Quasinormal modes of NUT-charged black branes in the AdS/CFT correspondence
QQuasinormal modes of NUT-charged black branes inthe AdS/CFT correspondence
Pablo A. Cano, a and David Pereñiguez b a Instituut voor Theoretische Fysica, KU Leuven.Celestijnenlaan 200D, B-3001 Leuven, Belgium b Instituto de Física Teórica UAM/CSIC.C/Nicolás Cabrera, 13-15, C.U. Cantoblanco, 28049 Madrid, Spain
E-mail: [email protected] , [email protected] Abstract:
We study the scalar, electromagnetic and gravitational perturbations of planarAdS black holes with NUT charge. In the context of the AdS/CFT correspondence, thesesolutions describe a thermal quantum field theory embedded in a Gödel-type universe withclosed time-like curves. For a given temperature and NUT charge, two different planar Taub-NUT solutions exist, but we show that only the one with a positive specific heat contributesto the Euclidean saddle point in the path integral. By using the Newman-Penrose formalism,we then derive the master equations satisfied by scalar, electromagnetic and gravitationalperturbations in this background, and show that the corresponding equations are separable.Interestingly, the solutions pile up in the form of Landau levels, and hence are characterizedby a single quantum number q . We determine the appropriate boundary conditions satisfiedby the master variables and using these we compute the quasinormal modes of scalar andgravitational perturbations. On the other hand, electromagnetic perturbations depend on afree parameter whose determination is problematic. We find that all the scalar and gravi-tational QNM frequencies lie in the lower half of the complex plane, indicating that theseTaub-NUT spacetimes are stable. We discuss the implications of these results in the light ofthe AdS/CFT correspondence. a r X i v : . [ h e p - t h ] F e b ontents Motivated by the AdS/CFT correspondence [1–3], the study of asymptotically anti-de Sitter(AdS) black holes has been a major field of research in the last two decades. According tothis correspondence, black hole solutions in the bulk of AdS are dual to a thermal quantumfield theory living in the boundary of the spacetime and whose temperature is given by theHawking’s temperature of the black hole. In this context, the holographic dictionary can beapplied to gain a great deal of information about the hydrodynamics of strongly-coupled plas-mas by studying the properties of the black hole solutions [4–7]. In particular, perturbationsof different fields in the background of a black hole geometry can be used to compute transport– 1 –oefficients and correlators in the dual theory, and thus providing us with valuable results thatcan be difficult to obtain by first principles in the quantum theory.In a black hole, the late-time behaviour of perturbations is ruled by the quasinormal modes(QNMs), which are solutions satisfying an outgoing boundary condition at the horizon ( i.e. ,absence of waves coming from the horizon) plus — in the context of AdS/CFT — Dirichletboundary conditions at infinity— see the reviews [8, 9]. Quasinormal modes only exist for adiscrete set of complex frequencies, called the QNM frequencies, and whose imaginary partdetermines the damping time. The QNMs of black holes defined in this way correspond tothe poles of the retarded Green functions of the dual theory and therefore they characterizethe response of the dual plasma under perturbations [10–17].A large part of the literature on this topic has focused on AdS solutions — see theprevious references — and especially on black holes with a planar horizon, since these aredual to a 4-dimensional CFT in flat space. In this paper, nonetheless, we are interested inAdS geometries. As a matter of fact, the AdS /CFT correspondence is well-motivated [18]and it can indeed be relevant for certain condensed-matter systems that behave effectivelyas dimensional [6, 7]. The quasinormal modes of 4-dimensional Schwarzschild-AdSblack holes were studied in Refs. [19–22], while those of black holes with planar, toroidal andcylindrical topologies were first computed in Refs. [23, 24]. The results on the latter werelater revised and extended in Ref. [25] by implementing the boundary conditions required byholography. On the other hand, the quasinormal modes of large Kerr-AdS black holes wereanalyzed in [26].In addition to these cases, there is a family of gravitational solutions that has not been yetfully exploited in holography: black holes with NUT charge [27–30]. Taub-NUT solutionshave the distinct property of being only locally asymptotically AdS, which translates intothe fact that the boundary is no longer (locally) conformally flat. Thus, NUT charge breaksconformal invariance of the dual theory, and this may allow us to probe non-trivial aspects ofthe CFT. For instance, Euclidean AdS-Taub-NUT solutions describe CFTs placed on squashedspheres [29, 30], and studying how the free energy depends on the NUT charge has led tointeresting results both in supersymmetric [31, 32] and non-supersymmetric [33–36] setups.Lorentzian Taub-NUT solutions, on the other hand, have been less studied in the contextof holography due to their seemingly pathological properties. Indeed, these solutions containMisner strings and closed time-like curves [37, 38], and they give rise to an apparent failure ofthe first law of thermodynamics [39]. However, there is a renewed interest in “rehabilitating”these spacetimes. On the one hand, Ref. [40] has shown that freely falling observers do notexperience any of these pathologies, since there are no closed time-like geodesics and Misnerstrings are invisible to those observers — see also [41]. On the other hand, the thermodynamicdescription of Taub-NUT solutions has been finally understood on the basis that Misner stringsare acceptable and that, accordingly, the NUT charge should be regarded as an independentthermodynamic variable [42–44] — see also [45]. We use the term “Taub-NUT” to refer indistinctly to both NUT-type and bolt-type solutions. – 2 –orentzian AdS-Taub-NUT solutions give indeed rise to interesting boundary theories. InRefs. [46, 47] it was noted that, unlike the Kerr-AdS solution, NUT-charged solutions describefluids with vorticity, and hence explore a qualitatively different aspect of the dual theory.More recently, Ref. [48] initiated the study of scalar perturbations of spherical Taub-NUTs inconnection to holography, finding that the result is dramatically dependent on whether theMisner string is regarded as physical or not. In this work, we will consider instead the case ofplanar Taub-NUT black holes [49] — we recall that, just like in the case of AdS black holes,NUT-charged solutions can have either spherical, planar or hyperbolic transverse sections. Weconsider this case to be particularly interesting for two main reasons. First, the planar NUTsolutions are free of Misner strings, so that one gets rid of all the difficulties and ambiguitiesintroduced by these objects. Second, these solutions are a generalization of the planar blackholes, and hence the boundary metric can be considered as a continuous deformation of flatspace. More precisely, the boundary of these geometries is similar to a Gödel universe [49],where the NUT charge controls the rotation. In this sense, it is interesting to see how theproperties of the dual strongly-coupled plasma change as we increase the NUT charge.In this paper, we explore this question by computing the quasinormal mode spectrum ofplanar Taub-NUT black holes. We shall perform an analysis of (massless) scalar, electromag-netic and gravitational perturbations, providing — to the best of our knowledge — the firstcomplete calculation of quasinormal modes of black holes with NUT charge.The paper is organized as follows• In Section 2 we review the planar Taub-NUT geometries, establishing their basic proper-ties, their thermodynamics description and introducing the Newmann-Penrose formalismthat we use in the next section.• In Section 3 we perform perturbation theory on these geometries. The case of a scalarfield is considered first and we note an interesting analogy between the angular separationof the QNMs and Landau quantization. We then use the Newmann-Penrose formalismto derive separable equations for the master electromagnetic and gravitational variables.• In Section 4 we study the boundary conditions for QNMs. Imposing Dirichlet boundaryconditions on the electromagnetic and gravitational perturbations, we derive the formof the boundary conditions on the master Newmann-Penrose variables. We find that,besides the QNM frequency, the QNMs depend on another parameter related to thepolarization, and which has to be determined by solving simultaneously the equations forboth NP variables. We show, however, that this method only works in the gravitationalcase, since the electromagnetic NP variables satisfy degenerate equations.• We compute the QNM frequencies of scalar and gravitational perturbations in Section 5.Despite the breaking of parity, the spectra of both types of perturbations is symmetricunder the change of sign of the NUT charge. We obtain an analytic approximationfor a special family of gravitational QNMs, that we call pseudo-hydrodynamic modes,– 3 –hose frequency vanishes in the zero NUT charge limit. In addition, we provide strongevidence that no unstable mode exists.• We present our conclusions in Section 6.
We consider Einstein gravity with a negative cosmological constant, S = 116 πG (cid:90) d x (cid:112) | g | (cid:20) R + 6 L (cid:21) (2.1)In this paper, we are interested in the following solution of Einstein’s theory, correspondingto a Taub-NUT black hole with planar topology [30], ds = − V ( r ) (cid:18) dt + 2 nL xdy (cid:19) + dr V ( r ) + r + n L (cid:0) dx + dy (cid:1) (2.2)where n is the NUT charge, the function V ( r ) is given by V ( r ) = ( r − r + ) (cid:0) n + 6 n rr + + rr + (cid:0) r + rr + + r (cid:1)(cid:1) L r + ( n + r ) , (2.3)and the coordinates ( x, y ) span R . For n = 0 this solution reduces to the AdS black brane,but nevertheless it has some remarkable properties that we review next. First of all, thissolution conserves all the symmetries of the black brane, corresponding to time translationsand the symmetries of R , with the difference that the latter now act non-trivially in the timevariable. The corresponding four Killing vectors read ξ ( t ) = ∂ t ,ξ (1) = − nL y∂ t + ∂ x ,ξ (2) = ∂ y ,ξ (3) = nL ( x − y ) ∂ t + y∂ x − x∂ y . (2.4)Note that these symmetries allow one to consider quotients of this solution by discrete groups.For instance one may take y to be periodic, in which case the black hole would have cylindricaltopology. We will restrict to the case of ( x, y ) spanning the plane.The event horizon of the black hole is located at r = r + > , which is a Killing horizonfor ξ ( t ) . The corresponding surface gravity reads κ = 12 V (cid:48) ( r + ) = 3( n + r )2 L r + . (2.5)– 4 –ne can see that the function V ( r ) is strictly positive for r + < r < ∞ and hence there areno other horizons for ∂ t . There are, however, horizons for the other Killing vectors, whichindicate the presence of closed timelike curves (CTCs). For instance, the norm of ξ (2) reads ξ = r + n L − V ( r ) (cid:18) nxL (cid:19) , (2.6)and hence it becomes timelike if x is large. The symmetries of this spacetime imply that thereare CTCs around any point (in the region r > r + ), but, however, there are no closed timelikegeodesics [40, 50], so that the solution is possibly less pathological than one would expect.On the other hand, unlike the spherical Taub-NUT solutions, these NUT black branes do notpossess Misner singularities.At infinity, the metric function V ( r ) behaves as V ( r ) = r /L + O (1) , and hence theboundary metric at r → ∞ is conformally equivalent to d ˆ s = − (cid:18) dt + 2 nL xdy (cid:19) + dx + dy . (2.7)This metric is not conformally flat, and therefore the solution is only asymptotically locallyAdS. In the boundary theory, this means that conformal invariance is broken. However,the boundary still has many symmetries — given by (2.4) — and one can see that it is ahomogeneous space corresponding to a Lorentzian continuation of Nil space — the groupmanifold of Heisenberg’s group. Indeed, note that the translational Killing vectors satisfy theHeisenberg’s algebra (cid:2) ξ ( t ) , ξ (1) (cid:3) = (cid:2) ξ ( t ) , ξ (2) (cid:3) = 0 , (cid:2) ξ (1) , ξ (2) (cid:3) = 2 nL ξ ( t ) . (2.8)On a more physical perspective, the metric (2.7) can be interpreted as a rotating universe,very similar to the non-trivial (2 + 1) -dimensional section of the famous Gödel solution [51],the paradigmatic example of a universe with closed timelike curves. Hence, when one appliesthe holographic dictionary to these solutions, one is probing the dynamics of a quantum theoryplaced in this exotic spacetime. Although the existence of a globally defined timelike Killingvector allows one to define a Hamiltonian, performing quantum field theory in this backgroundis challenging due to its unusual causal structure [50, 53–55]. In this sense, holography canbe used to gain some insight about the behaviour of a quantum theory in such spacetime.Besides, the dual CFT would be in a thermal state whose properties are determined by thethermodynamic quantities of the black hole, that we review next.
The temperature of the NUT-charged black branes is given by Hawking’s result T = κ/ (2 π ) ,so that T = 3( n + r )4 πL r + . (2.9) More precisely, the metric (2.7) is the equatorial section of the Som-Raychaudhuri solution [52], as originallynoted in [49]. Both metrics have qualitatively similar properties. – 5 –ne can see that, for a given n , the temperature reaches its minimum value for r + = | n | , inwhose case we have T = T ∗ , where T ∗ = 3 | n | πL . (2.10)On the other hand, the temperature diverges both for r + → and r + → ∞ . Hence, when T > T ∗ , there are two different black hole solutions with the same T and n . This allows us todistinguish three different families of solutions, corresponding to n < − r + , − r + < n < r + or n > r + . We can also identify the mass of the solution by analyzing the behaviour near infinity.In fact, one can just apply the usual the ADM result which tells us that the total energy E can be identified by looking at the /r term in the asymptotic expansion of V . In particular,the coefficient of that term should be equal to − πGL E/V , where V is the volume of thetransverse space, V = (cid:82) dxdy . Note that in this case V is infinite, and hence it is moreappropriate to talk about energy density ρ = E/V , rather than total energy. This quantity,in fact, can be interpreted as the energy density in the boundary CFT. The expansion of V ( r ) reads, 3 V ( r ) = r L + 5 n L − r + 6 n r − n L rr + + O (cid:18) r (cid:19) (2.11)and therefore, we get ρ = r + 6 n r − n πGL r + . (2.12)On the other hand, the entropy of the black hole is given by S = A/ (4 G ) , but since thisarea of the horizon is divergent, it is again convenient to work in terms if the entropy density, s = S/V , which reads s = r + n GL (2.13)Now, an apparent puzzle in the case of these solutions is that the first law of thermodynamicsdoes not seem to hold, i.e. , we get dρ (cid:54) = T ds when varying the previous expressions withrespect to r + . However, the reason is that the NUT charge should also be interpreted as athermodynamical variable which will modify the first law. For a long time this was a sourceof confusion because in the case of spherical Taub-NUT black holes, since regularity of theEuclidean geometry imposes a restriction between NUT charge and temperature [39]. Onlyrecently it was realized that one can achieve a full-cohomegeneity first law for spherical NUTsby allowing the NUT charge to vary independently. In the case of planar NUT black holes,however, there is no restriction between n and T , and it is natural to treat the NUT charge asan additional thermodynamic variable. To the best of our knowledge, the existence of a firstlaw in the case of planar Taub-NUT solutions was reported in [56].In order to complete the thermodynamic characterization of these planar NUT blackholes, we must compute the free energy from the Euclidean on-shell action. The Euclideansolution is obtained, not only by Wick-rotating the time coordinate, t = iτ , but also the NUTcharge, ˆ n = in . In that case the metric reads– 6 – s E = V ( r ) (cid:18) dτ + 2ˆ nL xdy (cid:19) + dr V ( r ) + r − ˆ n L (cid:0) dx + dy (cid:1) . (2.14)It is important to note that, in Euclidean signature, only the solutions with r ≥ ˆ n areregular, which means that the Lorentzian solutions with n > r do not have an Euclideandescription. This suggests that for a given T > T ∗ only the solution with r ≥ n should betaken into account in the path integral, and hence that it is the dominant saddle. Let us alsomention that, in the literature, the Euclidean solutions with r = ˆ n are called Taub-NUT,while the rest are Taub-bolt. However, we shall make no distinctions since the former can beconsidered as a limit of the latter.The free energy can be computed from the following well-posed and regularized Euclideanaction I E = − πG (cid:90) d x (cid:112) | g | (cid:20) R + 6 L (cid:21) − πG (cid:90) d x √ h (cid:20) K − L − L R (cid:21) , (2.15)where K is the extrinsic curvature of the boundary and R is the Ricci scalar of the boundary’sintrinsic. The free energy F = T I E reads F = − V ( r + 3ˆ n )16 πGL r + . (2.16)Let us then define the free-energy density ε = F/V and express this result in terms of theLorentzian NUT charge n , ε = − ( r + 3 n )16 πGL r + . (2.17)Now, it turns out that, instead of n , the thermodynamic relations are most naturally writtenin terms of the variable θ = n . Then, using the chain rule one can compute the derivatives ofthe free energy at constant θ and T , which read s = − (cid:18) ∂ε∂T (cid:19) θ = r + n GL , (2.18) ψ = − (cid:18) ∂ε∂θ (cid:19) T = 3 n ( r − n )8 πGL r + . (2.19)We check that s indeed coincides with the Bekenstein-Hawking entropy density. On the otherhand, ψ is a new thermodynamic potential conjugate to θ . Making use of these results, oneobserves that the energy ρ computed according to the ADM prescription, coincides with thedouble Legendre transform of the free energy with respect to T and θ . ρ = ε + T s + θψ . (2.20)– 7 –his is is not the standard definition of internal energy, which suggests that, in the presence ofNUT charge, the potentials ε and ρ probably have a different thermodynamic interpretation.In any case, this result implies that ρ satisfies the following first law, dρ = T ds + θdψ . (2.21)Finally, we can study the thermodynamic stability of these solutions. One can first cancompute the specific heat at constant θ , C θ = T (cid:18) ∂s∂T (cid:19) θ = r ( n + r )2 GL ( r − n ) , (2.22)and one can see that C θ > as long as r > n , implying thus stability when n is held fixed.More generally, one can study the concavity of the free energy, for which one may computethe second variation of ε , which reads δ ε = − πr G (cid:0) r − n (cid:1) δT − n (cid:0) n + r (cid:1) GL (cid:0) r − n (cid:1) δT δθ + 3 n (cid:0) r − n r + 3 n (cid:1) πGL r + (cid:0) r − n (cid:1) δθ . (2.23)The solution will be thermally stable if this is a negative-definite quadratic from, but we cansee that this never happens because the term with δθ is positive for r > n , while the oneof δT is only negative in that region. Therefore, these planar Taub-NUT black holes are onlythermodynamically stable under changes of the temperature but not under changes of n . The description of perturbations on a black hole spacetime is a task of extraordinary com-plexity. The linearized equations governing first order field components on local coordinatesare considerably involved already in the simplest backgrounds such as Schwarzschild’s blackhole, and almost intractable in more realistic cases like Kerr’s spacetime. In addition, it is farfrom obvious how the large amount of gauge symmetry should be fixed. Teukolsky’s seminalwork [57] constituted a major breakthrough in the clarification of these issues. Considering analgebraically special background space, of Petrov type D (e.g. Schwarzschild and Kerr space-times), he derived decoupled equations for perturbations of several kinds and, furthermore,these admit solutions in separable form. One of the elements underlying such a remarkablesuccess is the Newmann–Penrose (NP) formalism [58]. In particular, it provides a very nat-ural formulation of Petrov’s classification, as well as the Goldberg–Sachs theorem, and thistranslates into the vanishing of several NP variables of the background. It is in this situationthat the equations decouple and, in addition, become gauge invariant.The study of perturbations on the background (2.2) can be conveniently performed inthe NP formalism. A Newmann–Penrose frame on a pseudo–Riemannian space is a complextetrad e a , e = m , e = m , e = l , e = k , (2.24) We will be following the conventions in [59]. – 8 –omposed of two real, null vectors k and l , and one complex, null vector m together with itsconjugate m , so that k · k = l · l = m · m = 0 , (2.25)and these are further subject to the normalization conditions k · l = − m · m = − , k · m = l · m = 0 . (2.26)When acting as operators on functions ϕ , it is customary to give particular names to thevectors of the NP basis, Dϕ := k µ ∂ µ ϕ, ∆ ϕ := l µ ∂ µ ϕ, δϕ := m µ ∂ µ ϕ, δ ∗ ϕ := m µ ∂ µ ϕ. (2.27)A convenient choice for the space (2.2) is k = k µ ∂ µ = 1 V ( ∂ t + V ∂ r ) , l = l µ ∂ µ = 12 ( ∂ t − V ∂ r ) , (2.28) m = m µ ∂ µ = iL e − i arctan ( r/n ) (cid:112) n + r ) (cid:18) ∂ x + i∂ y − i nxL ∂ t (cid:19) . (2.29)The vectors k and l are geodesic and shear-free so that the following spin coefficients vanish κ = σ = ν = λ = 0 . (2.30)By the Goldberg–Sachs theorem it follows that the space must be of Petrov type D , so fourout of the five Weyl scalars vanish Ψ = Ψ = Ψ = Ψ = 0 . (2.31)In addition, the frame has been chosen to be parallelly propagated along k , i.e. ∇ k k = ∇ k l = ∇ k m = 0 , (2.32)a property that implies the vanishing of additional spin coefficients.As shown by Teukolsky [57], the vanishing of these quantities makes perturbation theorytractable on such a background — we will make use of those results in next section. For that,we need the spin connection, whose non-vanishing components read ρ = − Γ = − r + in , µ = Γ = − V r + in ) , γ = 12 (Γ − Γ ) = 14 (cid:18) V (cid:48) + 2 nin + r V (cid:19) . (2.33)On the other hand, the Ricci tensor has the same components as the metric R = − R = − /L , and the only non-vanishing Weyl scalar, Ψ , reads Ψ = − C µνρσ k µ m ν l ρ m σ = − C = − − (cid:15)i L (cid:18) i(cid:15) ( r/r + ) + i(cid:15) (cid:19) , (2.34)where (cid:15) = n/r + . This completes the enumeration of non-vanishing NP variables of the space(2.14), in the frame (2.28). – 9 – Perturbation theory
In this section we study scalar, electromagnetic and gravitational perturbations around theplanar NUT black holes introduced in the previous section. By using the Newmann-Penroseformalism, we will show that in all cases the perturbations can be analyzed through a fewmaster variables that satisfy decoupled equations. Once the problem is reduced to a decou-pled equation for a scalar variable Ψ , one can try to separate variables. Now, an importantdifference with respect to the NUT-neutral case is that the translational Killing vectors ξ (1) , ξ (2) do not commute, [ ξ (1) , ξ (2) ] (cid:54) = 0 , and as usual these do not commute with the rotationalvector ξ (3) . Hence, one cannot fully separate the equations a priori, and at best one can choosethe variable Ψ to be an eigenfunction for one of the sets of commuting Killing vectors (cid:8) ξ ( t ) , ξ (1) (cid:9) , (cid:8) ξ ( t ) , ξ (2) (cid:9) , (cid:8) ξ ( t ) , ξ (3) (cid:9) . (3.1)In the coordinates in which (2.2) is expressed, the vector ξ (2) = ∂ y is a coordinate vectorand hence it is appropriate to choose the set (cid:8) ξ ( t ) , ξ (2) (cid:9) . Due to the symmetries of themetric, this is completely equivalent to choosing the set (cid:8) ξ ( t ) , ξ (1) (cid:9) .To see this, notice thatthe transformation t (cid:48) = t + nL xy , x (cid:48) = − y , y (cid:48) = x leaves the metric invariant while setting ξ (1) = ∂ (cid:48) y . On the other hand, the analysis of quasinormal modes using the set (cid:8) ξ ( t ) , ξ (3) (cid:9) ismore obscure, but one expects again that the results would be equivalent. From now on weassume that our perturbations are eigenfunctions of ξ ( t ) and ξ (2) , and hence we have ξ ( t ) Ψ = − iω Ψ ξ (2) Ψ = ik Ψ (cid:41) ⇒ Ψ = e − i ( ωt − ky ) h ( r, x ) . (3.2)In addition, the dependence on the x and r coordinates can be further separated as we showbelow. Let us consider first the case of a massless scalar field φ in the background of (2.2) satisfyingthe wave equation, ∇ µ ∇ µ φ = 0 . (3.3)As we just discussed above, we separate the t and y coordinates according to φ = e − i ( ωt − ky ) ψ ( r, x ) √ r + n , (3.4)where the factor of / √ r + n is conventional. Then, we find that ψ satisfies the followingequation: − V ∂∂r (cid:18)
V ∂ψ∂r (cid:19) + ψ (cid:20) − ω + Vr + n (cid:18) V (cid:48) r + n V ( r + n ) (cid:19)(cid:21) (3.5)– 10 – V L ( n + r ) (cid:34) − ∂ ψ∂x + (cid:18) k + 2 nxωL (cid:19) ψ (cid:35) = 0 . (3.6)We then note that this equation admits for separable solutions ψ ( r, x ) = Y ( r ) H ( x ) , (3.7)where Y ( r ) and H ( x ) satisfy respectively the following equations − V ddr (cid:18)
V dYdr (cid:19) + ψ (cid:20) − ω + Vr + n (cid:18) L E + V (cid:48) r + n V ( r + n ) (cid:19)(cid:21) =0 , (3.8) − d H dx + 12 (cid:18) k + 2 nxωL (cid:19) H = EH , (3.9)where E is a constant. In the case of vanishing NUT charge, this constant can take anyvalue, as it is related to the wavenumber in the x direction, which can be chosen freely. Thissituation changes dramatically in the presence of NUT charge. Indeed, we observe a quiteremarkable fact: the equation (3.9) is identical to that of a quantum harmonic oscillator,where the point of equilibrium is located at x = − kL / (2 nω ) , and where the correspondingmass and frequency are m = 1 , ω os = (2 nω/L ) . There is an even more accurate analogywith Landau quantization that we explore below. Now we search for regular solutions suchthat H ( x ) → at x → ±∞ , and this leads to the familiar results for the eigenfunctions andeigenvalues of the harmonic oscillator. There is a catch, though, since we have to take intoaccount that ω is complex and that n can have either sign. Thus, we must distinguish betweenthe cases Re ( nω ) > and Re ( nω ) > . Introducing s = sign [ Re ( nω )] , (3.10)we have that the physically relevant solution of (3.9) reads H q ( x ) = e − s nωL (cid:16) x + kL nω (cid:17) H q (cid:18)(cid:114) snωL (cid:18) x + kL nω (cid:19)(cid:19) , q = 0 , , . . . , (3.11)where H q ( z ) are the Hermite’s polynomials. The eigenvalues E q read in turn E q = snωL (1 + 2 q ) . (3.12)Thus, unlike the NUT-neutral case, we obtain a quantization condition on the angular part ofthe perturbations, and hence the spectrum of quasinormal modes will be discrete. Also notethat the eigenvalues E q are independent from the wavenumber in the y direction, k , and hencethe quasinormal modes will be degenerate.Now we can bring this result to the radial equation (3.14), and it also proves useful toperform the following redefinitions z = r + r , (cid:15) = nr + , ˆ ω = L ωr + , ˆ V = L Vr . (3.13)– 11 –hen, the radial equation reads ˆ V z ddz (cid:18) ˆ V z dYdz (cid:19) + ψ (cid:34) ˆ ω − z ˆ V z (cid:15) (cid:32) s(cid:15) ˆ ω (1 + 2 q ) − ∂ z ˆ V z + (cid:15) z ˆ V (1 + (cid:15) ) (cid:33)(cid:35) =0 (3.14)Notice that the only free parameters in this equation are (cid:15) , the dimensionless frequency ˆ ω andthe index q . Relation to Landau quantization
Interestingly, the perturbations in the Taub-NUT backgrounds organize in an analogous wayto the Landau levels of a charged particle moving in a uniform magnetic field. In order toestablish this analogy, let us first note that we can write the metric (2.2) in a gauge-invariantform as ds = − V ( r ) ( dt + A ) + dr V ( r ) + r + n L (cid:0) dx + dy (cid:1) , (3.15)where A is a 1-form satisfying dA = nL dx ∧ dy . Thus, coordinate transformations of the form t → t + f ( x, y ) can be reabsorbed as gauge transformations A → A − df . Then, one can infact interpret this A as a uniform magnetic field with magnitude B = 2 n/L . Let us thenconsider a particle of charge e moving in the ( x, y ) plane (in flat space) in the background ofthis field. The Hamiltonian is given by H = 12 π x + 12 π y , (3.16)where, in the gauge A = Bxdy , the momenta read π x = − i∂ x , π y = − i∂ y − exB . (3.17)Then, the Schördinger equation H ψ = Eψ yields (cid:20) − ∂ x + 12 ( − i∂ y − exB ) (cid:21) ψ = Eψ , (3.18)and by using − i∂ y ψ = kψ we get the same equation (3.9) we got for the angular part of theperturbations in the Taub-NUT geometry, provided one identifies the charge of the particlewith the frequency of the perturbation as e = − ω . Thus, the transverse ( x, y ) part of thequasinormal modes of the Taub-NUT background are eigenfunctions of this Hamiltonian andhave the same quantization, which is given by the Landau levels q = 0 , , . . . . As we shownext, electromagnetic and gravitational perturbations organize in a similiar fashion. Clearly,this analogy can be traced back to the fact that NUT charge is the gravitational equivalentof magnetic charge. – 12 – .2 Electromagnetic and Gravitational perturbations Let us now address the study of perturbations electromagnetic and gravitational perturbations.Thus, we consider a vector field A µ satisfying Maxwell equations in the background of (2.2) ∇ µ F µν = 0 , F µν = 2 ∂ [ µ A ν ] , (3.19)and a metric perturbation ˜ g µν = g µν + h µν satisfying the linearized Einstein’s equations G Lµν [ h αβ ] + 3 L h µν = 0 . (3.20)While the symmetries of the (2.2) may still allow one to perform a complete decomposition of A µ and h µν — see [60–62] for the case of SU(2) symmetry and [63] for electromagnetic pertur-bations in the Kerr-NUT-(A)dS spacetime — we find that the Newmann-Penrose formalismoffers a possibly clearer way to compute perturbations.In the NP frame, the field strength of the Maxwell field is described by three independent(complex) components that are customarily denoted as, φ = F µν k µ m ν = F , φ = 12 F µν ( k µ l ν + m µ m ν ) = 12 ( F + F ) , φ = F µν m µ l ν = F . (3.21)Since the background considered here is neutral, the φ i correspond to linear electromagneticperturbations. In addition, since the metric (2.2) is of Petrov type D and our NP frame (2.28)has k and l aligned with the repeated principal null directions, we can apply directly theresults from Teukolsky [57]. These imply that φ and φ satisfy two decoupled equations thatread [( D − ρ − ρ ∗ ) (∆ + µ − γ ) − δδ ∗ ] φ =0 , [(∆ + γ − γ ∗ + 2 µ + µ ∗ ) ( D − ρ ) − δ ∗ δ ] φ =0 (3.22)On the other hand, gravitational perturbations are described by small changes in the NPframe, e a → e a + e (1) a + ... , which, in turn, induce changes in the NP variables introducedabove, e.g. Ψ a → Ψ a + Ψ (1) a + ... for the Weyl scalars Ψ a . Since Ψ , Ψ , Ψ and Ψ all havea vanishing background value, it follows that they are already linearized. In particular thescalars Ψ and Ψ , which are defined as Ψ = C µνρσ k µ m ν k ρ m σ , (3.23) Ψ = C µνρσ l µ m ν l ρ m σ , (3.24)satisfy two decoupled equations, [( D − ρ − ρ ∗ )(∆ − γ + µ ) − δδ ∗ − ] Ψ =0 , [(∆ + 3 γ − γ ∗ + 4 µ + µ ∗ ) ( D − ρ ) − δ ∗ δ − ] Ψ =0 (3.25)– 13 –e now search for separable solutions of the variables ( φ , φ ) and (Ψ , Ψ ) . For any ofthese — call it ψ — we can separate the dependence on t and y as in (3.2), so that we write ψ = e − i ( ωt − ky ) H ( x ) R ( r ) . On the other hand, the dependence on the coordinate x in (3.22)and (3.25) only appears in the operator δδ ∗ , which reads δδ ∗ = L n + r ) (cid:34) ∂ x + (cid:18) ∂ y − nxL ∂ t (cid:19) + i nL ∂ t (cid:35) . (3.26)Thus, demanding that δδ ∗ ψ = λ ( r ) ψ leads to the same equation for H as in the scalar case,given by (3.9). Likewise, by imposing regularity of H at infinity we obtain the Hermitefunctions (3.11) and therefore we get δδ ∗ ψ = − L n + r (cid:16) E q − nωL (cid:17) ψ , δ ∗ δψ = − L n + r (cid:16) E q + nωL (cid:17) ψ , (3.27)where the eigenvalues E q are those in (3.12). Finally, it is possible to write (3.22) and (3.25)in a symmetric form [64] by introducing the radial functions Y ± and Y ± as, φ = e − i arctan ( r/n ) V √ n + r e − iωt + iky H q ( x ) Y +1 ( r ) , φ = 1 √ n + r e − iωt + iky H q ( x ) Y − ( r ) , (3.28)and Ψ = e − i arctan ( r/n ) V √ n + r e − iωt + iky H q ( x ) Y +2 ( r ) , Ψ = 1 √ n + r e − iωt + iky H q ( x ) Y − ( r ) . (3.29)Then Eqs. (3.22) and (3.25) yield the following master equations for the radial variables Λ Y ± S ( r ) + SP ( r )Λ ± Y ± S ( r ) − (cid:18) L E q r + n + Q S ( r ) (cid:19) V ( r ) Y ± S ( r ) = 0 , (3.30)where S = 1 for electromagnetic perturbations and S = 2 for gravitational ones. Here wehave introduced the differential operators Λ ± = ddr ∗ ± iω, Λ = d dr ∗ + ω , where ddr ∗ = V ddr , (3.31)and P ( r ) and Q S ( r ) are functions given by P = − V (cid:48) + 2( r − in ) Vn + r , (3.32)and Q S = n V ( n + r ) ( S = 1) (cid:0) n + 8 inr − r (cid:1) V ( n + r ) − (4 in − r ) V (cid:48) n + r − V (cid:48)(cid:48) S = 2) (3.33)– 14 –hese equations can be written in a dimensionless way by introducing z , (cid:15) and ˆ ω as in (3.13),which implies that the dimensionless QNM frequencies ˆ ω will only depend on (cid:15) and the level q . In order to obtain these frequencies, the radial equations (3.30) must be supplemented withsuitable boundary conditions, which we determine in the following section. Once we have determined the master equations governing the perturbations of scalar, electro-magnetic and gravitational fields, we are interested in studying the corresponding quasinormalmodes, which are determined by a specific choice of boundary conditions. At the horizon ofthe black holes, these modes satisfy the condition of behaving as outgoing waves, while theconditions at the boundary of AdS can be chosen in different ways. For instance, one mightimpose the master variables to vanish at infinity [23, 24]. However, we are interested in mak-ing contact with the AdS/CFT correspondence, and in that case the boundary conditions areuniquely determined [65–67]. According to AdS/CFT, the different operators of the boundarytheory couple to the normalizable mode of the perturbations in the bulk, and therefore wemust make sure that only those modes are excited if we want to interpret the correspondingquasinormal modes as perturbations of a plasma in the dual theory.In order to study the boundary conditions it is useful to introduce first the coordinate z = r + r , (4.1)so that the metric can be written as ds = 1 z (cid:34) − r f ( z ) L (cid:18) dt + 2 nL xdy (cid:19) + dz L f ( z ) + r + z n L (cid:0) dx + dy (cid:1)(cid:35) (4.2) f ( z ) = L r z V ( r + /z ) . (4.3)In this way, infinity corresponds to z = 0 , while the horizon is placed at z = 1 . On the otherhand, the tortoise coordinate r ∗ is defined by r ∗ = − (cid:90) dzL r + f ( z ) , (4.4)and we note that near the horizon z = 1 it reads r ∗ ≈ πT log(1 − z ) , (4.5)where T is the Hawking temperature (2.9). – 15 – .1 Scalar field In the near-horizon region z = 1 , the solution to the radial scalar equation (3.14) can beexpanded in a Frobenius series Y ( z ) = (1 − z ) α (cid:2) c + + c (1 − z ) + c (1 − z ) + . . . (cid:3) . (4.6)The indicial equation has the following two solutions for α , α ± = ± i ˆ ω (cid:15) ) , (4.7)and taking into account (4.5) and that ˆ ω = L ω/r + , we get that the solution behaves as Y ∼ e πT α ± r ∗ = e ± iωr ∗ . Since the solution must behave as an outgoing wave at the horizon,we must choose the root α − .On the other hand, near the AdS boundary z = 0 we find that there are two independentmodes: Y ( z ) = az + bz − (1 + O ( z )) when z → . (4.8)We keep the normalizable mode, which is the one that couples to a scalar field in the dualtheory, and hence we have to impose that Y (0) = 0 . The conditions at infinity and at thehorizon can only be satisfied simultaneously by a discrete set of complex frequencies ω : thequasinormal mode frequencies. In the case of the electromagnetic field, the analysis of the boundary conditions in the near-horizon are analogous to the scalar case. Again one finds that the NP variables φ , φ can beexpanded in a Frobenius series near z = 1 , and imposing the condition of outgoing waves onefinds the following solutions for the radial functions Y ± : Y ± ∼ (1 − z ) α ± when z → , (4.9)where α +1 = − i ˆ ω (cid:15) ) , α − = 1 − i ˆ ω (cid:15) ) . (4.10)The analysis of boundary conditions at infinity, on the other hand, is much involved than inthe case of a scalar field. By analyzing the solutions of the radial equations (3.30) for Y ± , wesee that the two independent solutions behave near z = 0 as Y ± ( z ) = a ± + b ± z when z → , (4.11)where a ± and b ± are constants. Now, the boundary conditions are not imposed directly onthe NP variables but on the perturbation of the Maxwell field A µ , so we must study how theserelate. Let us for into account that we can always choose a gauge in which the z -componentthe vector vanishes A z = 0 . Then, the solutions to Maxwell equations near z = 0 behave– 16 –s A a ∼ A (1) a + zA (2) a , where a denotes the boundary indices a = t, x, y . Therefore, Dirichletboundary conditions imply that A (1) a = 0 , and we only keep the mode that decays at infinity.Separating variables, this means that we can write the vector asymptotically as A a = ze − i ( ωt − ky ) γ a ( x ) + O ( z ) , (4.12)where γ a are certain functions and one can check that the following term in the z -expansionis indeed O ( z ) . Now, the functions γ a are not arbitrary, but we find that Maxwell equationsimpose the following constraint, (cid:18) ω − nxL (cid:18) k + 2 nxωL (cid:19)(cid:19) γ t − iγ (cid:48) x + (cid:18) k + 2 nxωL (cid:19) γ y = 0 . (4.13)On the other hand, we are searching for solutions such that the NP variables φ and φ areseparated, and this will impose, too, conditions on the γ a . Computing φ and φ from thevector perturbation (4.12) we find that ˆ φ = e − i ( ωt − ky ) (cid:2) A +1 + B +1 z + O ( z ) (cid:3) , (4.14) ˆ φ = e − i ( ωt − ky ) (cid:2) A − + B − z + O ( z ) (cid:3) , (4.15)where ˆ φ , are defined as ˆ φ = V (cid:112) n + r e i arctan ( r/n ) φ , ˆ φ = (cid:112) n + r φ , (4.16)and the coefficients A ± , B ± read A ± = 2 ± / L (cid:18) − inxγ t L ± γ x + iγ y (cid:19) , (4.17) B ± = 2 ± / Lr + (cid:20) L γ (cid:48) t + (cid:18) ∓ k − n xL (cid:19) L γ t + i (cid:0) ∓ n + L ω (cid:1) γ x + (cid:0) n ∓ L ω (cid:1) γ y (cid:21) . (4.18)Now, on the other hand, if both φ and φ can be separated, then the result should read ˆ φ = e − i ( ωt − ky ) H q +1 ( x ) (cid:2) a +1 + b +1 z + O ( z ) (cid:3) , (4.19) ˆ φ = e − i ( ωt − ky ) H q − ( x ) (cid:2) a − + b − z + O ( z ) (cid:3) , (4.20)where we have taken into account (4.11) and where H q ± ( x ) are the eigenfunctions in (3.11),with two possibly different levels q +1 and q − for each of the variables. Thus, we obtain asystem of four equations for the variables γ a and the four constants a ± , b ± , A ± = a ± H q ± , B ± = b ± H q ± . (4.21)– 17 –ogether with (4.13), we have to solve a system of five equations which is not guaranteed tohave solutions. In order to simplify the computations, at this point it is interesting to notethat we can set k = 0 without loss of generality. In fact, the change of variables ˆ x = x − σ , ˆ t = t + 2 nL σy (4.22)leaves invariant the background metric and therefore is a symmetry of the linearized equations.On the other hand it transforms the perturbation A a as follows ˆ A a = ze − i ( ω ˆ t − ˆ ky )ˆ γ a (ˆ x ) , where ˆ k = k + 2 nωσL , (4.23)and ˆ γ ˆ t = γ t , ˆ γ ˆ x = γ x , ˆ γ y = γ y − nσL γ t . (4.24)Therefore, by choosing σ = − kL / (2 nω ) we get ˆ k = 0 . Equivalently, we can always work withthe solution with k = 0 and generate another solution with k (cid:54) = 0 by applying the isometrictransformation (4.22). Thus, from now on we set k = 0 .One can see that from the five equations in (4.13) and (4.21) it is possible to obtainexplicitly the values of γ t , γ (cid:48) t , γ x , γ (cid:48) x and γ y , but of course, in order for this to be an actualsolution, γ (cid:48) t and γ (cid:48) x should in fact be the derivatives of γ t and γ x . As it turns out, this onlyhappens when the following constraints meet. First, the two levels q +1 and q − must be relatedaccording to q +1 = q − + 2 s , (4.25)where we recall that s = sign [ Re ( nω )] . Thus we have q − = 0 , , , ... for s = 1 and q − =2 , , , ... for s = − . On the other hand, the ratios of the constants a ± , b ± , λ ± = b ± a ± (4.26)must be related according to λ − = λ +1 (2 q(cid:15) − ˆ ω + (cid:15) ) − i (cid:0) q + 3)ˆ ω(cid:15) − ˆ ω + 3 (cid:15) (cid:1) − iλ +1 + (2 q + 5) (cid:15) − ˆ ω . (4.27)where q is q = (cid:40) q − if s = 1 , − − q − if s = − (4.28)Note that this is all we need in order to characterize the boundary conditions, since theoverall normalization of Y ± is not relevant when searching for quasinormal modes. Now,consistency of the system of equations requires an additional constraint that involves suchoverall normalization, – 18 – (cid:15) a − a +1 = (cid:40) iλ +1 − (2 q − + 5) (cid:15) + ˆ ω if s = 1 , − ( iλ +1 +(2 q − − (cid:15) +ˆ ω )4( q − − q − if s = − . (4.29)In that case, the explicit solution for the γ a reads γ t = − ia +1 L ( − iλ +1 − ˆ ω + (cid:15) ) √ r + x ˆ ω(cid:15) (cid:2) q − ) H q − + H q − (cid:3) , (4.30) γ x = √ a +1 L(cid:15) (cid:2) ( − iλ +1 + (cid:15) (2 q − + 5) − ˆ ω ) H q − + (cid:15) H q − (cid:3) , (4.31) γ y = − i √ a +1 L ˆ ω(cid:15) (cid:104) (cid:0) q − + 1) (cid:15) + λ +1 ( i ˆ ω − i ( q − + 1) (cid:15) ) − (4 q − + 7)ˆ ω(cid:15) + ˆ ω (cid:1) H q − + (cid:15) ( (cid:15) − iλ +1 ) H q − (cid:105) , (4.32)for s = 1 , and there is a similar solution for s = − .Then, in order to find the electromagnetic quasinormal modes, the idea would be tosimultaneously solve the radial equations (3.30) for Y +1 and Y − with the levels q ± relatedaccording to (4.25) and with the boundary conditions given by (4.9), (4.11) and (4.27). Notethat, once (cid:15) and q − are specified, the problem only contains two parameters, ˆ ω and λ + , andthe hope is a solution exists only for discrete values of these quantities. Unfortunately, thisis not the case, since the equations for Y +1 and Y − are degenerate. In order to see this, wefirst note the following Maxwell equations in the NP formalism ( D − ρ ) φ = δ ∗ φ , ( D − ρ ) φ = δ ∗ φ . (4.33)Combining these it is possible to derive the following relation between φ and φ , ˆ δ ∗ ˆ δ ∗ φ = R ( D − ρ ) R ( D − ρ ) φ , (4.34)where R = i (cid:112) r + n ) L e − i arctan( r/n ) , ˆ δ ∗ = Rδ ∗ = ∂ x − i∂ y + i nxL ∂ t . (4.35)Then, by using the decomposition (3.28) one first derives the relation between the levels q ± given in (4.25) , and one also obtains a relation between Y +1 and Y − , Y +1 = − Y − (cid:16) (2 q − + 1) (cid:15) (cid:0) z (cid:15) − z (cid:15) + z (cid:0) − (cid:15) + 6 (cid:15) + 1 (cid:1) − (cid:1) + ˆ ω (cid:0) z (cid:15) + 1 (cid:1) (cid:17) q − + 1)( q − + 2)( z − (cid:15) (3 z (cid:15) + z (6 (cid:15) + 1) + z + 1) − i (cid:0) z (cid:15) + 1 (cid:1) Y (cid:48)− q − + 1)( q − + 2) (cid:15) , (4.36) Interestingly, the operators ˆ δ ∗ and ˆ δ act as the ladder operators of the harmonic oscillator, so they raiseand lower the Landau level q . – 19 – − = − Y +1 (cid:16) (2 q − + 5) (cid:15) (cid:0) z (cid:15) − z (cid:15) + z (cid:0) − (cid:15) + 6 (cid:15) + 1 (cid:1) − (cid:1) + ˆ ω (cid:0) z (cid:15) + 1 (cid:1) (cid:17) z − (cid:15) (3 z (cid:15) + z (6 (cid:15) + 1) + z + 1)+ i (cid:0) z (cid:15) + 1 (cid:1) Y (cid:48) +1 (cid:15) , (4.37)where we have used the master equations (3.30). One can see that these relations map thesolutions of Y ± with the boundary conditions (4.9) into each other and they imply that theasymptotic behaviour of these functions is always related according to (4.27) — independenlyof the boundary conditions imposed on the vector A µ . Therefore, both equations are degen-erate and the value of λ +1 (or λ − ) cannot be found in this way. In the case of vanishingNUT charge, one can decouple the electromagnetic perturbations in modes of definite parity,which are achieved only for two specific values of λ +1 ( λ − ). However, NUT charge breaks allreflection symmetries of the background, and therefore we do not have a similar decompositionof the perturbations. Hence, we seem to be unable to determine the polarization parameter λ ± , which would suggest that the spectrum of QNMs depends continuously on this param-eter. Clearly, more research in this direction is needed in order to understand the puzzlingproperties of electromagnetic perturbations in these geometries. By now, we will focus on thegravitational case, for which a similar method does work. Let us finally turn to the case of the boundary conditions for gravitational perturbations. Inthe near-horizon region we find that the outgoing-wave condition leads to the following formof the radial functions Y ± , Y ± ∼ (1 − z ) α ± when z → , (4.38)where α +2 = − i ˆ ω (cid:15) ) , α − = 2 − i ˆ ω (cid:15) ) . (4.39)On the other hand, the discussion on boundary conditions at infinity proceeds analo-gously to the electromagnetic case. First, by analyzing the solutions of the Newmann-Penrosevariables Y ± , one can see that near the boundary they behave as Y ± ( z ) = a ± + b ± z when z → . (4.40)The integration constants a ± and b ± will be then ultimately related to the boundaryconditions imposed on the metric perturbation. Let us consider a metric perturbation g µν → g µν + h µν in the geometry of these NUT black branes. Due to gauge freedom, wecan always choose a gauge in which h µz = 0 , so that the non-vanishing components are thosetransverse to the z direction, h ab . Then, near z = 0 , the metric perturbation h ab has twomodes, h ab = zh (1) ab + z − (cid:16) h (2) ab + O ( z ) (cid:17) when z → . (4.41)– 20 –he holographic dictionary tells us that the renormalizable mode is the one coupled to thedual stress-energy tensor, T ab , and therefore we set h (2) ab = 0 . Now we can use the fact that ∂ t and ∂ y are Killing vectors in order to separate variables, so that we have h ab = ze − i ( ωt − ky ) γ ab ( x ) + O ( z ) when z → . (4.42)However, just like in the case of electromagnetic perturbations, we can always set k = 0 byperforming the isometric transformation (4.22). For the sake of completeness let us point outthat the transformed metric perturbation reads ˆ h ab = ze − i ( ω ˆ t − ˆ ky )ˆ γ ab (ˆ x ) , ˆ k = k + 2 nωσL (4.43)where ˆ γ ˆ t ˆ t = γ tt , ˆ γ ˆ t ˆ x = γ tx , ˆ γ ˆ x ˆ x = γ xx (4.44) ˆ γ ˆ ty = γ ty − nσL γ tt , ˆ γ ˆ xy = γ xy − nσL γ tx , ˆ γ yy = γ yy − nσL γ ty + (cid:18) nσL (cid:19) γ tt . (4.45)so that, by choosing σ = − kL / (2 nω ) we get ˆ k = 0 . Thus, let us set k = 0 from now on.Next, we have to determine the equations satisfied by the “polarization matrix” γ ab . Byexpanding the linearized Einstein equations around z = 0 , we find that the components ofthis matrix satisfy four equations, corresponding to to G µz + 3 /L g µz = 0 . These yield nωL xγ tx + iγ (cid:48) tx − ω ( γ xx + γ yy ) = 0 , (cid:18) − n x L (cid:19) γ tt + 4 nxL γ ty − γ xx − γ yy = 0 , n xL γ tt − (cid:18) − n x L (cid:19) γ (cid:48) tt − iω (cid:18) − n x L (cid:19) γ tx − nL γ ty − nxL γ (cid:48) ty − inωxL γ xy + γ (cid:48) yy = 0 ,ω (cid:18) − n x L (cid:19) γ ty − iγxy (cid:48) + 2 nxωL γ yy = 0 , (4.46)where a prime denotes a derivative with respect to x . Let us now leave these equations fora moment to consider the NP variables Ψ and Ψ . These can be computed from the metricperturbation h µν according to their definition in (3.23) and (3.24). Using (4.42) and expandingnear z = 0 we find that ˆΨ = e − iωt (cid:2) A +2 + B +2 z + O ( z ) (cid:3) , (4.47) ˆΨ = e − iωt (cid:2) A − + B − z + O ( z ) (cid:3) , (4.48)– 21 –here ˆΨ and ˆΨ are the rescaled variables ˆΨ = V (cid:112) n + r e +4 i arctan ( r/n ) Ψ , ˆΨ = (cid:112) n + r Ψ , (4.49)and the coefficients A ± , B ± read A ± = − · ± L (cid:20) n x L γ tt ± inxL γ tx − nxL γ ty −
14 ( γ xx − γ yy ) ∓ i γ xy (cid:21) (4.50) B ± = ∓ · ± L r + (cid:34) ± in (cid:18) − n x L (cid:19) γ tt + nx (cid:18) nL ∓ ω (cid:19) ( γ tx ± iγ ty )+ 14 (cid:0) n ∓ L ω (cid:1) ( ± iγ xx − γ xy ∓ iγ yy ) + 14 (cid:0) ± inxγ (cid:48) tt − L (cid:0) γ (cid:48) tx + ± iγ (cid:48) ty (cid:1)(cid:1) (cid:35) . (4.51)Now, when searching for quasinormal modes, we demand that the variables Ψ , be sepa-rable, and this gives us additional equations for the metric perturbation. If these are separable,then we have seen that they have the form ˆΨ = e − iωt H q +2 ( x ) (cid:2) a +2 + b +2 z + O ( z ) (cid:3) , (4.52) ˆΨ = e − iωt H q − ( x ) (cid:2) a − + b − z + O ( z ) (cid:3) , (4.53)where we are using (4.40), and the levels q ± are allowed to be different. Thus, consistencywith separability demands the following constraints A ± = a ± H q ± , B ± = b ± H q ± . (4.54)In total, (4.54) and (4.46) form a system of eight equations for the six variables γ ab , andtherefore it is an overdetermined system; in order for a solution to exist, the parameters a ± , b ± and the levels q ± cannot be arbitrary. By analyzing those equations, one can see that asolution exists only if the levels q ± are related according to q +2 = q − + 4 s , (4.55)so that q − takes the values q − = 0 , , , . . . for s = 1 and q − = 4 , , , . . . for s = − . Inaddition, the ratios λ ± = b ± a ± , (4.56)must be related according to λ − = M q + P q λ +2 Q q + S q λ +2 , (4.57)where M q = − i (cid:0)(cid:0) q + 40 q + 41 (cid:1) ˆ ω (cid:15) − q + 5)ˆ ω (cid:15) + 7(2 q + 5)ˆ ω(cid:15) + ˆ ω + 2 (cid:15) (cid:1) , (4.58)– 22 – q = (cid:0) q + 2 q − (cid:1) ˆ ω(cid:15) − q + 3)ˆ ω (cid:15) + ˆ ω − (cid:15) , (4.59) Q q = (cid:0) q + 18 q + 35 (cid:1) ˆ ω(cid:15) − q + 7)ˆ ω (cid:15) + ˆ ω + 2 (cid:15) , (4.60) S q = i (cid:0) − (2 q + 5)ˆ ω(cid:15) + ˆ ω − (cid:15) (cid:1) , (4.61)and where q = (cid:40) q − if s = 1 , − − q − if s = − (4.62)There is also a relation between the normalizations of Y +2 and Y − , which reads (cid:15) ˆ ω a − a +2 = (cid:0) q − + 18 q − + 35 (cid:1) ˆ ω(cid:15) − q − +7)ˆ ω (cid:15) +ˆ ω +2 (cid:15) − iλ +2 (cid:0) (2 q − + 5)ˆ ω(cid:15) − ˆ ω + 2 (cid:15) (cid:1) (4.63)for s = 1 and (cid:15) ˆ ω a − a +2 = (cid:0) q − − q − + 19 (cid:1) ˆ ω(cid:15) + 2(2 q − − ω (cid:15) + ˆ ω + 2 (cid:15) + iλ +2 (cid:0) (2 q − − ω(cid:15) + ˆ ω − (cid:15) (cid:1) q − − q − − q − − q − (4.64)for s = − , but this is irrelevant for the computation of quasinormal modes. Finally, one canobtain an explicit solution for the γ ab in terms of Hermite functions H p , which we show inAppendix A.These results fix the boundary conditions up to the choice of the complex constant λ +2 (and up to trivial rescalings of Y ± ). In the case of vanishing NUT charge, there are twoadmissible values of λ +2 that give rise to quasinormal modes, and these correspond to choosingeither parity odd or parity even polarizations. However, in the case at hands the backgroundbreaks parity, and hence one cannot determine a priori the value of λ +2 . Then, in order tofind the quasinormal modes, one has to solve simultaneously the equations (3.30) for Y +2 and Y − with the boundary conditions discussed above. Unlike in the electromagnetic case, thereis no general relation between the variables Y ± , and one can explicitly check that differentboundary conditions on the metric perturbation give rise to different relations between theasymptotic behaviour of these functions. Therefore the equations are not degenerate in thiscase, and the problem will only have solutions for a discrete set of values of ω (the quasinormalfrequencies) and λ +2 (which determine the polarization). Having reduced the study of perturbations to a one-dimensional problem given by the radialequations (3.14) and (3.30) and having determined the boundary conditions that the corre-sponding variables must satisfy, we are now ready to compute the quasinormal modes. Beforeshowing the explicit results, we can first determine some general properties of the quasinor-mal mode frequencies ω . First, note that the dimensionless frequencies ˆ ω will only depend– 23 –n (cid:15) and on the level q (plus on the overtone number, which we omit). Therefore, the actualfrequencies scale linearly with the size of the black brane for fixed (cid:15) , ω = ˆ ω q ( (cid:15) ) L r + . (5.1)In other words, since (cid:15) = n/r + , we conclude that the QNM frequencies are homogeneousfunctions of degree 1 of r + and n . From the point of view of the dual CFT, however, thequantities r + and n do not have a direct interpretation, and instead the physically relevantquantities in the boundary theory are the ratio n/L — see (2.7) — and the temperature T given by (2.9). The QNM frequencies are then homogeneous functions of T and n/L , andthey can be conveniently expressed in terms of the dimensionless ratio ξ = 3 n πT L , (5.2)which satisfies − ≤ ξ ≤ . Then, the QNM frequencies read ω = 2 π ξ ˆ ω q ( (cid:15) ( ξ ))3 (cid:16) − (cid:112) − ξ (cid:17) T, (5.3)where (cid:15) and ξ are related by (cid:15) ( ξ ) = 1 ξ (cid:16) − (cid:112) − ξ (cid:17) . (5.4)Thus, we shall study ω/T as a function of ξ . The frequencies feature in addition a symmetryunder the exchange of sign of n (or ξ , equivalently). Namely, we have ω q ( − n ) = − ω ∗ q ( n ) , (5.5)meaning that given a QNM frequency ω q ( n ) of the solution with NUT charge n , then − ω ∗ q ( n ) is a frequency of the solution with charge − n . This result can be obtained by noticing thatthe complex-conjugate variables Y ∗± S satisfy the same equations and boundary conditions as Y ± S with ω → − ω ∗ and n → − n . Thus, there is a correspondence between the QNMs withRe ( ω ) > and NUT charge n and those with Re ( ω ) < and NUT charge − n , and vice-versa.Hence, it is sufficient to focus on studying the QNMs with Re ( ω ) > for both positive andnegative n . In the case of scalar QNMs, one can also see that the frequencies are actuallysymmetric under the change of sign of n , ω scalar ( n ) = ω scalar ( − n ) , because the radial equationis invariant under the change of sign of n . For the gravitational perturbations, however, onecan see that the replacement n → − n is not a symmetry of the master equations (3.30), norof the boundary conditions (4.57). Thus, in principle one should not expect the spectrum ofquasinormal modes to be identical for positive and negative n . Notice that given ξ , there are actually two compatible values of (cid:15) , given by (cid:15) ± ( ξ ) = ξ (cid:16) ± (cid:112) − ξ (cid:17) .However, only the ( − ) branch contributes to the Euclidean saddle point and thus we will focus on this case. – 24 –n order to compute the QNM frequencies (and, depending on case, the polarization pa-rameter λ +1 , λ +2 ), we use the following method. Taking into account the boundary conditionswe have determined, we first expand the corresponding variables Y S near the horizon using aFrobenius series and asymptotically using a Taylor expansion. This gives us two approximatesolutions Y + S ( z ) and Y ∞ S ( z ) valid in the regions z ∼ and z ∼ , respectively. One must thentry to glue both solutions, but this only will be possible if ˆ ω is a QNM frequency. One mayuse directly the asymptotic expansions Y + S ( z ) and Y ∞ S ( z ) to find the QNM frequencies byimposing the glueing condition Y ∞ S ∂ z Y + S − ∂ z Y ∞ S Y + S (cid:12)(cid:12) z joint = 0 , for some intermediate z joint .This yields an algebraic equation for ω , whose solutions should converge to the QNM fre-quencies when the number of terms in the asymptotic expansions tend to infinity, However,we have found that the convergence is not very good as we increase the NUT charge, and inorder to improve the accuracy of our results we use a numerical integration. Thus, we use thenear-horizon expansion Y + S ( z ) to initialize the numerical method at some z ini close to z = 1 ,and we numerically integrate the solution up to some z end close to z = 0 . Then, we computethe Wronskian W S = Y ∞ S ∂ z Y num S − ∂ z Y ∞ S Y num S (cid:12)(cid:12)(cid:12) z end , (5.6)and we search for solutions of W S = 0 . In the case of the scalar field, S = 0 , we have asingle equation, W = 0 , that determines the QNM frequencies. In the case of fields with spin S (cid:54) = 0 we have two such equations, namely W ± = 0 for electromagnetic perturbations and W ± = 0 for gravitational ones. Thus, we have to solve for two variables, which are the QNMfrequency ˆ ω and the polarization parameter λ +1 or λ +2 . However, as we discussed above, thismethod does not work for the electromagnetic case, since both equations W ± = 0 have infact the same solutions, and therefore it is not possible to determine λ +1 in this way. Thus,we will focus on the gravitational QNMs, which can be indeed determined from the equations W ± = 0 .In order to understand the structure of these quasinormal modes, it is important to seehow they relate to those of the black brane. One can see that, in the limit of vanishing NUTcharge, we should recover the quasinormal modes of vanishing momentum ( ˆ k = 0 ) of the blackbrane. Also, note that the spectrum becomes independent of the level q in that limit, andhence an infinite number of modes ω q of different q collapse to the same mode. On the otherhand, it is not clear that one can recover the QNMs of black branes with arbitrary momentumin the limit of n → . Note that this momentum can be identified as ˆ k = lim n → E q = lim n → snω (1 + 2 q ) /L , (5.7)thus, in order to get a non-vanishing value one must take simultaneously n → and q → ∞ in a way that qn remains finite in that limit. However, the resulting value of ˆ k would be ingeneral complex unless one chooses q to be complex as well, but in that case the connectionwith the QNMs of Taub-NUT black holes is broken. Hence, one should not expect to recoverall the QNMs of the planar black holes in a continuous way. In any case, as a test for ourmethod, we have checked that in this limit we reproduce the correct values for the axial– 25 – - - - - - - - - Figure 1 . Real and imaginary parts of the scalar QMN frequencies ω q,m /T as a function of ξ = n πT L .In order of increasing opacity the curves correspond to the levels q = 0 , , ..., . The fundamental mode ( m = 0) is shown in blue and the first overtone ( m = 1) in red. and polar gravitational QNM frequencies, as shown in tables 3 and 2 of Refs [23] and [25],respectively. Let us now present our results. We start with the simple case of a massless scalar field. For every value of ξ and the level q ,there is an infinite family of QNMs ω q,m , where, for decreasing order of the imaginary part welabel these modes by m = 0 , , . . . . The one with the largest imaginary part is the fundamentalmode ( m = 0) and the rest are overtones.In figure 1 we show the fundamental mode and the first overtone for the scalar QNMfrequencies for the levels q = 0 , , . . . . As discussed above, we see that in the limit ξ → allthe modes with different q collapse to the same corresponding mode of the black brane. As acheck, we get that ω q, (0) ≈ (7 . − . i ) T ≈ (1 . − . i ) r + /L , (5.8)which agrees with the fundamental mode of the black brane when r >> (cid:126)k [23]. As we cansee in figure 1, the real part of ω grows almost linearly with ξ (or n ), while the imaginary parthas a non-monotonic dependence. Also, note that these frequencies are symmetric for ξ → − ξ .For ξ ∼ ± we see that Im ( ω q,m ) ∼ for large q , but our numeric results suggest that it neverbecomes positive, and therefore, scalar perturbations are stable for the whole range of ξ . Werecall that the results in Fig. 1 refer to the branch of black holes with positive specific heat, r > n . We have briefly looked to case of r < n , and for those black holes our resultsindicate that all the quasinormal modes have very small imaginary parts Im ( ω q,m ) ∼ , butthat still do not cross 0. – 26 – .2 Gravitational Let us now turn to the most interesting case of gravitational modes. Here, in analogy withthe case of the black brane, we may distinguish two families of modes according to the theirbehaviour in the limit n → . We find that for every level q there is a special mode such that ω q → in the limit n → .We recall that, in the case of the black brane, both axial and polar perturbations contain ahydrodynamic mode, i.e , one whose frequency vanishes when (cid:126)k → [25]. In the presence ofNUT charge, one cannot talk about hydrodynamic modes because the spectrum of quasinormalmodes is discrete, and thus we refer to the modes ω q that vanish for n → as “pseudo-hydrodynamic”. These must be in fact related to the hydrodynamic modes of the black brane.We show these pseudo-hydrodynamic modes in Fig. 2 for the levels q = 0 , . . . (where q = q − if Re ( nω ) > and q = q +2 if Re ( nω ) < ). Let us stress that for every value of q we find only one of these modes and not two, unlike in the case of the black brane — wecomment on this below. As we can see, the real part behaves linearly with ξ near ξ = 0 ,while the imaginary part is quadratic in that region. For larger values of ξ the real part of ω q transitions to a different linear dependence, while the imaginary part has a non-monotonicbehaviour. Indeed, after reaching a minimum value, the imaginary part grows and becomesclose to 0 for ξ = ± . We observe that for larger q , the imaginary part becomes even smallernear ξ = ± , but interestingly it does not become positive, which indicates that there are nounstable modes — we study the stability of these solutions below. Another property of theseQNM frequencies that is worth remarking is that they are symmetric under the exchange ξ → − ξ . We also note that when we change the sign of ξ , the polarization parameters λ ± transform as ( λ +2 , λ − ) → ( − λ − , − λ +2 ) . All of this indicates that the exchange of sign ofthe NUT charge must be indeed a hidden symmetry of the equations (3.30) with the boundaryconditions (4.57).Let us now focus on the region ξ << . We can actually obtain analytic approximationsfor the pseudo-hydrodynamic QNMs in this limit. Taking into account the input from thenumerical result, we will have ˆ ω q = (cid:15)a q − ib q (cid:15) + O ( (cid:15) ) , (5.9)for some coefficients a q and b q (near ξ = 0 the relation between this variable and (cid:15) is simply ξ ≈ (cid:15) ). Analogously, we can expand the polarization parameter λ +2 as λ +2 = λ (0) + (cid:15)λ (1) + O ( (cid:15) ) . (5.10)Recalling now the boundary conditions (4.38), we write the functions Y ± as Y +2 ( z ) = (1 − z ) − i ˆ ω (cid:15) ∞ (cid:88) i =0 c (+2) i (1 − z ) i , (5.11)– 27 – - - - - - - - - - - - - - - - - Figure 2 . Pseudo-hydrodynamic QNM frequencies of gravitational perturbations as a function of ξ = n πT L . Top row: in order of increasing opacity we show the levels q = 0 , , ..., , where q = q − if Re ( nω ) > and q = q +2 if Re ( nω ) < . Bottom row: behaviour near ξ = 0 and comparison withthe analytic result (5.17). In order to facilitate the visualization in that case we only show the modeswith q = 0 , , , , . Y − ( z ) = (1 − z ) − i ˆ ω (cid:15) ∞ (cid:88) i =0 c ( − i (1 − z ) i . (5.12)Using the master equations (3.30) one can then find explicitly the values of all the coefficients c ( ± i in terms of the first one up to a given order i = i max . We can then implement a methodsimilar the one of Horowitz and Hubeny [19] and glue these expansions with the ones in (4.40)at z = 0 . This yields the equations λ +2 Y +2 (0) − Y (cid:48) +2 (0) = 0 , λ − Y − (0) − Y (cid:48)− (0) = 0 , (5.13)where in addition we recall that λ − is related to λ +2 according to (4.57). In general, thismethod can be used to obtain an approximate solution. However, when we use the expressions(5.9) and (5.10) and expand (5.13) around (cid:15) = 0 , it turns out that we can obtain the valuesof a q , b q and λ (0) exactly. Without loss of generality, let us consider n > and Re ( ω ) > .– 28 –hen, even for i max = 1 , one can see that λ − Y − (0) − Y (cid:48)− (0) = 0 does not admit a solutionwith c ( − (cid:54) = 0 unless λ − is non-zero for (cid:15) = 0 . For generic values of the parameters of theexpansion, the numerator in (4.57) is of order (cid:15) while the denominator is of order (cid:15) , so thatin this case λ − = 0 for (cid:15) = 0 . The denominator is order (cid:15) only if a q − (5 + 2 q ) a q − . (5.14)The positive root of this equation yields the following value for a q , a q = 12 (cid:16) q + 5 + (cid:112) q + 20 q + 33 (cid:17) . (5.15)With this choice, one can solve the equations (5.13) order by order in the (cid:15) expansion. For,say, i max = 3 , one finds λ (0) = 3 / and b q = 43 (cid:32) q + 5 q + 4 + ( q + 2)( q + 3)(2 q + 5) (cid:112) q + 20 q + 33 (cid:33) . (5.16)Then it is easy to check that these results do not change for larger values of i max , and thusare exact. This leads to the following expression for the physical frequency ωω q = 2 πT (cid:20) a q ξ − ib q ξ (cid:21) = na q L − i b q n πT L , (5.17)which is valid when q | ξ | << . As we show in the second row of Fig. 2, these expressions matchthe numeric results with great accuracy. Finally, it is interesting to analyze what happens inthe limit n → and q → ∞ such that qn remains finite. In that case we have ω ≈ nqL − i nq ) πT L . (5.18)On the other hand, we recall that in this limit we can identify a momentum for the perturba-tions ˆ k according to (5.7), which yields ˆ k = 4 nqωL . (5.19)Moreover, this is a real momentum when | n | q << . Combining this expression with (5.18)we obtain the following effective dispersion relation for small ˆ kω ≈ ˆ k √ − i ˆ k πT . (5.20)This is precisely the dispersion relation for the hydrodynamic mode of polar perturbations inthe absence of NUT charge [25]. Hence, the pseudo-hydrodynamic modes of the NUT-chargedblack holes are analogous to that mode of the black brane. One may wonder why we do notobtain other modes similar to the hydrodynamic mode of polar perturbations. The reason isthat such mode is purely damped, and according to the identification (5.7) we would need to– 29 –hoose q to be imaginary in order to recover a solution of the black brane with real momentum.Thus, that mode is simply not present in the Taub-NUT planar black holes.When ξ becomes larger, we cannot obtain an analytic result for the frequencies, but wecan obtain a reasonable good approximation for the real part. In fact, we observe that thereal part of the dimensionless frequencies ˆ ω q is a linear function of (cid:15) , and a fit to the numericaldata shows that the slope is proportional to q . Namely, we getRe (ˆ ω q ) ≈ . q + 3) (cid:15) , (5.21)plus a constant term that is much smaller. Interestingly, this seems to work not only for (cid:15) ≤ , but for arbitrarily large (cid:15) . Now, when we take into account (5.3), we deduce that thedimensionful frequencies ω q are also a linear function of ξ Re ( ω q ) ≈ πT ξ . q + 3) ≈ . nL ( q + 3) . (5.22) The rest of the gravitational quasinormal modes have frequencies that tend to a constant, non-vanishing value in the limit n → . As already remarked before, those values will correspondto the black brane’s QNM frequencies at vanishing momentum. It is known that the axial andpolar QNMs of the black brane become degenerate when the momentum tends to zero [25],which means that we should also expect to obtain an splitting of those modes as we increasethe value of n . Indeed, we observe that the ordinary QNMs come in two families that we labelas ω + q,m and ω − q,m , where m = 0 , , , .. denotes the overtone. For the same m , all the modesin the two families collapse to the same QNM lim n → ω + q,m ( n ) = lim n → ω − q (cid:48) ,m ( n ) ≡ ω m (0) ∀ q, q (cid:48) (5.23)Obviously, the two different families ω ± correspond to the two polarization modes of the gravi-ton, but unlike the case of the planar black hole, one cannot identify these modes accordingto their parity, since the background breaks all reflection symmetries.In Fig. 3 we show the lowest ( m = 0 ) QNMs for a few levels q , where the first thingwe notice is that the spectrum is again symmetric for ξ > and ξ < . The structureof the QNM frequencies as a function of ξ is somewhat similar to the one of the pseudo-hydrodynamic modes, with the real part scaling almost linearly with q for most of the rangeof ξ . In particular, we have the following fits to the real parts of the dimensionless frequenciesRe (ˆ ω + q ) ≈ (16 . . q ) (cid:15) + c + q , Re (ˆ ω − q ) ≈ (14 . . q ) (cid:15) + c − q , (5.24)where the constant terms are small. When we use (5.3), this produces an almost linear relationbetween ω q and ξ , although the non-vanishing constant terms introduce non-linearities near ξ = ± . On the other hand, the imaginary part becomes very small as ξ → ± , but asbefore, we do not observe any mode becoming unstable. In addition, for every value of ξ – 30 – - - - - - - - - - - - - - - - - Figure 3 . Ordinary gravitational quasinormal modes: we show the lowest overtones m = 0 of bothfamilies ω − q, (left) and ω + q, (right) as a function of ξ = n πT L . In order of increasing opacity thecurves correspond to the levels q = 0 , , . . . , . and q , the imaginary parts of these modes are larger (in absolute value) than those of thepseudo-hydrodynamic modes, and hence there is no level crossing. In the opposite limit, at ξ = 0 , all the modes collapse to ω ± q, (0) ≈ (1 . − . i ) r + /L , which agrees with the firstordinary mode of the black brane in the limit of vanishing momentum [23–25]. So far, all the modes we have found are stable, meaning that their associated frequencies lie inthe lower half of the complex plane. In order to show that the Taub-NUT solution is (linearly)stable one must prove that this property holds for every quasinormal modes. Here we provideevidence that this is indeed the case, but for future analyses it would be important to providea solid proof of this fact.As we have seen, the quasinormal modes with the lowest imaginary part are the pseudo-hydrodynamic ones, and the imaginary part becomes smaller as we increase q . Therefore, weshould analyze the behaviour of these modes when q → ∞ . In Fig. 4 we have plotted thetrajectories in the complex plane of these modes for many values of q and a some selected– 31 – - - Figure 4 . Trajectories in the complex plane of the QNM frequencies ω q of lowest imaginary part(corresponding to the pseudo-hydrodynamic modes) for a few values of (cid:15) . For large q the imaginarypart tends to zero exponentially, but it never becomes positive. values of (cid:15) = n/r + . Thanks to the logarithmic scale in the vertical axis, we can see clearlythat the imaginary part tends to zero exponentially with q and that it also decreases when (cid:15) grows. Indeed, a fit to the numerical data reveals that the imaginary part of the QNMfrequencies ω q for large q is well approximated byIm ( ω q ( (cid:15) )) ≈ − T A ( (cid:15) ) e − (2 . (cid:15) − . q , (5.25)which is valid as long as (cid:15) is not far from . For smaller values of (cid:15) , the imaginary part is larger(in absolute value) and therefore, the negativity of Im ( ω q ( (cid:15) )) for (cid:15) = 1 implies the stability ofall the modes with (cid:15) ≤ . However, the asymptotic behaviour for q → ∞ is difficult to accessfor small (cid:15) , since it requires going to larger and larger q , in which case our numeric methodbecomes less accurate. In any case, our data suggests that the imaginary part of ω q ultimatelydecays exponentially with q for any value of (cid:15) . Thus, the conclusion is that the lowest-lyingmodes for every q are stable for every | (cid:15) | ≤ , and by extension all the modes are. Thissignals that, despite the apparent pathological properties of the NUT-charged spacetimes,they actually give rise to stable and well-defined dynamics.Finally, although we have focused on the case | (cid:15) | ≤ because it is the relevant one forholography, one may wonder what happens if we take even larger values of the NUT charge | (cid:15) | ≥ . In fact, since those solutions do not posses an Euclidean continuation, one may thinkthat they could be unstable. In Fig. 5, we show the lowest gravitational QNM for a few valuesof q as a function of (cid:15) = n/r + , extended beyond (cid:15) = 1 . We observe nothing special going on atthat point, and in fact, the modes keep on being stable as we increase (cid:15) . Nevertheless, Fig. 5shows that Taub-NUT solutions with increasingly large NUT charge have more quasinormal– 32 – .0 0.2 0.4 0.6 0.8 1.0 1.2 1.4010203040 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 - - - - - Figure 5 . Pseudo-hydrodynamic quasinormal modes extended for n/r + > for the levels q = 0 , ..., .There is nothing special at the point n = r + and the modes keep on being stable beyond it. However,their imaginary parts become exponentially small as we increase n . modes with extremely small imaginary parts, and it would be interesting to study if this couldeventually give rise to a non-trivial instability when nonlinearities are taken into account. We have performed a thorough analysis of the quasinormal modes of the planar Taub-NUTspacetimes given by (2.2). As we discussed, these describe the linear response to perturbationsof a strongly-coupled plasma placed in the geometry (2.7), corresponding to a Gödel-typeuniverse with closed time-like curves.Our analysis revealed that QNMs in this background organize analogously to the Landaulevels of a charged particle in a uniform magnetic field. Thus, unlike in the case of planarblack holes, the spectrum of QNM frequencies is discrete and labelled by a unique quantumnumber q (the Landau level). On the other hand, the QNMs are infinitely degenerate in themomentum k along the isometric direction, which we chose to be y . Another novel aspectintroduced by the NUT charge is that all the reflection symmetries of the spacetime are broken,which implies that one cannot decompose the perturbations of fields with spin into modes ofdefinite parity. This leads to the appearance of an additional “polarization parameter” λ +2 characterizing the gravitational QNMs. We argued that such parameter has to be determinedtogether with the corresponding QNM frequency ω by solving simultaneously the equationsfor the two NP variables Ψ and Ψ . In the limit of vanishing NUT charge this procedure infact selects the values of λ +2 that give rise to modes of definite parity, but the value cannotbe obtained a priori when the NUT charge is nonzero. However, the numerical results providehints for the existence of some additional structure. We observe that the values of λ ± givingrise to QNMs satisfy Re ( λ − ) = − Re ( λ +2 ) . If this is assumed, then the relation (4.57) In the case of zero NUT charge, the parameters that give rise to modes of definite parity satisfy λ − = – 33 –ould imply that only either the real or the imaginary part has to be determined numerically.This also suggests that, by further analyzing the NP system, perhaps one could be able toobtain a closed expression for these parameters — it would be interesting to explore thiselsewhere. Finally, despite parity violation, we found, rather surprisingly, that the spectrumof gravitational QNM frequencies is symmetric under the change of sign of the NUT charge.In addition, there is a conjugation symmetry that relates the positive-frequency modes of thesolution with charge n to the negative-frequency ones of the solution with charge − n , andvice-versa — see Eq. (5.5).In the case of electromagnetic perturbations we have shown that a similar method doesnot work, since the equations for the NP variables φ and φ are degenerate. Thus thecorresponding polarization parameter λ +1 cannot be determined in this way or by parityarguments. This may lead to the conclusion that the spectrum of QNMs depends continuouslyon this parameter, but this issue certainly deserves further research. Perhaps analyzing theperturbations in terms of the vector field rather than in terms of the Newmann-Penrosevariables could shed light on this problem.Our numerical results on the scalar and gravitational QNM frequencies show that all ofthem lie in the lower half of the complex plane, and hence no instabilities are found despitethe exotic causal structure of these spacetimes. Thus, this constitutes yet another step intothe rehabilitation of Lorentzian spacetimes with NUT charge, in line with Refs. [40–44]. Ifwe now apply the AdS/CFT correspondence, this result tells us not only that one shouldbe able to perform quantum field theory in the background of the causality-violating metric(2.7), but that it should be possible to obtain sensible answers. Hence, it would now beinteresting to perform a direct QFT computation in (2.7) to try to reproduce the resultsobtained from holography. In particular, we managed to obtain an analytic expression forthe pseudo-hydrodynamic modes in the limit of small NUT charge — see Eq. (5.17). Aswe have shown, that result generalizes the standard dispersion relation for the sound modein flat space to the case of the background (2.7) when n/L << T . It would be extremelyinteresting to attempt a derivation of that relation by studying the perturbations of a fluid insuch background.Let us close our paper by commenting on other directions that should be considered. Aswe already mentioned, one should try to better understand the properties of electromagneticQNMs. On the other hand, we have focused mainly on the scalar and gravitational modeswith lowest imaginary part, but it would be interesting to complete the classification of QNMsby analyzing the overtone structure and the highly damped modes. In addition, even thoughwe have provided compelling numerical evidence that no unstable QNMs exist, it would beimportant to offer a mathematical proof of this fact. Finally, it would also be worth extendingthese results to the case of Taub-NUT solutions of different topologies — the spherical case isparticularly interesting due to the interplay with the Misner string [48] — or to higher dimen-sions. Hopefully these will offer further insight on the role of NUT charge in the AdS/CFT − λ +2 , and from this observation and (4.57) one can obtain their explicit values. – 34 –orrespondence. Acknowledgements
We are glad to thank Vitor Cardoso and Karl Landsteiner for insightful discussions andcomments. The work of PAC is supported by a postdoctoral fellowship from the ResearchFoundation - Flanders (FWO grant 12ZH121N). DP is funded by a “Centro de ExcelenciaInternacional UAM/CSIC” FPI pre-doctoral grant
A Asymptotic form of the metric perturbation
As we have seen, the metric perturbation satisfying Dirichlet boundary conditions can bewritten near the boundary as h ab = ze − iωt γ ab ( x ) + O ( z ) , (A.1)where we are already setting k = 0 without loss of generality. The equations of motion allowone to express the component γ xx in terms of the rest as γ xx = (cid:18) − n x L (cid:19) γ tt + 4 nxL γ ty − γ yy . (A.2)Then, it is convenient to introduce a new matrix σ ab as follows γ ab = e − snωx /L σ ab . One findsthat the equations of motion together with the separability conditions on the NP variablesimply that σ ab is given by a finite sum of Hermite polynomials. In the case s = 1 it reads σ tt = − a +2 L r + ˆ ω(cid:15) H q +2 (ˆ x ) (cid:0) − (2 q + 7)ˆ ω(cid:15) + ˆ ω + iλ +2 ˆ ω − (cid:15) ( (cid:15) − iλ +2 ) (cid:1) , (A.3) σ tx = − ia +2 L √ r + (ˆ ω(cid:15) ) / (cid:104) (cid:15)H q − +3 (ˆ x ) ( − iλ +2 − ˆ ω + (cid:15) )+ 2 H q − +1 (ˆ x ) (cid:0) i ˆ ωλ +2 + ( q − + 1) (cid:15) ( (cid:15) − iλ +2 ) − (3 q − + 10) ˆ ω(cid:15) + ˆ ω (cid:1) (cid:105) , (A.4) σ ty = 5 a +2 L √ r + (ˆ ω(cid:15) ) / (cid:104) H q − +1 (ˆ x ) (cid:0) q − + 2) (cid:15) ( (cid:15) − iλ +2 ) + ˆ ω (( − q − − (cid:15) + iλ +2 )+ˆ ω(cid:15) (cid:0)(cid:0) q − + 23 q − + 29 (cid:1) (cid:15) − i (3 q − + 5) λ +2 (cid:1) + ˆ ω (cid:1) + (cid:15)H q − +3 (ˆ x ) (cid:0) − i ˆ ωλ +2 + 4 (cid:15) ( (cid:15) − iλ +2 ) + (4 q − + 13) ˆ ω(cid:15) − ˆ ω (cid:1) (cid:105) , (A.5) σ xy = 5 ia +2 L r + ˆ ω(cid:15) (cid:104) H q − (ˆ x ) (cid:0) − q − ( q − + 2) (cid:15) ( (cid:15) − iλ +2 ) + ˆ ω ( − q − + 8) (cid:15) + iλ +2 )+ˆ ω(cid:15) (cid:0)(cid:0) q − + 44 q − + 55 (cid:1) (cid:15) − i (4 q − + 7) λ +2 (cid:1) + ˆ ω (cid:1) − (cid:15) ( (cid:15) − iλ +2 ) H q − +4 (ˆ x ) − (cid:15)H q − +2 (ˆ x ) (cid:0) i ˆ ωλ +2 + 2 ( q − + 2) (cid:15) ( (cid:15) − iλ +2 ) − (4 q − + 13) ˆ ω(cid:15) + ˆ ω (cid:1) (cid:105) , (A.6)– 35 – yy = − a +2 L r + ˆ ω (cid:15) (cid:104) H q − ( x ) (cid:0) − ω(cid:15) (cid:0)(cid:0) q − + 28 q − + 54 q − + 29 (cid:1) (cid:15) − i (cid:0) q − + 10 q − + 5 (cid:1) λ +2 (cid:1) (A.7) − (cid:0) q − + 3 q − + 2 (cid:1) (cid:15) ( (cid:15) − iλ +2 ) + ˆ ω ( − q − + 9) (cid:15) + iλ +2 )+ˆ ω (cid:15) (cid:0)(cid:0) q − + 82 q − + 83 (cid:1) (cid:15) − i (2 q − + 3) λ +2 (cid:1) + ˆ ω (cid:1) + (cid:15) H q − +4 ( x ) (cid:0) − (cid:15) ( (cid:15) − iλ +2 ) − q − + 3) ˆ ω(cid:15) + ˆ ω (cid:1) − (cid:15)H q − +2 ( x ) (ˆ ω − q − + 5) (cid:15) ) (cid:0) i ˆ ωλ +2 − (cid:15) ( (cid:15) − iλ +2 ) − (2 q − + 7) ˆ ω(cid:15) + ˆ ω (cid:1) (cid:105) , (A.8)while for s = − the solution is σ tt = 17 a +2 L H q +2 +2 (ˆ x ) (cid:0) i ˆ ωλ +2 + ˆ ω(cid:15) (2 q +2 + 3) − (cid:15) ( (cid:15) − iλ +2 ) + ˆ ω (cid:1) r + ˆ ω(cid:15) ( q +2 + 1) ( q +2 + 2) , (A.9) σ tx = 17 ia +2 L ˆ ω √ r + ( q +2 + 1) ( q +2 + 2) ( q +2 + 3) ( − ˆ ω(cid:15) ) / (cid:104) (cid:15) (cid:0) q + 5 q +2 + 6 (cid:1) ( iλ +2 + ˆ ω − (cid:15) ) × H q +2 +1 (ˆ x ) + H q +2 +3 (ˆ x ) (cid:0) i ˆ ωλ +2 − (cid:15) ( q +2 + 4) ( (cid:15) − iλ +2 ) + ˆ ω(cid:15) (3 q +2 + 5) + ˆ ω (cid:1) (cid:105) , (A.10) σ ty = − a +2 L √ r + ( q +2 + 1) ( q +2 + 2) ( q +2 + 3) ( − ˆ ω(cid:15) ) / (cid:104) H q +2 +3 (ˆ x ) (cid:0) − (cid:15) ( q +2 + 3) ( (cid:15) − iλ +2 )+ˆ ω ( (cid:15) (5 q +2 + 11) + iλ +2 ) + ˆ ω(cid:15) (cid:0) (cid:15) (cid:0) q + 17 q +2 + 14 (cid:1) + iλ +2 (3 q +2 + 10) (cid:1) + ˆ ω (cid:1) + 2 (cid:15) (cid:0) q + 5 q +2 + 6 (cid:1) H q +2 +1 (ˆ x ) (cid:0) i ˆ ωλ +2 + ˆ ω(cid:15) (4 q +2 + 7) − (cid:15) ( (cid:15) − iλ +2 ) + ˆ ω (cid:1) (cid:105) , (A.11) σ xy = 17 ia +2 L r + ˆ ω(cid:15) ( q +2 + 1) ( q +2 + 2) ( q +2 + 3) ( q +2 + 4) (cid:104) H q +2 +4 (ˆ x ) (cid:0) − (cid:15) (cid:0) q + 8 q +2 + 15 (cid:1) × ( (cid:15) − iλ +2 ) + ˆ ω (2 (cid:15) (3 q +2 + 7) + iλ +2 ) + ˆ ω(cid:15) (cid:0) (cid:15) (cid:0) q + 36 q +2 + 35 (cid:1) + iλ +2 (4 q +2 + 13) (cid:1) +ˆ ω (cid:1) − (cid:15) (cid:0) q + 10 q + 35 q + 50 q +2 + 24 (cid:1) ( (cid:15) − iλ +2 ) H q +2 (ˆ x )+ 4 (cid:15) (cid:0) q + 7 q +2 + 12 (cid:1) H q +2 +2 (ˆ x ) (cid:0) i ˆ ωλ +2 − (cid:15) ( q +2 + 3) ( (cid:15) − iλ +2 ) + ˆ ω(cid:15) (4 q +2 + 7) + ˆ ω (cid:1) (cid:105) , (A.12) σ yy = 17 a +2 L r + ˆ ω (cid:34) − H q +2 +4 (ˆ x ) (cid:15) ( q +2 + 1) ( q +2 + 2) ( q +2 + 3) ( q +2 + 4) (cid:16) ω(cid:15) (cid:0) (cid:15) (cid:0) q + 32 q + 74 q +2 + 41 (cid:1) + iλ +2 (cid:0) q + 30 q +2 + 55 (cid:1)(cid:1) − (cid:15) (cid:0) q + 7 q +2 + 12 (cid:1) ( (cid:15) − iλ +2 ) + ˆ ω ( (cid:15) (8 q +2 + 22) + iλ +2 )+ ˆ ω (cid:15) (cid:0) (cid:15) (cid:0) q + 98 q +2 + 123 (cid:1) + 3 iλ +2 (2 q +2 + 7) (cid:1) + ˆ ω (cid:17) − H q +2 +2 (ˆ x ) (2 (cid:15) (2 q +2 + 5) + ˆ ω ) (cid:0) i ˆ ωλ +2 + ˆ ω(cid:15) (2 q +2 + 3) − (cid:15) ( (cid:15) − iλ +2 ) + ˆ ω (cid:1) (cid:15) ( q +2 + 1) ( q +2 + 2) – 36 – H q +2 (ˆ x ) (cid:0) ω(cid:15) ( q +2 + 2) − (cid:15) ( (cid:15) − iλ +2 ) + ˆ ω (cid:1) (cid:35) , (A.13)where in each case ˆ x = x (cid:113) snωL . References [1] J. M. Maldacena,
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