping the angular momentum fixed.Expanding the mass variation δ M g of the Kerr metric in the decoupling limit (2.12) gives δ M g [ λ ] = δM (cid:16) λ − h (-2)M + λ − h (-1)M + h (0)M + · · · (cid:17) . (5.60)43ence, to have a finite limit as λ → , the mass perturbation δM should be of order λ . Thesurviving perturbation is given by h (-2)M = 4 J / (1 + cos θ ) (cid:18) d t + 16(4 r − τ ) d r (cid:19) . (5.61)From the near horizon point of view, the near-NHEK background with temperature τ H = τ π is perturbed to a nearby near-NHEK with temperature τ (cid:48) H = τ H (cid:18) √ J λ − δM π τ H (cid:19) . (5.62)as can be directly seen from the near extremal limit (2.11). In AdS language, mass pertur-bation of order λ change the AdS temperature without turning on any dynamics. Indeed,one can explicitly check that this mode carries no sl (2) × u (1) charges.In order to have dynamical fields in NHEK emerging in the λ → limit, we turn to ascenario with δM ∼ λ or δM ∼ λ . In this case, the mass perturbation cannot survive the limitby itself. However, (5.32) allows us to combine the δM perturbation with a diffeomorphism,that we shall specifically denote by ˜ ξ M , so that the combined perturbation is finite in thedecoupling limit.Given the conditions we found around (5.42) and the order of the divergences appearingin (5.60) when δM is order λ or λ , the expansion of the diffeomorphism must be of the form ξ M [ λ ] = δM (cid:16) λ − ξ M(-2) + λ − ξ M(-1) + ξ M(0) + · · · (cid:17) , (5.63)The resulting metric perturbation from the combined effect of (5.63) and (5.60) is δg [ λ ] = δM (cid:16) λ − (cid:16) h (-2)M + L ξ M(-2) g NHEK (cid:17) + λ − (cid:16) h (-1)M + L ξ M(-2) g (1) + L ξ M(-1) g NHEK (cid:17) + · · · (cid:17) . (5.64)The leading divergent term requires h (-2)M + L ξ M(-2) g NHEK = 0 . (5.65)This determines ξ M(-2) = 2 √ Jτ (cid:18) − t ∂ t + 4 r + τ r − τ r ∂ r (cid:19) + ζ , (5.66)up to a Killing vector ζ of the near-NHEK background. Since we already discussed thecontribution of ζ to the NHEK perturbation in the preceding paragraph on the diffeomorphismsector, we will set ζ = 0 in the following.The vanishing of the leading order λ − coefficient in (5.64) requires −L ξ M(-1) g NHEK = h (-1)M + L ξ M(-2) g (1) . (5.67)44owever, since the source in the right hand side is not pure gauge, there exists no vector field ξ M(-1) mapping the near-NHEK metric to this physical perturbation. Thus, for this combinedmetric and diffeomorphism perturbation to have a finite decoupling limit, δM must be at leastof order λ . The corresponding surviving perturbation in NHEK is given by lim λ → δg [ λ ] = δM λ − (cid:16) h (-1)M + L ξ M(-2) g (1) + L ξ M(-1) g NHEK (cid:17) ≡ h M . (5.68)To compare with the low-lying modes in Sec. 3, we transform h M µν to the appropriate gaugeas described in (5.46)-(5.49). This imposes ( ξ M(-1) ) θ = 0 . Furthermore, comparing h M θθ and h M φφ with (5.47) and (5.48), we learn (cid:15) χ ( x , θ ) = (cid:15) Φ( x ) = 4 δMλτ (cid:18) r + τ r (cid:19) . (5.69)This allows us to identify the constants c i , defined in (4.33), that determine the low-lyingmode perturbation in (3.44). We learn that c ± = 0 , c = 4 δM(cid:15)λτ . (5.70)Finally, the remaining vector field ξ M(-1) can be determined by requiring h M to satisfy (5.48)and (5.49). This gives ξ M(-1) = 2 r r − τ ∂ r + ξ res , (5.71)where ξ res is given by (5.58).It is reassuring to check the value of δM in (5.70) can also be determined using the relationbetween charges in Kerr and in NHEK. According to the coordinate transformation (2.3), wehave ∂ t = 2 Jλ ∂ ˜ t + √ Jλ ∂ ˜ φ , ∂ φ = ∂ ˜ φ . (5.72)Therefore, the charges in Kerr and near-NHEK are related by δ Q ∂ t = 2 Jλ δ Q ∂ ˜ t + √ Jλ δ Q ∂ ˜ φ , δ Q ∂ φ = δ Q ∂ ˜ φ . (5.73)A mass perturbation of Kerr corresponds to δ Q ∂ ˜ t = δM and δ Q ∂ ˜ φ = 0 , giving the NHEKcharges δ Q ∂ t = 2 Jλ δM , δ Q ∂ φ = 0 . (5.74)Comparing with (4.34), we recover δM = (cid:15)λ c τ . (5.75)45o summarize, Kerr mass perturbations δM ∼ λ can be interpreted as changing thetemperature in the near-NHEK metric. When δM ∼ λ , they can be combined with anadditional diffeomorphism, as in (5.63), and explicitly given by (5.66) and (5.71), to give a χ = Φ JT mode in near NHEK with coefficients given by (5.70). Note that the mass+diffeoperturbation (5.70) has the same structure as the anabasis mode (5.38). However, while thelatter has (cid:15) ∼ λ , the mass perturbation has (cid:15) ∼ λ . Hence, they belong to different parameterregions. One intuition for the similarity is that adding finite energy to AdS corresponds toan RG flow from the conformal fixed point at the IR towards UV, and hence corresponds toa deviation from the near horizon throat.As a final remark, we restricted ourselves to the minimal diffeomorphism necessary tocancel the divergent part of (5.60). Note that adding a pure diffeomorphism of the form (5.59)still leads to a perturbation surviving the decoupling limit as λ → . The resulting matchingwould be modified accordingly and in general will be different from (5.70). Angular momentum perturbation.
The discussion of angular momentum perturbationsis technically similar to the mass perturbation one. Let us just summarize the conclusion. Tomake δ J ˜ g finite in the decoupling limit, we need δJ ∼ λ . This perturbation changes the IRtemperature to δτ H = τ H (1 − δJλ τ ) .When δJ ∼ λ , δ J ˜ g is divergent in the decoupling limit. However, we can combine thelatter with a diffeomorphism ξ J to cancel this divergence. Choosing ξ J appropriately, we canalso match the JT mode which nonzero energy, δ Q ∂ t , in near-NHEK. In particular, we have δJ = (cid:15)λ c τ √ J , (5.76)which agrees with (5.73). Note that δ Q ∂ φ is zero as we take λ → . When δJ ∼ λ , thedivergence can not be removed by a diffeomorphism, as in the discussion of mass perturbations. Marginal deformation.
A general Kerr perturbation with vanishing Weyl scalars (5.32)can be glued to a linear combination of perturbations in near-NHEK. In the following, wediscuss the particular case δJ = 2 √ J δM (5.77)preserving the extremality condition J = M . This property is responsible for the cancellationof the leading order term in the λ -expansion, leading to a perturbation of the form ( δ M + δ J ) g [ λ ] = δM ( λ − h (-1) + h (0) + · · · ) . (5.78)When δM ∼ λ , the perturbation surviving the limit is given by the leading term h (-1) = J (1 + cos θ )(4 r + τ )2 r (cid:18) d t + 16(4 r − τ ) d r (cid:19) . (5.79)46hich can be written as a diffeomorphism with h (-1) + L ξ (-1) g NHEK = 0 , with ξ (-1) = 4 r r − τ ∂ r + t ∂ φ . (5.80)Such perturbation keeps the temperature invariant and carries no charges. Thus, this is atrivial diffeomorphism.When δM ∼ λ , the marginal perturbation is divergent in the near horizon limit, but wecan cancel the leading divergence with a diffeomorphism ξ [ λ ] = δM (cid:0) λ − ξ (-1) + ξ (0) + · · · (cid:1) , (5.81)where ξ (-1) is given by (5.80), and ξ (0) is chosen such that the perturbation surviving the limit δM (cid:16) h (0) + L ξ (-1) g (1) + L ξ (0) g NHEK (cid:17) , (5.82)satisfies the gauge conditions (5.46)-(5.49). As in previous discussions, analyzing the h θθ and h φφ components enables us to identify this mode as χ ( x , θ ) = c ( φ ) θ ) , Φ( x ) = c ( φ ) , c ( φ ) = 4 δM(cid:15) √ J . (5.83)Furthermore preservation of the remaining gauge conditions determines the subleading diffeo-morphism to be ξ (0) = τ r √ J (4 r − τ ) ∂ r + ξ res , (5.84)where ξ res is again given by (5.58).One can easily recognize that (5.83) has angular dependence (cid:96) = 0 and is just the marginaldeformation (3.53) discussed in Sec. 3. This mode carries u (1) NHEK charge according to(4.34), in agreement with the general map between Kerr and NHEK charges (5.73), sincethe choice (5.77) forces the NHEK mass to vanish. Therefore an extremal perturbation with δJ = 2 √ J δM ∼ λ in Kerr corresponds to a NHEK mode with (cid:96) = 0 as described by (5.83).This completes our analysis of all smooth and axisymmetric gravitational perturbationsinterpolating between NHEK and Kerr. A summary of these modes appears in the first twoblocks in table 1. This last section is devoted to explore further properties of the low-lying modes in cases wherewe allow for angular singularities. Clearly this takes us to unknown territory, where the rulesare less clear and we might encounter more pathologies as we study them. A fair objection topursue this direction is that most likely there is no physical process inducing these singular47erturbations. Hence, a conservative view might be that if the singularity is not balanced—aswe did in Sec. 3.2.1—one should just discard these configurations as unphysical.Nevertheless, there are some formal (theoretical) reasons why it is interesting to considerthese singular perturbations. And there are at least three directions that are worth mentioning:1. The potential relation these singular modes might have with superrotations that appearin the asymptotic symmetry analysis of Minkowski space. It has been shown in [58]that finite superrotations act on isolated defects on the celestial sphere, and these areclosely related to the C -metric deformations we described in Sec. 3.2.2. It would be veryinteresting to understand the interpretation of other low-lying modes in the context ofcelestial holography, and if they have a role in gravitational scattering.2. The regularity conditions on low-lying modes arose from the angular dependence ofthe modes, and not from physics on the AdS portion. These conditions on modeswith ∆ = 2 , are given by (3.30). In this context it is interesting to explore if theseconditions are universal or only required for specific black holes. More concretely, arethere examples of nearly-AdS holography for which ∆ = 2 , are not constrained by(3.30) and still well-behaved? or could one prove that regardless of the gravitationaltheory and the origin of the AdS background these modes are always constrained?3. As we will show the matching procedure of the singular perturbations involves non-separable solutions to the Teukolsky’s master equations. From a mathematical perspec-tive it is interesting to investigate how the non-separable solutions behave (and contrastto separable solutions).Answering these questions is outside the scope of this work. Our intention here is to initiatea discussion regarding how one would describe these modes and their properties in the wholeKerr geometry.The subsequent discussion will be divided in two parts. We first focus on building theHertz potential associated with singular perturbations, starting from the perturbations aroundNHEK. The advantage of transcribing the information of these modes to the Hertz potential istwofold: first, it illustrates some stark differences on the behaviour of Ψ H for (cid:96) = 0 , relativeto the propagating modes; second, it allows us to be more systematic as we attempt to extendthese modes to the full Kerr geometry via a matching procedure. This second feature isparticularly important for singular modes that have vanishing Weyl scalars on NHEK. Thematching procedure is the focus of the second portion of this section, where we highlight placeswhere we can make some progress and also potential obstacles to reconstruct these modes. The task is to build a Hertz potential Ψ H for the low-lying modes χ ( x , θ ) = sin θ (cid:88) (cid:96) =0 , S (cid:96) ( θ ) χ (cid:96) ( x ) , (cid:3) χ (cid:96) ( x ) = (cid:96) ( (cid:96) + 1) χ (cid:96) ( x ) . (6.1)48he first steps follow the analysis in Sec. 5.1. In particular we want to use results from thatsection that do not assume any property of χ nor Ψ H . This singles out (5.29) and (5.30), andfrom them we want to determine Ψ H .It is important to stress that if we assume a separable ansatz for Ψ H , such as (5.22), itleads to pathologies for (cid:96) = 0 , . This pathology is clear in (5.31). Actually, we will show thatHertz potentials associated to low-lying modes take the form Ψ H = (cid:88) (cid:96) =0 , (cid:16) S (cid:96) ( θ ) U (cid:96) ( x ) − S inhom (cid:96) ( θ ) n a n b ∇ a ∇ b χ (cid:96) ( x ) (cid:17) . (6.2)Crucially this Hertz potential is not a separable solution in contrast to (5.22). Our task is todetermine S inhom (cid:96) and U (cid:96) such that Ψ H solves (5.15), while being compatible with (5.29) and(5.30). This will give us a reversible map between Ψ H and χ .Let us first determine S inhom (cid:96) . Acting with L θ ( L θ + 2) on (6.2) gives zero on the first termbecause L θ S = 0 , and ( L θ + 2) S = 0 . (6.3)Comparing its action on the second term of (6.2) with (5.30) gives us the relation L θ ( L θ + 2) S inhom (cid:96) = S (cid:96) . (6.4)Using (6.3), the above equation can be integrated to L θ S inhom = S , and ( L θ + 2) S inhom = − S , (6.5)up to an homogenous piece that can be re-absorbed into the second term in (6.2). Notice wecan combine both equations (6.5) as ( L θ + (cid:96) ( (cid:96) + 1)) S inhom (cid:96) = − (cid:0) (cid:96) − (cid:1) S (cid:96) , (cid:96) = 0 , . (6.6)Next we require our ansatz (6.2) to solve the Hertz potential master equation (5.15). Thisgives an equation for U (cid:96) ( L x − (cid:96) ( (cid:96) + 1)) U (cid:96) ( x ) = 2(2 (cid:96) − n a n b ∇ a ∇ b χ (cid:96) ( x ) , (6.7)where we used the identity ( L x − (cid:96) ( (cid:96) + 1)) n a n b ∇ a ∇ b χ (cid:96) ( x ) = 0 , (6.8)holding for any χ (cid:96) satisfying the Klein-Gordon equation (6.1). Furthermore, acting with l a l b ∇ a ∇ b on (6.2) and plugging it into (5.29), we get the further relation l a l b ∇ a ∇ b U (cid:96) = − χ (cid:96) , (6.9)49here we used (A.43) and the Klein-Gordon equation for χ (cid:96) .Given the above analysis, our task of mapping any Hertz potential of the form (6.2) to alower lying χ mode (6.1), and vice-versa, reduces to solving (6.7) and (6.9) while χ (cid:96) satisfyingthe Klein-Gordon equation. We claim that this reconstruction always has a solution and weprovide a detailed derivation for both (cid:96) = 0 , in App. A.5. The bottom line is that startingfrom the low-lying χ − modes there is a corresponding Hertz potential of the form (6.2) whichobeys the master equation (5.15).Within the singular perturbations there are special cases when the Hertz potential isactually separable. From (6.2), requiring that the Hertz potential is separable, i.e. Ψ H = (cid:88) (cid:96) =0 , S (cid:96) ( θ ) U (cid:96) ( x ) , (6.10)implies that n a n b ∇ a ∇ b χ (cid:96) ( x ) = 0 . (6.11)In this situation the master equation becomes ( L x − (cid:96) ( (cid:96) + 1)) U (cid:96) ( x ) = 0 , (6.12)which originates from inserting (6.10) into (5.15). And there is as well the relation (6.9) thatrelates U (cid:96) ( x ) to χ (cid:96) ( x ) —note that (6.9) is compatible with (6.11)-(6.12). It is straightforwardto construct solutions in this case. One way is to first solve for U (cid:96) ( x ) from (6.12) and thenbuild the resulting χ (cid:96) from (6.9) which can be shown to comply with the constraints (6.11)after using (A.43). Another option is to first solve for χ (cid:96) from its equation of motion in(6.1) and the constraint (6.11), and then determine U (cid:96) from (6.9) and (6.12), similarly to thediscussion in App. A.5.One interesting aspect of solutions obeying (6.10)-(6.12) is the behaviour of the Weylscalars evaluated on them. Recalling the relation (3.20), we see that (6.11) implies Ψ = 0 . Ifwe further set l a l b ∇ a ∇ b χ (cid:96) ( x ) = 0 , (6.13)then Ψ = 0 . However, the Hertz potential associated to these modes is non-zero , despitehaving trivial Weyl scalars. The fact that Ψ H is separable and non-vanishing establishes clearrules on how we should match this special class of singular modes to the whole geometry. Wewill discuss this matching procedure in the next subsection. We have shown that Ψ H for these low-lying modes in (6.2) takes the form of sum of two terms,and hence does not comply with the usual basis of solutions that is typically used to describelinearized solutions to the Teukolsky’s master equation. We also take this as an indication thatthe Hertz potential ˜Ψ H in Kerr, which reduces to Ψ H in the decoupling limit (5.14), cannot bea single term either. It is not clear to us how to find the most general solutions to the master50quation (5.4) in the Kerr geometry. Despite this significant obstruction to reconstruct thesesingular low-lying modes in the entire geometry, we make the following exploratory ansatz ˜Ψ H = e − i ˜ ω ˜ t (cid:16) ˜ S (1) (cid:96) ˜ ω ( θ ) ˜ R (1) (cid:96) ˜ ω (˜ r ) + ˜ S (2) (cid:96) ˜ ω ( θ ) ˜ R (2) (cid:96) ˜ ω (˜ r ) (cid:17) , (6.14)where the angular functions ˜ S (1) (cid:96) ˜ ω and ˜ S (2) (cid:96) ˜ ω should satisfy (cid:16) L θ + K (cid:96) ˜ ω (cid:17) S (1) (cid:96) ˜ ω + (cid:16) a ˜ ω cos θ + a ˜ ω cos θ (cid:17) ˜ S (1) (cid:96) ˜ ω = − (cid:0) (cid:96) − (cid:1) ˜ S (2) (cid:96) ˜ ω , (cid:16) L θ + K (cid:96) ˜ ω (cid:17) S (2) (cid:96) ˜ ω + (cid:16) a ˜ ω cos θ + a ˜ ω cos θ (cid:17) ˜ S (2) (cid:96) ˜ ω = 0 . (6.15)Plugging (6.14) into the master equation (5.4), and using (6.15), we get two coupled equationsfor the radial functions (cid:16) L ˜ ω, ˜ r − K (cid:96) ˜ ω (cid:17) ˜ R (2) (cid:96) ˜ ω = − (cid:0) (cid:96) − (cid:1) ˜ R (1) (cid:96) ˜ ω , (cid:16) L ˜ ω, ˜ r − K (cid:96) ˜ ω (cid:17) ˜ R (1) (cid:96) ˜ ω = 0 . (6.16)where L ˜ ω, ˜ r is given by (5.6) by replacing ∂ ˜ t → − i ˜ ω .Without solving this system of equations explicitly, we will show that ˜Ψ H would complywith (5.14) by matching equations in the decoupling limit. Matching of the angular part isstraightforward. In the low frequency limit ˜ ω → , it is easy to see that ˜ S (2) (cid:96) ˜ ω is just thespherical harmonic function S (cid:96) , and ˜ S (1) (cid:96) ˜ ω satisfies the same differential equation as S inhom (cid:96) in(6.6). Therefore, in the low frequency limit, we have lim λ → ˜ S (1) (cid:96) ˜ ω = S inhom (cid:96) lim λ → ˜ S (2) (cid:96) ˜ ω = S (cid:96) . (6.17)Matching of the radial part is also similar to the discussion in Sec. 5.1: the radial equationcan be approximated in the near region with ˜ r − r + r + (cid:28) in the parameter regime (5.10). Notethat the radial differential operator L ˜ ω, ˜ r appearing in (6.16) reduces to L ω,r expressed infrequency space, as discussed in (5.17). Using this relation and comparing (6.16) with (6.7)and (6.8), it is straightforward to see that in the decoupling limit e − i ˜ ω ˜ t ˜ R (1) (cid:96) ˜ ω and e − i ˜ ω ˜ t ˜ R (2) (cid:96) ˜ ω satisfy the same equation as n a n b ∇ a ∇ b χ (cid:96) ( x ) and U (cid:96) , respectively.One can study the behaviour of ˜ R (1) (cid:96) ˜ ω and ˜ R (2) (cid:96) ˜ ω in the very far region of Kerr. Frompreliminary results, they seem to have a reasonable asymptotic series expansion, but a moredetailed analysis that includes the matching region is undoubtedly required. The angularfunction S inhom (cid:96) carries logarithmic terms: these do not affect the h µν in (5.1) for NHEK, butwe haven’t explored if the logarithmic pieces enter in the metric perturbation for the Kerrgeometry. We leave a more systematic study of these non-separable solutions for future work.Finally, we discuss how to glue modes of the form (6.10)-(6.13), i.e. carrying trivial Weylscalars on NHEK. The natural ansatz for the Hertz potential on Kerr is ˜Ψ H = e − i ˜ ω ˜ t ˜ S (cid:96) ˜ ω ( θ ) ˜ R (cid:96) ˜ ω (˜ r ) , (6.18)51ith ˜ S (cid:96) ˜ ω ( θ ) and ˜ R (cid:96) ˜ ω (˜ r ) satisfying (5.8) and (5.9). Similar to the discussion in Sec. 5.1, theradial equation (5.9) in Kerr can actually be solved in the near and far region as long as a | ˜ ω | (cid:28) , ˜ τ H (cid:28) , with ˜ ω ˜ τ H fixed. (6.19)A very interesting remark in this case regards the behaviour of the Weyl scalars between NHEKand Kerr. While we imposed the vanishing of the Weyl scalars in NHEK, this condition doesnot have to hold in Kerr. In fact, we can construct explicit solutions where it does not.This is due to the λ → limit in (5.21): the leading contributions vanish as we go nearthe horizon, but subleading contributions in ˜Ψ , are not necessarily zero. That is, (5.21)allows for singular perturbations where ˜Ψ , are non-zero, while as we take the decouplinglimit one still gets Ψ , = 0 . This is a striking feature that we have explored by studyingsome solutions to (6.10)-(6.13) and applying the same matching procedure as in Sec. 5.1. Itwould be interesting to investigate the properties of these metric perturbations in Kerr morecarefully and understand their imprints in the far region. Acknowledgements
AC, WS and BY would like to thank KITP, and in particular the program “GravitationalHolography,” for its hospitality during the completion of this work. The work of AC andVG is supported by the Delta ITP consortium, a program of the NWO that is funded by theDutch Ministry of Education, Culture and Science (OCW). VG acknowledges the postdoctoralprogram at ICTS for funding support through the Department of Atomic Energy, Governmentof India, under project no. RTI4001. WS and BY are supported by the NFSC Grant No.11735001. This research was supported in part by the National Science Foundation underGrant No. NSF PHY-1748958.
A Aspects of Teukolsky formalism
In this appendix we start by reviewing some of the basic elements of gravitational perturba-tions, gathering definitions and well known results that are specific to the Kerr backgroundand its near horizon geometry. Readers can find an excellent review in Appendix C of [43],and more recently in [60]. This discussion includes a summary of Wald’s theorem [30] char-acterizing the subset of Kerr perturbations with vanishing Weyl scalars. Also, appendix A.5presents a set of identities involving differential operators constructed out of the near horizonAdS tetrad which are used in Sec. 3 and Sec. 5. A.1 Overview
Any on-shell metric perturbation h of the Kerr black hole must solve the linearized Einstein’sequations E · h = 0 , (A.1)52here E is a self-adjoint linear partial differential operator (PDO). This is a coupled systemof partial differential equations typically written in a non-gauge invariant way. Hence, it ishard to solve and to extract the two polarizations carried by the perturbation.Using the Newman-Penrose formalism, Press and Teukolsky [24–26, 54] used gauge in-variant quantities, the Weyl scalars Ψ , defined in terms of h by Ψ =
T · h , (A.2)where T is a linear PDO, to show that any on-shell perturbation satisfied the Teukolsky’smaster equation O ·
Ψ = 0 . (A.3)A further achievement of the Teukolsky’s formalism was that these equations can be solvedby separation of variables. Hence, the original problem is mapped to a set of ODEs.The reconstruction of the gravitational perturbation h from the Weyl scalars is solvedusing the Cohen-Kegeles formalism [66, 67] (see also [68]). This is typically performed in theingoing (outgoing) radiation gauge IRG (ORG) l µ h µν = g µν h µν = 0 (IRG) , n µ h µν = g µν h µν = 0 (ORG) , (A.4)and involves the Hertz potential Ψ H whose existence is best understood following the robustmathematical formulation of this reconstruction problem given by Wald [69].Wald constructed two linear PDO, S and O , satisfying the operator equation [69] S · E = O · T (A.5)Applying this to h shows that E · h = 0 implies that Ψ must satisfy the master equation(A.3), in agreement with Teukolsky’s work. Furthermore, taking the adjoint (in the sense ofoperators) of (A.5) gives E ·S † = T † ·O † , since E † = E is self-adjoint. Thus, we can reconstructthe perturbation h h = S † · Ψ H (A.6)in terms of a potential Ψ H satisfying O † · Ψ H = 0 . (A.7)If Ψ H is a Hertz potential, i.e. it satisfies O † · Ψ H = 0 , we get a solution of Teukolsky equation O ·
Ψ = 0 given by
Ψ =
T · S † · Ψ H . This is simply obtained by reconstructing the metricand computing Ψ from it. This shows that there is a unique Weyl scalar Ψ for a given Hertzpotential. However, notice this conclusion doesn’t go the other way: different Hertz potentialscan give the same Ψ . 53 .2 Newman-Penrose formalism & master equations In this Appendix, we collect useful formulas in the Newman-Penrose (NP) formalism [70].For book keeping purposes, we introduce the parameter ι to denote different conventions inthe literature : ι = 1 corresponds to the mostly positive signature for the metric, namely ( − , + , + , +) , used in this paper, whereas ι = − corresponds to the (+ , − , − , − ) signatureused in, e.g., [43].The Newman-Penrose (NP) formalism [70] decomposes the metric components g µν = ι ( − l µ n ν − l ν n µ + m µ ¯ m ν + ¯ m µ m ν ) (A.8)in terms of three complex valued tetrads satisfying l · m = l · ¯ m = n · m = n · ¯ m = 0 ,l · l = n · n = m · m = ¯ m · ¯ m = 0 ,l · n = − ι, m · ¯ m = ι . We will using the mostly-plus signature conventions, ι = 1 , which is the opposite of thatfollowed by in [24–26].There are five inequivalent gauge invariant Weyl scalars built from contractions of theWeyl tensor with these tetrads Ψ = ι C µναβ l µ m ν (cid:96) α m β , Ψ = ι C µναβ l µ n ν (cid:96) α m β , Ψ = ι C µναβ l µ m ν ¯ m α n β , Ψ = ι C µναβ l µ n ν ¯ m α n β , Ψ = ι C µναβ n µ ¯ m ν n α ¯ m β . (A.9)In our manipulations we will denote with lower case, ψ i , the value of the Weyl scalars onthe background geometry, and with upper case, Ψ i , the linear contribution due to the metricperturbation. Only Ψ and Ψ are invariant under tetrad rotations and diffeomorphisms (atlinear order). Moreover, for the Kerr background, only one, either Ψ or Ψ is needed toestablish the dynamical properties of the linearized solutions. Hence, the remaining Weylscalars are not needed to describe gravitational waves on Kerr at this order.Introducing the differential operators D = l µ ∇ µ , ˆ∆ = n µ ∇ µ , δ = m µ ∇ µ , ¯ δ = ¯ m µ ∇ µ , (A.10) We have added hats on some of the commonly used symbols to avoid conflict with notation used in themain text. κ = − ιm µ Dl µ , ˆ τ = − ιm µ ˆ∆ l µ , (A.11) σ = − ιm µ δl µ , ρ = − ιm µ ¯ δ(cid:96) µ , (A.12) π = ι ¯ m µ Dn µ , ν = ι ¯ m µ ˆ∆ n µ , (A.13) µ = ι ¯ m µ δn µ , ˆ λ = ι ¯ m µ ¯ δn ν , (A.14) ε = − ι ( n µ D(cid:96) µ − ¯ m µ Dm µ ) , γ = − ι (cid:16) n µ ˆ∆ l µ − ¯ m µ ˆ∆ m µ (cid:17) , (A.15) β = − ι ( n µ δl µ − ¯ m µ δm µ ) , α = − ι (cid:0) n µ ¯ δl µ − ¯ m µ ¯ δm µ (cid:1) . (A.16)Teukolsky’s master equation (A.3) for the gravitational perturbations Ψ and Ψ can bewritten as O · Ψ = 0 , O · Ψ = 0 , (A.17)where O = ( D − ε + ¯ ε − ρ − ¯ ρ )( ˆ∆ + µ − γ ) − ( δ + ¯ π − ¯ α − β − τ )(¯ δ + π − α ) − ψ , O = ( ˆ∆ + 3 γ − ¯ γ + 4 µ + ¯ µ )( D + 4 ε − ρ ) − (¯ δ − ¯ˆ τ + ¯ β + 3 α + 4 π )( δ − ˆ τ + 4 β ) − ψ . The adjoint equations (A.7) satisfied by the Hertz potentials Ψ H and Ψ H reconstructingthe perturbation h are O † · Ψ H = 0 , O † · Ψ H = 0 , (A.18)with O † = ( ˆ∆ + 3 γ − ¯ γ + ¯ µ )( D + 4 ε + 3 ρ ) − (¯ δ + ¯ β + 3 α − ¯ τ )( δ + 4 β + 3ˆ τ ) − ψ , (A.19) O † = ( D − ε + ¯ ε − ¯ ρ )( ˆ∆ − γ − µ ) − ( δ − β − ¯ α + ¯ π )(¯ δ − α − π ) − ψ . (A.20)The reconstructed metric in IRG is then given by h IRG µν = (cid:15) (cid:110) l ( µ m ν ) [( D − ρ + ¯ ρ )( δ + 4 β + 3ˆ τ ) + ( δ + 3 β − ¯ α − ˆ τ − ¯ π )( D + 3 ρ )] − l µ l ν ( δ + 3 β + ¯ α − ˆ τ )( δ + 4 β + 3ˆ τ ) − m µ m ν ( D − ρ )( D + 3 ρ ) (cid:111) Ψ H + c.c. . (A.21)To reconstruct the metric in ORG one would use Ψ H instead. Notice the existence of a typo(shown in red) in the formula given in [43]. The one above agrees with, for example, Table 1in [68]. The relation between the Hertz potential and the defining Weyl scalar is given in [69],55hich we summarize below Ψ = (cid:15) D − ε + ¯ ε − ¯ ρ )( D − ε + 2¯ ε − ¯ ρ )( D − ε + 3¯ ε − ¯ ρ )( D + 4¯ ε + 3 ¯ ρ ) ¯Ψ H , Ψ = (cid:15) (cid:8) (¯ δ + 3 α + ¯ β − ¯ˆ τ )(¯ δ + 2 α + 2 ¯ β − ¯ˆ τ )(¯ δ + α + 3 ¯ β − ¯ˆ τ )(¯ δ + 4 ¯ β + 3¯ˆ τ ) ¯Ψ H +3Ψ [ˆ τ (¯ δ + 4 α ) − ρ (∆ + 4 γ ) − µ ( D + π (ˆ δ + 4 β ) + 2Ψ ]Ψ H (cid:111) . (A.22) A.3 Kerr & NHEK specifics
All previous equations hold for any type D vacuum solution. When restricting to the Kerrmetric (2.1), the Newman-Penrose tetrad usually used is the Kinnersley tetrad [71] ˜ l = ˜ r + a ∆ ∂ ˜ t + ∂ ˜ r + a ∆ ∂ ˜ φ , ˜ n = 12Σ (cid:16) (˜ r + a ) ∂ ˜ t − ∆ ∂ r + a ∂ ˜ φ (cid:17) , ˜ m = 1 √ (cid:18) i a sin θ ∂ ˜ t + ∂ θ + i sin θ ∂ ˜ φ (cid:19) , (A.23)where we defined Γ ≡ ˜ r + ia cos θ . The only non-vanishing Weyl scalar is ˜ ψ = − M ¯Γ , and the spin coefficients are κ = σ = ˆ λ = ν = (cid:15) = 0 , ρ = − ,β = cot θ / Γ , π = ia sin θ √ , α = π − ¯ β , (A.24) ˆ τ = − ia sin θ √ , µ = − ∆2Γ¯Γ , γ = µ + ˜ r − M . Notice we introduced a tilde for these tetrads (A.23) and ˜ ψ to identify them as full Kerrquantities, in agreement with the conventions used in the main text. Near-NHEK tetrad.
In the near-extremal near horizon limit ˜ r = √ J + λ (cid:18) r + τ r (cid:19) , ˜ t = 2 J tλ , ˜ φ = φ + √ J tλ , (A.25)56eading to (2.10), there is a finite NHEK tetrad l = 1(1 − τ r ) (cid:18) r ∂ t + (cid:18) − τ r (cid:19) ∂ r − r (cid:18) τ r (cid:19) ∂ φ (cid:19) ,n = 12 J (1 + cos θ ) (cid:18) ∂ t − r (cid:18) − τ r (cid:19) ∂ r − r (cid:18) τ r (cid:19) ∂ φ (cid:19) ,m = 1 √ √ J (1 + i cos θ ) (cid:18) ∂ θ + i (cid:18) θ − sin θ (cid:19) ∂ φ (cid:19) , (A.26)related to (A.23) by l = lim λ → ( λ ˜ l ) , n = lim λ → ( λ − ˜ n ) , m = lim λ → ˜ m . (A.27)These tetrads satisfy (A.9) and, once more, the only non-vanishing Weyl scalar is ψ = iJ ( i + cos θ ) . (A.28) Master equations (Kerr & NHEK).
Teukolsky’s master equation (A.3) for a spin- s field ˜Ψ ( s ) in the full Kerr geometry is [24] (cid:20) (˜ r + a ) ∆ − a sin θ (cid:21) ∂ t ˜Ψ ( s ) + 4 M a ˜ r ∆ ∂ ˜ t ∂ ˜ φ ˜Ψ ( s ) + (cid:20) a ∆ − θ (cid:21) ∂ φ ˜Ψ ( s ) − ∆ − s ∂ ˜ r (cid:16) ∆ s +1 ∂ ˜ r ˜Ψ ( s ) (cid:17) − θ ∂ θ (cid:16) sin θ ∂ θ ˜Ψ ( s ) (cid:17) − s (cid:20) a (˜ r − M )∆ + i cos θ sin θ (cid:21) ∂ ˜ φ ˜Ψ ( s ) − s (cid:20) M (˜ r − a )∆ − ˜ r − i a cos θ (cid:21) ∂ ˜ t ˜Ψ ( s ) + ( s cot θ − s ) ˜Ψ ( s ) = 0 . (A.29)Working in ingoing radiation gauge (IRG), the relevant Hertz potential is ˜Ψ H and using theadjoint properties of the operator O , we learn that it satisfies the master equation for s = − , i.e. ˜Ψ H = ˜Ψ (-2) .To solve (A.29) for normalizable solutions on the 2-sphere, we can use standard separationof variables ˜Ψ ( s ) = (cid:90) d ˜ ω (cid:88) (cid:96),m e − i ˜ ω ˜ t + im ˜ φ ˜ R (cid:96)m ˜ ω (˜ r ) ˜ S (cid:96)m ˜ ω ( θ ) , (A.30)57o obtain the two decoupled ODEs d ˜ R (cid:96)m ˜ ω d ˜ r + 2( s + 1)(˜ r − M ) d ˜ R (cid:96)m ˜ ω d ˜ r + (cid:18) C − is (˜ r − M ) C ∆ + 4 is ˜ ω ˜ r − B (cid:19) ˜ R (cid:96)m ˜ ω , θ ddθ (cid:32) sin θ d ˜ S (cid:96)m ˜ ω dθ (cid:33) (A.31) + (cid:18) a ˜ ω cos θ − m sin θ − a ˜ ω s cos θ − ms cos θ sin θ − s sin θ − m K (cid:96)m ˜ ω (cid:19) ˜ S (cid:96)m ˜ ω , where we defined C ≡ (˜ r + a ) ˜ ω − am , B ≡ K (cid:96)m ˜ ω − m a ˜ ω − am ˜ ω − s ( s + 1) , (A.32)in terms of the separation constant K (cid:96)m ˜ ω .For axisymmetric perturbations, i.e. with m = 0 , the above constants become C = (˜ r + a )˜ ω , B = K (cid:96) ˜ ω + a ˜ ω − s ( s + 1) , (A.33)and the angular ODE reduces to θ ddθ (cid:32) sin θ d ˜ S (cid:96) ˜ ω dθ (cid:33) + (cid:18) a ˜ ω cos θ − a ˜ ω s cos θ − s sin θ + K (cid:96) ˜ ω (cid:19) ˜ S (cid:96) ˜ ω = 0 . (A.34)The radial ODE for axisymetric perturbations can be written in a Schrödinger like way ∆ − s dd ˜ r (cid:32) ∆ s +1 d ˜ R (cid:96) ˜ ω d ˜ r (cid:33) = V (˜ r ) ˜ R (cid:96) ˜ ω , V (˜ r ) ≡ B − is ˜ ω ˜ r − C − is (˜ r − M ) C ∆ . (A.35)In the near horizon limit (A.25), the master equation (A.29) becomes r (4 r − τ ) ∂ t Ψ ( s ) − r ∂ r Ψ ( s ) − r (4 r + s (4 r + τ ))4 r − τ ∂ r Ψ ( s ) − θ ∂ θ ( sin θ ∂ θ Ψ ( s ) ) − s r (4 r + τ )(4 r − τ ) ∂ t Ψ ( s ) + ( s cot θ − s )Ψ ( s ) = 0 . (A.36)Performing some partial separation of variables Ψ ( s ) = (cid:88) (cid:96) U (cid:96) ( t, r ) S (cid:96) ( θ ) , (A.37)58he resulting angular ODE reduces to S (cid:48)(cid:48) (cid:96) + cot θ S (cid:48) (cid:96) − θ S (cid:96) = − K (cid:96) S (cid:96) , s = ± . (A.38)This is the same equation as the one appearing from Einstein’s equations in (3.9) and definesspin-weighted spherical harmonics with K (cid:96) = (cid:96) ( (cid:96) + 1) . The resulting equation for U (cid:96) is ∂ t U (cid:96) − r s r − τ ) s − ∂ r (cid:0) r − s (4 r − τ ) s +1 ∂ r U (cid:96) (cid:1) − s r (4 r + τ ) ∂ t U (cid:96) + 116 r (4 r − τ ) ( K − s ( s + 1)) U (cid:96) = 0 . (A.39) A.4 Wald’s theorem [30].
In this portion we summarise the results of [30] regarding gravitational perturbations on Kerr.The content of this work is a theorem which states that for well-behaved perturbations ona Kerr black hole, ˜Ψ and ˜Ψ uniquely determine each other. To understand this theorem,we need to decode two pieces of it: what it is meant by “well-behaved,” and the implicationsbehind determining ˜Ψ from ˜Ψ (or viceversa).In [30], “well-behaved” means verbatim:“ . . . that at some initial “time” ( i.e. , on an initial spacelike hypersurface which in-tersects the future event horizon) the perturbation (1) vanishes sufficiently rapidlyat infinity, (2) has no angular singularities, and (3) is regular on the future eventhorizon.”Our low-lying perturbations in Sec. 3.2 can carry angular singularities, which are interestingfor our applications. These cases therefore will lay outside the scope of the theorem.Determining ˜Ψ in terms of ˜Ψ , or the reverse, is clearly the important outcome. Thisimplies that the essential information of a gravitational perturbation is encoded in one Weylscalar, which was particularly useful for the stability analyses of Kerr and astrophysical ob-servations. The key is to show that if ˜Ψ = 0 it implies ˜Ψ = 0 , and viceversa. The proofis relatively straightforward from the Newman-Penrose formalism and uses the well-behavedproperties stated above.The results in [30] also proceed to characterise the solutions to ˜Ψ = Ψ = 0 , and thereare four linearly independent solutions:1. change in mass, from M to M + δM ,2. change in angular momentum, from J to J + δJ ,3. a perturbation towards Kerr-NUT,4. a perturbation towards the rotating C -metric.59he first two are viewed as trivial perturbations, and the last two as physically unacceptablesince they are excluded by the boundary conditions deemed as physical in [30]. The analysisimplicitly treats all diffeomorphisms as trivial transformations. A.5 Further identities and proofs
In this subsection we first collect some useful identities regarding differential operators actingon AdS that are used in Sec. 3 and Sec. 5. We then present details of the proofs of therelations between the Hertz potential and the low-lying modes discussed in Sec. 6. AdS tetrad. Using the notation of Sec. 2.1, AdS tetrads are two-dimensional vectorsobeying n · n = l · l = 0 , l · n = − . (A.40)In the coordinate system (2.15), they are explicitly given by l = 1 r ∂ t + ∂ r , n = 12 (cid:0) ∂ t − r ∂ r (cid:1) . (A.41)For thermal AdS with metric (D.2), the corresponding tetrad is l = 1(1 − τ r ) (cid:18) r ∂ t + (cid:18) − τ r (cid:19) ∂ r (cid:19) , n = 12 (cid:18) ∂ t − r (cid:18) − τ r (cid:19) ∂ r (cid:19) . (A.42)Given an arbitrary scalar function U ( x ) on AdS , the following identities, used in Sec. 3 andSec. 5, hold (cid:3) U ( x ) = − l a ∇ a )( n b ∇ b ) U ( x )= − n a ∇ a )( l b ∇ b ) U ( x ) + 4 n b A b ( l a ∇ a ) U ( x ) , l a l b ∇ a ∇ b U ( x ) = ( l a ∇ a )( l b ∇ b ) U ( x ) , n a n b ∇ a ∇ b U ( x ) = ( n a ∇ a )( n b ∇ b ) U ( x ) + 2 n b A b ( n a ∇ a ) U ( x ) , (cid:3) ( (cid:3) − U ( x ) = 4 l a l b ∇ a ∇ b (cid:16) n c n d ∇ c ∇ d U ( x ) (cid:17) , L x ( L x − U ( x ) = 4 n a n b ∇ a ∇ b (cid:16) l c l d ∇ c ∇ d U ( x ) (cid:17) , (A.43)where L x is defined in (5.15). Construction of Hertz potential for low-lying modes.
In this discussion, we give thedetails of the derivation on how to build Ψ H from a low-lying χ -mode discussed in Sec. 6.More precisely, given the decompositions Ψ H in (6.2) and χ ( x , θ ) = sin θ χ (cid:96) ( x ) S (cid:96) ( θ ) , ourmatching analysis in Sec. 6 showed [ L x − (cid:96) ( (cid:96) + 1)] U (cid:96) ( x ) = 2(2 (cid:96) − n a n b ∇ a ∇ b χ (cid:96) ( x ) , (A.44) l a l b ∇ a ∇ b U (cid:96) = − χ (cid:96) . (A.45)60ur task is to show that we can find a U (cid:96) ( x ) provided that χ (cid:96) satisfies the Klein-Gordonequation (cid:3) χ (cid:96) = (cid:96) ( (cid:96) + 1) χ (cid:96) , (A.46)for (cid:96) = 0 , .Consider the (cid:96) = 1 case first. A solution to (A.45) is l a ∇ a U ( x ) = n a ∇ a χ ( x ) , (A.47)since l a l b ∇ a ∇ b U ( x ) = ( l a ∇ a ) U ( x )= ( l a ∇ a )( n b ∇ b ) χ ( x )= − (cid:3) χ ( x )= − χ ( x ) , (A.48)where in the last line we used (A.46). Furthermore, it is easy to check that (A.47) solves(A.44). Hence, any solution to the linear operator equation (A.47) determines the relationbetween both modes.For the case (cid:96) = 0 , given an on-shell mode χ , the general solution to (A.45) is U = U p + U h , l a ∇ a U h = 0 , (A.49)with U p a given solution of (A.45), i.e. l a l b ∇ a ∇ b U p = − χ . What we show next is that givena solution χ and any particular solution U p , we can find a zero mode U h such that the Hertzequation (A.44) is satisfied.To prove this, we eliminate χ in (A.44) by plugging (A.45) and use (A.43) to get thedifferential equation for U , ( L x − U = 0 . (A.50)Inserting (A.49) into (A.50) gives U h = −
14 ( L x − U p , (A.51)where we used that L x U h = 0 . (A.51) enables us to solve for U h for any given U p determinedby a given χ . The last step is to show that the right hand side of (A.51) is necessarily a zeromode of l a ∇ a whenever (cid:3) χ = 0 . Indeed, applying l a ∇ a to (A.51), we get ( l a ∇ a ) U h = −
14 ( l a ∇ a )( L x − U p . (A.52)61owever, since ( l b ∇ b )( L x −
2) = 2( n a ∇ a )( l b ∇ b )( l c ∇ c ) , it follows ( l b ∇ b )( L x − U p = 4( n a ∇ a )( l b ∇ b )( n d ∇ d )( l c ∇ c )( l e ∇ e ) U p = − n a ∇ a )( l b ∇ b )( n d ∇ d ) χ = 2( n a ∇ a ) (cid:3) χ . (A.53)Thus, the right hand side of (A.52) vanishes as long as (cid:3) χ = 0 and U h given by (A.51) isindeed a zero mode of l a ∇ a . B Nearly-AdS holography In this appendix we review some aspects of nearly-AdS holography; for a more comprehensiveand detailed review we recommend [72].The holographic understanding of AdS was always more challenging than its higherdimensional cousins, mainly because of the lack of finite energy excitations in a pure gravitytheory above the AdS vacuum [73, 74]. However, it was observed in [22] that nearly AdS was a sensible theory by including the leading corrections away from pure AdS .One natural way to think of these leading corrections is to embed the AdS geometry into adifferent spacetime with potentially different asymptotics. This view appears naturally whendiscussing near-extremal black hole physics, because their near horizon geometry developsa local AdS throat [75]. Using the terminology of the AdS/CFT correspondence, suchembedding provides a UV completion of the AdS physics. It was argued in [22] that theseleading gravitational effects are captured by the Jackiw-Teitelboim (JT) theory [27, 28] inAdS with action I JT [ g ab , Φ JT ] = Φ πG (cid:90) Σ d x √− g R + Φ πG (cid:90) ∂ Σ d t √− γ K + 116 πG (cid:90) Σ d x √− g Φ JT ( R + 2) + 18 πG (cid:90) ∂ Σ d t √− γ Φ b ( K − , (B.1)where we have set (cid:96) AdS = 1 here. This is a particular case of two dimensional dilaton gravitytheory with potential V (Φ JT ) = 2Φ JT , where Φ JT measures the deviations in the size of theextremal black hole horizon Φ , i.e. Φ (cid:29) Φ JT . The first line is purely topological. It encodesthe entropy of the extremal black hole through the size of the extremal horizon captured by Φ . Variation with respect to the scalar JT-field Φ JT gives rise to R + 2 = 0 , forcing the twodimensional metric g ab to be locally AdS . Variation with respect to g ab gives rise to ∇ a ∇ b Φ JT − g ab (cid:3) Φ JT + g ab Φ JT = 0 . (B.2) This result can be proved under mild and reasonable assumptions, applies to fairly broad effective actionsemerging from string theory dynamics and survives higher order corrections [75] (see also [76] for a review onthe allowed such near horizon geometries). The original JT theory corresponds to the second line, where the boundary term was added to have awell defined variational principle. We included the first line to consider this review in light of the more recentperspective on this theory, as we discuss below. It equals − S χ Euler , where S is the extremal entropy and χ Euler is the Euler number of the manifold Σ .
62e will refer to these as the JT equations and to any scalar field, such as Φ JT , satisfying themas JT mode.To extract the effective action describing the low energy excitations of nearly-AdS , wevery briefly review the arguments in [23]. Given the UV completion provided by a near-extremal black hole, we want to identify the degrees of freedom responsible for the leadinggravitational corrections to pure AdS . To be definite, describe the latter in Poincaré coordi-nates ds AdS = 1 z (cid:0) − d t + d z (cid:1) . (B.3)The general solution to the JT equation (1.5) is given by Φ JT = 1 z (cid:0) α + β t + γ ( t − z ) (cid:1) . (B.4)Inspired by holography, we glue the JT geometry to the UV spacetime close to the AdS boundary ( z = δ → across a cut-off surface ( t ( u ) , z ( u )) , where u stands for the boundarytime, by requiring the boundary conditions g | bdy = − δ = − ( t (cid:48) ) + ( z (cid:48) ) z , Φ JT | bdy = Φ b = Φ r δ (B.5)In the absence of the dilaton field (pure AdS situation), the first condition can always besolved by an arbitrary t ( u ) , modulo the SL(2) isometry of pure AdS , with the choice z ( u ) = δ t (cid:48) . An alternative way of reaching this conclusion is to consider the asymptotic symmetriesof AdS [77–79]. These are generated by ζ t = f ( t ) , ζ z = z f (cid:48) ( t ) , (B.6)and would map the cut-off t ( u ) = u to t ( u ) = u + f ( u ) . Either way, the reparameterisationsymmetry is spontaneously broken by pure AdS giving rise to an infinite number of pseudo-Goldstone modes parameterised by t ( u ) . In the presence of the dilaton field, the secondboundary condition correlates the shape of the cut-off surface t ( u ) with the source Φ r ( u ) α + β t ( u ) + γ t ( u )( t (cid:48) ( u )) = Φ r ( u ) . (B.7)If we interpret the latter as an equation of motion for a dynamical field t ( u ) , it was noticedin [23] that it can be obtained from varying the effective action I Schwarzian = − πG (cid:90) ∂ Σ du Φ r ( u ) { t ( u ) , u } , (B.8) It is important to stress that to keep the near-extremal black hole UV interpretation of the construction,the boundary value Φ b must satisfy Φ b ∝ δ − (cid:28) φ . { t ( u ) , u } ≡ (cid:18) t (cid:48)(cid:48) t (cid:48) (cid:19) (cid:48) − (cid:18) t (cid:48)(cid:48) t (cid:48) (cid:19) . (B.9)In fact, if we ignore the topological terms, the action (B.8) originates from the boundary termin (B.1) by using the Φ JT equation of motion, i.e. R = − , and evaluating the extrinsiccurvature K using the boundary conditions (B.5) [23]. We conclude that the zero modes geta non-vanishing action, proportional to the Schwarzian (B.9), whenever there is a source, i.e. a non-trivial boundary condition for the dilaton field which is also responsible for explicitlybreaking the AdS asymptotic symmetry group. From purely kinematic considerations, theeffective action (B.8) is the simplest local boundary action linear in the source Φ r ( u ) andinvariant under the global SL(2) acting on the space of pseudo-Goldstone modes t ( u ) .Before closing this review, we would like to stress two further points. The first one isconcerned with the energy of the excitations described by the JT action (B.1) M ( u ) = − Φ r πG { t ( u ) , u } . (B.10)In the absence of a source Φ r , it vanishes, as it should for pure AdS . In the presence ofa source, it can be finite. This is a consequence of appropriately embedding the leadinggravitational corrections to pure AdS in a UV complete scheme, as originally envisioned in[22]. In fact, in the absence of matter, conservation of energy is equivalent to the equation ofmotion for t ( u ) . The second one has to do with the expected universality of the physics justreviewed. The Schwarzian action (B.8), together with the addition of relevant matter degreesof freedom, captures the thermodynamics of near-extremal black holes at low temperaturesand is expected to arise as a universal low energy sector in near-extremal black hole physics. C Plebański–Demiański type D solutions
The complete family of type D spacetimes in Einstein-Maxwell theory was given by Plebańskiand Demiański [80]. We refer to [81] for a review and reinterpretation of these geometries.In this Appendix, we will focus on the solutions of pure Einstein gravity with
Λ = 0 , whichare summarized in Fig. 1 of [81]. In addition to the mass and angular momentum, the lineelement also contains a NUT parameter n and an acceleration parameter α . In the notationsof [81], the line element is d s = 1Ω (cid:18) Qρ (cid:2) d t − (cid:0) a sin θ + 4 n sin ( θ ) (cid:1) d φ (cid:3) − ρ Q d r − ρ (cid:101) P sin θ d θ (C.1) − (cid:101) Pρ (cid:2) a d t − ( r + ( a + n ) )d φ (cid:3) (cid:33) , ρ = r + ( n + a cos θ ) , Ω = 1 − αr ( n + a cos θ ) , (C.2) Q = α ( a − n ) r −
11 + 3 α n ( a − n ) (2 r ( M + αn ( M αn − a − n )) − (1 + 2 M αn )( r + a − n )) , (cid:101) P = (1 − aαM cos θ ) sin θ + α a ( a − n )(1 + 2 αnM )1 + 3 α n ( a − n ) (4 n + a cos θ ) cos θ sin θ . C.1 Changing extremal mass
The Kerr metric is obtained after setting α = n = 0 and extremality is achieved with a = M .The near horizon geometry is obtained taking the limit (2.3) which leads to the NHEK metric(2.4). We can consider the perturbation which changes the extremal mass M J → J + (cid:15) δJ + O ( (cid:15) ) , (C.3)This gives the perturbation (3.44) with χ ( x , θ ) = δJJ (1 + cos θ ) , Φ( x ) = 2 δJJ . (C.4)which corresponds to the even ∆ = 1 mode. We see that the constant Φ mode is turned onand cancels the two conical singularities at θ = 0 and θ = π carried by χ . C.2 Adding NUT charge
We obtain the Kerr-NUT metric after setting the acceleration parameter α to zero. Theextremal limit is achieved for M ≡ M = a − n . (C.5)The near horizon geometry can then be obtained using t → a ( a + n ) tλ , r → M + λr , φ → φ + a tλ , (C.6)in the limit λ → . We obtain the NHEK-NUT metric ds = (cid:0) a (1 + cos θ ) + 2 an cos θ (cid:1) (cid:18) − r d t + d r r + d θ (cid:19) (C.7) + 4 a sin θa (1 + cos θ ) + 2 n cos θ (( a + n )d φ + M r d t ) , This formula is obtained from (17) of [81] with e = g = Λ = 0 and with ω = 1 . The perturbation corresponding to adding NUT charge to theNHEK is obtained by writing n = (cid:15) δn + O ( (cid:15) ) , (C.8)which gives the perturbation (3.44) with χ ( x , θ ) = 2 δnM cos θ , Φ( x ) = 2 δnM , (C.9)corresponding to an (cid:96) = 0 mode. C.3 Accelerated NHEK
After setting the NUT parameter n to zero, we obtain the spinning C -metric [56, 57]. For theextremal case M = a , we can take the near horizon limit using t → J − α J tλ , r → r + + λr , φ → φ + √ J − α J tλ , (C.10)with λ → . This leads to the accelerated NHEK geometry: ds = J (1 + cos θ )(1 − αJ cos θ ) (cid:20) − α J (cid:18) − r d t + d r r (cid:19) + d θ (1 − αJ cos θ ) (cid:21) + 4 J sin θ θ (cid:18) d φ + r − α J d t (cid:19) , (C.11)which was studied in [83]. We can then consider the perturbation of NHEK corresponding toadding acceleration: α = (cid:15) δα + O ( (cid:15) ) . (C.12)We have to redefine θ → θ − (cid:15) δα sin θ to fit this perturbation into the ansatz (3.44) and weobtain an (cid:96) = 0 mode: χ ( x , θ ) = 4 J δα cos θ , Φ( x ) = 0 . (C.13) D Isometries of (near-)NHEK
Starting from the sl (2) Killing vectors of the NHEK geometry given in (2.5), the sl (2) Killingvectors of the near-NHEK geometry at temperature τ / (2 π ) are obtained using the diffeomor-phism (2.7) with f ( t ) satisfying { f ( t ) , t } = − τ . (D.1)The metric of the AdS factor takes the form ds = − r (cid:18) − τ r (cid:19) d t + d r r . (D.2) Our formula differs from their by a rescaling of the angle φ → aa + n φ . f ( t ) = a e τt + bc e τt + d , (cid:0) a bc d (cid:1) ∈ GL(2 , R ) . (D.3)In the main text, we use the choice f ( t ) = e τt which is the only choice for which ζ isproportional to ∂ t . This leads to the Killing vectors ζ = 1 τ ∂ t , ζ ± = (cid:18) r + τ τ (4 r − τ ) ∂ t ∓ r∂ r − τ r r − τ ∂ φ (cid:19) e ± τt , (D.4)satisfying the sl (2) algebra [ ζ , ζ ± ] = ± ζ ± , [ ζ − , ζ + ] = 2 ζ , (D.5)and corresponding to the dual scalars Φ ζ = rτ (cid:18) τ r (cid:19) , Φ ζ ± = rτ (cid:18) − τ r (cid:19) e ± τt . (D.6)Another natural choice is f ( t ) = τ tanh (cid:0) τt (cid:1) , (D.7)which has a smooth limit lim τ → f ( t ) = t corresponding to the Poincaré basis of NHEK. Thisleads to the Killing vectors ξ = 4 r + τ τ (4 r − τ ) sinh( τ t ) ∂ t − r cosh( τ t ) ∂ r − τ r r − τ sinh( τ t ) ∂ φ ,ξ − = 12 (cid:18) r + τ r − τ cosh( τ t ) (cid:19) ∂ t − τ r τ t ) ∂ r − rτ r − τ cosh( τ t ) ∂ φ ,ξ + = − τ (cid:18) − r + τ r − τ cosh( τ t ) (cid:19) ∂ t − rτ sinh( τ t ) ∂ r − r r − τ cosh( τ t ) ∂ φ , corresponding to the dual scalars Φ ξ − = r (cid:18) τ r (cid:19) + r (cid:18) − τ r (cid:19) cosh( τ t ) , Φ ξ = rτ (cid:18) − τ r (cid:19) sinh( τ t ) , (D.8) Φ ξ + = − rτ (cid:18) τ r (cid:19) + 2 rτ (cid:18) − τ r (cid:19) cosh( τ t ) .
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