Boundary Conditions for Kerr-AdS Perturbations
BBoundary Conditions for Kerr-AdS Perturbations ´Oscar J. C. Dias (cid:63) , Jorge E. Santos ‡ (cid:63) Institut de Physique Th´eorique, CEA Saclay,CNRS URA 2306, F-91191 Gif-sur-Yvette, France ‡ Department of Physics, UCSB, Santa Barbara, CA 93106, USA [email protected], [email protected]
Abstract
The Teukolsky master equation and its associated spin-weighted spheroidal harmonicdecomposition simplify considerably the study of linear gravitational perturbations of theKerr(-AdS) black hole. However, the formulation of the problem is not complete before weassign the physically relevant boundary conditions. We find a set of two Robin boundary con-ditions (BCs) that must be imposed on the Teukolsky master variables to get perturbationsthat are asymptotically global AdS, i.e. that asymptotes to the Einstein Static Universe. Inthe context of the AdS/CFT correspondence, these BCs allow a non-zero expectation valuefor the CFT stress-energy tensor while keeping fixed the boundary metric. When the rota-tion vanishes, we also find the gauge invariant differential map between the Teukolsky andthe Kodama-Ishisbashi (Regge-Wheeler − Zerilli) formalisms. One of our Robin BCs maps tothe scalar sector and the other to the vector sector of the Kodama-Ishisbashi decomposition.The Robin BCs on the Teukolsky variables will allow for a quantitative study of instabilitytimescales and quasinormal mode spectrum of the Kerr-AdS black hole. As a warm-up forthis programme, we use the Teukolsky formalism to recover the quasinormal mode spectrumof global AdS-Schwarzschild, complementing previous analysis in the literature. a r X i v : . [ h e p - t h ] F e b ontents a = 0 ) 14 vs Kodama-Ishibashi (Regge-Wheeler − Zerilli) . . . . . . . . . . . . . 17
A An overview of perturbations in Kerr(-AdS) 27
It is unquestionable that few systems are isolated in Nature and we can learn a lot from studyingtheir interactions. Black holes are no exception and the study of their perturbations and inter-actions reveals their properties (see e.g. the recent roadmap [1] and review [2] on the subject).The simplest deformation we can introduce in a background is a linear perturbation, which of-ten encodes interesting physics such as linear stability of the system and its quasinormal modespectrum. Moreover, it also anticipates some non-linear level properties. For example, in thecollision of two black holes, such as in the coalescence of a binary system, after the inspiral andmerger phase, the system undergoes a ring down phase where gravitational wave emission is dic-tated by the quasinormal mode frequencies. The linear perturbation fingerprints are thereforevaluable from a theoretical and gravitational-wave detection perspective [1, 2]. Perhaps moresurprisingly, linear analysis of black holes in AdS can be used to infer properties about theirnonlinear stability [3, 4, 5]. Linear analysis can also infer some properties of (nonlinear) blackhole collisions and associated gravitational wave emission in the close-limit approximation [6].To study linear gravitational perturbations of a black hole we need to solve the linearizedEinstein equation. `A priori this is a remarkable task involving a coupled system of PDEs.Fortunately, for the Kerr(-AdS) black holes (which are Petrov type D backgrounds), Teukolskyemployed the Newman-Penrose formalism to prove that all the gravitational perturbation in-formation is encoded in two decoupled complex Weyl scalars [7, 8]. These are gauge invariantquantities with the same number of degrees of freedom as the metric perturbation. Moreover,there is a single pair of decoupled master equations governing the perturbations of these Weylscalars (one element of the pair describes spin s = 2 and the other s = − asymptotically global AdS . To make this statement precise, recall that once we havethe solution of the Teukolsky pair of master variables ( s = ± s = −
2) and the other in the outgoing radiation gauge (ORG; s = 2). By asymptotically global AdS perturbations we mean that we want the BCs in theTeukolsky scalars that yield metric perturbations that decay at asymptotic infinity accordingto the power law found by Henneaux and Teiltelboim [31, 32]. Our task is thus very well defined.We have to work out the inverse Hertz map and find how the Henneaux-Teiltelboim metric BCstranslate into the Teukolsky scalars.Before arguing further that this choice should be the physically relevant option, it is illu-minating to recall what is the situation in an asymptotically flat system. In this case, the BCchoice in the Teukolsky scalars amounts to choosing the purely outgoing traveling mode. Intu-itively, this is because we are not interested in scattering experiments (where an ingoing modecomponent would be present). Formally, this is because this is the choice that yields a metricperturbation preserving the asymptotic flatness of the original Kerr black hole, i.e. conservingasymptotically the Poincar´e group of the Minkowski spacetime.A similarly reasoning justifies why the Henneaux-Teiltelboim BCs should be the physicallyrelevant boundary condition to be imposed in the Kerr-AdS system [32]. These are the BCsthat preserve asymptotically the global AdS symmetry group O (3 ,
2) and yield finite surfaceintegral charges associated with the O (3 ,
2) generators. Yet an additional reason to single outthis BC is justified by the AdS/CFT duality. The Kerr-AdS is asymptotically global AdSand the CFT lives on the boundary of this bulk spacetime. As desired in this context, theHenneaux-Teiltelboim BCs are such that they allow for a non-zero expectation value for theCFT stress-energy tensor while keeping fixed the boundary metric. This criterion to select theBCs of gauge invariant variables was first emphasized in the context of the Kodama-Ishibashi(KI) formalism [14] by Michalogiorgakis and Pufu [33]. They pointed out that previous analysisof quasinormal modes of the 4-dimensional global AdS-Schwarzschild black hole using the KImaster equations were preserving the boundary metric only in the KI vector sector, but not inthe KI scalar sector of perturbations (here, vector/sector refer to the standard KI classificationof perturbations). Indeed, previous studies in the literature had been imposing Dirichlet BCson the KI gauge invariant variables. It turns out that in the KI scalar sector keeping the1oundary metric fixed requires a Robin BC (which relates the field with its derivate) [33].Still in the context of AdS/CFT on a sphere, other boundary conditions that might be calledasymptotically globally AdS (and that promote the boundary graviton to a dynamical field)were proposed in [34]. However, they turn out to lead to ghosts (modes with negative kineticenergy) and thus make the energy unbounded below [35]. So, the Henneaux-Teiltelboim BCsare also the physically relevant BCs for the AdS/CFT where the CFT lives in the EinsteinStatic Universe.So, a global AdS geometry with Henneaux-Teitelboim BCs does not deform the boundarymetric. This is the mathematical statement materializing the pictoric idea that a global AdSbackground behaves like a confining box with a reflecting wall. An interesting observation thatemerges from our study is that these BCs require that we consider a particular linear combina-tion of the Teukolsky IRG and ORG metric contributions. We can interpret this property asbeing a manifestation of the common lore that only a standing wave with a node at the AdSboundary can fit inside the confining box. This pictorial notion of a standing wave and nodeis very appealing but, what is the formal definition of a node in the present context? Does itmean that we have to impose a Dirichlet BC on the Teukolsky scalars? No. Instead we will findthe Robin BC (3.9)-(3.12), much like what happens in the scalar sector of the aforementioned4-dimensional KI system. An inspection of this Robin BC (pair) immediately convinces us thatwe hardly could guess it without the actual computation.At first sight the fact the asymptotically global AdS BC requires a sum of the TeukolskyIRG and ORG metric components is rather surprising and, even worrying. Surprising becausein the asymptotically flat case we just need to use the outgoing contribution. Eventually wor-rying because it is known that in Petrov type D backgrounds the two Teukolsky families ofperturbations ( s = ±
2) encode the same information, once we use the Starobinsky-Teukolskyidentities [36, 37, 38, 9, 10] that fix the relative normalization between the s = ± s = ± s = 2 (say) Teukolsky sectorof perturbations. We believe that an infinitesimal rotation of the tetrad basis should allow toderive our results using only the outgoing gauge (say), although at the cost of loosing contactwith the standing wave picture.We have already mentioned that perturbations for static backgrounds (with global AdSand global AdS-Schwarzschild being the relevant geometries here) can be studied using theKodama-Ishibashi (KI) gauge invariant formalism [14] (i.e. the Regge-Wheeler − Zerilli formal-ism [12, 13]). On the other hand, the Teukolsky formalism also describes these cases whenrotation is absent. Therefore, the two formalisms must be related in spite of their differences,although this one-to-one map has not been worked out to date. We fill this gap in the literature.The difference that stands out the most is that the KI formalism decomposes the gravitationalperturbations in scalar and vector spherical harmonics while the Teukolsky formalism uses in-stead a harmonic decomposition with respect to the spin-weighted spherical harmonics. Theseharmonics are distinct and ultimately responsible for the different routes taken by the two for-malisms. However, both the KI spherical harmonics and the spin-weighted spherical harmonicscan be expressed in terms of the standard scalar spherical harmonic (associated Legendre poly-nomials) and their derivatives. These two maps establish the necessary bridge between theangular decomposition of the two formalisms. We then need to work out the radial map, whichfollows from the fact that the metric perturbations of the two formalisms must be the samemodulo gauge transformations. This gauge invariant differential map expresses the KI master2ariables (for the KI scalar and vector sector) in terms of the s = 2 (say) Teukolsky masterfield and its first radial derivative and is given in (4.15)-(4.16). To have the complete mapbetween the KI and Teukolsky ( a = 0) formalisms we also need to discuss the relation betweenthe asymptotically global AdS KI BCs and the global AdS Teukolsky BCs. This is done in(4.18)-(4.22). The fact that our BCs for the Teukolsky variables match the Michalogiorgakis-Pufu BCs for the KI variables is a non-trivial check of our computation in the limit a → Our results and presentation contribute to complementthese analysis by plotting the spectrum as a function of the horizon radius, and not just a fewpoints of the spectrum. Our analysis focus on the parameter space region of r + /L (the horizonradius in AdS units) where the spectrum meets the normal modes of AdS and where it variesthe most. We will not discuss the asymptotic (for large overtone [41]-[44] and for large harmonic[5]) behaviour of the QN mode spectrum.The plan of this paper is the following. Section 2 discusses the Kerr-AdS black hole in theChambers-Moss coordinate frame [46] (instead of the original Carter frame [45]) that simplifiesconsiderably our future discussion of the results. We discuss the Teukolsky formalism, theassociated Starobinsky-Teukolsky identities and the Hertz map in a self-contained expositionbecause they will be fundamental to derive our results. In Section 3 we find the BCs on theTeukolsky variables that yields asymptotically global AdS perturbations. Section 4 constructsthe gauge invariant differential map between the Teukolsky and Kodama-Ishibashi (Regge-Wheeler − Zerilli) gauge invariant formalisms. Finally, in Section 5 we study the QNM spectrum/ CFT timescales of the global AdS(-Schwarzschild) background.
The Kerr-AdS geometry was originally written by Carter in the Boyer-Lindquist coordinatesystem { T, r, θ, φ } [45]. Here, following Chambers and Moss [46], we introduce the new timeand polar coordinates { t, χ } related to the Boyer-Lindquist coordinates { T, θ } by t = Ξ T , χ = a cos θ , (2.1) As noticed in [33] the analysis done in [39]-[44] in the KI scalar case does not impose asymptotically global AdSBCs and thus we will not discuss further the scalar results of these studies. ds = − ∆ r ( r + χ ) Ξ (cid:18) dt − a − χ a dφ (cid:19) + ∆ χ ( r + χ ) Ξ (cid:18) dt − a + r a dφ (cid:19) + (cid:0) r + χ (cid:1) ∆ r dr + (cid:0) r + χ (cid:1) ∆ χ dχ (2.2)where∆ r = (cid:0) a + r (cid:1) (cid:18) r L (cid:19) − M r , ∆ χ = (cid:0) a − χ (cid:1) (cid:18) − χ L (cid:19) , Ξ = 1 − a L . (2.3)The Chambers-Moss coordinate system { t, r, χ, φ } has the nice property that the line elementtreats the radial r and polar χ coordinates at an almost similar footing. One anticipates that thisproperty will naturally extend to the radial and angular equations that describe gravitationalperturbations in the Kerr-AdS background. In this frame, the horizon angular velocity andtemperature are given byΩ H = ar + a , T H = 1Ξ (cid:20) r + π (cid:18) r (cid:96) (cid:19) r + a − πr + (cid:18) − r (cid:96) (cid:19)(cid:21) . (2.4)The Kerr-AdS black hole obeys R µν = − L − g µν , and asymptotically approaches globalAdS space with radius of curvature L . This asymptotic structure is not manifest in (2.2), one ofthe reasons being that the coordinate frame { t, r, χ, φ } rotates at infinity with angular velocityΩ ∞ = − a/ ( L Ξ). However, if we introduce the coordinate change T = t Ξ , Φ = φ + aL t Ξ ,R = (cid:112) L ( a + r ) − ( L + r ) χ L √ Ξ , cos Θ = L √ Ξ r χa (cid:112) L ( a + r ) − ( L + r ) χ , (2.5)we find that as r → ∞ (i.e. R → ∞ ), the Kerr-AdS geometry (2.2) reduces to ds AdS = − (cid:18) R L (cid:19) dT + dR R L + R (cid:0) d Θ + sin Θ d Φ (cid:1) , (2.6)that we recognize as the line element of global AdS. In other words, the conformal boundaryof the bulk spacetime is the static Einstein universe R t × S : ds bdry = lim R →∞ L R ds AdS = − dT + d Θ + sin Θ d Φ . This is the boundary metric where the CFT lives in the context ofthe AdS /CFT correspondence.The ADM mass and angular momentum of the black hole are related to the mass M androtation a parameters through M ADM = M/ Ξ and J ADM = M a/ Ξ , respectively [48, 49]. Thehorizon angular velocity and temperature that are relevant for the thermodynamic analysis arethe ones measured with respect to the non-rotating frame at infinity [48, 49] and given in termsof (2.4) by T h = Ξ T H and Ω h = Ξ Ω H + aL . The event horizon is located at r = r + (the largestreal root of ∆ r ), and it is a Killing horizon generated by the Killing vector K = ∂ T + Ω h ∂ Φ .Further properties of the Kerr-AdS spacetime are discussed in Appendix A of [47].4 .2 Teukolsky master equations The Kerr-AdS geometry is a Petrov type D background and therefore perturbations of thisgeometry can be studied using the Teukolsky formalism, which uses the Newman-Penrose (NP)framework [7, 8, 11].The building blocks of this formalism are: • the NP null tetrad e a = { (cid:96) , n , m , m } (the bar demotes complex conjugation) obeying thenormalization conditions (cid:96) · n = − , m · m = 1; • the NP spin connection γ cab = e µb e νc ∇ µ e a ν (with γ cab = − γ acb ); • the associated NP spin coeficients defined in terms of γ cab as κ = − γ , λ = γ , ν = γ , σ = − γ , α = ( γ − γ ) , β = ( γ − γ ) ,µ = γ , ρ = − γ , π = γ , τ = − γ , γ = ( γ − γ ) , (cid:15) = ( γ − γ );(2.7) • the five complex Weyl scalars ( C abcd are the Weyl tensor components in the NP null basis)Ψ = C , Ψ = C , Ψ = C , Ψ = C , Ψ = C ; (2.8) • and the NP directional derivative operators D = (cid:96) µ ∂ µ , ∆ = n µ ∂ µ , δ = m µ ∂ µ , ¯ δ = m µ ∂ µ .The complex conjugate of any complex NP quantity can be obtain through the replacement3 ↔ ,vanish: Ψ = Ψ = Ψ = Ψ = 0 and Ψ = − M ( r − iχ ) − . Due to the Goldberg-Sachs theoremthis further implies that κ = λ = ν = σ = 0 . In addition, we might want to set (cid:15) = 0 bychoosing (cid:96) to be tangent to an affinely parametrized null geodesic (cid:96) µ ∇ µ (cid:96) ν = 0. This was theoriginal choice of Teukolsky and Press (when studing perturbations of the Kerr black hole) whoused the the outgoing (ingoing) Kinnersly tetrad that is regular in the past (future) horizon [38].In the Kerr-AdS case we can work with the natural extension of Kinnersly’s tetrad to AdS, andthis was the choice made in [50]. However, here we choose to work with the Chambers-Mossnull tetrad defined as [46], (cid:96) µ ∂ µ = 1 √ (cid:112) r + χ (cid:18) Ξ a + r √ ∆ r ∂ t + (cid:112) ∆ r ∂ r + a Ξ √ ∆ r ∂ φ (cid:19) , n µ ∂ µ = 1 √ (cid:112) r + χ (cid:18) Ξ a + r √ ∆ r ∂ t − (cid:112) ∆ r ∂ r + a Ξ √ ∆ r ∂ φ (cid:19) , m µ ∂ µ = − i √ (cid:112) r + χ (cid:32) Ξ a − χ (cid:112) ∆ χ ∂ t + i (cid:112) ∆ χ ∂ χ + a Ξ (cid:112) ∆ χ ∂ φ (cid:33) , (2.9)which is not affinely parametrized ( (cid:15) (cid:54) = 0). The motivation for this choice is two-folded. First,the technical analysis of the angular part of the perturbation equations and solutions will bemuch simpler because this Chambers-Moss tetrad explores the almost equal footing treatmentof the r, χ coordinates much more efficiently than Kinnersly’s tetrad. Second, to completeour analysis later on we will have to discuss how the metric perturbations h ab (built out ofthe NP perturbed scalars) transform both under infinitesimal coordinate transformations andinfinitesimal change of basis. It turns out that if we work in the Chambers-Moss tetrad, the5esults will be achieved without requiring a change of basis, while the Kinnersly’s option woulddemand it. Again, this simplifies our exposition.Teukolsky’s treatment applies to arbitrary spin s perturbations. Here, we are interested ingravitational perturbations so we restrict our discussion to the s = ± i and their perturbations by δ Ψ i with i = 1 , · · · ,
5. Theimportant quantities for our discussion are the scalars δ Ψ and δ Ψ . They are invariant bothunder infinitesimal coordinate transformations and under infinitesimal changes of the NP basis.A remarkable property of the Kerr-AdS geometry is that all information on the most general linear perturbation of the system is encoded in these gauge invariant variables δ Ψ and δ Ψ .That is, the perturbation of the leftover NP variables can be recovered once δ Ψ and δ Ψ areknown. The later are the solutions of the Teukolsky master equations.For s = 2 perturbations the Teukolsky equation is (cid:26) [ D − (cid:15) + ¯ (cid:15) − ρ − ¯ ρ ] (∆ + µ − γ ) − [ δ + ¯ π − ¯ α − β − τ ] (cid:0) ¯ δ + π − α (cid:1) − (cid:27) δ Ψ = 4 π T , if s = 2 , (2.10)while s = − (cid:26) [∆ + 3 γ − ¯ γ + 4 µ + ¯ µ ] ( D + 4 (cid:15) − ρ ) − (cid:2) ¯ δ − ¯ τ + ¯ β + 3 α + 4 π (cid:3) ( δ − τ + 4 β ) − (cid:27) δ Ψ = 4 π T , if s = − . (2.11)The explicit form of the source terms T ( s ± ) , that vanish in our analysis, can be found in [8].Next we introduce the separation ansatz (cid:40) δ Ψ = ( r − iχ ) − e − iωt e imφ R ( − ω(cid:96)m ( r ) S ( − ω(cid:96)m ( χ ) , for s = − ,δ Ψ = ( r − iχ ) − e − iωt e imφ R (2) ω(cid:96)m ( r ) S (2) ω(cid:96)m ( χ ) , for s = 2 . (2.12)Also, define the radial {D n , D † n } and angular {L n , L † n } differential operators D n = ∂ r + i K r ∆ r + n ∆ (cid:48) r ∆ r , D † n = ∂ r − i K r ∆ r + n ∆ (cid:48) r ∆ r L n = ∂ χ + K χ ∆ χ + n ∆ (cid:48) χ ∆ χ , L † n = ∂ χ − K χ ∆ χ + n ∆ (cid:48) χ ∆ χ , (2.13)where the prime (cid:48) represents derivative wrt the argument and K r = Ξ (cid:2) ma − ω (cid:0) a + r (cid:1)(cid:3) , K χ = Ξ (cid:2) ma − ω (cid:0) a − χ (cid:1)(cid:3) . (2.14)With the ansatz (2.12), the Teukosky master equations separate into a pair of equations for theradial R ( s ) ω(cid:96)m ( r ) and angular S ( s ) ω(cid:96)m ( χ ) functions, (cid:16) D †− ∆ r D + 6 (cid:16) r L − i Ξ ω r (cid:17) − λ (cid:17) R ( − ω(cid:96)m ( r ) = 0 , (cid:16) L †− ∆ χ L + 6 (cid:16) χ L + Ξ ω χ (cid:17) + λ (cid:17) S ( − ω(cid:96)m ( χ ) = 0 , for s = − , (2.15) Excluding the exceptional perturbations that simply change the mass or angular momentum of the background[55]. The Teukolsky formalism does not address these modes. See Appendix A for a detailed discussion. (cid:16) D − ∆ r D † + 6 (cid:16) r L + i Ξ ω r (cid:17) − λ (cid:17) R (2) ω(cid:96)m ( r ) = 0 , (cid:16) L − ∆ χ L † + 6 (cid:16) χ L − Ξ ω χ (cid:17) + λ (cid:17) S (2) ω(cid:96)m ( χ ) = 0 , for s = 2 , (2.16)where we introduced the separation constant λ ≡ λ (2) ω(cid:96)m = λ ( − ω(cid:96)m . (2.17)Some important observations are in order: • First note that the radial operators obey D † n = ( D n ) ∗ (where ∗ denotes complex conjuga-tion) while the angular operators satisfy L † n ( χ ) = −L n ( − χ ). • Consequently, the radial equation for R ( − ω(cid:96)m is the complex conjugate of the radial equa-tion for R (2) ω(cid:96)m , but the angular solutions S ( ± ω(cid:96)m are instead related by the symmetry S (2) ω(cid:96)m ( χ ) = S ( − ω(cid:96)m ( − χ ). The later statement implies that the separation constants aresuch that λ ( − ω(cid:96)m = λ (2) ω(cid:96)m ≡ λ with λ being real. • The eigenfunctions S ( s ) ω(cid:96)m ( χ ) are spin-weighted AdS spheroidal harmonics, with positiveinteger (cid:96) specifying the number of zeros, (cid:96) − max {| m | , | s |} (so the smallest (cid:96) is (cid:96) = | s | = 2).The associated eigenvalues λ can be computed numerically. They are a function of ω, (cid:96), m and regularity imposes the constraints that − (cid:96) ≤ m ≤ (cid:96) must be an integer and (cid:96) ≥ | s | . • We have the freedom to choose the normalization of the angular eigenfunctions. A naturalchoice is (cid:90) − (cid:16) S ( s ) ω(cid:96)m (cid:17) dχ = 1 . (2.18) Suppose we solve the radial and angular equations (2.16), (2.15) for the spin s = ±
2. Thesesolutions for R ( s ) ω(cid:96)m and S ( s ) ω(cid:96)m ( χ ), when inserted in (2.12), are not enough to fully determine theNP gauge invariant Weyl scalars δ Ψ , δ Ψ . The reason being that the relative normalizationbetween δ Ψ and δ Ψ remains undetermined, and thus our linear perturbation problem isyet not solved [38, 9, 10]. Given the natural normalization (2.18) chosen for the weightedspheroidal harmonics, the completion of the solution for δ Ψ , δ Ψ requires that we fix the relativenormalization between the radial functions R (+2) ω(cid:96)m and R ( − ω(cid:96)m . This is what the Starobinsky-Teukolsky (ST) identities acomplish [36, 37, 38, 9, 10]. A detailed analysis of these identitiesfor the Kerr black hole is available in the above original papers or in the seminal textbook ofChandrasekhar [11]. Here, we present these identities for the Kerr-AdS black hole.Act with the operator D †− ∆ r D † D † ∆ r D † on the Teukolsky equation (2.16) for R (2) ω(cid:96)m anduse the equation of motion for R ( − ω(cid:96)m . This yields one of the radial ST identities. Similarly,to get the second, act with the operator D − ∆ r D D ∆ r D on the Teukolsky equation (2.15)for R ( − ω(cid:96)m , and make use of the equation obeyed by R ( − ω(cid:96)m . These radial ST identities for theKerr-AdS background relate R (2) ω(cid:96)m to R ( − ω(cid:96)m , (cid:40) D †− ∆ r D † D † ∆ r D † R (2) ω(cid:96)m = C st R ( − ω(cid:96)m , D − ∆ r D D ∆ r D R ( − ω(cid:96)m = C ∗ st R (2) ω(cid:96)m , (2.19)where we have chosen the radial ST constants {C st , C ∗ st } to be related by complex conjugation.This is possible because, as noted before, the R ( ± ω(cid:96)m solutions are related by complex conjugation.7o get the angular ST identities, act with the operator L †− ∆ µ L † L † ∆ µ L † on the Teukolskyequation (2.16) for S (2) ω(cid:96)m (and use the equation of motion for S ( − ω(cid:96)m ), and act with L − ∆ µ L L ∆ µ L on the equation (2.15) for S ( − ω(cid:96)m (and use the equation for S ( − ω(cid:96)m ). This yields the pair of STidentities, (cid:40) L †− ∆ µ L † L † ∆ µ L † S (2) ω(cid:96)m = K st S ( − ω(cid:96)m , L − ∆ µ L L ∆ µ L S ( − ω(cid:96)m = K st S (2) ω(cid:96)m . (2.20)Since the equations for S ( ± ω(cid:96)m are related by the symmetry χ → − χ , the ST constant K st on theRHS of these ST identities is real. Moreover, because S ( ± ω(cid:96)m are both normalized to unity − see(2.18) − the ST constant is the same in both angular ST identities.To determine |C st | we act with the operator of the LHS of the first equation of (2.19) on thesecond equation and evaluate explicitly the resulting 8 th order differential operator. A similaroperation on equations (2.20) fixes K . We find that |C st | = K + 144 M ω Ξ , (2.21) K = λ ( λ + 2) + 8 λ Ξ aω [(6 + 5 λ ) ( m − aω ) + 12 aω ] + 144Ξ a ω ( m − aω ) + 4 a L (cid:2) λ ( λ + 2) ( λ −
6) + 12Ξ ( m − aω ) [2 mλ − aω ( λ − (cid:3) + 4 a ( λ − L . (2.22)This fixes completely the real constant K st (we choose the positive sign when taking the squareroot of K to get, when a →
0, the known relation between the s = ± C st . However, we emphasize that to find the asymp-totically global AdS boundary conditions in next section, we do not need to know C st , just K st . Moreover, we do not need the explicit expression for C st to construct the map between theKodama-Ishibashi and the a = 0 Teukolsky formalisms of Section 4.Neverthless we can say a bit more about the phase of C st . Recall that in the Kerr case,finding the real and imaginary parts of C st requires a respectful computational effort whichwas undertaken by Chandrasekhar [10] (also reviewed in sections 82 to 95 of chapter 9 of thetextbook [11]). `A priori we would need to repeat the computations of [10], this time in theAdS background, to find the phase of C st in the Kerr-AdS background (which was never doneto date). However, if we had to guess it we would take the natural assumption that C st is givenby the solution of (2.21) that reduces to the asymptotically flat partner of [10] when L → ∞ , C st = C + i C with C = K st , C = − M ω Ξ . (2.23)However, we emphasize again that this expression must be read with some grain of salt andneeds a derivation along the lines of [10] to be fully confirmed.Having fixed the ST constants we have specified the relative normalization between theTeukolsky variables δ Ψ and δ Ψ . We ask the reader to see Appendix A for a further discussionof this issue. In the previous subsections we found the solutions of the Teukolsky master equations for thegauge invariant Weyl scalars of the Newman-Penrose formalism. We will however need to knowthe perturbations of the metric components, h µν = δg µν . These are provided by the Hertz map, h µν = h µν ( ψ H ), which reconstructs the perturbations of the metric tensor from the associatedscalar Hertz potentials ψ H (in a given gauge) [24]-[29]. The later are themselves closely relatedto the NP Weyl scalar perturbations δ Ψ and δ Ψ .8n the Kerr-AdS background, the Hertz potentials are defined by the master equations theyobey to, namely, (cid:2) (∆ + 3 γ − γ + µ )( D + 4 (cid:15) + 3 ρ ) − ( δ + β + 3 α − τ )( δ + 4 β + 3 τ ) − (cid:3) ψ ( − H = 0 , (2.24a) (cid:2) ( D − (cid:15) + (cid:15) − ρ )(∆ − γ − µ ) − ( δ − β − α + π )( δ − α − π ) − (cid:3) ψ (2) H = 0 . (2.24b)Introducing the ansatz for the Hertz potential ψ ( s ) H = (cid:40) e − iωt e imφ ( r − iχ ) R ( − ω(cid:96)m ( r ) S ( − ω(cid:96)m ( χ ) , s = − ,e − iωt e imφ ( r − iχ ) R (2) ω(cid:96)m ( r ) S (2) ω(cid:96)m ( χ ) , s = 2 , (2.25)into (2.24) (in the Kerr-AdS background), we find that R ( s ) ω(cid:96)m and S ( s ) ω(cid:96)m are exactly the solutionsof the radial and angular equations (2.15) and (2.16). This fixes the precise map between theHertz potentials and the NP Weyl scalar perturbations.The Hertz map is such that the Hertz potentials ψ ( − H and ψ (2) H generate the metric pertur-bations in two different gauges, namely the ingoing (IRG) and the outgoing (ORG) radiationgauge, defined byIRG : (cid:96) µ h µν = 0 , g µν h µν = 0 , ORG : n µ h µν = 0 , g µν h µν = 0 . (2.26)The Hertz map is finally given by h IRG µν = (cid:110) (cid:96) ( µ m ν ) (cid:2) ( D + 3 (cid:15) + ¯ (cid:15) − ρ + ¯ ρ ) ( δ + 4 β + 3 τ ) + ( δ + 3 β − ¯ α − τ − ¯ π ) ( D + 4 (cid:15) + 3 ρ ) (cid:3) − (cid:96) µ (cid:96) ν ( δ + 3 β + ¯ α − τ ) ( δ + 4 β + 3 τ ) − m µ m ν ( D + 3 (cid:15) − ¯ (cid:15) − ρ ) ( D + 4 (cid:15) + 3 ρ ) (cid:111) ψ ( − H +c.c. , (2.27) h ORG µν = (cid:110) n ( ν m µ ) (cid:2)(cid:0) ¯ δ + ¯ β − α + ¯ τ + π (cid:1) (∆ − γ − µ ) + (∆ − γ − ¯ γ + µ − ¯ µ ) (cid:0) ¯ δ − α − π (cid:1)(cid:3) − n µ n ν (cid:0) ¯ δ − ¯ β − α + π (cid:1) (cid:0) ¯ δ − α − π (cid:1) − m µ m ν (∆ − γ + ¯ γ + µ ) (∆ − γ − µ ) (cid:111) ψ (2) H +c.c. . (2.28)We have explicitly checked that (2.27) and (2.28) satisfy the linearized Einstein equation (seealso footnote 3).It is important to emphasize that the Hertz map provides the most general metric pertur-bation with (cid:96) ≥ We start this section with a brief recap of the Teukolsky system which emphasizes some of itsproperties that are essential to discuss the asymptotic boundary conditions.The gravitational Teukolsky equations are described by a set of two families of equations,one for spin s = 2 and the other for s = −
2. In Petrov type D backgrounds, these twofamilies encode the same information, once we use the Starobinsky-Teukolsky identities that fixthe relative normalization between both spin-weighted spheroidal harmonics S ( s ) ω(cid:96)m and between Note that (2.28), whose explicit derivation can be found in an Appendix of [29], corrects some typos in the mapfirst presented in [25]. R ( s ) ω(cid:96)m . Indeed, modulo the ST relative normalization, the two radialfunctions are simply the complex conjugate of each other, and the two angular functions arerelated by S (2) ω(cid:96)m ( χ ) = S ( − ω(cid:96)m ( − χ ). This is a consequence of the fact that the Teukolsky operatoracting on δ Ψ is the adjoint of the one acting on δ Ψ .The upshot of these observations, with relevance for practical applications, is that theTeukosky system in Petrov type D geometries is such that we just need to analyze the s = 2sector (for example) to find all the information, except BCs, on the gravitational perturbations(excluding modes that just shift the mass and angular momentum). In other words, given R (2) ω(cid:96)m , S (2) ω(cid:96)m and the ST constants we can reconstruct all the s = − s = 2 Teukolskysystem of equations.The situation is far less trivial when we look into perturbations of Kerr-AdS. This time thesecond order differential system has two independent asymptotic solutions that are power lawsof the radial variable. The BC to be chosen selects the relative normalization between thesetwo solutions. What is the criterion to make this choice? This will be made precise in thenext subsection. Before such a formal analysis we can however describe it at the heuristic level.Basically we want the perturbed background to preserve the asymptotic global AdS characterof the Kerr-AdS background. Global AdS asymptotic structure means that the system behavesas a confining box were the only allowed perturbations are those described by standing waves.Standing waves on the other hand can be decomposed as a fine-tuned sum of IRG and ORGmodes such that we have a node at the asymptotic AdS wall. With this brief argument weconclude that to find the asymptotic global AdS BC we necessarily need to use the informationon both the IRG and ORG Teukolsky metric perturbations, i.e. the BC discussion will requireusing information on both spins. Once we find it, it is still true that the spin s = 2 sector ofthe Teukolsky system encodes the same information as the s = − s = 2 sector (say). (Notethat an infinitesimal rotation of the tetrad basis should allow to derive our results using onlythe ORG, say).So we take the most general gravitational perturbation of the Kerr-AdS black hole to begiven by the sum of the ingoing and outgoing radiation gauge contributions as written in (2.27)and (2.28). (By diffeomorphism invariance, this solution can be written in any other gaugethrough a gauge transformation). The physically relevant perturbations are those that areregular at the horizon and asymptotically global AdS. In this section we find one of our mostfundamental results, namely the BCs we need to impose on our perturbations. When considering linear perturbations of a background we have in mind two key properties:the perturbations should keep the spacetime regular and they should be as generic as pos-sible, but without being so violent that they would destroy the asymptotic structure of thebackground. To make this statement quantitative, in the familiar case of an asymptoticallyMinkowski background, the appropriate boundary condition follows from the requirement thatthe perturbations preserve asymptotically the Poincar´e group of the Minkowski spacetime [30].For the AdS case, Boucher, Gibbons, and Horowitz [31] and Henneaux and Teiltelboim [32]10ave defined precisely what are the asymptotic BC we should impose to get perturbations thatapproach at large spacelike distances the global AdS spacetime. The main guideline is that per-turbations in a global AdS background must preserve asymptotically the global AdS symmetrygroup O (3 , O (3 , O (3 ,
2) generators.If we work in the coordinate system { T, R, Θ , Φ } , where the line element of global AdS isgiven by (2.6), the metric perturbations that obey the above BCs behave asymptotically as [32]: h T µ = 1
R F
T µ ( T, Θ , Φ) + O (cid:0) R − (cid:1) , for µ = T, Θ , Φ , (3.1a) h T R = 1 R F T R ( T, Θ , Φ) + O (cid:0) R − (cid:1) , (3.1b) h RR = 1 R F RR ( T, Θ , Φ) + O (cid:0) R − (cid:1) , (3.1c) h Rµ = 1 R F Rµ ( T, Θ , Φ) + O (cid:0) R − (cid:1) , for µ = Θ , Φ , (3.1d) h µν = 1 R F µν ( T, Θ , Φ) + O (cid:0) R − (cid:1) , for µ, ν = Θ , Φ , (3.1e)where F µν ( T, Θ , Φ) are functions of { T, Θ , Φ } only.These BCs are defined with respect to a particular coordinate system. Consider a genericinfinitesimal coordinate transformation x µ → x µ + ξ µ , where ξ is an arbitrary gauge vector field.Under this gauge transformation the metric perturbation transforms according to h µν → h µν − ∇ ( µ ξ ν ) , (3.2)which we can use to translate the BCs (3.1) in the { T, R, Θ , Φ } frame into any other coordinatesystem, so long as ξ decays sufficiently fast at infinity. Modulo gauge transformations, the most general perturbation of linearized Einstein equationsin the Kerr-AdS background can be written as h µν = h IRG µν + h ORG µν , (3.3)where h IRG µν and h ORG µν are determined by the Hertz maps (2.27) and (2.28), with the Hertz po-tentials ψ ( ± H defined in (2.25) and the associated Teukolsky functions obeying the equations ofmotion (2.16) and (2.15). Note that the relative normalization between these two contributionsis fixed by the Starobinsky-Teukolsky treatment.Solving the radial Teukolsky equations (2.16) and (2.15) at infinity, using a standard Frobe-nius analysis, we find that the two independent asymptotic decays for R ( ± ω(cid:96)m are R (2) ω(cid:96)m (cid:12)(cid:12) r →∞ ∼ A (2)+ Lr + A (2) − L r + O (cid:18) L r (cid:19) ,R ( − ω(cid:96)m (cid:12)(cid:12) r →∞ ∼ B ( − Lr + B ( − − L r + O (cid:18) L r (cid:19) , (3.4)11here the amplitudes A ( s ) ± ≡ { A (2) ± , B ( − ± } are, at this point, independent arbitrary constants.Our task is to find the BC we have to impose in order to get a perturbation h µν that isasymptotically global AdS. That is, we must find the constraints, A ( s ) − = A ( s ) − (cid:16) A ( s )+ (cid:17) , thatthese amplitudes have to obey to get the Henneaux-Teiltelboim decay (3.1). We will find thatthe most tempting condition, where we set to zero the leading order term in the expansion, A ( s )+ = 0, is too naive and does not do the job. Note that it follows from (2.5) that for large R ,or r , one has R ∼ r (cid:2) ( L − χ ) /L Ξ (cid:3) / and cos Θ ∼ χ (cid:2) L Ξ / ( L − χ ) (cid:3) / . Therefore, to getthe asymptotically global AdS decay of h µν in the { t, r, χ, φ } coordinate system we can simplyreplace { T, R, Θ , Φ } → { t, r, χ, φ } in (3.1).In h IRG µν , we can express S ( − ω(cid:96)m ( χ ) as a function of S (2) ω(cid:96)m ( χ ) and its derivative using the firstStarobinsky-Teukolsky (ST) identity in (2.20) and the angular equation of motion (2.16). Thiseliminates S ( − ω(cid:96)m from (3.3). At this stage, we could also immediately replace the ST constant K st by its expression (2.22). Instead, we choose to keep it unspecified until a later stage in ourcomputation.The explicit expression of h IRG µν + h ORG µν when we introduce (3.4) into (2.27) and (2.28)contains order r terms but no other higher power of r . Our first task is to use all the gaugefreedom (3.2) to eliminate, if possible, these O ( r ) terms and all lower power law terms thatare absent in the asymptotically global AdS decay (3.1). The gauge parameter compatible withthe background isometries is ξ = e − iωt e imφ ξ µ ( r, χ ) dx µ . A simple inspection of ∇ ( µ ξ ν ) concludesthat the most general components of the gauge vector field, that can contribute up to O ( r )terms, can be written as the power law expansion in r : ξ t = n (cid:88) j =0 ξ ( j ) t ( χ ) r − j , ξ r = n (cid:88) j =0 ξ ( j ) r ( χ ) r − (1+ j ) , ξ χ = n (cid:88) j =0 ξ ( j ) χ ( χ ) r − j , ξ φ = n (cid:88) j =0 ξ ( j ) φ ( χ ) r − j . (3.5)Inserting this expansion and (3.4) into (3.3), we find that there is a judicious choice of thefunctions ξ ( i ) µ ( χ ) such that we can eliminate most of the radial power law terms that are absentin the several metric components of (3.1) (the expressions are long and not illuminating). Moreconcretely, we are able to gauge away all desired terms but the O ( r ) contribution in thecomponents h χχ , h χφ and h φφ .At this point, having used all the available diffeomorphism (3.2), we find ourselves at a keystage of the analysis. To eliminate the undesired leftover O ( r ) contributions we will have to fixthe BCs that the amplitudes introduced in (3.4) have to obey to guarantee that the perturbationis asymptotically global AdS. There are two conditions that eliminate simultaneously the O ( r )terms in h χχ , h χφ and h φφ . One is the coefficient of a term proportional to S (2) ω(cid:96)m ( χ ), and the otheris proportional to ∂ χ S (2) ω(cid:96)m ( χ ). Clearly, these two contributions have to vanish independently.We can use them to express, for example, the amplitude B ( − and the ST constant K st interms of the other amplitudes B ( − − , A (2) ± , perturbation parameters ω, m, λ and the rotation Note that in this computation we use the angular equation of motion (2.16) to get rid of second and higherderivatives of S (2) ω(cid:96)m . a (the mass parameter M is absent in these expressions):0 = W B ( − − B ( − − L (cid:20) − a m L Ξ (cid:16) i A (2) − + 5 L Ξ ωA (2)+ (cid:17) + 2 a (cid:0) − λ + 6 L Ξ ω (cid:1) A (2)+ + λ L (cid:0) λ − L Ξ ω (cid:1) A (2)+ − L Ξ ω (cid:16) iA (2) − + L Ξ ωA (2)+ (cid:17) (cid:18) λ − L Ξ ω (cid:19) (cid:21) , (3.6) K st = − B ( − − L W (cid:26) λ (2 + λ ) L + 8 a λ (6 + 5 λ ) m L Ξ ω − a m L Ξ ω (cid:0) − λ + 2 L Ξ ω (cid:1) +4 a L (cid:20) λ (cid:2) −
12 + ( − λ ) λ + 24 m (cid:3) + 2 (cid:2) (6 − λ ) λ + 18 m (cid:3) L Ξ ω (cid:21) +4 a L (cid:20) − λ + λ − λm + 12 (cid:2) λ − (cid:0) m (cid:1)(cid:3) L Ξ ω + 36 L Ξ ω (cid:21) +48 a m (cid:0) λ + 3 L Ξ ω (cid:1) (cid:27) , (3.7)where we have defined W ≡ L (cid:104) λ (2 + λ ) A (2) − − λ ) L Ξ ω (cid:16) iλA (2)+ + 2 L Ξ ωA (2) − (cid:17) + 4 i (2 + 3 λ ) L Ξ ω A (2)+ (cid:105) +8 L Ξ ω (cid:16) A (2) − − i L Ξ ωA (2)+ (cid:17) − a m L Ξ (cid:104) i λA (2)+ + L Ξ ω (cid:16) A (2) − − i L Ξ ωA (2)+ (cid:17)(cid:105) +2 a L (cid:104) (cid:16) i L Ξ ωA (2)+ − A (2) − (cid:17) (cid:0) λ − L Ξ ω (cid:1) + 3 λA (2) − (cid:105) . (3.8)At this stage we finally introduce the explicit expression for the angular Starobinski-Teukolskyconstant, namely, K st is given by the positive square root of (2.22). In addition, we also usethe property that the radial R ( ± ω(cid:96)m solutions are the complex conjugate of each other. In theseconditions we find that conditions (3.6)-(3.7) are obeyed if and only if A (2) − = − i ηA (2)+ , B ( − − = i ηB ( − , (3.9)with two possible solutions for η , that we call η s and η v , for reasons that will become clear inthe next section. These define the BCs we look for. To sum up, the two possible BCs on theTeukolsky amplitudes, defined in (3.4), that yield an asymptotically global AdS perturbation,take the form (3.9) with 1) η = η s = Λ − √ Λ Λ , or (3.10)2) η = η v = Λ + √ Λ Λ , (3.11)where we have introducedΛ ≡ a ( λ − − λ + 1) L ω Ξ + 8 L ω Ξ + L (cid:2) λ ( λ + 2) − aω [5( m − aω ) + 2 aω ] (cid:3) , Λ ≡ a ( λ − + L λ ( λ + 2) + 48( λ + 6) a Ξ L ω ( m − aω ) + 8 λ (5 λ + 6)( m − aω ) L Ξ aω +4 a L (cid:20) λ (cid:2) −
12 + ( λ − λ + 24( m − aω ) Ξ (cid:3) + 12Ξ L ω (cid:2) λ + 3( m − aω ) Ξ (cid:3) (cid:21) , Λ ≡ L Ξ (cid:2) am + L ω (cid:0) λ − L ω Ξ (cid:1)(cid:3) . (3.12)13ote that the BCs do not depend on the mass parameter M of the background black hole(neither does the ST constant K st ).The metric of the Kerr-AdS black hole asymptotically approaches that of global AdS. Theboundary conditions (3.9)-(3.10) are the most fundamental result of our study: perturbationsobeying these BCs are the ones that preserve the asymptotically global AdS behavior of thebackground. These are also natural BCs in the context of the AdS/CFT correspondence: theyallow a non-zero expectation value for the CFT stress-energy tensor while keeping fixed theboundary metric. The reader interested in different BCs that allow, e.g. for a dynamicalboundary metric, can start from the respective asymptotic metric decay that replaces (3.1) andwork out the above procedure to get the associate BCs on the Teukoslsky variables. At the horizon, the BCs must be such that only ingoing modes are allowed.A Frobenious analysis at the horizon gives the two independent solutions, R ( s ) ω(cid:96)m ∼ A in ( r − r + ) − s − i ω − m Ω H πTH + A out ( r − r + ) s + i ω − m Ω H πTH + O (cid:0) ( r − r + ) (cid:1) , (3.13)where A in , A out are arbitrary amplitudes and Ω H , T H are the angular velocity and temperaturedefined in (2.4). The BC is determined by the requirement that the solution is regular in ingoingEddington-Finkelstein coordinates (appropriate to extend the analysis through the horizon) anddemanding regularity of the Teukolsky variable in this coordinate system. This requires thatwe set A out = 0 in (3.13): R ( s ) ω(cid:96)m (cid:12)(cid:12)(cid:12)(cid:12) r → r + = A in ( r − r + ) − s − i ω − m Ω H πTH + O (cid:0) ( r − r + ) (cid:1) . (3.14) a = 0 ) In the previous section we found the boundary conditions we need to impose on the solutionof the Teukolsky master equation to get gravitational perturbations of the Kerr-AdS blackhole that preserve the asymptotically global AdS behavior of the background. The Kerr-AdSfamily includes the global AdS-Schwarzchild black hole and the global AdS geometry as specialelements when we set, respectively, a = 0 and a = 0 = M . Thus, our BCs also apply toperturbations of these static backgrounds.On the other hand, perturbations of the global AdS(-Schwarzchild) backgrounds were al-ready studied in great detail in the literature using the Kodama-Ishibashi (KI) gauge invariantformalism. In four dimensions, the KI formalism reduces exactly to the analysis firstly done byZerilli and Regge and Wheeler (in the L → ∞ case). Indeed, the KI vector master equation isthe Regge-Wheeler master equation for odd (also called axial) perturbations [12], and the KIscalar master equation is the Zerilli master equation for even (also called polar) perturbations[13] .Clearly, there must be a one-to-one map between the Kodama-Ishibashi and the Teukolskyformalisms (when a = 0). This map was never worked out so we take the opportunity to findit . Actually, this task will reveal to be quite fruitful since we will find some remarkably simpleconnections. For an earlier discussion of the relation between the Teukolsky and the Regge-Wheeler − Zerilli variables in theasymptotically flat case see [51]. K . If thisis the case, we can do a harmonic decomposition of the perturbations h ab according to how theytransform under coordinate transformations on K . This is certainly the case of the global AdS(-Schwarzchild) backgrounds where the base space is a sphere, K = S , and we can introduce a spherical harmonic decomposition of gravitational perturbations. (Unfortunately, the Kerr-AdSgeometry cannot be written as a local product of two such spaces and the KI formalism doesnot apply to it). On the other hand, the Teukolsky formalism uses a harmonic decompositionwith respect to the spin-weighted spherical harmonics S ( s ) ω(cid:96)m .These two harmonic decompositions are distinct and responsible for the differences betweenthe Teukolsky and KI formalisms. We can however write uniquely the scalar/vector KI har-monics in terms of standard scalar spherical harmonics (associated Legendre polynomials), andthere is another unique differential map that generates the spin-weighted spherical harmonicsalso from standard spherical harmonics. This provides the necessary bridge between the twoformalisms that leads to their unique map. To appreciate this we will discuss these harmonicsin detail. Also, to make the KI discussion self-contained, we will briefly review the KI formalismin the next subsection, before constructing the desired map in subsection 4.2.So far we have not discussed the role of the boundary conditions (BCs) in this map. Wefound a set of two distinct BCs for the Teukolsky solution. Quite interestingly, we will seethat Teukolsky perturbations with BC (3.10) maps to the KI scalar modes while Teukolskyperturbations with BC (3.11) generates the KI vector modes. In the Kodama-Ishibashi (KI) formalism [14], the most general perturbation of the global AdS(-Schwarzchild) geometries is decomposed into a superposition of two classes of modes: scalar andvector. Scalar and vector modes are expanded in terms of the scalar S ( x, φ ) and vector V j ( x, φ )harmonics that we review next ( x = cos θ ; see (2.1)).Scalar perturbations are given by [14] h s ab = f ab S , h s ai = rf a S i , h s ij = 2 r ( H L γ ij S + H T S ij ) , (4.1)where ( a, b ) are components in the orbit spacetime parametrized by { t, r } , ( i, j ) are legs onthe sphere, { f ab , f a , H T , H L } are functions of ( t, r ), and S is the KI scalar harmonic, S i = − λ − / s D i S , S ij = λ − s D i D j S + γ ij S , γ jk is the unit radius metric on S and D j is the associatedcovariant derivative. Assuming the ansatz S ( x, φ ) = e imφ Y m(cid:96) s ( x ), the scalar harmonic equation( (cid:52) S + λ s ) S = 0 ( (cid:52) S = γ jk D j D k ) reduces to ∂ x (cid:2)(cid:0) − x (cid:1) ∂ x Y m(cid:96) s ( x ) (cid:3) + (cid:18) λ s − m − x (cid:19) Y m(cid:96) s ( x ) = 0 . (4.2)Its regular solutions, with normalization (cid:82) π dφ (cid:82) − dx | S | = 1, are S ( x, φ ) = (cid:115) (cid:96) s + 14 π ( (cid:96) s − m )!( (cid:96) s + m )! P m(cid:96) s ( x ) e imφ ≡ Y m(cid:96) s ( x, φ ) , (4.3)with λ s = (cid:96) s ( (cid:96) s + 1) , (cid:96) s = 0 , , , · · · , | m | ≤ (cid:96) s . where P m(cid:96) ( x ) is the associated Legendre polynomial. Hence, the KI scalar harmonic S ( x, φ ) isthe standard scalar spherical harmonic Y m(cid:96) s ( x, φ ).15n the other hand, the KI vector perturbations are given by [14] h v ab = 0 , h v ai = rh a V i , h v ij = − λ − v r h T D ( i V j ) , (4.4)where { h a , h T } are functions of ( t, r ) and the KI vector harmonics V j are the solutions of( (cid:52) S + λ v ) V j = 0 , D j V j = 0 (Transverse condition) . (4.5)The regular vector harmonics can be written in terms of the spherical harmonic Y m(cid:96) v ( x, φ ) as V j dx j = − i m − x Y m(cid:96) v ( x, φ ) dx + (cid:0) − x (cid:1) ∂ x Y m(cid:96) v ( x, φ ) dφ , (4.6)with λ v = (cid:96) v ( (cid:96) v + 1) − , (cid:96) v = 1 , , · · · , | m | ≤ (cid:96) v . With this harmonic decomposition, the linearized Einstein equation reduces to a set of twodecoupled gauge invariant KI master equations for the KI master fields Φ ( j ) (cid:96) j which can be writtenin a compact form as, (cid:18) (cid:3) − U j f (cid:19) Φ ( j ) (cid:96) j ( t, r ) = 0 , where f = 1 + r L − Mr , and j = { s , v } , (4.7)for scalar ( s ), and vector ( v ) perturbations. Here, (cid:3) is the d’Alembertian operator in the2-dimensional orbit spacetime, and the expression for the potentials { U s , U v } can be foundin equations (3.2)-(3.8) and (5.15) of KI [14], respectively. They depend on the propertiesof the background, namely on the mass parameter M and cosmological length L , and on theeigenvalues λ s , λ v of the associated (regular) vector, and scalar harmonics defined above. Sincethe background is time-translation invariant, the fields can be further Fourier decomposed intime as Φ ( j ) (cid:96) j ( t, r ) = e − iω j t Φ ( j ) ω j (cid:96) j ( r ).We will need to express KI master fields in terms of the metric functions. Going through[14], for ω (cid:54) = 0, one finds that the gauge invariant KI scalar master field is given byΦ (s) ω(cid:96) s = − ω r ( f tt + 4 f H L ) + f (2 if tr + ω rf rr ) ω f ( λ s − f + rf (cid:48) ) + 2 i (cid:2) rf f (cid:48) t + (cid:0) r ω + f − rf f (cid:48) (cid:1) f t (cid:3) √ λ s ω f ( λ s − f + rf (cid:48) )+ 2 rλ s f ( λ s − f + rf (cid:48) ) (cid:20) (cid:2) r ω + 2 f − f (cid:0) λ s + rf (cid:48) (cid:1)(cid:3) H T + rf (cid:0) f + rf (cid:48) (cid:1) H (cid:48) T + r f H (cid:48)(cid:48) T (cid:21) + 2 r ( f f (cid:48) r + f (cid:48) f r ) √ λ s ( λ s − f + rf (cid:48) ) . (4.8)On the other hand the gauge invariant KI vector master field isΦ (v) ω(cid:96) v = if √ λ v ω (cid:16)(cid:112) λ v h r + rh (cid:48) T (cid:17) . (4.9)The KI master variables have the asymptotic expansion,Φ ( j ) ω j (cid:96) j (cid:12)(cid:12) z → ∼ Φ + Φ Lr + · · · where j = { s , v } . (4.10)The linear differential map h ( j ) ab = h ( j ) ab (cid:0) Φ ( j ) (cid:1) that reconstructs the metric perturbations (ina given gauge) can be read from (4.1) and (4.8) (scalar case) and from (4.4) and (4.9) (vector16ase), if we follow [14]. The requirement that these metric perturbations are asymptoticallyglobal AdS in the sense described in Section 3 imposes the conditions [33, 4]:Scalar BC: Φ = − Mλ s − , (4.11)Vector BC: Φ = 0 . (4.12)We can of course consider different asymptotic BCs. For example, past studies on (quasi)normalmodes have considered the BC Φ = 0 for scalar modes, instead of (4.11). However, these BCsare not asymptotically global AdS, i.e. they do not preserve the boundary metric, as first ob-served in [33]. We thus we do not consider them. Other BCs that might be called asymptoticallyglobally AdS were studied in [34], but turn out to lead to ghosts (modes with negative kineticenergy) and thus make the energy unbounded below [35]. vs Kodama-Ishibashi (Regge-Wheeler − Zerilli)
Equations (4.3) and (4.6) express the KI scalar S ( x, φ ) and vector V ( x, φ ) harmonics as afunction of the scalar spherical harmonic Y m(cid:96) ( x, φ ) defined in (4.3). The spin-wheighted sphericalharmonics S ( s ) (cid:96)m ( x, φ ) used in the Teukolsky harmonic decomposition are also related to the scalarspherical harmonic Y m(cid:96) ( x, φ ) through the differential map [52, 53] S ( − (cid:96)m ( x, φ ) = S (2) (cid:96)m ( − x, φ ) ; S (2) (cid:96)m ( x, φ ) = (cid:115) ( (cid:96) − (cid:96) + 2)! (cid:115) ( (cid:96) + 1)!( (cid:96) − (cid:20)(cid:0) − x (cid:1) ∂ x (cid:18) √ − x S (1) (cid:96)m ( x, φ ) (cid:19) − i √ − x ∂ φ S (1) (cid:96)m ( x, φ ) (cid:21) ,S (1) (cid:96)m ( x, φ ) = (cid:115) ( (cid:96) − (cid:96) + 1)! (cid:20)(cid:112) − x ∂ x Y (cid:96)m ( x, φ ) − i √ − x ∂ φ Y (cid:96)m ( x, φ ) (cid:21) . (4.13)These (regular) harmonics obey the angular Teukolsky equations (2.15)-(2.16) with a = 0 and λ = (cid:96) ( (cid:96) + 1) − h s µν + h v µν = h IRG µν + h ORG µν , (4.14)where recall that the KI metric on the LHS is given by (4.1), (4.3), (4.4) and (4.6). On theother hand, the Teukolsky metric on the RHS is given by (2.27), (2.28), (2.25) and (4.13).Fix the LHS of (4.14) to have fixed values of KI quantum numbers (cid:96) s and (cid:96) v . Then, the mostnatural expectation is that such a KI perturbation is described by a (possibly infinite) sum, inthe quantum number (cid:96) , of Teukolsky harmonics (the background is spherically symmetric sowe can set wlog m = 0 in our discussion; see below). In fact, a mode by mode analysis (usingproperties of internal products) reveals that (cid:96) s = (cid:96) v = (cid:96) (with no sum involved). This simplifiesconsiderably the construction of the map.Take (4.14) with the identification (cid:96) s = (cid:96) v = (cid:96) with integer (cid:96) ≥
2. The later inequalityrequires a discussion before proceeding. The KI formalism describes all scalar modes withinteger (cid:96) s ≥ (cid:96) s = 0 are perturbations that just shift the mass of the solution and (cid:96) s = 1 is a pure gauge mode), and all vector modes with integer (cid:96) v ≥ (cid:96) v = 1 areperturbations that generate just a shift in the angular momentum of the solution) [14, 54].However, the Teukosky quantum number is constrained to be an integer (cid:96) ≥ | s | = 2, so the17eukolsky formalism is blind to the modes that generate deformations in the mass and angularmomentum of the geometry [27, 55]. The simplest way to confirm this is to note that the map(4.13) would be trivial for the “ (cid:96) = 0 ,
1” modes.In these conditions, each metric component on the LHS of (4.14) is proportional to thespherical harmonic Y m(cid:96) ( x ), or to its first derivative, or to a linear combination of both suchcontributions. The same applies to the RHS of (4.14). Matching all the coefficients of theseangular contributions we can find the radial KI metric functions { f ab , f a , H T , H L } , { h a , h T } that describe the Teukolsky perturbations. Finally, inserting these radial KI functions into (4.8)and (4.9) we express the KI variables Φ (s , v) ω(cid:96) ( r ) as a function of the radial Teukolsky functions R ( s ) ω(cid:96)m ( r ) and their first derivative. Note that the above KI metric functions are in a particulargauge induced by the Hertz construction. However, (4.8) and (4.9), and thus the associatedfinal map, are gauge invariant. To write Φ (s , v) ω(cid:96) ( r ) as a function of a single Teukolsky function,e.g. R (2) ω(cid:96)m ( r ), we further use the radial Starobinsky-Teukolsky identity (2.19) (which, recall,expresses R ( − ω(cid:96)m ( r ) as a function of R (2) ω(cid:96)m ( r ) and its first derivative), and the equation of motion(2.16) for R ( − ω(cid:96)m ( r ). We finally end up with the desired gauge invariant map between the KIscalar and vector master fields Φ (s , v) ω(cid:96)m and the Teukolsky radial function R (2) ω(cid:96)m (and its derivative):Φ ( s ) ω(cid:96) = 1 K st C st (cid:112) ( (cid:96) + 2)! (cid:112) ( (cid:96) − r f ( λr + 6 M ) (cid:18) K st C st ( (cid:96) − (cid:96) + 2)! + ( (cid:96) + 2)!( (cid:96) − i M ω (cid:19) × (cid:26) rf (cid:2)(cid:0) iω + f (cid:48) (cid:1) ( λr + 6 M ) − λf (cid:3) (cid:16) R (2) ω(cid:96) (cid:17) (cid:48) + (cid:20) f − f (cid:0) λ + 4 − irω + rf (cid:48) (cid:1) + r (cid:0) λ + 2 + rf (cid:48) (cid:1) (cid:0) f (cid:48) + 3 iωf (cid:48) − ω (cid:1) + f (cid:2) ( λ + 2 − iωr ) + 4 rω (2 rω + i ) − rf (cid:48) (cid:0) λ − irω + 2 rf (cid:48) (cid:1)(cid:3) (cid:21) R (2) ω(cid:96) (cid:27) , (4.15)Φ ( v ) ω(cid:96) = i K st C st (cid:112) ( (cid:96) + 2)! (cid:112) ( (cid:96) − r f (cid:18) −K st C st ( (cid:96) − (cid:96) + 2)! − ( (cid:96) + 2)!( (cid:96) − i M ω (cid:19) × (cid:26) (cid:2) f (cid:0) λ + 2 − irω − rf (cid:48) (cid:1) + r (cid:0) f (cid:48) + 3 iωf (cid:48) − ω (cid:1)(cid:3) R (2) ω(cid:96) + rf (cid:0) rf (cid:48) − f + 2 iωr (cid:1) (cid:16) R (2) ω(cid:96) (cid:17) (cid:48) (cid:27) . (4.16)In these expressions, λ = (cid:96) ( (cid:96) + 1) − a = 0 and M is themass parameter of the black hole. Note that for a = 0, the case we are discussing in thisand following sections, the background spacetime is spherically symmetric. Consequently, theradial Teukolsky equations and solutions are independent of the azimuthal quantum number m . Therefore, henceforth we droped the associated subscript in R (2) ω(cid:96)m in the maps (4.15)-(4.16)(and henceforward). The Starobinsky-Teukolsky angular and radial constants K st and C st aregiven by (2.21)-(2.23), which for a = 0 boil down to K st = ( (cid:96) + 2)!( (cid:96) − , C st = K st − i M ω. (4.17)To check our matching we explicitly verify that our KI master fields obey the KI master equa-tions (4.7) when R (2) ω(cid:96) ( r ) satisfies the radial Teukolsky equation (2.16), with λ = (cid:96) ( (cid:96) + 1) − L → ∞ . We ask the reader to revisit the discussion leading to (2.23). For that reason, and without any loss exceptcompactness, we prefer to present our forthcoming results without explicitly giving the expression for C st .
18o have the complete map between the Teukolsky and KI formalism we still need to discussthe relation between the asymptotically global AdS KI BCs (4.10)-(4.12) and the global AdSTeukolsky BCs (3.9)-(3.11). The later, for a = 0, simply reduce to1) A (2) − = − i η A (2)+ , η = η s = − Lω (cid:18) λλ − L ω − (cid:19) , (4.18)2) A (2) − = − i η A (2)+ , η = η v = λ Lω − Lω . (4.19)It follows from (3.4) and (3.9) that asymptotically R (2) ω(cid:96) (cid:12)(cid:12) r →∞ ∼ A (2)+ Lr (cid:20) − i η Lr + 12 (cid:0) (cid:96) + (cid:96) − − L ω (cid:1) L r (cid:21) + O (cid:18) L r (cid:19) , (4.20)Consider first the scalar case described by (4.15). Choose the BC to be such that η = η s as defined in (4.18). In these conditions, inserting (4.20) into the scalar map (4.15) and takingits asymptotic expansion we find that it reduces exactly to the KI expression (4.10), Φ ( s ) ∼ Φ + Φ Lr , withΦ = − M(cid:96) ( (cid:96) + 1) − , (4.21)Φ = L A (2)+ C st (cid:112) ( (cid:96) − (cid:112) ( (cid:96) + 2)! ( (cid:96) +2)!( (cid:96) − − iM ω(cid:96) ( (cid:96) + 1) − L ω (cid:18) C st + ( (cid:96) + 2)!( (cid:96) − iM ω (cid:19) , which matches the global AdS KI BC (4.11) for scalar modes once we use λ s = (cid:96) ( (cid:96) + 1). Wesee this as one of the most non-trivial tests of our calculations.Next, take the vector case described by (4.16). This time select the BC η = η v defined in(4.19). Plug (4.20) into the vector map (4.16) and take its asymptotic expansion. We get theKI expression (4.10) withΦ = 0 , Φ = L A (2)+ ωC st (cid:112) ( (cid:96) − (cid:112) ( (cid:96) + 2)! (cid:18) ( (cid:96) + 2)!( (cid:96) − iM ω (cid:19) (cid:18) C st + ( (cid:96) + 2)!( (cid:96) − − iM ω (cid:19) , (4.22)which is the global AdS KI BC (4.12) for vector modes.So one of the Teukolsky Robin BCs selects the scalar sector of KI perturbations and theother selects the vector KI sector. This is the simplest map we could have predicted!To sum this section, we found the differential map between the Teukolsky/KI variables andBCs. For scalar modes, the differential map is given by (4.15) with the global AdS Teukolsky BC(4.18) mapping into the scalar KI BC (4.11) via (4.21). For vector modes, the differential mapis instead (4.16), and the global AdS Teukolsky BC (4.19) maps into the vector KI BC (4.12)through (4.22). By continuity, when we turn on the rotation a , we can say that the TeukolskyBC (3.9),(3.10) generates the “rotating scalar modes”, while the BC (3.9),(3.11) selects the“rotating vector modes”. The normal modes of global AdS can be studied using the Teukolsky equations and the boundaryconditions (3.9)-(3.11). We take the opportunity to find these normal mode frequencies sincethe scalar modes are not explicitly derived in the literature. We also revisit, from a Teukolskyperspective, the quasinormal mode spectrum of the global AdS-Schwarzschild black hole.19ntroducing the differential operator definitions (2.13), the s = +2 the Teukolsky equations(5.1) read ∂ χ (cid:16) ∆ χ ∂ χ S (2) ω(cid:96)m (cid:17) + (cid:34) − (cid:0) K χ − ∆ (cid:48) χ (cid:1) ∆ χ + (cid:18) χ L − K (cid:48) χ + ∆ (cid:48)(cid:48) χ (cid:19) + λ (cid:35) S (2) ω(cid:96)m = 0 ,∂ r (cid:16) ∆ r ∂ r R (2) ω(cid:96)m (cid:17) + (cid:34) ( K r + i ∆ (cid:48) r ) ∆ r + (cid:18) r L − iK (cid:48) r + ∆ (cid:48)(cid:48) r (cid:19) − λ (cid:35) R (2) ω(cid:96)m = 0 . (5.1)It follows from (2.1) that in the static case it is appropriate to work in the coordinate system { t, r, x, φ } where χ = a x . In these conditions equations (2.16) with a = 0 reduces to ∂ x (cid:104)(cid:0) − x (cid:1) ∂ x S (2) ω(cid:96)m ( x ) (cid:105) + (cid:20) λ − − ( m + 2 x ) − x (cid:21) S (2) ω(cid:96)m ( x ) = 0 ,∂ r (cid:16) ∆ r ∂ r R (2) ω(cid:96) ( r ) (cid:17) + (cid:34) (cid:0) ωr − i ∆ (cid:48) r (cid:1) ∆ r + 2 (cid:18) r L (cid:19) + 8 irω − λ (cid:35) R (2) ω(cid:96) ( r ) = 0 . (5.2)The spin weighted spherical harmonic is independent of the mass parameter M and cosmologicalradius L , and can be found analytically. Next, we discuss in detail the regularity analysis thatleads to the solution (4.13).The regular solution at the north pole x = 1 is ( F is the standard Hypergeometric function) S (2) ω(cid:96)m = (1 − x ) | m +2 | (1 + x ) | m − | F (cid:18) (cid:16) ˜ a − ˜ b (cid:17) , (cid:16) ˜ a + ˜ b (cid:17) , | m + 2 | + 1 , − x (cid:19) , with ˜ a = | m − | + | m + 2 | + 1 , ˜ b = √ λ + 9 . (5.3)This solution diverges at the south pole x = − x ) − (or as ln(1 + x )in the special case of m = 2) unless we quantize the angular eigenvalue and quantum numbersas λ = (cid:96) ( (cid:96) + 1) − , with (cid:96) = 2 , , , · · · , | m | ≤ (cid:96) , (5.4)where we have introduced the quantum number (cid:96) with (cid:96) − max {| m | , | s | = 2 } giving the numberof zeros of the eigenfunction along the polar direction. The regular spin s = 2 spherical harmonicthat solves the angular equation (5.2) is finally S (2) ω(cid:96)m = (1 − x ) | m +2 | (1 + x ) | m − | (5.5) × F (cid:18)
12 ( − (cid:96) + | m − | + | m + 2 | ) ,
12 (2 + 2 (cid:96) + | m − | + | m + 2 | ) , | m + 2 | + 1 , − x (cid:19) . with the quantum numbers (cid:96), m constrained by the conditions (5.4). This harmonic is validboth for the global AdS-Schwarzchild and global AdS backgrounds since the angular equation isindependent of M . Using the relation between the Hypergeometric function and the AssociatedLegendre polynomial we can rewrite (5.5) as (4.13).To study the (quasi)normal modes of these backgrounds we now need to study the radialequation (5.2). Since this equation depends on the mass parameter M we need to study thecases M = 0 and M > .1 Normal modes of global AdS
In the global AdS background ( M = 0), the radial Teukolsky equation (5.2) has an exactsolution. The solution that is regular at the origin ( r = 0) is R (2) ω(cid:96) = A (cid:18) − i rL (cid:19) ( Lω − (cid:18) i rL (cid:19) − ( Lω +2 (cid:96) ) (cid:16) rL (cid:17) (cid:96) F (cid:18) (cid:96) − , (cid:96) + 1 + Lω, (cid:96) + 1) , rr − i L (cid:19) . (5.6)where A is an arbitrary amplitude. Asymptotically this solution behaves as R (2) ω(cid:96) (cid:12)(cid:12)(cid:12)(cid:12) r →∞ ∼ − A e − iπ ( Lω − (cid:96) ) (cid:20) − i Lr F ( (cid:96) − , (cid:96) + 1 + Lω, (cid:96) + 1) , L r (cid:18) ( Lω + (cid:96) − F ( (cid:96) − , (cid:96) + 1 + Lω, (cid:96) + 1) , (cid:96) − Lω + (cid:96) + 1) (cid:96) + 1 F ( (cid:96), (cid:96) + 2 + Lω, (cid:96) + 3 , (cid:19)(cid:21) + O (cid:18) L r (cid:19) . (5.7)Comparing this decay with (3.4) we can read the expressions for two amplitudes A (2)+ and A (2) − . These amplitudes are `a priori independent but the requirement that the perturbation isasymptotically global AdS constrains them to be related by the BCs (4.19). These BCs quantizethe frequencies of the perturbations that can fit in the global AdS box, respectively, as1) Scalar normal modes of global AdS: ωL = 1 + (cid:96) + 2 p , (5.8)2) Vector normal modes of global AdS: ωL = 2 + (cid:96) + 2 p , (5.9)where the non-negative integer p is the radial overtone that gives the number of nodes alongthe radial direction and recall that (cid:96) ≥ = 0 in the scalar case [33, 4]and Φ = 0 in the vector case [41, 5]. In this subsection we study some properties of the gravitational quasinormal mode (QNM)spectrum of the global AdS-Schwarzschild black hole (GAdSBH). In the AdS /CFT duality,this spectrum is dual to the thermalization timescales of the perturbed thermal states of theCFT living on the sphere, as discussed in [33] (following the detailed analysis of the AdS /CFT case presented in [23]). We focus our attention in the low-lying QNMs (small radial overtone p and harmonic (cid:96) ) because they are expected to dominate the late-time behavior of the timeevolution.Many properties of this gravitational QNM spectrum were already studied with some detailin the past. The low-lying KI vector QNMs with global AdS BCs were discussed in [39, 40, 33].The asymptotic behavior of these vector modes for large overtone were further analyzed in[39]-[44] (see footnote 1). On the other hand, the low-lying KI scalar QNMs with global AdSBCs were studied in [33] (see also [23]). Finally, the asymptotic behavior of the vector/scalarQNMs for large harmonic (cid:96) was found in a WKB analysis in [5].Our results agree with the vector results of Cardoso-Lemos [39] and with the results ofMichalogiorgakis-Pufu [33]. Our conclusions and presentation contribute to complement these21revious analysis mainly by plotting the QNM spectrum as a function of the horizon radius,and not just a few points of the spectrum. Our discussion will always focus on the parameterspace region of r + /L where the relevant physics is and/or where the spectrum varies the most.Given the t − φ symmetry of the GAdSBH, the QNM frequencies always come in trivial pairsof { ω, − ω ∗ } . We just plot the element of the pair with positive real frequency.To find the QNM spectrum we solve the Teukolsky radial equation (5.2) numerically subjectto the asymptotically global AdS BCs, namely, (4.18) in the scalar case and (4.19) in thevector case. We use spectral methods to solve the numerical problem, which uses a Chebyshevdiscretization of the grid. We work with the compact radial coordinate 0 ≤ y ≤ q j defined asVector: y = (cid:114) − r + r , R ( − ω(cid:96) = (cid:0) − y (cid:1) y − i ωL πTH q v ; (5.10)Scalar: y = 1 − r + r , R ( − ω(cid:96) = (1 − y ) y − iωL πTH (cid:18) − L r (1 − πT H r + / L )6 πT H (1 − y ) (cid:19) − iωL q s . The horizon BC (3.14) translates to a simple Neumann BC and the asymptotic BC (4.19)yields a Robin BC, in the vector case. For the scalar case, both the horizon BC (3.14) and theasymptotic BC (4.18) translate to a Robin BC relating q s and its derivative. In both cases, weget a generalized quadratic eingenvalue problem in the (complex) QNM frequencies ω . We givethe harmonic (cid:96) and run the code for several dimensionless horizon radius r + /L .To discuss the results consider first the vector QNM spectrum. We will either plot theimaginary part of the dimensionless frequency Im( ωL ) as a function of the real part Re( ωL ) orthe QNM real/imaginary parts as a function of the horizon radius in AdS units ( r + /L ).Figure 1: Left panel: Hydrodynamic QNM (which ends in the black point) and the first fourmicroscopic QNM curves (that start at the vector normal modes of AdS pinpointedas red dots) of the (cid:96) = 2 harmonic of the vector QNM spectrum of GAdSBH.
Right panel: Imaginary part of the hydrodynamic vector QNM as a function of the horizonradius in AdS units. For large r + /L , the data approaches the black curve whichis the analytical prediction (5.11). The green dots in the hydrodynamic and in thelowest-lying ( p = 0) microscopic QNM curves are exactly the values taken from Table2 of [39]. See text for detailed discussion of these plots.22igure 2: Lowest-lying ( p = 0) microscopic vector QNM of the (cid:96) = 2 harmonic of GAdSBH.The green dots are exactly the values taken from Table 2 of [39]. The blue (magenta)dots are obtained solving numerically the Teukolsky (KI) equations. (See text fordetailed discussion of these plots).On the Left panel of Fig. 1 we plot the first five low-lying QNMs of the (cid:96) = 2 vectorharmonic. The points in the vertical line have pure imaginary frequencies and, as r + /L growslarge, it approaches the black point (0 , hydrodynamic modes since, in thelimit r + (cid:29) L , they can be found solving the perturbed Navier-Stokes equation that describesthe hydrodynamic regime of the CFT on the sphere (the associated plasma is conformal, henceit has zero bulk viscosity and shear to entropy density ratio η/s = 1 / (4 π )) [33, 23]. To leadingorder in the inverse of the horizon radius, this hydrodynamic computation yields the frequency[33] ωL (cid:12)(cid:12) hydro = i (cid:96) + 2 (cid:96) − Lr + + O (cid:18) L r (cid:19) . (5.11)Recall that the hydrodynamic regime requires that the perturbation wavelength is much largerthan the thermal scale (the inverse of the temperature) of the theory. So this regime is achievedwhen r + /L (and thus the temperature T H L ) grows without bound and the perturbation fre-quency becomes arbitrary small. This is indeed what happens as we approach the black pointmoving from bottom to top along the vertical line of Fig. 1. On the Right panel of this fig-ure we plot the (imaginary) frequency of the hydrodynamic mode as a function of r + /L for0 < r + /L < r + /L grows the black curve indeed approaches the numericaldata.Returning to the Left panel of Fig. 1, the red points describe the first 4 radial overtones( p = 0 , , ,
3) of the (cid:96) = 2 vector normal modes of AdS; see (5.9). The associated four vectorQNMs in the
Left panel of Fig. 1 are microscopic modes (as oppose to hydrodynamic) becauseas these curves move away from the red points, i.e. as r + /L (and T H L ) grows so does Im( ωL ).So we never reach the hydrodynamic regime ω (cid:28) T H and we need the microscopic theory todescribe them. This tower of overtones which are continuously connected to the normal modesof AdS are often said to form the main series or main sequence of the vector QNM spectrum.In the plot we also pinpoint with green dots the points with fixed r + /L = 0 . r + /L = 1 . p = 0 and microscopic mode curves we also identify the greenmode with r + /L = 2). The green dots in the hydrodynamic and in the lowest-lying ( p = 0)microscopic QNM curves are also exactly the values taken from Table 2 of [39]. Modes with fixed r + /L in the main sequence and for p ≥ r + /L . To illustrate this property weconnect with auxiliary dashed black straight line the two set of modes with r + /L = 0 . , p = 0) QNM curve of the main sequencedo not however fit in these lines as is visible in the plot, but apart from this curiosity, the p = 0overtone curve is similar to the higher overtone curves.Figure 3: Lowest-lying ( p = 0) microscopic vector QNMs of the first 8 harmonics of theGAdSBH. From bottom to top we have: (cid:96) = 2 , , · · · ,
10. The red dots give thevector normal mode frequencies (5.9). The green dots are exactly taken from Table2 of [39].Further properties of this lowest-lying ( p = 0) microscopic QNM curve are displayed in Fig.2. The Left ( Right ) panel plots the real (imaginary) part of the QNM frequency as a function ofthe horizon radius for r + /L ≤
2. The curve starts at the vector AdS normal mode frequency (redpoint) and the frequency stays very close to the real axis for r + /L < .
2. The real (imaginary)part keeps (increasing) decreasing monotonically for larger values of r + /L (e.g. for r + = 100 L one has ωL ∼ . − . i [39]). The same 3 green points for r + /L = 0 . , , p = 0)microscopic QNM of the next seven vector harmonics (cid:96) . More concretely, from bottom to topwe have the harmonics (cid:96) = 2 , , · · · ,
10. From the
Left panel, we conclude that there are regionsin the parameter space (i.e. windows in the range of r + /L ) where the real part of the frequencyspectrum is in a first approximation isospectral (i.e. difference between consecutive harmonics atconstant r + /L is approximately constant) but there are also others where the lowest harmonics (cid:96) = 2 , Right panel, we see that the window of r + /L around24he global AdS case ( r + /L = 0; red dot (0,0)) where the imaginary part of the spectrum isapproximately flat increases as the harmonic (cid:96) grows. The green dots in the (cid:96) = 2 , (cid:96) = 2 harmonic of the scalar QNM spectrum of GAdSBH. The green dots haveexactly the values taken from Table 1 and 2 of [33] (we have added the r + = 0 . p ≥ (cid:96) = 2 scalar harmonic. In this case, the spectrum has no pure imaginaryfrequencies, and all the QNM curves are continuously connected to the scalar normal modesfrequencies (5.8) of AdS here coloured as red dots for the overtones p = 0 , , ,
3. Clearly, the p = 0 overtone curve on the left is a special curve. Indeed, in this case as r + /L (and T H L ) growsthe imaginary part of the frequency first decreases but then it has a minimum after which itapproaches zero (see the black point). That is, we approach the hydrodynamic regime ω (cid:28) T H .This is thus the scalar hydrodynamic QNM. For the p ≥ r + /L : these are microscopic scalar QNMs. Solving the linearizedhydrodynamic equations on R t × S for a conformal plasma, [33] finds that to leading order thescalar hydrodynamic QNM is described by ωL (cid:12)(cid:12) hydro = (cid:112) (cid:96) ( (cid:96) + 1) √ − i (cid:96) + 2 (cid:96) − Lr + + O (cid:18) L r (cid:19) , (5.12)and this fixes the black point in Fig. 4 when r + /L → ∞ . This analytical result was alreadycompared against numerical data for large radius in [33]. Much like in the vector case, inthe main sequence ( p ≥ r + /L scale linearly with r + /L . Toillustrate this property we connect with auxiliary dashed black straight line the two set of modeswith r + /L = 0 . , .
5. The green dots have exactly the values taken from Table 1 and 2 of [33](we have added the r + = 0 . p ≥ (cid:96) = 2 harmonic. Weplot the real (imaginary) part of the frequency as a function of r + /L in the window of valueswhere the frequency varies the most, namely r + /L <
7. The red point is the scalar normal25igure 5: Hydrodynamic scalar QNM of the (cid:96) = 2 harmonic of GAdSBH. The green dots areexactly the values taken from taken from Table 1 and 2 of [33]. The blue (magenta)dots are obtained solving numerically the Teukolsky (KI) equations. (See text fordetailed discussion of these plots).mode frequency (5.8) for p = 0. This plot complements the data in Table 1 and 2 of [33]which is also represented as green dots in these plots (these are the QNM for r + /L = 0 . , , r + /L the real (imaginary) part keeps decreasing (increasing) monotonically(for reference, for r + = 100 L one has ωL ∼ . − . i [33]). In the limit r + /L → ∞ itapproaches the real value, ωL = 1 . (cid:96) . More concretely, from bottom to top (on the Left panel and on the right side of the
Right panel) we have the harmonics (cid:96) = 2 , , · · · ,
10. Onthe
Left panel we normalize the real frequency to the respective p = 0 normal mode frequencyof AdS (5.8). We see that the real part of the frequency spectrum is in a first approximationisospectral at each fixed r + /L . From the Right panel, we conclude that the imaginary part ofthe frequency always has a minimum, unlike the vector case. Like in the vector QNM case, theinset plot shows that there is a window of r + /L around the global AdS case ( r + /L = 0; reddot (0,0)) where the imaginary part of the spectrum is approximately flat increases, and thiswindow increases as the harmonic (cid:96) grows. The green dots are exactly the values taken fromTable 1,2 and 3 of [33].As a final remark, note that as discussed below (4.14), the Teukolsky formalism describesonly the harmonics with (cid:96) ≥ | s | = 2. So, it misses the Kodama-Ishibashi vector mode with (cid:96) v = 1 and scalar mode with (cid:96) s = 0. The QNM spectrum of these KI modes is very specialbecause it only contains a zero-mode, i.e. a mode with zero frequency. The scalar zero-modeproduces a shift in the mass of the solution, while the vector zero-mode generates angularmomentum (thus connecting perturbatively global AdS-Schwarzschild to Kerr-AdS) [14, 54].26igure 6: Hydrodynamic scalar QNMs of the first 8 harmonics of the GAdSBH. Viewing fromthe right side of the plots, from bottom to top we have: (cid:96) = 2 , , ,
10. The red dotsgive the scalar normal mode frequencies (5.8) with p = 0. The green dots are exactlytaken from Table 1-3 of [33].The Teukolsky formulation is blind to these modes that generate deformations in the conservedcharges [27, 55]. Acknowledgments
It is a pleasure to warmly thank Gary Horowitz, Don Marolf and Harvey Reall for helpful dis-cussions. OD thanks the Yukawa Institute for Theoretical Physics (YITP) at Kyoto University,where part of this work was completed during the YITP-T-11-08 programme “Recent advancesin numerical and analytical methods for black hole dynamics”, and the participants of the work-shops “Iberian Strings 2013”, Lisbon (Portugal), “Holography, gauge theory and black holes”,Amsterdam (Netherlands), “XVIII IFT Xmas Workshop”, Madrid (Spain), “The HolographicWay: String Theory, Gauge Theory and Black Holes”, Nordita (Sweden), “Spanish RelativityMeeting in Portugal”, “Exploring AdS-CFT Dualities in Dynamical Settings”, Perimeter Insti-tute (Canada), and “Numerical Relativity and High Energy Physics”, Madeira (Portugal) fordiscussions. JS acknowledges support from NSF Grant No. PHY12-05500.
A An overview of perturbations in Kerr(-AdS)
The Teukolsky solutions δ Ψ and δ Ψ , the Starobinski-Teukolsky identities, and the Hertz mapthat constructs the associated metric perturbations provide the complete information about the most general metric perturbation of the Kerr(-AdS) black hole [8, 9, 10] (the only exceptionbeing the “ (cid:96) = 0 ,
1” modes that simply add mass or angular momentum to the background [55];onwards we omit these exceptional modes from our discussion).In this appendix we provide a brief historical overview of the studies that culminated withthe above conclusion. In this discussion we assume the background to be Petrov type D andwe highlight some facts that are sometimes not duly appreciated.In the Newman-Penrose formalism, the fundamental gravitational variables are the 4 com-ponents of the NP tetrad basis e a = { (cid:96) , n , m , m } , the 12 complex NP spin coeficients (2.2)27nd the 5 complex NP Weyl scalars { Ψ , · · · Ψ } . These variables are governed by a set of threesystems of equations, namely, the Bianchi identities, the Ricci identities, and the commutationrelations for the basis vectors. The metric is determined once we fix the tetrad basis by g µν = − (cid:96) ( µ n ν ) + 2 m ( µ m ν ) . (A.1)The most general perturbation of this system requires determining 10+24+16=50 real func-tions to specify the perturbations of the 5 complex Weyl scalars, 12 complex spin coeficientsand the 16 matrix components A ba that describe the deformations of the tetrad via δ e a = A ba e b .In the Kerr(-AdS) background, this general perturbation system divides into two sectors[9, 10]: I ) δ Ψ , δ Ψ , δ Ψ , δ Ψ , δκ, δσ, δλ, δν ; (A.2) II ) δ Ψ , δα, δβ, δ(cid:15), δγ, δπ, δρ, δµ, δτ, δ (cid:96) , δ n , δ m , δ m . (A.3)The first family describes perturbations of those variables that vanish in the Kerr(-AdS) back-ground because it is a Petrov type D geometry. The second involves all other quantities thatare not required to vanish in such a Petrov background. A remarkable property is that thesetwo sectors of perturbations “almost decouple” in the sense that we can solve the perturba-tion sector I ) without solving the NP equations involving the perturbations of sector II ). Thesolutions of sector I ) are however a prerequisite to then search for the solutions of sector II )[9, 10].Teukolsky [8], and later Chandrasekhar [9] following an independent computation, foundthe solutions of the perturbation sector I ). One just needs to solve the Teukolsky masterequation that gives the solution for δ Ψ , say. The Starobinski-Teukolsky identities fix therelative normalization between these two variables [36, 37, 38, 9, 10]. These scalars δ Ψ and δ Ψ are gauge invariant, i.e. invariant both under infinitesimal coordinate transformations andinfinitesimal changes of NP basis. We can then set δ Ψ = 0 = δ Ψ by an infinitesimal rotationof the tetrad basis. Finally, the perturbations of the spin coefficients δκ, δσ, δλ, δν are obtainedby applying differential operators to δ Ψ and δ Ψ . So the information on the linear perturbationof our system I ) is encoded in the gauge invariant variables δ Ψ and δ Ψ . (This is not the fullstory concerning perturbation sector I ); we will come back and complete it in the end of thisAppendix).The most general perturbation problem is however not yet solved since we still need tofind the solutions for the perturbation sector II ) in (A.3). In a tour de force computation thatrequires starting with sector I ) solutions, Chandrasekhar did a direct and complete integration ofthe remaining linearized NP equations to find the sector II ) solutions (A.3) [10]. Remarkably, inthe end of the day, all sector II ) perturbations are determined also only as a function of δ Ψ , δ Ψ .This justifies the statement that the Teukolsky master equations and the Starobinski-Teukolskyidentities encode the complete information about general perturbations of the Kerr(-AdS) blackhole. Note in particular that with the knowledge of the basis vector perturbations we can alsoeasily construct the perturbations of the metric components through the variation of (A.1) [10].An astonishing twist in this story is that we do not need the major effort of integrating thefull system of NP equations to get what is often the most desired result, namely the metricperturbations h ab . This was first realized by Cohen and Kegeles [24, 26] and Chrzanowski [25]who have assumed some ad hoc , but smart guessed, hypothesis to build the Hertz map. At thispoint in time, this map was a prescription to reconstruct the most general perturbations of themetric tensor only from the knowledge of the Teukolsky master solutions δ Ψ , δ Ψ , i.e. without This includes the spin coefficient (cid:15) that would vanish if we worked with an appropriate tetrad basis, but nototherwise. In particular it is not required to vanish by the algebraically special character of the spacetime. δ Ψ , δ Ψ ; 2)the fact that the Teukolsky operator for δ Ψ is the adjoint of the one for δ Ψ , which on theother hand is in the end of the day responsible for the existence of the Starobinsky-Teukolskydifferential identities (2.19)-(2.20); 3) the fact that the equations (2.24) defining the Hertz mapare the adjoint of the original Teukolsky master equations (2.10)-(2.11). Ultimately all theseproperties are due to the algebraically special character of the background.The upshot of Wald’s proof of the Hertz map prescription of [24]-[26] is that we can use thismap (2.27)-(2.28) to obtain the complete, most general, metric perturbation of the Kerr(-AdS)black hole (with the exception of the modes that change the mass and angular momentum),without needing to integrate the extra NP equations that would be necessary to find the solutions(A.3). It is worth to look back and appreciate this result: `a priori we had to find a total of 50variables. However, in the end of the day we just need to solve the Teukolsky master equationfor δ Ψ (the equation for δ Ψ is its adjoint and their relative normalization set by the STidentities) to get through the Hertz map the most general metric perturbation. An importantcorollary of this result is that the global AdS boundary conditions we find in Section 3 applyto generic perturbations of the Kerr-AdS black hole (with (cid:96) ≥ I ).Strickly speaking there is however some residual incompleteness in this solution since [9, 10]:1) at this point we just know the absolute value but not the real and imaginary parts of theangular Starobinski-Teukolsky constant C st , and 2) there is still an unknown numerical factorneeded to fully determine the perturbations δλ , and δν . Both these gaps in our knowledge arefilled once we solve perturbation sector II ) via an integrability condition [9, 10] (also reviewedin sections 82 to 95 of chapter 9 of the textbook [11]). To our knowledge, the analogouscomputation of [10] that determines this information in the Kerr-AdS case was never done, andit would be interesting to undergo this task. However, to determine the asymptotically globalAdS BCs of Section 3 we do not need this knowledge at all. Moreover, we do not need theexplicit expression for C st to construct the map between the Kodama-Ishibashi and the a = 0Teukolsky formalisms of Section 4. Nevertheless, and for completeness, in the main text weconjectured the expression for C st to be the solution of |C st | as given in (2.23) that reduces tothe asymptotically flat expression of [10] when L → ∞ . This expression is written in (2.23) or(4.17) when a →
0. This is a reasonable expectation but it would nevertheless be important toconfirm this expression with a computation similar to the one done in [10]. This would closethe Kerr-AdS linear gravitational perturbation programme.It is believed that Einstein’s equation is not a special system of coupled PDEs. On the otherhand, when these equations are linearized around a Petrov type D background and written inthe Newman-Penrose formalism it is astonishing to find how special the linearized PDE systemis.
References [1] V. Cardoso, L. Gualtieri, C. Herdeiro, U. Sperhake, P. M. Chesler, L. Lehner, S. C. Parkand H. S. Reall et al. , “NR/HEP: roadmap for the future,” arXiv:1201.5118 [hep-th].[2] E. Berti, V. Cardoso and A. O. Starinets, “Quasinormal modes of black holes and blackbranes,” Class. Quant. Grav. (2009) 163001 [arXiv:0905.2975 [gr-qc]].293] P. Bizon, A. Rostworowski, “On weakly turbulent instability of anti-de Sitter space,” Phys.Rev. Lett. (2011) 031102. [arXiv:1104.3702 [gr-qc]].[4] O. J. C. Dias, G. T. Horowitz and J. E. Santos, “Gravitational Turbulent Instability ofAnti-de Sitter Space,” Class. Quant. Grav. (2012) 194002 [arXiv:1109.1825 [hep-th]].[5] O. J. C. Dias, G. T. Horowitz, D. Marolf and J. E. Santos, “On the Nonlinear Stabil-ity of Asymptotically Anti-de Sitter Solutions,” Class. Quant. Grav. (2012) 235019[arXiv:1208.5772 [gr-qc]].[6] R. H. Price and J. Pullin, “Colliding black holes: The Close limit,” Phys. Rev. Lett. (1994) 3297 [gr-qc/9402039].[7] S. A. Teukolsky, “Rotating black holes - separable wave equations for gravitational andelectromagnetic perturbations,” Phys. Rev. Lett. (1972) 1114.[8] S. A. Teukolsky, “Perturbations of a rotating black hole. 1. Fundamental equations forgravitational electromagnetic and neutrino field perturbations,” Astrophys. J. (1973)635.[9] S. Chandrasekhar, “The Gravitational Perturbations of the Kerr Black Hole. I. The Per-turbations in the Quantities which Vanish in the Stationary State,” Proc. R. Soc. Lond. A358 (1978), 421.[10] S. Chandrasekhar, “The Gravitational Perturbations of the Kerr Black Hole. II. The Per-turbations in the Quantities which are Finite in the Stationary State,” Proc. R. Soc. Lond.
A 358 (1978), 441.[11] S. Chandrasekhar,
The mathematical theory of black holes , (Oxford University Press, NewYork, 1992).[12] T. Regge and J. A. Wheeler, “Stability of a Schwarzschild singularity,” Phys. Rev. (1957) 1063.[13] F. J. Zerilli, “Effective potential for even parity Regge-Wheeler gravitational perturbationequations,” Phys. Rev. Lett. (1970) 737.[14] H. Kodama, A. Ishibashi, “A Master equation for gravitational perturbations of maximallysymmetric black holes in higher dimensions,” Prog. Theor. Phys. (2003) 701-722. [hep-th/0305147].[15] S. W. Hawking and H. S. Reall, “Charged and rotating AdS black holes and their CFTduals,” Phys. Rev. D (2000) 024014 [arXiv:hep-th/9908109].[16] V. Cardoso and O. J. C. Dias, “Small Kerr-anti-de Sitter black holes are unstable,” Phys.Rev. D (2004) 084011 [arXiv:hep-th/0405006].[17] H. K. Kunduri, J. Lucietti and H. S. Reall, “Gravitational perturbations of higher dimen-sional rotating black holes: Tensor Perturbations,” Phys. Rev. D (2006) 084021.[18] V. Cardoso, O. J. C. Dias and S. Yoshida, “Classical instability of Kerr-AdS black holesand the issue of final state,” Phys. Rev. D (2006) 044008 [arXiv:hep-th/0607162].[19] J. Lucietti and H. S. Reall, “Gravitational instability of an extreme Kerr black hole,” Phys.Rev. D (2012) 104030 [arXiv:1208.1437 [gr-qc]].3020] G. T. Horowitz and V. E. Hubeny, “Quasinormal modes of AdS black holes and the ap-proach to thermal equilibrium,” Phys. Rev. D (2000) 024027 [hep-th/9909056].[21] D. Birmingham, I. Sachs and S. N. Solodukhin, “Conformal field theory interpretation ofblack hole quasinormal modes,” Phys. Rev. Lett. (2002) 151301 [hep-th/0112055].[22] P. K. Kovtun and A. O. Starinets, “Quasinormal modes and holography,” Phys. Rev. D (2005) 086009 [hep-th/0506184].[23] J. J. Friess, S. S. Gubser, G. Michalogiorgakis and S. S. Pufu, “Expanding plasmas andquasinormal modes of anti-de Sitter black holes,” JHEP (2007) 080 [hep-th/0611005].[24] J. M. Cohen and L. S. Kegeles, “Space-time perturbations”, Phys. Lett. A (1975) 5.[25] P. L. Chrzanowski, “Vector potential and metric perturbations of a rotating black hole”,Phys. Rev. D (1975) 2042.[26] J. M. Cohen and L. S. Kegeles, “Constructive procedure for perturbations of spacetimes,”Phys. Rev. D (1979) 1641.[27] R. M. Wald, “Construction of solutions of gravitational, electromagnetic, or other per-turbation equations from solutions of decoupled equations”, Phys. Rev. Lett. (1978)203.[28] J. M. Stewart, “Hertz-Bromwich-Debye-Whittaker-Penrose potentials in general relativ-ity”, Proc. Roy. Soc. Lond. A (1979), 527.[29] O. J. C. Dias, H. S. Reall and J. E. Santos, “Kerr-CFT and gravitational perturbations,”JHEP (2009) 101 [arXiv:0906.2380 [hep-th]].[30] R. Geroch, “Structure of the gravitational field at spatial infinity”, J. Math. Phys. 13, 956(1972).[31] W. Boucher, G. W. Gibbons and G. T. Horowitz, “A Uniqueness Theorem For Anti-deSitter Space-time,” Phys. Rev. D (1984) 2447.[32] M. Henneaux and C. Teitelboim, “Asymptotically anti-De Sitter Spaces,” Commun. Math.Phys. (1985) 391.[33] G. Michalogiorgakis and S. S. Pufu, “Low-lying gravitational modes in the scalar sector ofthe global AdS(4) black hole,” JHEP (2007) 023 [hep-th/0612065].[34] G. Compere and D. Marolf, “Setting the boundary free in AdS/CFT,” Class. Quant. Grav. , 195014 (2008) [arXiv:0805.1902 [hep-th]].[35] T. Andrade and D. Marolf, “AdS/CFT beyond the unitarity bound,” JHEP , 049(2012) [arXiv:1105.6337 [hep-th]].[36] A.A. Starobinsky, “Amplification of waves during reflection from a rotating black hole,”Sov. Phys. - JETP (1973) 28.[37] A.A. Starobinsky and S.M. Churilov, “Amplification of electromagnetic and gravitationalwaves scattered by a rotating black hole,” Sov. Phys. - JETP (1973) 1.3138] S. A. Teukolsky and W. H. Press, “Perturbations of a Rotating Black Hole. III - Interac-tion Of The Hole With Gravitational And Electromagnetic Radiation,” Astrophys. J. (1974) 443.[39] V. Cardoso and J. P. S. Lemos, “Quasinormal modes of Schwarzschild anti-de Sitter blackholes: Electromagnetic and gravitational perturbations,” Phys. Rev. D (2001) 084017[gr-qc/0105103].[40] E. Berti and K. D. Kokkotas, “Quasinormal modes of Reissner-Nordstrom-anti-de Sitterblack holes: Scalar, electromagnetic and gravitational perturbations,” Phys. Rev. D (2003) 064020 [gr-qc/0301052].[41] J. Natario and R. Schiappa, “On the classification of asymptotic quasinormal frequenciesfor d-dimensional black holes and quantum gravity,” Adv. Theor. Math. Phys. (2004)1001 [hep-th/0411267].[42] V. Cardoso, R. Konoplya and J. P. S. Lemos, “Quasinormal frequencies of Schwarzschildblack holes in anti-de Sitter space-times: A Complete study on the asymptotic behavior,”Phys. Rev. D (2003) 044024 [gr-qc/0305037].[43] S. Musiri, S. Ness and G. Siopsis, “Perturbative calculation of quasi-normal modes of AdSSchwarzschild black holes,” Phys. Rev. D (2006) 064001 [hep-th/0511113].[44] G. Siopsis, “Low frequency quasi-normal modes of AdS black holes,” JHEP (2007)042 [hep-th/0702079].[45] B. Carter, “Hamilton-Jacobi and Schrodinger separable solutions of Einstein’s equations,”Commun. Math. Phys. (1968) 280.[46] C. M. Chambers and I. G. Moss, “Stability of the Cauchy horizon in Kerr-de Sitter space-times,” Class. Quant. Grav. (1994) 1035 [gr-qc/9404015].[47] O. J. C. Dias, R. Monteiro, H. S. Reall and J. E. Santos, “A Scalar field condensationinstability of rotating anti-de Sitter black holes,” JHEP (2010) 036 [arXiv:1007.3745[hep-th]].[48] M. M. Caldarelli, G. Cognola and D. Klemm, “Thermodynamics of Kerr-Newman-AdSblack holes and conformal field theories,” Class. Quant. Grav. (2000) 399 [hep-th/9908022].[49] G. W. Gibbons, M. J. Perry and C. N. Pope, “The first law of thermodynamicsfor Kerr − anti-de Sitter black holes,” Class. Quant. Grav. (2005) 1503 [arXiv:hep-th/0408217].[50] O. J. C. Dias, J. E. Santos and M. Stein, “Kerr-AdS and its Near-horizon Geometry: Per-turbations and the Kerr/CFT Correspondence,” JHEP (2012) 182 [arXiv:1208.3322[hep-th]].[51] C. O. Lousto and B. F. Whiting, “Reconstruction of black hole metric perturbations fromweyl curvature”, Phys. Rev. D 66 (2002), 024026.[52] E. T. Newman and R. Penrose, “Note on the Bondi-Metzner-Sachs group,” J. Math. Phys. (1966) 863. 3253] J. N. Goldberg, A. J. MacFarlane, E. T. Newman, F. Rohrlich and E. C. G. Sudarshan,“Spin s spherical harmonics and edth,” J. Math. Phys. (1967) 2155.[54] O. J. C. Dias and H. S. Reall, “Algebraically special perturbations of the Schwarzschildsolution in higher dimensions,” arXiv:1301.7068 [gr-qc].[55] R. Wald, “On perturbations of a Kerr black hole,” J. Math. Phys.14