Contents
Abstract

The Teukolsky master equation and its associated spin-weighted spheroidal harmonic decomposition simplify considerably the study of linear gravitational perturbations of the Kerr(-AdS) black hole. However, the formulation of the problem is not complete before we assign the physically relevant boundary conditions. We find a set of two Robin boundary conditions (BCs) that must be imposed on the Teukolsky master variables to get perturbations that are asymptotically global AdS, i.e. that asymptotes to the Einstein Static Universe. In the context of the AdS/CFT correspondence, these BCs allow a non-zero expectation value for the CFT stress-energy tensor while keeping fixed the boundary metric. When the rotation vanishes, we also find the gauge invariant differential map between the Teukolsky and the Kodama-Ishisbashi (Regge-WheelerZerilli) formalisms. One of our Robin BCs maps to the 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 instability timescales and quasinormal mode spectrum of the Kerr-AdS black hole. As a warm-up for this programme, we use the Teukolsky formalism to recover the quasinormal mode spectrum of global AdS-Schwarzschild, complementing previous analysis in the literature.

Óscar J. C. Dias, Jorge E. Santos

Institut de Physique Théorique, CEA Saclay,

CNRS URA 2306, F-91191 Gif-sur-Yvette, France

Department of Physics, UCSB, Santa Barbara, CA 93106, USA

oscar.dias@cea.fr, jss55@physics.ucsb.edu

1 Introduction and summary

It is unquestionable that few systems are isolated in Nature and we can learn a lot from studying their interactions. Black holes are no exception and the study of their perturbations and interactions 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 often encodes interesting physics such as linear stability of the system and its quasinormal mode spectrum. Moreover, it also anticipates some non-linear level properties. For example, in the collision of two black holes, such as in the coalescence of a binary system, after the inspiral and merger phase, the system undergoes a ring down phase where gravitational wave emission is dictated by the quasinormal mode frequencies. The linear perturbation fingerprints are therefore valuable from a theoretical and gravitational-wave detection perspective [1, 2]. Perhaps more surprisingly, linear analysis of black holes in AdS can be used to infer properties about their nonlinear stability [3, 4, 5]. Linear analysis can also infer some properties of (nonlinear) black hole 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 linearized Einstein equation. À 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), Teukolsky employed the Newman-Penrose formalism to prove that all the gravitational perturbation information is encoded in two decoupled complex Weyl scalars [7, 8]. These are gauge invariant quantities 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 Weyl scalars (one element of the pair describes spin and the other modes). In a mode by mode analysis, each master equation further separates into a radial and angular equation which makes the analysis technically tractable [8, 9, 10, 11]. (In the absence of rotation, and only in this case, we can instead use a similar pair of decoupled master equations for a distinct pair of gauge invariant variables proposed by Regge and Wheeler [12] and Zerilli [13], and later recovered and extended by Kodama and Ishibashi [14]).

Solving these master equations is not our only task. Like in any PDE system, it is also important to assign physically relevant boundary conditions. Without the later, the formulation of the problem is not complete.

In this paper we are interested in linear gravitational perturbations of the Kerr-AdS black hole, with a focus on its boundary conditions (BCs). The (extra) motivation to put into a firmer basis the linear perturbation problem of the Kerr-AdS system is two-folded. First, the Kerr-AdS black hole is known to have linear gravitational instabilities sourced by superradiance [15]-[18] and by extremality [19]. Second, in the AdS/CFT duality context, perturbing a (Kerr-AdS) black hole in the bulk is dual to perturbing the associated CFT thermal state (with a chemical potential) living in bulk boundary. The time evolution of the perturbed black hole maps into the time evolution of the thermal state fluctuations and the quasinormal mode spectrum of the black hole is dual to the thermalization timescale in the CFT (see e.g [20]-[23], [2]).

From the mathematical perspective, the boundary condition choice is arbitrary. We need physical input to fix it. Not always but quite often, this leads to a unique choice. We establish what are the BCs we need to impose in the Teukolsky master solutions to get perturbations that are asymptotically global AdS. To make this statement precise, recall that once we have the solution of the Teukolsky pair of master variables (), we can reconstruct the metric perturbations using the Hertz map [24]-[29]. We get a pair of metric perturbations, one in the ingoing radiation gauge (IRG; ) and the other in the outgoing radiation gauge (ORG; ). By asymptotically global AdS perturbations we mean that we want the BCs in the Teukolsky scalars that yield metric perturbations that decay at asymptotic infinity according to 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 BCs translate into the Teukolsky scalars.

Before arguing further that this choice should be the physically relevant option, it is illuminating to recall what is the situation in an asymptotically flat system. In this case, the BC choice in the Teukolsky scalars amounts to choosing the purely outgoing traveling mode. Intuitively, this is because we are not interested in scattering experiments (where an ingoing mode component would be present). Formally, this is because this is the choice that yields a metric perturbation preserving the asymptotic flatness of the original Kerr black hole, i.e. conserving asymptotically the Poincaré group of the Minkowski spacetime.

So, a global AdS geometry with Henneaux-Teitelboim BCs does not deform the boundary metric. This is the mathematical statement materializing the pictoric idea that a global AdS background behaves like a confining box with a reflecting wall. An interesting observation that emerges from our study is that these BCs require that we consider a particular linear combination of the Teukolsky IRG and ORG metric contributions. We can interpret this property as being a manifestation of the common lore that only a standing wave with a node at the AdS boundary can fit inside the confining box. This pictorial notion of a standing wave and node is very appealing but, what is the formal definition of a node in the present context? Does it mean that we have to impose a Dirichlet BC on the Teukolsky scalars? No. Instead we will find the Robin BC (3.9)-(3.2), much like what happens in the scalar sector of the aforementioned 4-dimensional KI system. An inspection of this Robin BC (pair) immediately convinces us that we hardly could guess it without the actual computation.

At first sight the fact the asymptotically global AdS BC requires a sum of the Teukolsky IRG and ORG metric components is rather surprising and, even worrying. Surprising because in the asymptotically flat case we just need to use the outgoing contribution. Eventually worrying because it is known that in Petrov type D backgrounds the two Teukolsky families of perturbations () encode the same information, once we use the Starobinsky-Teukolsky identities [36, 37, 38, 9, 10] that fix the relative normalization between the Weyl scalars perturbations. Our result is however not in contradiction with this property. Indeed, the previous statement implies that the most general solution of the master equations contain the same information but says nothing about the BCs. This just highlights that the differential equations and the BCs are two distinct aspects of the problem, which is not a surprise. Once we find our BCs, in practical applications, we just need to study the (say) Teukolsky sector of perturbations. We believe that an infinitesimal rotation of the tetrad basis should allow to derive our results using only the outgoing gauge (say), although at the cost of loosing contact with the standing wave picture.

We have already mentioned that perturbations for static backgrounds (with global AdS and global AdS-Schwarzschild being the relevant geometries here) can be studied using the Kodama-Ishibashi (KI) gauge invariant formalism [14] (i.e. the Regge-WheelerZerilli formalism [12, 13]). On the other hand, the Teukolsky formalism also describes these cases when rotation 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 gravitational perturbations in scalar and vector spherical harmonics while the Teukolsky formalism uses instead a harmonic decomposition with respect to the spin-weighted spherical harmonics. These harmonics are distinct and ultimately responsible for the different routes taken by the two formalisms. However, both the KI spherical harmonics and the spin-weighted spherical harmonics can be expressed in terms of the standard scalar spherical harmonic (associated Legendre polynomials) and their derivatives. These two maps establish the necessary bridge between the angular decomposition of the two formalisms. We then need to work out the radial map, which follows from the fact that the metric perturbations of the two formalisms must be the same modulo gauge transformations. This gauge invariant differential map expresses the KI master variables (for the KI scalar and vector sector) in terms of the (say) Teukolsky master field and its first radial derivative and is given in (4.15)-(4.16). To have the complete map between the KI and Teukolsky () formalisms we also need to discuss the relation between the 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 . Yet this exercise reveals to be more profitable. Indeed, an interesting outcome is that there is a Teukolsky solution/BC that maps to a KI scalar mode/BC and a second one that maps to a KI vector mode/BC. This is the simplest possible map between the two formalisms that could have been predicted, yet still a surprise.

The plan of this paper is the following. Section 2 discusses the Kerr-AdS black hole in the Chambers-Moss coordinate frame [46] (instead of the original Carter frame [45]) that simplifies considerably our future discussion of the results. We discuss the Teukolsky formalism, the associated Starobinsky-Teukolsky identities and the Hertz map in a self-contained exposition because they will be fundamental to derive our results. In Section 3 we find the BCs on the Teukolsky variables that yields asymptotically global AdS perturbations. Section 4 constructs the gauge invariant differential map between the Teukolsky and Kodama-Ishibashi (Regge-WheelerZerilli) gauge invariant formalisms. Finally, in Section 5 we study the QNM spectrum / CFT timescales of the global AdS(-Schwarzschild) background.

2 Gravitational perturbations of the Kerr-AdS black hole

The Kerr-AdS geometry was originally written by Carter in the Boyer-Lindquist coordinate system [45]. Here, following Chambers and Moss [46], we introduce the new time and polar coordinates related to the Boyer-Lindquist coordinates by

 t=ΞT,χ=acosθ, (2.1)

where is to be defined in (2.3). In this coordinate system the Kerr-AdS black hole line element reads [46]

 +(r2+χ2)Δrdr2+(r2+χ2)Δχdχ2 (2.2)

where

 Δr=(a2+r2)(1+r2L2)−2Mr,Δχ=(a2−χ2)(1−χ2L2),Ξ=1−a2L2. (2.3)

The Chambers-Moss coordinate system has the nice property that the line element treats the radial and polar coordinates at an almost similar footing. One anticipates that this property will naturally extend to the radial and angular equations that describe gravitational perturbations in the Kerr-AdS background. In this frame, the horizon angular velocity and temperature are given by

 ΩH=ar2++a2,TH=1Ξ[r+2π(1+r2+ℓ2)1r2++a2−14πr+(1−r2+ℓ2)]. (2.4)

The Kerr-AdS black hole obeys , and asymptotically approaches global AdS space with radius of curvature . This asymptotic structure is not manifest in (2.1), one of the reasons being that the coordinate frame rotates at infinity with angular velocity . However, if we introduce the coordinate change

 T=tΞ,Φ=ϕ+aL2tΞ, R=√L2(a2+r2)−(L2+r2)χ2L√Ξ,cosΘ=L√Ξrχa√L2(a2+r2)−(L2+r2)χ2, (2.5)

we find that as (i.e. ), the Kerr-AdS geometry (2.1) reduces to

that we recognize as the line element of global AdS. In other words, the conformal boundary of the bulk spacetime is the static Einstein universe : . This is the boundary metric where the CFT lives in the context of the AdS/CFT correspondence.

The ADM mass and angular momentum of the black hole are related to the mass and rotation parameters through and , respectively [48, 49]. The horizon angular velocity and temperature that are relevant for the thermodynamic analysis are the ones measured with respect to the non-rotating frame at infinity [48, 49] and given in terms of (2.4) by and . The event horizon is located at (the largest real root of ), and it is a Killing horizon generated by the Killing vector . Further properties of the Kerr-AdS spacetime are discussed in Appendix A of [47].

2.2 Teukolsky master equations

The Kerr-AdS geometry is a Petrov type D background and therefore perturbations of this geometry 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 (the bar demotes complex conjugation) obeying the normalization conditions ;

• the NP spin connection (with );

• the associated NP spin coeficients defined in terms of as {fleqn}

 κ= −γ311, λ= γ424, ν= γ422, σ= −γ313, α= 12(γ124−γ344), β= 12(γ433−γ213), (2.7) μ= γ423, ρ= −γ314, π= γ421, τ= −γ312, γ= 12(γ122−γ342), ϵ= 12(γ431−γ211);
• the five complex Weyl scalars ( are the Weyl tensor components in the NP null basis)

 Ψ0=C1313,Ψ1=C1213,Ψ2=C1342,Ψ3=C1242,Ψ4=C2424; (2.8)
• and the NP directional derivative operators . The complex conjugate of any complex NP quantity can be obtain through the replacement .

The Kerr-AdS background is a Petrov type D spacetime since all Weyl scalars, except , vanish: and . Due to the Goldberg-Sachs theorem this further implies that . In addition, we might want to set by choosing to be tangent to an affinely parametrized null geodesic . This was the original choice of Teukolsky and Press (when studing perturbations of the Kerr black hole) who used 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, and this was the choice made in [50]. However, here we choose to work with the Chambers-Moss null tetrad defined as [46],

 ℓμ∂μ=1√2√r2+χ2(Ξa2+r2√Δr∂t+√Δr∂r+aΞ√Δr∂ϕ), nμ∂μ=1√2√r2+χ2(Ξa2+r2√Δr∂t−√Δr∂r+aΞ√Δr∂ϕ), mμ∂μ=−i√2√r2+χ2(Ξa2−χ2√Δχ∂t+i√Δχ∂χ+aΞ√Δχ∂ϕ), (2.9)

which is not affinely parametrized (). The motivation for this choice is two-folded. First, the technical analysis of the angular part of the perturbation equations and solutions will be much simpler because this Chambers-Moss tetrad explores the almost equal footing treatment of the coordinates much more efficiently than Kinnersly’s tetrad. Second, to complete our analysis later on we will have to discuss how the metric perturbations (built out of the NP perturbed scalars) transform both under infinitesimal coordinate transformations and infinitesimal change of basis. It turns out that if we work in the Chambers-Moss tetrad, the results will be achieved without requiring a change of basis, while the Kinnersly’s option would demand it. Again, this simplifies our exposition.

Teukolsky’s treatment applies to arbitrary spin perturbations. Here, we are interested in gravitational perturbations so we restrict our discussion to the spins. Let us denote the unperturbed NP Weyl scalars by and their perturbations by with . The important quantities for our discussion are the scalars and . They are invariant both under 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 222Excluding 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. linear perturbation of the system is encoded in these gauge invariant variables and . That is, the perturbation of the leftover NP variables can be recovered once and are known. The later are the solutions of the Teukolsky master equations.

For perturbations the Teukolsky equation is

 { [D−3ϵ+¯ϵ−4ρ−¯ρ](Δ+μ−4γ) −[δ+¯π−¯α−3β−4τ](¯δ+π−4α)−3Ψ2}δΨ0=4πT0,ifs=2, (2.10)

while perturbations are described by the Teukolsky equation

 { [Δ+3γ−¯γ+4μ+¯μ](D+4ϵ−ρ) −[¯δ−¯τ+¯β+3α+4π](δ−τ+4β)−3Ψ2}δΨ4=4πT4,ifs=−2. (2.11)

The explicit form of the source terms , that vanish in our analysis, can be found in [8].

Next we introduce the separation ansatz

 (2.12)

Also, define the radial and angular differential operators

 Dn=∂r+iKrΔr+nΔ′rΔr,D†n=∂r−iKrΔr+nΔ′rΔr Ln=∂χ+KχΔχ+nΔ′χΔχ,L†n=∂χ−KχΔχ+nΔ′χΔχ, (2.13)

where the prime represents derivative wrt the argument and

 Kr=Ξ[ma−ω(a2+r2)],Kχ=Ξ[ma−ω(a2−χ2)]. (2.14)

With the ansatz (2.12), the Teukosky master equations separate into a pair of equations for the radial and angular functions,

 ⎧⎪ ⎪⎨⎪ ⎪⎩(D†−1ΔrD1+6(r2L2−iΞωr)−λ)R(−2)ωℓm(r)=0,(L†−1ΔχL1+6(χ2L2+Ξωχ)+λ)S(−2)ωℓm(χ)=0,fors=−2, (2.15)
 ⎧⎪ ⎪⎨⎪ ⎪⎩(D−1ΔrD†1+6(r2L2+iΞωr)−λ)R(2)ωℓm(r)=0,(L−1ΔχL†1+6(χ2L2−Ξωχ)+λ)S(2)ωℓm(χ)=0,fors=2, (2.16)

where we introduced the separation constant

 λ≡λ(2)ωℓm=λ(−2)ωℓm. (2.17)

Some important observations are in order:

• First note that the radial operators obey (where denotes complex conjugation) while the angular operators satisfy .

• Consequently, the radial equation for is the complex conjugate of the radial equation for , but the angular solutions are instead related by the symmetry . The later statement implies that the separation constants are such that with being real.

• The eigenfunctions are spin-weighted AdS spheroidal harmonics, with positive integer specifying the number of zeros, (so the smallest is ). The associated eigenvalues can be computed numerically. They are a function of and regularity imposes the constraints that must be an integer and .

• We have the freedom to choose the normalization of the angular eigenfunctions. A natural choice is

 ∫1−1(S(s)ωℓm)2dχ=1. (2.18)

2.3 Starobinsky-Teukolsky identities

Suppose we solve the radial and angular equations (2.16), (2.15) for the spin . These solutions for and , when inserted in (2.12), are not enough to fully determine the NP gauge invariant Weyl scalars . The reason being that the relative normalization between and remains undetermined, and thus our linear perturbation problem is yet not solved [38, 9, 10]. Given the natural normalization (2.18) chosen for the weighted spheroidal harmonics, the completion of the solution for requires that we fix the relative normalization between the radial functions and . This is what the Starobinsky-Teukolsky (ST) identities acomplish [36, 37, 38, 9, 10]. A detailed analysis of these identities for the Kerr black hole is available in the above original papers or in the seminal textbook of Chandrasekhar [11]. Here, we present these identities for the Kerr-AdS black hole.

Act with the operator on the Teukolsky equation (2.16) for and use the equation of motion for . This yields one of the radial ST identities. Similarly, to get the second, act with the operator on the Teukolsky equation (2.15) for , and make use of the equation obeyed by . These radial ST identities for the Kerr-AdS background relate to ,

 ⎧⎨⎩D†−1ΔrD†0D†0ΔrD†1R(2)ωℓm=CstR(−2)ωℓm,D−1ΔrD0D0ΔrD1R(−2)ωℓm=C∗stR(2)ωℓm, (2.19)

where we have chosen the radial ST constants to be related by complex conjugation. This is possible because, as noted before, the solutions are related by complex conjugation.

To get the angular ST identities, act with the operator on the Teukolsky equation (2.16) for (and use the equation of motion for ), and act with on the equation (2.15) for (and use the equation for . This yields the pair of ST identities,

 (2.20)

Since the equations for are related by the symmetry , the ST constant on the RHS of these ST identities is real. Moreover, because are both normalized to unity see (2.18) the ST constant is the same in both angular ST identities.

To determine we act with the operator of the LHS of the first equation of (2.19) on the second equation and evaluate explicitly the resulting order differential operator. A similar operation on equations (2.20) fixes . We find that

 |Cst|2=K2st+144M2ω2Ξ2, (2.21) K2st=λ2(λ+2)2+8λΞ2aω[(6+5λ)(m−aω)+12aω]+144Ξ4a2ω2(m−aω)2 +4a2L2[λ(λ+2)(λ−6)+12Ξ2(m−aω)[2mλ−aω(λ−6)]]+4a4(λ−6)2L4. (2.22)

This fixes completely the real constant (we choose the positive sign when taking the square root of to get, when , the known relation between the spin-weighted spherical harmonics) but not the complex constant . However, we emphasize that to find the asymptotically global AdS boundary conditions in next section, we do not need to know , just . Moreover, we do not need the explicit expression for to construct the map between the Kodama-Ishibashi and the Teukolsky formalisms of Section 4.

Neverthless we can say a bit more about the phase of . Recall that in the Kerr case, finding the real and imaginary parts of requires a respectful computational effort which was undertaken by Chandrasekhar [10] (also reviewed in sections 82 to 95 of chapter 9 of the textbook [11]). À priori we would need to repeat the computations of [10], this time in the AdS background, to find the phase of in the Kerr-AdS background (which was never done to date). However, if we had to guess it we would take the natural assumption that is given by the solution of (2.21) that reduces to the asymptotically flat partner of [10] when ,

 Cst=C1+iC2withC1=Kst,C2=−12MωΞ. (2.23)

However, we emphasize again that this expression must be read with some grain of salt and needs a derivation along the lines of [10] to be fully confirmed.

Having fixed the ST constants we have specified the relative normalization between the Teukolsky variables and . We ask the reader to see Appendix A for a further discussion of this issue.

2.4 Metric perturbations: the Hertz potentials

In the previous subsections we found the solutions of the Teukolsky master equations for the gauge invariant Weyl scalars of the Newman-Penrose formalism. We will however need to know the perturbations of the metric components, . These are provided by the Hertz map, , which reconstructs the perturbations of the metric tensor from the associated scalar Hertz potentials (in a given gauge) [24]-[29]. The later are themselves closely related to the NP Weyl scalar perturbations and .

In the Kerr-AdS background, the Hertz potentials are defined by the master equations they obey to, namely,

 [(Δ+3γ−¯¯¯γ+¯¯¯μ)(D+4ϵ+3ρ)−(¯¯¯δ+¯¯¯β+3α−¯¯¯τ)(δ+4β+3τ)−3Ψ2]ψ(−2)H=0, (2.24a) [(D−3ϵ+¯ϵ−¯¯¯ρ)(Δ−4γ−3μ)−(δ−3β−¯¯¯¯α+¯¯¯π)(¯¯¯δ−4α−3π)−3Ψ2]ψ(2)H=0. (2.24b)

Introducing the ansatz for the Hertz potential

 ψ(s)H=⎧⎨⎩e−iωteimϕ(r−iχ)2R(−2)ωℓm(r)S(−2)ωℓm(χ),s=−2,e−iωteimϕ(r−iχ)2R(2)ωℓm(r)S(2)ωℓm(χ),s=2, (2.25)

into (2.24) (in the Kerr-AdS background), we find that and are exactly the solutions of the radial and angular equations (2.15) and (2.16). This fixes the precise map between the Hertz potentials and the NP Weyl scalar perturbations.

The Hertz map is such that the Hertz potentials and generate the metric perturbations in two different gauges, namely the ingoing (IRG) and the outgoing (ORG) radiation gauge, defined by

 IRG:ℓμhμν=0,gμνhμν=0,ORG:nμhμν=0,gμνhμν=0. (2.26)

The Hertz map is finally given by333Note that (2.28), whose explicit derivation can be found in an Appendix of [29], corrects some typos in the map first presented in [25].

 −ℓμℓν(δ+3β+¯α−τ)(δ+4β+3τ)−mμmν(D+3ϵ−¯ϵ−ρ)(D+4ϵ+3ρ)}ψ(−2)H +c.c., (2.27) hORGμν={n(ν¯¯¯¯¯mμ)[(¯δ+¯β−3α+¯τ+π)(Δ−4γ−3μ)+(Δ−3γ−¯γ+μ−¯μ)(¯δ−4α−3π)] −nμnν(¯δ−¯β−3α+π)(¯δ−4α−3π)−¯¯¯¯¯mμ¯¯¯¯¯mν(Δ−3γ+¯γ+μ)(Δ−4γ−3μ)}ψ(2)H +c.c.. (2.28)

We have explicitly checked that (2.4) and (2.28) satisfy the linearized Einstein equation (see also footnote 3).

It is important to emphasize that the Hertz map provides the most general metric perturbation with of the Kerr-AdS black hole [8, 9, 10, 27]. We defer a detailed discussion of this observation to Appendix A.

We start this section with a brief recap of the Teukolsky system which emphasizes some of its properties 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 and the other for . In Petrov type D backgrounds, these two families encode the same information, once we use the Starobinsky-Teukolsky identities that fix the relative normalization between both spin-weighted spheroidal harmonics and between both radial functions . Indeed, modulo the ST relative normalization, the two radial functions are simply the complex conjugate of each other, and the two angular functions are related by . This is a consequence of the fact that the Teukolsky operator acting on is the adjoint of the one acting on .

The upshot of these observations, with relevance for practical applications, is that the Teukosky system in Petrov type D geometries is such that we just need to analyze the sector (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 , and the ST constants we can reconstruct all the Teukolsky quantities.

Were we discussing perturbations of the asymptotically flat Kerr black hole and this section on the boundary conditions would end with the following single last observation. Being a second order differential system, the gravitational field has two independent asymptotic solutions, namely, the ingoing and outgoing traveling modes. Since we are not interested in scattering experiments, the BC (that preserves asymptotic flatness) would be fixed by selecting the purely outgoing BC. For practical purposes, we would definitely just need to study the Teukolsky system of equations.

The situation is far less trivial when we look into perturbations of Kerr-AdS. This time the second order differential system has two independent asymptotic solutions that are power laws of the radial variable. The BC to be chosen selects the relative normalization between these two solutions. What is the criterion to make this choice? This will be made precise in the next 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 character of the Kerr-AdS background. Global AdS asymptotic structure means that the system behaves as 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 ORG modes such that we have a node at the asymptotic AdS wall. With this brief argument we conclude that to find the asymptotic global AdS BC we necessarily need to use the information on both the IRG and ORG Teukolsky metric perturbations, i.e. the BC discussion will require using information on both spins. Once we find it, it is still true that the spin sector of the Teukolsky system encodes the same information as the one, and we will be able to study the properties of perturbations in Kerr-AdS using only the sector (say). (Note that an infinitesimal rotation of the tetrad basis should allow to derive our results using only the ORG, say).

So we take the most general gravitational perturbation of the Kerr-AdS black hole to be given by the sum of the ingoing and outgoing radiation gauge contributions as written in (2.4) and (2.28). (By diffeomorphism invariance, this solution can be written in any other gauge through a gauge transformation). The physically relevant perturbations are those that are regular at the horizon and asymptotically global AdS. In this section we find one of our most fundamental results, namely the BCs we need to impose on our perturbations.

3.1 Definition of asymptotically global AdS 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 possible, but without being so violent that they would destroy the asymptotic structure of the background. To make this statement quantitative, in the familiar case of an asymptotically Minkowski background, the appropriate boundary condition follows from the requirement that the perturbations preserve asymptotically the Poincaré group of the Minkowski spacetime [30].

For the AdS case, Boucher, Gibbons, and Horowitz [31] and Henneaux and Teiltelboim [32] have defined precisely what are the asymptotic BC we should impose to get perturbations that approach at large spacelike distances the global AdS spacetime. The main guideline is that perturbations in a global AdS background must preserve asymptotically the global AdS symmetry group , much like perturbations in a flat background must preserve asymptotically the Poincaré group of the Minkowski spacetime. More concretely, asymptotically global AdS spacetimes are defined by BCs on the gravitational field which obey the following three requirements [32]:

(1) they should contain the asymptotic decay of the Kerr-AdS metric;

(2) they should be invariant under the global AdS symmetry group ;

(3) they should make finite the surface integral charges associated with the generators.

If we work in the coordinate system , where the line element of global AdS is given by (2.6), the metric perturbations that obey the above BCs behave asymptotically as [32]:

 hTμ=1RFTμ(T,Θ,Φ)+O(R−2),forμ=T,Θ,Φ, (3.1a) hTR=1R4FTR(T,Θ,Φ)+O(R−5), (3.1b) hRR=1R5FRR(T,Θ,Φ)+O(R−6), (3.1c) hRμ=1R4FRμ(T,Θ,Φ)+O(R−5),forμ=Θ,Φ, (3.1d) hμν=1RFμν(T,Θ,Φ)+O(R−2),forμ,ν=Θ,Φ, (3.1e)

where are functions of only.

These BCs are defined with respect to a particular coordinate system. Consider a generic infinitesimal coordinate transformation , where is an arbitrary gauge vector field. Under this gauge transformation the metric perturbation transforms according to

 hμν→hμν−2∇(μξν), (3.2)

which we can use to translate the BCs (3.1) in the frame into any other coordinate system, so long as decays sufficiently fast at infinity.

3.2 Boundary conditions for asymptotically global AdS perturbations

Modulo gauge transformations, the most general perturbation of linearized Einstein equations in the Kerr-AdS background can be written as

 hμν=hIRGμν+hORGμν, (3.3)

where and are determined by the Hertz maps (2.4) and (2.28), with the Hertz potentials defined in (2.25) and the associated Teukolsky functions obeying the equations of motion (2.16) and (2.15). Note that the relative normalization between these two contributions is fixed by the Starobinsky-Teukolsky treatment.

Solving the radial Teukolsky equations (2.16) and (2.15) at infinity, using a standard Frobenius analysis, we find that the two independent asymptotic decays for are

 R(2)ωℓm∣∣r→∞∼A(2)+Lr+A(2)−L2r2+O(L3r3), R(−2)ωℓm∣∣r→∞∼B(−2)+Lr+B(−2)−L2r2+O(L3r3), (3.4)

where the amplitudes are, at this point, independent arbitrary constants. Our task is to find the BC we have to impose in order to get a perturbation that is asymptotically global AdS. That is, we must find the constraints, , that these amplitudes have to obey to get the Henneaux-Teiltelboim decay (3.1). We will find that the most tempting condition, where we set to zero the leading order term in the expansion, , is too naive and does not do the job. Note that it follows from (2.1) that for large , or , one has and . Therefore, to get the asymptotically global AdS decay of in the coordinate system we can simply replace in (3.1).

In , we can express as a function of and its derivative using the first Starobinsky-Teukolsky (ST) identity in (2.20) and the angular equation of motion (2.16). This eliminates from (3.3). At this stage, we could also immediately replace the ST constant by its expression (2.22). Instead, we choose to keep it unspecified until a later stage in our computation.

The explicit expression of when we introduce (3.2) into (2.4) and (2.28) contains order terms but no other higher power of . Our first task is to use all the gauge freedom (3.2) to eliminate, if possible, these terms and all lower power law terms that are absent in the asymptotically global AdS decay (3.1). The gauge parameter compatible with the background isometries is . A simple inspection of concludes that the most general components of the gauge vector field, that can contribute up to terms, can be written as the power law expansion in :

 ξt=n∑j=0ξ(j)t(χ)r2−j,ξr=n∑j=0ξ(j)r(χ)r−(1+j),ξχ=n∑j=0ξ(j)χ(χ)r2−j,ξϕ=n∑j=0ξ(j)ϕ(χ)r2−j. (3.5)

Inserting this expansion and (3.2) into (3.3), we find that there is a judicious choice of the functions such that we can eliminate most of the radial power law terms that are absent in the several metric components of (3.1) (the expressions are long and not illuminating). More concretely, we are able to gauge away all desired terms but the contribution in the components , and .

At this point, having used all the available diffeomorphism (3.2), we find ourselves at a key stage of the analysis. To eliminate the undesired leftover contributions we will have to fix the BCs that the amplitudes introduced in (3.2) have to obey to guarantee that the perturbation is asymptotically global AdS. There are two conditions that eliminate simultaneously the terms in , and . One is the coefficient of a term proportional to , and the other is proportional to .444Note that in this computation we use the angular equation of motion (2.16) to get rid of second and higher derivatives of . Clearly, these two contributions have to vanish independently. We can use them to express, for example, the amplitude and the ST constant in terms of the other amplitudes , perturbation parameters and the rotation background parameter (the mass parameter is absent in these expressions):

 0=WB(−2)+−B(−2)−L[−4amLΞ(2iA(2)−+5LΞωA(2)+)+2a2(−6+λ+6L2Ξ2ω2)A(2)+ +λL2(2+λ−4L2Ξ2ω2)A(2)+−8L3Ξω(iA(2)−+LΞωA(2)+)(1+12λ−L2Ξ2ω2)], (3.6) Kst=−B(−2)−L3W{λ2(2+λ)2L6+8aλ(6+5λ)mL6Ξ2ω−144a3mL4Ξ2ω(−2+λ+2L2Ξ2ω2) +4a2L4[λ[−12+(−4+λ)λ+24m2]+2[(6−5λ)λ+18m2]L2Ξ2ω2] +4a4L2[36−12λ+λ2−48λm2+12[λ−6(1+m2)]L2Ξ2ω2+36L4Ξ4ω4] +48a6m2(2λ+3L2Ξ2ω2)}, (3.7)

where we have defined

 W≡L3[λ(2+λ)A(2)−−4(1+λ)LΞω(iλA(2)++2LΞωA(2)−)+4i(2+3λ)L3Ξ3ω3A(2)+] +8L7Ξ4ω4(A(2)−−iLΞωA(2)+)−4amL2Ξ[3iλA(2)++LΞω(5A(2)−−8iLΞωA(2)+)] +2a2L[2(2iLΞωA(2)+−A(2)−)(3+λ−3L2Ξ2ω2)+3λA(2)−]. (3.8)

At this stage we finally introduce the explicit expression for the angular Starobinski-Teukolsky constant, namely, is given by the positive square root of (2.22). In addition, we also use the property that the radial solutions are the complex conjugate of each other. In these conditions we find that conditions (3.6)-(3.7) are obeyed if and only if

 A(2)−=−iηA(2)+,B(−2)−=iη