Superradiance in Kerr-like black holes
SSuperradiance in deformed Kerr black holes
Edgardo Franzin,
1, 2, 3, 4
Stefano Liberati,
2, 3, 4 and Mauro Oi
5, 6 Department of Astrophysics, Cosmology and Fundamental Interactions (COSMO), Centro Brasileirode Pesquisas F´ısicas (CBPF), rua Dr. Xavier Sigaud 150, Urca, Rio de Janeiro – RJ, 22290-180 Brazil SISSA, International School for Advanced Studies, via Bonomea 265, 34136 Trieste, Italy IFPU, Institute for Fundamental Physics of the Universe, via Beirut 2, 34014 Trieste, Italy INFN, Sezione di Trieste, via Valerio 2, 34127 Trieste, Italy Dipartimento di Fisica, Universit`a di Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy INFN, Sezione di Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy
Recent strong-field regime tests of gravity are so far in agreement with general relativity. In particular,astrophysical black holes appear all to be consistent with the Kerr spacetime, but the statistical error on currentobservations allows for small yet detectable deviations from this description. Here we study superradiance ofscalar and electromagnetic test fields in deformed Kerr spacetimes and we observe that for large deformationssuperradiance is highly suppressed with respect to the Kerr case. Surprisingly, for small deformations thereexists a range of values for the deformation parameter for which the maximum amplification factor is largerthan the Kerr one. We also provide a first result about the superradiant instability of these deformed spacetimesagainst massive scalar fields.
I. INTRODUCTION
General relativity has been extensively and successfullytested [1, 2] from the weak to the strong regime — the mostrecent results being the detection of gravitational waves pro-duced by the merger of two black holes [3] and the observa-tion of the shadow of the supermassive black hole M87* [4].Nowadays black holes are widely accepted as astrophysicalobjects [5, 6] compatible with the Kerr metric [7], yet, we stilldo not have the ultimate evidence for such black holes to ex-actly match this general-relativistic solution, as their definingproperty — the event horizon — is intrinsically not directly observable [8–11].There exists a number of alternative theories of gravity aswell as exotic compact objects proposed to compete or substi-tute black holes. These black-hole mimickers typically sharethe same features at large distances, while they present qual-itative di ff erences close to the event horizon. Current and fu-ture gravitational-wave observations are and will be able totest general relativity, the no-hair theorem, the near-horizongeometry, distinguish the Kerr spacetime from putative altern-atives, and even probe quantum gravity e ff ects [11–17]. Thesee ff ects in a consistent setup are commonly invoked to regular-ize spacetime singularities, which are inevitable in classicalgeneral relativity [18].While nowadays observations agree with numerical simu-lations based on Einstein gravity, the current uncertainties onthe measurements of the black-hole parameters leave room foralternatives. A possible framework is to describe this freedomby introducing suitable parametrized deviations from the Kerrgeometry. The observed interval values for the black-holemass M and angular momentum J = aM can be thereforetranslated in an allowed range for the deformation paramet-ers. Of course, we do not expect these deviations to be largeor they would be observable in the weak-field regime as well.But, for instance, one can consider non-negligible deviationsfrom Kerr and obtain the same quasinormal frequencies. Ifthe geometry of the spacetime is di ff erent from Kerr only ina small region near the would-be horizon, asymptotically the geometry would be barely distinguishable from Kerr, leavinga weak signature in the form of gravitational-wave echoes atlate times [10, 19–23].From this point of view, instead of testing a specific the-ory against general relativity case by case and / or a specificblack-hole alternative, it could be more convenient to workin a model-independent framework describing the most gen-eric black holes in any metric theory of gravity. The idea ofthis framework is similar to the parametrized post-Newtonian(PPN) formalism [1] but in this case it is valid in the wholespace outside the event horizon.In Refs. [24–26], deviations from general relativity and thegeneral-relativistic black-hole geometry are written in termsof an expansion in M / r being r some radial coordinate. Somecoe ffi cients are easily constrained with the PPN parameters,while a very large number of equally important coe ffi cientsremains undetermined in the near-horizon region, with the ad-ditional drawback of a lack of a hierarchy among them. Evenif this formulation works well for small deviations from gen-eral relativity, it fails for e.g. Einstein–dilaton–Gauss–Bonnetwith large coupling constants [27].A more robust general parametrization to describe, respect-ively, spherically symmetric and axisymmetric asymptoticallyflat black holes has been introduced in Refs. [28, 29], andtested to constrain deviations from the Kerr hypothesis withthe iron-line method [30–32] and to produce black-hole shad-ows simulations [33, 34]. In this framework, deviations fromgeneral relativity and the Kerr metric are given again as anexpansion whose coe ffi cient values can be fixed from obser-vations in the strong-gravity regime (close to the horizon) andin the post-Newtonian region (far from the black hole). Thisparametrization also allows for non-spherical deformations ofthe horizon, provides a faster convergence of the series, andtypically requires a small number of parameters to approxim-ate known solutions to the desired precision. Besides, thereexists a hierarchy among the parameters.A di ff erent perspective is to modify each mass and spinterm in the Kerr metric and test whether the magnitude of thespacetime curvature matches with that predicted by general a r X i v : . [ g r- q c ] F e b relativity [35]. More recently, the work of Ref. [25] has beenextended to the most general stationary, axisymmetric andasymptotically flat spacetime with separable geodesic equa-tions [36].However, even if these parametrizations may depend on alarge number of parameters to be fixed with data, it is nat-ural to think that astrophysical observables — e.g. quasinor-mal frequencies, orbits of particles, accretion, parameters ofthe shadow, electromagnetic radiation — depend only on afew of them [37].A common feature of rotating spacetimes is the multifa-ceted phenomenon of superradiance [38–40]: in a gravita-tional system and under certain conditions, the scattering ofradiation o ff absorbing rotating objects produces waves withamplitude larger than the incident one. For a monochromaticwave of frequency ω scattering o ff a body with angular ve-locity Ω , the superradiant condition is satisfied as long as ω < m Ω , being m the azimuthal number with respect to therotation axis.When rotating black holes are surrounded with matter, su-perradiance gives rise to exponentially growing modes, i.e.black-hole bombs [41, 42]. The scattering of massive fieldsproduces a similar e ff ect: the mass term can e ff ectively con-fine the field giving rise to floating orbits and superradiant in-stabilities which extract rotational energy away from the blackhole [43–45]. The observation or the absence of e ff ects relatedto these instabilities can be used to impose bounds on the massof ultralight bosons, see e.g. Refs. [46–50].Similarly to the Kerr black hole, deformed-Kerr space-times dissipate energy as well as any classical dissipative sys-tem, and the aim of this paper is to investigate di ff erencesand analogies for these objects with respect to the super-radiant scattering around Kerr black holes. We stress thatthese spacetimes are not solutions to the field equations ofany specific gravitational theory, meaning that we can onlystudy test fields propagating in these backgrounds while thegravitational-wave dynamics is excluded. However, in exten-ded theories of gravity exact rotating solutions are di ffi cult toderive and in some cases they are known only perturbativelyin the spin parameter, or numerically. To our knowledge, thereare no studies of superradiant amplification in these exten-ded theories, neither for those which admit general-relativisticsolutions [51, 52] but predict di ff erent dynamics.In the most general parametrization, there is no reasonto believe that the separability property of the Kerr metricis guaranteed, not even for the Klein–Gordon equation. Inparticular, the class of deformed-Kerr spacetimes which al-lows for the separation of variables in the Klein–Gordon andHamilton–Jacobi equations has been derived in Ref. [53],which is a subclass of the Johannsen metrics [25]. In thispaper we show that under given conditions, a subclass of themetrics presented in Ref. [53] also allows for the separationof variables in the Maxwell equation.The results presented in this paper are mostly relative to aminimally deformed spacetime, i.e. introducing a single extraparameter, the Konoplya–Zhidenko black hole [54]. Despiteits simplicity, this model preserves a lot of features of the Kerrspacetime: the asymptotic properties, the post-Newtonian ex- pansion coe ffi cients, the relation between quadrupole momentand mass, the spherical horizon, and the mirror symmetry.Yet, it allows for significant di ff erences in the near-horizonregion [55–57].The scope of this paper is twofold: first we analyze thestructure of the Konoplya–Zhidenko spacetime, and secondwe study superradiant scattering of test fields. In particular,the paper is organized as follows. In Section II we reviewthe family of spacetime which admits separability of the per-turbative equations for massless spin-0 and spin-1 fields, witha particular focus on the Konoplya–Zhidenko rotating blackhole. In Section III we present our results regarding the super-radiant emission in the Konoplya–Zhidenko minimal deform-ation for massless and massive bosonic test fields. Finally,we conclude with a discussion and prospects in Section IV.In Appendix A we derive the angular and radial equations fora general non-Kerr black-hole parametrization and we studytheir boundary conditions. In Appendix B we provide usefulformulas for the Konoplya–Zhidenko spacetime, namely theEinstein tensor, the geodesic equations and the four-velocityof a zero-angular-momentum observer. In Appendix C westudy the instability of the Konoplya–Zhidenko black holeagainst massive scalar fields in the low-frequency, small-massand small-deformation limit. Throughout this work we use G = c = II. DEFORMED KERR SPACETIMES AND THEKONOPLYA–ZHIDENKO BLACK HOLE
The metric of a generic axially symmetric, stationary andasymptotically flat spacetime can be written asd s = − N − W sin θ K d t − Wr sin θ d t d ϕ + K r sin θ d ϕ + Σ r (cid:32) B N d r + r d θ (cid:33) , (1)where N , W , K , Σ and B are in general functions of r and θ . Inthis paper we focus on deformed Kerr spacetimes which pos-sess Kerr-like symmetries and admit separable Klein–Gordonequations for test fields [53]. As in Ref. [53], we are not in-terested in the general conditions for the separability of vari-ables, which are related to the symmetry of the backgroundand the choice of appropriate coordinates. Being our prag-matic objective to test strong-gravity e ff ects in an asymptot-ically flat and axisymmetric spacetime describing a Kerr-likeblack hole, we can simplify the above spacetime metric leav-ing only three arbitrary functions of the radial coordinate, sothat B ( r , θ ) = R B ( r ) , Σ ( r , θ ) = r R Σ ( r ) + a cos θ, (2a) W ( r , θ ) = aR M ( r ) Σ ( r , θ ) , N ( r , θ ) = R Σ ( r ) − R M ( r ) r + a r , (2b) K ( r , θ ) = Σ ( r , θ ) (cid:104) r R Σ ( r ) + a R Σ ( r ) + a cos θ N ( r , θ ) (cid:105) + a W ( r , θ ) r . (2c)For further convenience we define ∆ ≡ r N = r R Σ − R M r + a and we observe that for this class of spacetimes the eventhorizon is defined by the largest positive root of ∆ = R M → M + O (1 / r ) as r → ∞ . With a suitable change ofthe radial coordinate it is possible to set R B or R Σ to 1, so onlytwo of the three radial functions are independent. The Kerrmetric is recovered for R Σ = R B = R M = M . Equa-tions (2) describe a Petrov D spacetime, and as a consequence,the Hamilton–Jacobi equation is separable with a generalizedCarter constant [53] — see also Appendix B. In Appendix A,we show that the subclass of this spacetime such that R B = R Σ = (1 + ξ/ r ) also admits separable Maxwell equationsfor test fields.A minimal deformation for the Kerr spacetime was in-troduced by Konoplya and Zhidenko in Ref. [54] and canbe obtained from Eqs. (2) by setting R Σ = R B = R M = M + η/ r . For the rest of the paper we consider thisbackground geometry. A. Event horizons and causal structure
For the Konoplya–Zhidenko metric the event horizon radiusis given by the largest positive real root of ∆ = r − Mr + a − η/ r =
0, which in general admits three (possibly complex-valued) solutions r k = M + √ M − a cos (cid:32) β − k π (cid:33) , (3) β =
13 cos − M − Ma + η (cid:0) M − a (cid:1) / , k = , , . We immediately notice that the Kerr limit η → ∆ = a < M and in the small η/ M limit, we have r = r + + η r + ( r + − r − ) − η (2 r + − r − ) r + ( r + − r − ) + O (cid:16) η (cid:17) , (4)where r ± = M ± √ M − a are the radii of the event andCauchy horizon for the Kerr spacetime. For | η | / M (cid:46) / ff erence between r calculated as a linear correction to r + and the exact value as in Eq. (3) is less than 1% for values of a (cid:46) . M . Equation (4) does not apply in the extremal limit,which must be treated separately, as in this case the leadingorder correction is O (cid:16) η / (cid:17) and r is given by r = M + (cid:114) η M − η M + O (cid:16) η / (cid:17) . (5)Under these assumptions, the compactness of the spacetimefor a < M is C = C Kerr (cid:32) − η r + ( r + − r − ) (cid:33) + O (cid:16) η (cid:17) , (6) being C Kerr = M / r + the compactness of the Kerr black hole,while in the extremal case ( C Kerr = C = − (cid:114) η M + η M + O (cid:16) η / (cid:17) . (7)Equation (6) indicates that positive (negative) values of η cor-responds to less (more) compact configurations.For a < M , instead of working with η , deviations from theKerr spacetime can be parametrized in terms of the quantity δ r , such that the position of the event horizon can be writtenas r = r + + δ r — cfr. Eq. (4), although δ r can account forlarge values of η/ M and it is not limited to a perturbative ex-pansion. This writing is obviously coordinate-dependent butsince we are using asymptotic Boyer–Lindquist coordinates,a significant deviation from Kerr should be similarly acknow-ledged by di ff erent observers.Di ff erently from the Kerr case, in the Konoplya–Zhidenkospacetime there exists no maximum value for a beyond whichthe spacetime always describes a naked singularity. As a al-ways enters quadratically in Eq. (3), without loss of generality,in the following we consider positive a .Although this spacetime belongs to a class of metrics whichare constructed to describe the spacetime outside the event ho-rizon, it is instructive to explore the implications inside thehorizon. This should be taken with great care and interpretedprudently, but it might give insights about what a small dif-ference at infinity entails about the structure of the spacetimeinside the horizon. This being said, in what follows we do notlimit our analysis to the largest positive real root of ∆ = R = η/ (cid:104) r (cid:16) r + a cos θ (cid:17)(cid:105) , from which we inferthat r = ∆ = η ± = (cid:20) Ma − M ± (cid:16) M − a (cid:17) / (cid:21) , (8)and to define three separate parameter regions as (I) η < η − ;(II) η − (cid:54) η (cid:54) η + ; (III) η (cid:62) η + . Then we sort configurationsaccording to the value of the spin parameter: below the Kerrbound, a < M ; highly spinning M (cid:54) a < a ∗ ≡ M / √
3; andultra spinning a (cid:62) a ∗ . Below the Kerr bound:
In region (I), there is only one realsolution given by r in Eq. (3) which is always negative andhence the spacetime describes a naked singularity.In region (II), the equation ∆ = r . The root r is always positivewhile r is negative (positive) for η − < η < < η < η + ). Inparticular for η = η − , r = (1 / M + √ M − a ), whilefor η = η + , r = (2 / M + √ M − a ).In region (III), r is the only positive-definite real root. Notice, however, that ∂ r /∂η diverges as η → η − . r r r t * (a) Region (II) with η − < η < r r r r t * (b) Region (II) with 0 < η < η + . r r t * (c) Region (III). Figure 1. Light-cone structure in advanced coordinates for a Konoplya–Zhidenko black hole below the Kerr bound.
Highly spinning:
For a = M and η > r = (2 / (cid:104) + cos (cid:16) arccos (27 η/ − (cid:17)(cid:105) . The othersolutions r and r are generically complex-valued but for0 < η < M /
27 the imaginary part goes to zero and thereal part is positive.For M < a < a ∗ , η − is positive and in the subregion of region(I) such that 0 < η < η − , the only real positive root is r . Inregion (II) the three real roots are positive and the event hori-zon is given by r , while in region (III) the only real solutionis r .Notice that for each value of η in the range 0 < η < M / aa + = M √ + (cid:114) + η M cos β + / , (9)with β + =
13 cos − M − η M − η (cid:0) M + η M (cid:1) / , for which the largest root of ∆ = r to r dis-continuously. Alternatively, for a fixed a , the largest root of ∆ = r to r at η = η − . Depending on the spe-cific values of the parameters the ratio r / r can be of severalorders of magnitude, and the compactness of the black holechanges accordingly. Ultra spinning:
For the particular case a = a ∗ with η > r = M / + (cid:112) η − M /
27 and r and r are complex-valuedunless η = M /
27, for which r = r = r = M /
3. For a > a ∗ , η ± in Eq. (8) become complex-valued and independentlyon the value of η , r and r are complex-valued, while r ispositive for η > ff erent than that of a Kerr black hole.As an example, consider a Konoplya–Zhidenko black holebelow the Kerr bound. Following a standard procedure, we define advanced coordinates and we plot null rays in Fig. 1,where t ∗ = t + r − r ∗ being r ∗ a tortoise coordinate defined byd r ∗ / d r = ( r + a ) / ∆ .In the external regions, i.e. for r > r , we observe a peelingstructure, typical of black-hole horizons. In region (II), for η − < η <
0, the light-cone structure is nearly similar to thatof a Kerr black hole, there are an outer and an inner horizonand a timelike singularity. In region (II) but for 0 < η <η + , a null trajectory encounters a black-hole horizon, a white-hole-like horizon and then again a black-hole-like horizon toeventually reach a spacelike singularity. In region (III) thereis only one horizon and the light-cone structure looks like theSchwarzschild one with a spacelike singularity.For further convenience, the angular velocity Ω = − g t ϕ / g ϕϕ at the horizon reads Ω k = ar k + a = a Mr k + η/ r k , (10)where the value of k depends on the specific values of theblack-hole parameters. B. Ergoregions
An ergosurface is a static limit surface, i.e. no static ob-server is allowed beyond this surface. Ergosurfaces in theseblack-hole spacetimes are defined as the roots of the equation g tt =
0, or equivalently r − Mr + a cos θ − η/ r =
0, whichread r erg k = M + √ M − a cos θ cos (cid:32) β erg − k π (cid:33) , (11) β erg =
13 cos − η + M − Ma cos θ (cid:0) M − a cos θ (cid:1) / , k = , , . For configurations below the Kerr bound in regions (II) and(III), the location of the ergosurface is r erg0 .For highly spinning configurations, the ergosurface is again r erg0 in regions (II) and (III), but it is piecewise and non-continuous in region (I): it is given by r erg0 in the angu-lar interval [ θ , θ ] and r erg2 in the complementary interval,[0 , θ ) ∪ ( θ , π ] where θ , ( θ = π − θ ) are the solutions of η = (cid:20) Ma cos θ − M − (cid:16) M − a cos θ (cid:17) / (cid:21) , (12)once the values of M , a and η are fixed; the maximum value of θ is cos − ( M / a ), attained for η → + . This means that whenpassing from a configuration in region (I) to one in region (II),the volume between the ergosurface and the event horizon, theergoregion, can change dramatically.Notice that for configurations below the Kerr bound andhighly spinning and values of η in regions (II) and (III) thevolume of the ergoregion is maximum for η = η − and it de-creases for larger values.For the particular case a = a ∗ , the location of the ergos-urface is r erg0 as long as η (cid:62) M /
27, but piecewise and dis-continuous for 0 < η < M /
27 as described above. For su-perspinning configurations, let θ ∗ the smallest root of cos θ = a ∗ / a . For 0 < η < M /
27 the ergoregion is piecewise anddiscontinuous: it is given by r erg1 for [0 , θ ∗ ) ∪ ( π − θ ∗ , π ], r erg2 for[ θ ∗ , θ ) ∪ ( θ , π − θ ∗ ], and r erg0 for [ θ , θ ] where θ , are againthe solutions of Eq. (12). For η (cid:62) M /
27 the ergoregion isstill piecewise but no longer discontinuous: it is given by r erg0 in the interval [ θ , θ ] and r erg1 in the complementary interval[0 , θ ) ∪ ( θ , π ].The fact that superspinning configuration for some valuesof the deformation parameter can have a piecewise and non-continuous ergoregion, i.e. no longer an ergosurface, poses aserious problem on the viability of these particular configur-ations as black-hole mimickers. We expect these particularsolutions to be dynamically unstable, but this analysis is bey-ond the scope of this paper and is left for future work. C. Photon orbits
Photon orbits for the Konoplya–Zhidenko black hole canbe studied starting from the geodesic equations derived in Ap-pendix B. In particular, the radial null geodesic in the equat-orial plane is˙ r = E + a E − L r + M ( L − aE ) r + η ( L − aE ) r , (13)where the dot indicates derivative with respect to an a ffi neparameter, while E and L are, respectively, the energy and theangular momentum of the photon, although it is more con-venient to characterize the geodesic by the impact parameter D ≡ L / E .The radius of photon orbits r c and its corresponding impactparameter D c are determined by Eq. (13) and its derivativeevaluated at r = r c = const. The problem is well-known forthe Kerr black hole [58], but the term introduced by the de-formation parameter η makes the equation no longer amenableto analytical methods for all values of L and E . Therefore, wedecide to adopt a small η/ M approximation and work belowthe Kerr bound. This guarantees some level of analyticity and exploits known results to be compared with. In what follows,the sign of a is important to distinguish between direct ( a > a <
0) orbits, so uniquely for the remainderof this subsection we allow a ∈ [ − M , M ].In practice, we expand the light ring radius r c and the im-pact parameter D c around the Kerr values in powers of η/ M .Here we report the leading-order corrections for the most fa-miliar cases, i.e. a = − M , , M . When a = − M we find r c ≈ M + η M , D c ≈ M + η M . (14)In the non-rotating limit, i.e. for a =
0, we get r c ≈ M + η M , D c ≈ √ M + √ η M . (15)For a = M the leading order correction is milder, r c ≈ M + (cid:114) η M , D c ≈ M + (cid:114) η M . (16)For general values of the deformation parameter, and to al-low the spin parameter above the Kerr bound, the radius ofthe photon orbits and the corresponding impact parameter canbe determined numerically. For | η | / M (cid:46) / r c and D c have maximum deviations from the Kerr values, respectively,of ∼
3% and ∼
4% for 0 (cid:54) a < . M , which reduce to lessthan 1% for − M (cid:54) a <
0. We have also checked that the lightring is always outside the horizon for η > η − and a (cid:54) a ∗ . D. The Konoplya–Zhidenko black hole as a solution of generalrelativity
Although these parametrically deformed metrics are built not to be exact solutions to any gravitational theory, it isan interesting exercise to figure out what kind of matter dis-tribution one would need in general relativity to obtain theKonoplya–Zhidenko black hole as an exact solution, andwhich energy conditions must be violated.We start by defining the stress-energy tensor out of the Ein-stein tensor, i.e., T µν = G µν / π , whose non-zero componentsare given in Appendix B.To characterize the would-be matter content of this space-time, a first possibility is to compute the eigenvalues of T µν .In particular, we identify the energy density with the oppositeof the eigenvalue relative to the timelike eigenvector, ρ = − η π r (cid:0) r + a cos θ (cid:1) . (17) In Refs. [59, 60] it is shown that the Konoplya–Zhidenko metric is an exactsolution of a (non-analytical) mixed scalar- f ( R ) gravitational theory. Being v t = { a + r / a , , , } , v r = { , , , } , v θ = { , , , } , and v ϕ = { a sin θ, , , } the eigenvectors of T µν , the timelike vector is v t for ∆ > v r otherwise. This matter distribution is concentrated close to the singularityand mainly along the equatorial plane, but it extends beyondthe event horizon although it decays quite fast for large valuesof the radius.Alternatively, the distribution of energy can be character-ized in an observer-dependent way by analysing the contrac-tion of the stress-energy tensor with the velocity of a physicalobserver, i.e., ρ = T µν u µ u ν . In view of the angular distributionof Eq. (17), for simplicity, consider a zero-angular-momentumobserver (ZAMO) in the equatorial plane, whose four-velocityin given in Appendix B. It can be verified that ρ ZAMO | θ = π/ = − η (cid:16) r + a (cid:17) π r . (18)Inspection of Eqs. (17) and (18) reveals that the sign ofthese energy densities is purely determined by the sign of η :negative (positive) values of η correspond to a positive (neg-ative) energy density; assuming a < M and in the small η/ M regime, they also correspond to configurations more (less)compact than a Kerr black hole with the same spin — cfr.Eq. (6). These results further imply that, for positive valuesof η , this matter distribution violates — at least — the weakenergy condition.Within this e ff ective description, it is possible to relate theabove matter distribution to the flux contribution to the Komarmass [61], 2 (cid:90) Σ d y √ h (cid:32) T µν − T g µν (cid:33) n µ ξ ν , (19)where Σ is a spacelike hypersurface that extends from theevent horizon to infinity, n µ the unit normal, h the determinantof the induced metric on Σ , T the trace of the stress-energytensor, and ξ ν the timelike Killing vector. Explicit evaluationof Eq. (19) indicates that this contribution can be of the samemagnitude of M for some specific values of the black-holeparameters, although for configurations below the Kerr boundit is typically of order ±
20% of M , where the sign depends onthe sign of η . It would be interesting to explore whether thisamount of putative matter can be used to model “dirty” blackholes as well.Configurations on the edge of η = η − , i.e. configurationsbetween regions (I) and (II) — which describe black holesfor a > M — seem particularly unstable. As the radiusof the event horizon and the volume of the ergoregion canchange abruptly and widely, one passes from small to enorm-ous violations of the energy conditions. Together with the oddpiecewise and disconnected ergosurface for some values ofthe parameter space, this might suggest that not every config-uration can mimic actual Kerr black holes.Nonetheless, if we drop the assumption that general relativ-ity is the correct gravitational theory, the discussion abovemight be extremely di ff erent. III. SUPERRADIANCE FROM THEKONOPLYA–ZHIDENKO BLACK HOLE
In the Konoplya–Zhidenko background, the scalar ( s = s = ±
1) wave equations are separablewith the angular part described by the spin-weighted spher-oidal harmonics equation and the radial part by ∆ − s dd r (cid:32) ∆ s + d R s d r (cid:33) + K − s (cid:16) r − M + η r (cid:17) K ∆+ s ω r − λ − s ( s + η r (cid:33) R s = , (20)where K ≡ ( r + a ) ω − am and λ ≡ A + a ω − ma ω , being A the eigenvalue of the angular equation, ω the frequency ofthe perturbation and m its azimuthal number. The angular ei-genvalue is also characterized by the harmonic number l . Asdiscussed in Appendix A, the physical information containedin the solution with spin-weight s is equivalent to that withspin-weight − s . This property will be particularly importantwhen computing the energy fluxes of electromagnetic wavesat infinity. A. Boundary conditions
To integrate Eq. (20) we need to supply it with bound-ary conditions. We first introduce a tortoise-like coordin-ate d r ∗ / d r ≡ ( r + a ) / ∆ and a new radial function Y s ( r ) = √ r + a ∆ s / R s ( r ) such that the radial equation becomesd Y s d r ∗ + K − s (cid:16) r − M + η r (cid:17) K + (4i rs ω − λ ) ∆ (cid:0) r + a (cid:1) − d G d r ∗ − G − s ( s + η ∆ r (cid:0) r + a (cid:1) Y s = , (21)where G = r ∆ / ( r + a ) + s ∆ (cid:48) / r + a ). Asymptotically( r → ∞ ), Eq. (21) becomesd Y s d r ∗ + (cid:32) ω + s ω r (cid:33) Y s ≈ , (22)whose solutions are Y s ∼ r ± s e ∓ i ω r ∗ where the plus (minus) signrefers to outgoing (ingoing) waves.Near the event horizon r ( r ∗ → −∞ ), let k ≡ ω − m Ω , Ω being defined in Eq. (10), then Eq. (21) becomesd Y s d r ∗ + ( k − i s σ ) Y s ≈ , σ = a + r (3 r − M )2 r (cid:16) r + a (cid:17) , (23)and the purely ingoing solution at the horizon is Y s ∼ exp [ − i ( k − i s σ ) r ∗ ] ∼ ∆ − s / e − i kr ∗ . B. Amplification factors
The asymptotic solutions to Eq. (22) can be used to definethe energy fluxes of bosonic fields at infinity. Since the η = η = η = η = - - - - - ω M Z , , η = η = - η = η = - - - - - ω M Z , , Figure 2. Spectra of the amplification factor for a scalar (left panel) and electromagnetic (right panel) field with l = m = ff a Konoplya–Zhidenko black hole with a = . M for selected values of η in units of M . Inset: Zoom in the superradiant region. Konoplya–Zhidenko spacetime shares the same asymptoticbehaviour and symmetries of the Kerr spacetime, the de-rivation of this section is very similar to what happens forKerr [62].Consider an incident wave of amplitude I from infinity pro-ducing a reflected wave of amplitude R , the asymptotic solu-tion to Eq. (22) can be written as Y s ∼ I e − i ω r ∗ r s + R e i ω r ∗ / r s . (24)The total energy flux at infinity per unit solid angle can becomputed out of the stress-energy tensor of the test fields asd E d t d Ω = d d t d Ω ( E in + E out ) = lim r →∞ r T rt , (25)where the ingoing and outgoing fluxes d E in / out / d t are propor-tional, respectively, to |I| and |R| [62]. When energy is ex-tracted from the black hole, the flux of energy through the ho-rizon is negative and energy conservation implies d E in / d t < d E out / d t . It is then possible to define the quantity Z s , l , m = d E out / d E in − s and quantum num-bers ( l , m ) o ff a black hole.In our case of interest, the amplification factors are Z , l , m = |R| |I| − , Z ± , l , m = |R| |I| (cid:32) ω B (cid:33) ± − , (26)where B = [ λ + s ( s + + ma ω − a ω . Notice thatthe expressions in Eq. (26) are the same as for Kerr as theasymptotic behaviour and the symmetries of the Konoplya–Zhidenko black hole are the same. However, the deformationparameter η changes the geometry of the near-horizon regionand it is responsible for a di ff erent amplification factor, as weshow in the next section. C. Numerical results
For general ω we need to numerical integrate the angularand radial equations. Our numerical routine works as follows. For each value of the spin-weight s , the quantum numbers( l , m ) and a ω , we first determine the angular eigenvalue usingthe Leaver method [63]. Second, fixed a value for η , we in-tegrate Eq. (21) from the horizon onwards until a su ffi cientlylarge radius. Then we compare our numerical solution andits radial derivative to the asymptotic behaviour in Eq. (24)and its derivative to extract the coe ffi cients I and R . Finally,we compute the amplification factor using Eq. (26). To in-crease the accuracy of this numerical procedure, we consider ahigher-order expansion near the horizon and in the asymptoticregion which reduces to those reported in the previous sectionat the leading order. The routine is repeated for several valuesof the frequency (typically) in the interval 0 < ω < m Ω .Modes with m (cid:54) Z s , l , m ( ω ) = Z s , l , − m ( − ω ) wecan consider positive frequencies only.We now define our working assumptions for what follows.We allow the deformation parameter in the range η (cid:62) η − andwe mainly exclude superspinning configurations from our in-vestigation, i.e., we focus on black holes below the Kerr boundand highly spinning in regions (II) and (III) introduced above.This has a practical advantage: the event horizon and the er-gosurface are always given by r and r erg0 . Despite the lackof observational evidence for rotating black holes beyond theKerr bound [64], it cannot be excluded that some highly spin-ning objects can be produced in high-energy astrophysicalphenomena that dynamically evolve in less spinning config-urations. Hence it makes sense to explore a bit this parameterregion.Some of our results are presented in Fig. 2 and more resultsare available online [65]. Both for scalar and electromagneticfields with quantum numbers l = m =
1, scattered o ff a blackhole with spin a = . M , we observe in the insets of Fig. 2that the position of the maximum of the amplification factor isclose to the superradiant threshold ω = m Ω where the curvebecomes very steep, as in the Kerr case.In absolute values, the maximum amplification factor isabout 0.4% and 4.4% for scalar and electromagnetic waves,as for Kerr. However, in the left panel of Fig. 3 we noticethat for scalar waves scattering o ff a deformed black hole with η/ M ≈ /
100 the maximum amplification factor is about 6%larger than in the non-deformed Kerr case, while for electro-magnetic waves, we observe a maximum amplification factorroughly 1% larger than in the Kerr case for η/ M ≈ − / η / M Z s , , m a x / Z s , , m a x , K e rr [ % ] sc a l a r e l e c t r o m agne t i c η / M I s , , / I s , , K e rr [ % ] - sc a l a r e l e c t r o m agne t i c Figure 3. Maximum value of the amplification factor Z s , , (leftpanel) and integral of the superradiant spectrum I s , , (right panel)for a scalar and electromagnetic field with l = m = η , normalized to the maximum value in the Kerr case, i.e. η = a = . M . These values of η/ M are not universal, but depend on thevalue of a / M . For smaller values of a / M , the maximum valueof Z s , l , m gets smaller, the position of the peak moves towardssmaller values of η/ M and the frequency range for whichthe amplification factor is positive shrinks. For configura-tions with higher spin, say at the Thorne limit a = . M ,the scalar (electromagnetic) amplification factor can be up to15% (1%) larger than in the Kerr case. This bigger amplifica-tion factor does not mean that these deformed spacetimes aremore superradiant than the Kerr spacetime, as the quantity I s , l , m = (cid:90) m Ω d ω Z s , l , m , (27)is always smaller than in Kerr, for positive values of η , asshown in the right panel of Fig. 3. However, a bosonic wavewith frequency close to the superradiant threshold can be sig-nificantly more enhanced in a deformed Kerr background. Fornegative values of η , which correspond to more compact con-figurations, I s , l , m is typically bigger than in Kerr and maximalclose to η = η − . For large enough positive values of the de-formation parameter the maximum value of the amplificationfactor and the range of superradiant frequencies are alwayssmaller than in the Kerr case. The physical explanation to thisresult is that, typically, for values of η/ M (cid:44) a =
0, superradiancedisappears and we recover the recent results on absorption indeformed Schwarzschild backgrounds [66, 67].In the inset of the left panel of Fig. 3, we observe that thesame maximum value of the amplification factor for a scalarfield is obtained for Kerr ( η =
0) and for η/ M ≈ / ff erent.In Fig. 4 it is evident that the most superradiant mode cor-responds to the minimum allowed value of l = m , as in theKerr case. Modes with di ff erent values of ( l , m >
0) qualit-atively share the same behaviour with the l = m = l = m = l = m = l = , m = - - - - - ω M Z , l , m Figure 4. Typical spectra of the amplification factor Z , l , m for di ff erentsuperradiant scalar field modes o ff a Konoplya–Zhidenko black holewith a = . M and η = M / though the maximum amplification factor is hierarchicallysmaller than the dominant one. For example, in the range0 . M (cid:46) a < M , for both scalar and electromagnetic fields wefind Z max s , , / Z max s , , ∼ − while Z max s , , / Z max s , , ∼ . a (cid:38) . M .For the l = m = Z max s , , and I s , , are always smallerthan in the Kerr case for positive values of η and a < M , butfor negative values the amplification factor can be bigger thanin Kerr. Again, this could be interpreted as a consequence ofthe fact that, for a given a , the ergoregion is larger than theKerr ergoregion for negative values of η . On the other hand,the l = m = η when a (cid:38) . M . For the remaining modes, i.e. with m (cid:54)
0, we have verified that the amplification factor is alwaysnegative, meaning that these modes are not superradiant.As previously discussed, the Konoplya–Zhidenko blackhole also admits superspinning configurations, i.e. with spinparameter a > M . If the rotation parameter is (slightly) abovethe Kerr bound, in principle, such energy extraction could rap-idly spin down these configurations to produce a black holewith a < M . a = M a = M a = M ω M Z , , Figure 5. Spectra of the amplification factor for a scalar field with l = m = ff a superspinning Konoplya–Zhidenko black hole with η = M for selected values of a . For completeness, we consider the scattering of a scalarfield o ff a superspinning black hole. We observe in Fig. 5that for η/ M = < η < M /
27 the position of the event horizon is not al-ways given by r for all values of a , and perhaps even moregravely, the ergosurface can be piecewise and non-continuous. D. Massive scalar fields
The extension to a massive scalar field with mass µ s (cid:126) isquite simple: such mass term in the Klein–Gordon equationintroduces, after separation, a quantity − µ s r ∆ / (cid:16) r + a (cid:17) inthe coe ffi cient of Y in Eq. (21) and shifts the frequency of theangular equation as ω → ω − µ s .The boundary conditions are slightly modified. In par-ticular, purely ingoing solutions at the horizon still require Y ∼ e − i kr ∗ , while the asymptotic behaviour at infinity is Y ∼ r − M µ s /(cid:36) e (cid:36) r ∗ ∼ r M ( µ s − ω ) /(cid:36) e (cid:36) r , (cid:36) = ± (cid:113) µ s − ω . (28)Massive waves can be superradiant for frequencies in therange µ s < ω < m Ω , while they are trapped near the horizonand exponentially suppressed at infinity for ω < µ s .The numerical routine for the computation of the ampli-fication factor is adapted from that used for massless waves,correcting the asymptotic behaviours accordingly. We limitthis analysis to the l = m = µ s < ω < Ω − µ s . Our results, as those inthe top panel of Fig. 6, show that superradiance grows withthe spin parameter a and is less pronounced for more massivefields, as in the Kerr spacetime. The bottom panel of Fig. 6shows that massive waves can be more amplified than in aKerr background with the same spin parameter for some val-ues of the deformation parameter, analogously to what wefound for massless fields, though waves with larger massesare still less enhanced. Even in this case, for positive valuesof η , the Konoplya–Zhidenko black hole is less superradiantthan Kerr in the sense of Eq. (27) with the interval of integra-tion adapted to [ µ s , m Ω ].Kerr black holes develop superradiant instabilities againstmassive fields [68] which can be used to constraint the exist-ence and the mass of ultralight bosons, i.e. using black holesas “particle detectors” [69]. In addition, the bosonic cloudcan produce long-lasting, monochromatic gravitational-wavesignals observable, in principle, in the sensitive band of cur-rent detectors [47, 48, 70]. We do not expect this picture tobe considerably changed for deformed rotating black holes. M μ = μ = μ = μ = μ = ω M Z , , M μ = μ = μ = μ = μ = η / M Z , , m a x / Z , , m a x , K e rr [ % ] Figure 6. (Top panel) Spectra of the amplification factor for massivescalar fields with l = m = ff a Konoplya–Zhidenko black hole with a = . M and η = M / Z , , for a massive scalar field with l = m = η normalized to the maximum value in the Kerr case, i.e. η =
0, for a = . M and for selected values of the mass parameter. Small values of the deformation parameter unveiled an inter-esting feature in the massless case, and we also expect a goodblack-hole mimicker not to turn upside-down the Kerr metric.This motivates us to investigate the stability of the Konoplya–Zhidenko spacetime against massive scalar fields in the small η/ M limit. Remarkably, in this limit, in the low-frequencyregime, i.e. for ω M (cid:28) a ω (cid:28) M µ s (cid:28)
1, the problem can be tackled withanalytical methods — see Appendix C for details.At leading order, the growth time of instability τ for a Kerrblack hole perturbed by an axion with mass m axion = µ axion (cid:126) = − eV is τ = (cid:16) . · s (cid:17) (cid:32) µ axion µ s (cid:33) Ma M µ s ) , (29)to which the deformation parameter adds the contribution(valid as long as η/ M is small) δτ = − (cid:16) . · s (cid:17) (cid:32) µ axion µ s (cid:33) (cid:32) η/ M . (cid:33) (cid:18) Ma (cid:19) M µ s ) . (30)Equation (30) implies that for positive (negative) values ofthe deformation parameter (within a perturbative regime),the growth time of instability is shorter (longer), i.e. theKonoplya–Zhidenko black hole is more (less) unstable than0Kerr. For an axion cloud around a supermassive black holewith M = M (cid:12) , M µ axion ≈ − , and the growth time ofinstability would be shorter but comparable with the age ofthe Universe. Yet, this timescale should also be shorter thanthe decay time of the particle for the instability to be reallye ff ective.As this preliminary result relies on several assumptions, it isto be confirmed by an exhaustive computation of quasi-normalmodes and bound states, which is left for future work. In fact,this result is valid for slowly rotating black holes hence wecannot conclude whether highly spinning configurations aremore unstable or not. IV. DISCUSSION
In this paper we have studied the superradiant scatteringof scalar and electromagnetic test fields o ff a deformed-Kerrblack hole. In these deformed-Kerr spacetimes, the best thatwe can do is to study test fields propagating in a fixed back-ground, but often test fields are a good proxy and the resultsfor scalar and electromagnetic waves are similar to those forgravitational waves. However, this is not always true and thecase of superradiance in general relativity is illustrative: themaximum amplification factors are approximately 0.4% formassless scalar fields with l = m =
1, 4.4% for electromag-netic waves with l = m = l = m = ffi ciently fast, provided the presence ofan ergoregion, but this would most likely require a full nu-merical simulation. In this sense, our results represent a firststep in the investigation of the phenomenon of superradiancein deformed rotating spacetimes.Before exploring superradiant scattering around these de-formed black holes, we have studied their structure thor-oughly. The simple Konoplya–Zhidenko metric, which shareswith Kerr the same symmetries and asymptotic behaviour,translates into a complicated causal structure. Depending onthe values of the parameters, these configurations can havefrom zero up to three horizons. When the spin parameteris above the Kerr bound, the ergoregion can be piecewiseand non-continuous. To consider this model as a valuableKerr black-hole mimicker we might probably need to ex-clude some regions of the parameter space. Moreover, whenconsidered as non-vacuum general-relativistic solutions, theseconfigurations require to be sustained by some exotic mat-ter. Yet, in the small-deformation limit and below the Kerrbound, we have shown that the horizon and the light ringradii are slightly modified with respect to the Kerr values ofonly a few percents. Optimistically, future observations ofe.g. black-hole shadows could set bounds on the deformationparameter [71, 72].Regarding superradiance, we have found maximum ampli- fication for the minimum value allowed of l = m (for scalarand electromagnetic fields l = m =
1) and highest values ofthe spin parameter, similarly to what happens for Kerr blackholes. Our numerical results show that for large deformationsand considering the same spin, superradiance is highly sup-pressed with respect to the Kerr black hole. This can be inter-preted in terms of the volume of the ergoregion: for a Kerr anda deformed-Kerr black hole with same a , for positive valuesof the deformation parameter the ergoregion is smaller in thelatter case as well as the amount of energy that can be extrac-ted, and the e ff ect of superradiance is damped. This seemsin agreement with the fact that the proper volume of the er-goregion of slowly rotating black holes in quadratic gravitydecreases with respect to the general-relativistic case, sug-gesting a smaller amplification factor [73]. Our results forsuperspinning configurations shown in Fig. 5 are compatiblewith the fact that the energy extraction by the Penrose processfor the superspinning Johannsen–Psaltis [24] and Konoplya–Zhidenko metric can be significantly larger than for a Kerrblack hole [74, 75]. Analogously, for negative values of thedeformation parameter, which correspond to more compactconfigurations, the volume of the ergoregion can be largerthan that of a Kerr black hole with the same mass and spin,and as a consequence, the superradiant phenomenon can beenhanced.The most interesting feature that we have found is the ex-istence of an interval of small values of the deformation para-meter for which the maximum of the amplification factor islarger than in Kerr. This interval contains positive values of η for scalar fields when a > . M , while for electromagneticfields it requires a ≈ M , meaning that there are configurationsless compact than a Kerr black hole with the same mass andangular momentum for which the superradiant scattering canbe larger. For di ff erent values of the spin parameter, to havemore superradiance one needs more compact configurations,i.e. with M / r >
1. If this trend is respected by gravitationalwaves, then a higher amplification factor for less compact de-formed spacetimes would occur for values of a extraordinaryclose to the extremal case.Besides, we have also presented some initial results onmassive scalar fields. Roughly, their behaviour is similar tothe massless and Kerr cases. Under some approximations —small frequency, small spin parameter, small scalar mass andsmall deformation parameter — the frequency eigenvalue canbe determined analytically with asymptotic matching tech-niques. For the expected most unstable mode l = m =
1, pos-itive values of the deformation parameter shorten the growthtime of instability, meaning that these spacetimes are more un-stable than Kerr against massive scalar fields. The validity ofthis result is limited: it needs to be taken with great care andone should not infer too much information, as it is based on alarge number of assumptions. A complete numerical investig-ation is left for future work.Within this context, knowledge of superradiant instabilit-ies can be used to put bounds on the existence and mass ofultralight particles. Nonetheless, if a black hole acts as a“particle detector”, the presence of (dark) matter around itmight modify the geometry and spoil superradiant e ff ects. In1Section II D we interpreted the Konoplya–Zhidenko metric interms of an exotic matter distribution and show that the mat-ter flux contribution to the Komar mass of the spacetime canbe a significant fraction of the black-hole mass M . To havethis contribution less than, say, 10% of M , and not to suppresssuperradiant e ff ects, the deformation parameter should takevalues | η | / M (cid:46) /
10, indicating once again that the most in-teresting phenomenology corresponds to small deformationsof the Kerr geometry.The spectra of the amplification factor can look very similarwhen comparing a Kerr black hole and a Konoplya–Zhidenkoblack hole with a small deformation parameter, as well as amassless scalar and a massive scalar with very small massparameter. In addition to this fact, there might be a sim-ilar “degeneracy” when comparing the spectra of a masslessscalar o ff a slightly deformed black hole with the spectra ofa little massive scalar o ff a Kerr black hole. We have veri-fied that this could actually happen in a number of cases.Our criterion for degenerate spectra is when both the max-imum value of the amplification factor, its corresponding fre-quency, its integral as in Eq. (27) and the threshold frequencyare the same within a tolerance of 5%. Since superradianceis suppressed for massive scalars, we expect this degeneracyto correspond to positive values of η . This is true for inter-mediate values of the spin parameter, but for a (cid:38) . M theparameter space also include small negative values of η/ M .As an example, for a = . M the spectra of a massivescalar field with M µ s ≈ .
025 and M µ s ≈ .
05 o ff a Kerrblack hole resemble the spectra of a massless scalar field o ff a Konoplya–Zhidenko black hole with − . (cid:46) η/ M (cid:46) . . (cid:46) η/ M (cid:46) . ff er ergoregion instabil-ity. The presence of the horizon guarantees from the begin-ning the key ingredient for superradiance: dissipation. In viewof testing the Kerr hypothesis, it could be interesting to usethis same parametrization and substitute the horizon with apartially reflective surface and see in which limits superradi-ance disappears. Other possible developments might includeconsidering non-minimally coupled scalar fields, or use theresults in Ref. [29] to study superradiance of test fields o ff Kerr–Sen and Einstein–dilaton–Gauss–Bonnet black holes.
ACKNOWLEDGMENTS
We are grateful to Vitor Cardoso for several valuable com-ments and a thorough reading of this manuscript, and to Ma-riano Cadoni for discussions. EF acknowledges partial fin-ancial support by CNPq Brazil, process no. 301088 / Appendix A: The Klein–Gordon and Maxwell equations indeformed backgrounds
Being s the spin weight of the test field, in linear per-turbation theory the scalar ( s =
0) and Maxwell ( s = ± Φ is easily obtained from (cid:3) Φ =
0, where the D’Alambert operator is built out of themetric (2). To derive the Maxwell equations in such space-time, we follow the method proposed in Ref. [76].First we choose a suitable null tetrad e µ ( a ) = { l µ , n µ , m µ , ¯ m µ } that easily reduces to the Kinnersley tetrad [77] in the Kerrspacetime, i.e. l µ = ∆ (cid:34) r R Σ + a , ∆ R B , , a (cid:35) , (A1a) n µ = Σ (cid:34) r R Σ + a , − ∆ R B , , a (cid:35) , (A1b) m µ = √ ρ [i a sin θ, , , i csc θ ] , (A1c)where ∆ = r R Σ − R M r + a , Σ = r R Σ + a cos θ and ρ = r √ R Σ − i a cos θ . The tetrad vectors satisfy e µ ( a ) e ( b ) µ = − − . (A2)The non-vanishing spin coe ffi cients are (cid:37) = − (cid:16) r R Σ (cid:17) (cid:48) R B Σ − i a cos θ Σ , (A3a) (cid:15) = − i a cos θ rR B Σ rR B − (cid:16) r R Σ (cid:17) (cid:48) √ R Σ , (A3b) µ = − ∆ R B Σ (cid:104)(cid:16) r R Σ (cid:17) (cid:48) + aR B cos θ (cid:105) , (A3c) γ = ∆ (cid:48) R B Σ − ∆ Σ (cid:16) ¯ ρ + r √ R Σ (cid:17) rR B √ R Σ (cid:16) r R Σ (cid:17) (cid:48) + a cos θ , (A3d) τ = a sin θ √ R B Σ ¯ ρ (cid:104) aR B cos θ − i (cid:16) r R Σ (cid:17) (cid:48) (cid:105) , (A3e) α = √ Σ ρ (cid:104) cot θ (cid:16) Σ − a − r R Σ (cid:17) + a sin θ R B (cid:16) r R Σ (cid:17) (cid:48) + sin θ (cid:16) ar (cid:112) R Σ − a ρ (cid:17) (cid:105) , (A3f) π = i a sin θ √ Σ ρ r (cid:112) R Σ + (cid:16) r R Σ (cid:17) (cid:48) R B − ρ , (A3g)2 β = √ Σ ¯ ρ i a sin θ r (cid:112) R Σ − (cid:16) r R Σ (cid:17) (cid:48) R B + Σ cot θ . (A3h)The sourceless decoupled Newman–Penrose equations forthe massless spin-1 field are given by [76] (cid:2) ( D − (cid:15) + ¯ (cid:15) − (cid:37) − ¯ (cid:37) ) ( ∆ + µ − γ ) − ( δ − β − ¯ α − τ + ¯ π ) (cid:16) ¯ δ + π − α (cid:17)(cid:105) φ = , (A4a) (cid:2) ( ∆ + γ − ¯ γ + µ + ¯ µ ) ( D − (cid:37) + (cid:15) ) − (cid:16) ¯ δ + α + ¯ β + π − ¯ τ (cid:17) ( δ − τ + β ) (cid:105) φ = , (A4b)where D = l µ ∇ µ , ∆ = n µ ∇ µ and δ = m µ ∇ µ , and the complexfields are defined as φ = F µν l µ m ν and φ = F µν ¯ m µ n ν , being F µν the electromagnetic field tensor.Di ff erently from the Kerr case, the spin coe ffi cient (cid:15) is gen-erally non-zero and as a consequence Eqs. (A4) are not separ-able into a radial and angular part. However, one can alwaysperform a null rotation of the tetrad to set (cid:15) = R B = (cid:15) = R Σ , R Σ = (cid:18) + ξ r (cid:19) , (A5)where ξ is a constant parameter.Under the above assumptions, decomposing the test fieldsas e − i ω t e i m ϕ S ( θ ) R s ( r ), the scalar and electromagnetic waveequations separate, with the angular part described by thespin-weighted spheroidal harmonics equation1sin θ dd θ (cid:32) sin θ d S d θ (cid:33) + (cid:32) a ω cos θ − m sin θ − a ω s cos θ − ms cos θ sin θ − s cot θ + s + A (cid:33) S = , (A6)while the radial part by the following equation, ∆ − s dd r (cid:32) ∆ s + d R s d r (cid:33) + (cid:34) K − i s ∆ (cid:48) K ∆ + srR Σ ω − λ + s ( s +
1) ( ∆ (cid:48)(cid:48) − (cid:35) R s = , (A7)where K = (cid:16) r R Σ + a (cid:17) ω − am and λ = A + a ω − am ω .The radial functions R , R and R − correspond to Φ , φ and φ /ρ .Equation (A6) together with regular boundary conditions at θ = { , π } is an eigenvalue problem for the separation con-stant A . For each value of s , m and a ω , the eigenvalues areidentified by a number l , whose smallest value is max ( | m | , | s | ).The eigenfunctions form a complete and orthonormal set in θ ∈ [0 , π ]. For a ω =
0, Eq. (A6) reduces to the spin-weightedspherical harmonics equation and A = ( l − s )( l + s +
1) [79];for a ω (cid:28)
1, Eq. (A6) can be solved perturbatively [80], but ingeneral it must be solved numerically [81].To integrate Eq. (A7) it is necessary to give boundary con-ditions at the horizon and at infinity. Therefore, we first in-troduce a tortoise-like coordinate given by dr ∗ / dr ≡ ( r R Σ + a ) / ∆ and the radial function Y s ( r ) = (cid:112) r R Σ + a ∆ s / R s ( r ).With these substitutions, Eq. (A7) becomesd Y s d r ∗ + ∆ (cid:2) s ( s +
1) ( ∆ (cid:48)(cid:48) − / − λ + s ω ( r + ξ ) (cid:3)(cid:2) ( r + ξ ) + a (cid:3) + K − i K s ∆ (cid:48) (cid:2) ( r + ξ ) + a (cid:3) − d G d r ∗ − G Y s = , (A8)where G = s ∆ (cid:48) / r R Σ + a ) + r √ R Σ ∆ / ( r R Σ + a ) and r isan implicit function of r ∗ .At infinity ( r ∗ → ∞ ), Eq. (A8) can be approximated asd Y s d r ∗ + (cid:32) ω + s ω r (cid:33) Y s = , (A9)from which we see that Y s ∼ r ± s e ∓ i ω r ∗ , where the upper (lower)sign refers to outgoing (ingoing) waves.Near the event horizon r ( r ∗ → −∞ ), Eq. (A8) becomesd Y s d r ∗ + ( k − i s σ ) Y s = , (A10)where k = ω (cid:32) + ξ (2 r + ξ )( r + ξ ) + a (cid:33) − m Ω , (A11) σ = r (cid:16) − R (cid:48) M ( r ) (cid:17) − ξ − a r (cid:2) ( r + ξ ) + a (cid:3) . (A12)The purely ingoing solution at the horizon is given by Y s ∼ exp [i( k − i σ ) r ∗ ] ∼ ∆ − s / e − i kr ∗ .Teukolsky and Press showed that one solution of the Teuk-olsky equation with spin-weight s contains the same physicalinformation of that with spin-weight − s [62]. This result is aconsequence of the fact that the Kerr spacetime is stationaryand axisymmetric. This fact holds for this class of deformedmetrics too, in fact, repeating the same derivation for Eq. (A7)but starting with the tetrad˜ l µ = − Σ∆ n µ , ˜ n µ = − ∆ Σ l µ , ˜ m µ = r √ R Σ − i a cos θ r √ R Σ + i a cos θ ¯ m µ , (A13)related to Eqs. (A1) by the simultaneous transformation ϕ →− ϕ , t → − t , one finds that, after the separation of the radialand angular variables, the radial function ˜ R s satisfies Eq. (A7)with s → − s and it is related to R − s through˜ R s = (cid:32) ∆ (cid:33) s R − s . (A14) Appendix B: Einstein tensor, geodesic equations and zeroangular momentum observers for the Konoplya–Zhidenkospacetime
The non-zero components of the Einstein tensor for theKonoplya–Zhidenko metric read3 G tt = η r (cid:16) θ − (cid:17) a + r (cid:16) − r + Mr + η (cid:17) − a cos θ sin θ r (cid:0) r + a cos θ (cid:1) , (B1a) G t ϕ = a η sin θ a (cid:16) r + a (cid:17) cos θ + r (cid:16) r − Mr + a r − η (cid:17) r (cid:0) r + a cos θ (cid:1) , (B1b) G rr = η r ∆ (cid:0) r + a cos θ (cid:1) , (B1c) G θθ = − η (cid:16) r + a cos θ (cid:17) r (cid:0) r + a cos θ (cid:1) , (B1d) G ϕϕ = − η sin θ a (cid:16) a − r + Mr + r η (cid:17) cos θ + r (cid:104) r + a r − a (cid:16) (2 M − r ) r + η (cid:17)(cid:105) r (cid:0) r + a cos θ (cid:1) . (B1e)The geodesic equations can be obtained via the Euler–Lagrange equations from the Lagrangian L = g µν ˙ x µ ˙ x ν ,where a dot indicates di ff erentiation with respect to an a ffi neparameter λ . However, it is simpler to use the integrals ofmotion, two of which are related to the obvious symmetriesof the metric, i.e. stationarity and axisymmetry, that can beexpressed respectively by p t ≡ g tt ˙ t + g t ϕ ˙ ϕ = − E , p ϕ ≡ g ϕϕ ˙ ϕ + g t ϕ ˙ t = L z , (B2)where E and L z represent the energy and the angular mo-mentum along the ϕ axis of the particle. Another constantof motion can be obtained observing that the Hamiltonian H = g µν p µ p ν , where p µ = ∂ L /∂ ˙ x µ , is independent of thea ffi ne parameter. Therefore we can write H = − (cid:15) , where (cid:15) is a constant parameter that can be + , , −
1, respectively,for timelike, null and spacelike geodesics. The last integral ofmotion is less obvious and it is related to the separability ofthe Hamilton–Jacobi equation˙ S = g µν ∂ S ∂ x µ ∂ S ∂ x ν , (B3)where S is a function of λ and the coordinates. In fact, withthe ansatz S = − (cid:15) λ − Et + S θ ( θ ) + S r ( r ) + L z ϕ , Eq. (B3)separates into an angular and a radial part. The (generalized)Carter constant Q = K − ( aE − L z ) is related to the separationconstant K associated to the hidden symmetry of the metricgenerated by a second-order Killing tensor K µν that satisfies ∇ ( ρ K µν ) =
0, where the round parentheses denote symmetriz-ation with respect to the indices. The explicit form of K µν is K µν = Σ l ( µ n ν ) + r g µν , (B4)where Σ = r + a cos θ while l µ and n µ are the vectors definedin Eqs. (A1) with R Σ = R B = R M = M + η/ r .Using these four integrals of motion it is possible to writethe geodesic equations as˙ t = E + (cid:16) Mr + η (cid:17) (cid:16) ( r + a ) E − aL z (cid:17) r ∆Σ , (B5a) ˙ ϕ = r Σ a (cid:16) Mr + η (cid:17) E − a L z r ∆ + r L z sin θ , (B5b) Σ ˙ r = (cid:104) aL z − E (cid:16) a + r (cid:17)(cid:105) − ∆ (cid:104) ( aE − L z ) + Q + r (cid:15) (cid:105) , (B5c) Σ ˙ θ = a cos θ ( E − (cid:15) ) − L z cot θ + Q . (B5d)The four-velocity of a zero-angular-momentum observer inthe equatorial plane is readily obtained, u t = r + a (cid:16) r + Mr + η (cid:17) r ∆ , (B6a) u r = − (cid:114) (cid:0) a + r (cid:1) (cid:0) η + Mr (cid:1) r , (B6b) u ϕ = a (cid:16) Mr + η (cid:17) r ∆ . (B6c) Appendix C: Frequency eigeinvalues in the low-frequency,small-mass and small-deformation limit
In the low-frequency regime, i.e., ω M (cid:28) a ω (cid:28) ff a Kerr blackhole can be computed analytically [82–84]. The angular equa-tion reduces to the scalar spherical harmonics equation andthe angular eigenvalue λ can be approximated as l ( l + M µ s (cid:28) η/ M (cid:28) r limit the radial equation for a massive scalarfield in the Konoplya–Zhidenko background becomes R (cid:48)(cid:48) ( r ) + r R (cid:48) ( r ) + (cid:32) − l ( l + r + M µ s r + ω − µ s (cid:33) R ( r ) = . (C1)4Defining k = µ s − ω , ν = M µ s / k , and x = kr the aboveequation reads xR (cid:48)(cid:48) ( x ) + R (cid:48) ( x ) + (cid:32) − l ( l + x + ν − x (cid:33) R ( x ) = , (C2)i.e., the same equation which governs an electron in the hy-drogen atom. For large x the two independent solutions ofEq. (C2) behave as R ( x ) ∼ x ± ( ν + e ∓ x / . Since we are in-terested in the unstable modes we take the solution with theupper signs, and the complete solution to Eq. (C2) with suchasymptotic behaviour is R ( x ) = e − x / x l U ( l − ν + , l + , x ) (C3)being U the confluent hypergeometric function.The regularity of the electron wave-function in x = ν as ν = l + + n with n positive. As theboundary conditions in this case are slightly di ff erent fromthe quantum mechanics problem (ingoing waves at the hori-zon) we guess ν = l + + n + δν where δν is a small complexnumber.In the small x limit, Eq. (C3) is R ( x ) ≈ Γ ( − l − Γ ( − l − ν ) x l + Γ (2 l + Γ ( l − ν + x − l − . (C4)In terms of the coordinate r and in the small δν limit R ( r ) ≈ ( − n (2 l + n + l + kr ) l + ( − n + δν (2 l )! n !(2 kr ) − l − . (C5)In the near-horizon region we write R ( r ) = ˚ R ( r ) + η δ R ( r )and we solve order by order in η/ M . We define a new dimen-sionless coordinate x ≡ ( r − r + ) / ( r + − r − ) and the quantity q ≡ ( am − Mr + ω ) / ( r + − r − ) where r ± = M ± √ M − a arethe radial location of the Kerr event and Cauchy horizon.At zeroth order, the radial equation reduces to x ( x + ˚ R (cid:48)(cid:48) ( x ) + x (2 x + x +
1) ˚ R (cid:48) ( x ) + (cid:16) q − l ( l + x ( x + (cid:17) ˚ R ( x ) = , (C6)whose general solution is a combination of the associated Le-gendre functions c P l q (1 + x ) + c Q l q (1 + x ) which repres-ent, respectively, the ingoing and outgoing waves at the hori-zon.Now assume there exists an intermediate region in whichthe two solutions calculated asymptotically and close to thehorizon overlap. Then the small x limit of the asymptotic solu-tion (C5) must be equal to the large x limit of the near-horizonsolution, supplied with the requirement of no outgoing wavesat the horizon ( c = P l q (1 + x ) ∼ (2 l )! x l l ! Γ ( l + − q ) + ( − − − l l ! x − l − (2 l + Γ ( − l − q ) . (C7) The constant c can be determined by comparing the r l terms, c = (2 k ) l ( r + − r − ) l ( − n l !(2 l + n + Γ ( l + − q )(2 l + l )! , (C8)while by comparing the r − l − terms we get δν = q [2 k ( r + − r − )] l + (cid:32) l !(2 l )!(2 l + (cid:33) × (2 l + n + n ! l (cid:89) j = (cid:16) j + q (cid:17) . (C9)Finally, the relation among n , δν and ω = σ + i γ gives σ ≈ µ s from the real part, while from the imaginary parti γ = (cid:18) M µ s l + + n (cid:19) δν M . (C10)Now we are able to give an estimate for the growth timeof the instability. At zeroth order, combining Eqs. (C9)and (C10) we notice that for m > l = m = n = γ = µ s aM ( M µ s ) , (C11)and the growth time, for an axion with mass m axion = µ axion (cid:126) = − eV, τ ≡ /γ = (cid:16) . · s (cid:17) (cid:32) µ axion µ s (cid:33) Ma M µ s ) . (C12)At first order, the zeroth-order solution enters as a “sourceterm”, x ( x + δ R (cid:48)(cid:48) ( x ) + x (2 x + x + δ R (cid:48) ( x ) + (cid:16) q − l ( l + x ( x + (cid:17) δ R ( x ) = T ( x ) , (C13)where r + ( r + − r − ) T ( x ) = − ˚ R (cid:48) ( x ) − q x + q (cid:16) r − − r − r + + r + (cid:17) Mr + − l ( l + ˚ R ( x ) , (C14)with R = c P l q (1 + x ) and c given by Eq. (C8).The homogenous problem associated to Eq. (C13) for δ R is the same as in Eq. (C6) for ˚ R , meaning that its generalsolution is again a combination of the associated Legendrefunctions, c P l q (1 + x ) + c Q l q (1 + x ). Again, c can beset to zero by the request of no outgoing waves at the horizon.The particular solution can be obtained with the method ofvariation of constants, δ R , p = − δ R , (cid:90) d z T ( z ) δ R , ( z ) z (1 + z ) W ( z )5 + δ R , (cid:90) d z T ( z ) δ R , ( z ) z (1 + z ) W ( z ) , (C15)where W ( x ) is the Wronskian associated with δ R , ( x ) = P l q (1 + x ) and δ R , ( x ) = Q l q (1 + x ).As in the zeroth-order calculation, assume that there ex-ists an intermediate overlapping region in which the solutionin Eq. (C5) is glued with the large r behaviour of the near-horizon solution.At this stage, we focus on the l = m = n = l = δ R , p ∼ − c x (cid:104) R Mr + + R r − + R r − r + + R r + (cid:105) Mr + ( r + − r − ) q (1 + q ) (1 − q ) Γ (1 − q ) , (C16)where R = q (1 − q ) (cid:16) q + (cid:17) (cid:2) ψ ( − q ) + γ E (cid:3) , (C17a) R = q (1 − q ) (cid:16) q − q + (cid:17) , (C17b) R = q − q + q − q + q − , (C17c) R = − q + q + q − q + q − , (C17d)being ψ ( z ) the digamma function, γ E the Euler–Mascheroniconstant and q is now meant to be computed for m =
1, we repeat what we have done for the zeroth-order solution, butmatching Eq. (C5) with c ˚ R + η (cid:16) c δ R , + δ R , p (cid:17) . We firstsolve for c and find that δν gains a correction proportionalto η , whose imaginary part sums up to γ computed at zerothorder, δγ = η k M µ s ( r + − r − )48 r + q (cid:0) q + (cid:1) (cid:104) g Mr + + g r − + g r − r + + g r + (cid:105) , (C18)where g = q (cid:16) q + (cid:17) (cid:16) q + (cid:17) (cid:61) ψ ( − q ) , (C19a) g = − q (cid:16) q + (cid:17) (cid:16) q + (cid:17) , (C19b) g = (cid:16) q + q + q + (cid:17) , (C19c) g = − (cid:16) q − q − q − (cid:17) . (C19d)We can now evaluate how this correction contributes to thegrowth time of the instability. At leading order, for an axion, δτ = − (cid:16) . · s (cid:17) (cid:32) µ axion µ s (cid:33) (cid:32) η/ M . (cid:33) (cid:18) Ma (cid:19) M µ s ) . (C20) [1] C. M. Will, The Confrontation between General Relativity andExperiment, Living Rev. 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