Superradiance in Kerr-Like Black Holes

Superradiance in Kerr-like black holes

Edgardo Franzin,1, 2, 3, 4 Stefano Liberati,2, 3, 4 and Mauro Oi5, 6 1Department of Astrophysics, Cosmology and Fundamental Interactions (COSMO), Centro Brasileiro de Pesquisas F´ısicas(CBPF), rua Dr. Xavier Sigaud 150, Urca, Rio de Janeiro – RJ, 22290-180 Brazil 2SISSA, International School for Advanced Studies, via Bonomea 265, 34136 Trieste, Italy 3IFPU, Institute for Fundamental Physics of the Universe, via Beirut 2, 34014 Trieste, Italy 4INFN, Sezione di Trieste, via Valerio 2, 34127 Trieste, Italy 5Dipartimento di Fisica, Universitàdi Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy 6INFN, 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 current observations allows for small yet detectable deviations from this description. Here we study superradiance of scalar and electromagnetic test fields around the Kerr-like Konoplya–Zhidenko black hole and we observe that for large values of the deformation parameter superradiance is highly suppressed with respect to the Kerr case. Surprisingly, there exists a range of small values of the deformation parameter for which the maximum amplification factor is larger than the Kerr one. We also provide a first result about the superradiant instability of these non-Kerr spacetimes against massive scalar fields.

I. INTRODUCTION geometry would be barely distinguishable from Kerr, leaving a weak signature in the form of gravitational-wave echoes at General relativity has been extensively and successfully late times [10, 19–23]. tested [1,2] from the weak to the strong regime — the most From this point of view, instead of testing a specific the- recent results being the detection of gravitational waves pro- ory against general relativity case by case and/or a specific duced by the merger of two black holes [3] and the observa- black-hole alternative, it could be more convenient to work tion of the shadow of the supermassive black hole M87* [4]. in a model-independent framework describing the most gen- Nowadays black holes are widely accepted as astrophysical eric black holes in any metric theory of gravity. The idea of objects [5,6] compatible with the Kerr metric [7], yet, we still this framework is similar to the parametrized post-Newtonian do not have the ultimate evidence for such black holes to ex- (PPN) formalism [1] but in this case it is valid in the whole actly match this general-relativistic solution, as their defining space outside the event horizon. property — the event horizon — is intrinsically not directly In Refs. [24–26], deviations from general relativity and the observable [8–11]. general-relativistic black-hole geometry are written in terms There exists a number of alternative theories of gravity as of an expansion in M/r being r some radial coordinate. Some well as exotic compact objects proposed to compete or substi- coefficients are easily constrained with the PPN parameters, tute black holes. These black-hole mimickers typically share while a very large number of equally important coefficients the same features at large distances, while they present qual- remains undetermined in the near-horizon region, with the ad- itative differences close to the event horizon. Current and fu- ditional drawback of a lack of a hierarchy among them. Even ture gravitational-wave observations are and will be able to if this formulation works well for small deviations from gen- test general relativity, the no-hair theorem, the near-horizon eral relativity, it fails for e.g. Einstein–dilaton–Gauss–Bonnet geometry, distinguish the Kerr spacetime from putative altern- with large coupling constants [27]. atives, and even probe quantum gravity effects [11–17]. These A more robust general parametrization to describe, re- effects in a consistent setup are commonly invoked to regular- spectively, spherically symmetric and axisymmetric asymp- ize spacetime singularities, which are inevitable in classical totically flat black holes has been introduced by Konoplya, arXiv:2102.03152v2 [gr-qc] 16 Apr 2021 general relativity [18]. Rezzolla and Zhidenko in Refs. [28, 29], and tested to con- While nowadays observations agree with numerical simu- strain deviations from the Kerr hypothesis with the iron-line lations based on Einstein gravity, the current uncertainties on method [30–32] and to produce black-hole shadows simula- the measurements of the black-hole parameters leave room for tions [33, 34]. In this framework, deviations from general alternatives. A possible framework is to describe this freedom relativity and the Kerr metric are given again as an expan- by introducing suitable parametrized deviations from the Kerr sion whose coefficient values can be fixed from observations geometry. The observed interval values for the black-hole in the strong-gravity regime (close to the horizon) and in the mass M and angular momentum J = aM can be therefore post-Newtonian region (far from the black hole). This para- translated in an allowed range for the deviation parameters. metrization also allows for non-spherical deformations of the Of course, we do not expect these deviations to be large or horizon, provides a faster convergence of the series, and typ- they would be observable in the weak-field regime as well. ically requires a small number of parameters to approximate But, for instance, one can consider non-negligible deviations known solutions to the desired precision. Besides, there exists from Kerr and obtain the same quasinormal frequencies. If a hierarchy among the parameters. the geometry of the spacetime is different from Kerr only in A different perspective is to modify each mass and spin a small region near the would-be horizon, asymptotically the term in the Kerr metric and test whether the magnitude of the 2 spacetime curvature matches with that predicted by general properties, the post-Newtonian expansion coefficients, the re- relativity [35]. More recently, the work of Ref. [25] has been lation between quadrupole moment and mass, the spherical extended to the most general stationary, axisymmetric and horizon, and the mirror symmetry. Yet, it allows for signific- asymptotically flat spacetime with separable geodesic equa- ant differences in the near-horizon region [55–57]. tions [36]. The scope of this paper is twofold: first we analyze the However, even if these parametrizations may depend on a structure of the Konoplya–Zhidenko spacetime, and second large number of parameters to be fixed with data, it is nat- we study superradiant scattering of test fields. In particu- ural to think that astrophysical observables — e.g. quasinor- lar, the paper is organized as follows. In SectionII we re- mal frequencies, orbits of particles, accretion, parameters of view the family of spacetimes which admits separability of the the shadow, electromagnetic radiation — depend only on a perturbative equations for massless spin-0 and spin-1 fields, few of them [37]. with a particular focus on the Konoplya–Zhidenko rotating A common feature of rotating spacetimes is the multifa- black hole. In Section III we present our results regarding ceted phenomenon of superradiance [38–40]: in a gravita- the superradiant emission in the Konoplya–Zhidenko space- tional system and under certain conditions, the scattering of time for massless and massive bosonic test fields. Finally, we radiation off absorbing rotating objects produces waves with conclude with a discussion and prospects in SectionIV. In amplitude larger than the incident one. For a monochromatic AppendixA we derive the angular and radial equations for wave of frequency ω scattering off a body with angular ve- a general non-Kerr black-hole parametrization and we study locity Ω, the superradiant condition is satisfied as long as their boundary conditions. In AppendixB we provide useful ω < mΩ, being m the azimuthal number with respect to the formulas for the Konoplya–Zhidenko spacetime, namely the rotation axis. Einstein tensor, the geodesic equations and the four-velocity When rotating black holes are surrounded with matter, su- of a zero-angular-momentum observer. In AppendixC we perradiance gives rise to exponentially growing modes, i.e. study the instability of the Konoplya–Zhidenko black hole black-hole bombs [41, 42]. The scattering of massive fields against massive scalar fields in the low-frequency, small-mass produces a similar effect: the mass term can effectively con- and small-deformation limit. Throughout this work we use fine the field giving rise to floating orbits and superradiant in- G = c = 1 units. stabilities which extract rotational energy away from the black hole [43–45]. The observation or the absence of effects related to these instabilities can be used to impose bounds on the mass II. PARAMETRIZED KERR-LIKE SPACETIMES AND THE KONOPLYA–ZHIDENKO BLACK HOLE of ultralight bosons, see e.g. Refs. [46–50]. Similarly to the Kerr black hole, Kerr-like spacetimes dis- sipate energy as well as any classical dissipative system, and The metric of a generic axially symmetric, stationary and the aim of this paper is to investigate differences and analogies asymptotically flat spacetime can be written as for these objects with respect to the superradiant scattering N2 − W2 sin2 θ around Kerr black holes. We stress that these spacetimes are ds2 = − dt2 − 2Wr sin2 θ dt dϕ K2 not solutions to the field equations of any specific gravitational ! Σ B2 theory, meaning that we can only study test fields propagating +K2r2 sin2 θ dϕ2 + dr2 + r2 dθ2 , (1) in these backgrounds while the gravitational-wave dynamics r2 N2 is excluded. However, in extended theories of gravity exact rotating solutions are difficult to derive and in some cases they where N, W, K, Σ and B are in general functions of r and θ. are known only perturbatively in the spin parameter, or nu- In this paper we focus on parametrized Kerr-like space- merically. To our knowledge, there are no studies of super- times which possess Kerr-like symmetries and admit sep- radiant amplification in these extended theories, neither for arable Klein–Gordon equations for test fields [53]. As in those which admit general-relativistic solutions [51, 52] but Ref. [53], we are not interested in the general conditions for predict different dynamics. the separability of variables, which are related to the symmetry of the background and the choice of appropriate co- In the most general parametrization, there is no reason to ordinates. Being our pragmatic objective to test strong-gravity believe that the separability property of the Kerr metric is effects in an asymptotically flat and axisymmetric spacetime guaranteed, not even for the Klein–Gordon equation. In par- describing a Kerr-like black hole, we can simplify the above ticular, the class of Kerr-like spacetimes which allows for the spacetime metric leaving only three arbitrary functions of the separation of variables in the Klein–Gordon and Hamilton– radial coordinate, so that Jacobi equations has been derived in Ref. [53], which is a subclass of the Johannsen metrics [25]. In this paper we show 2 2 2 B(r, θ) = RB(r), Σ(r, θ) = r RΣ(r) + a cos θ, (2a) that, under given conditions, a subclass of the metrics presen- 2 ted in Ref. [53] also allows for the separation of variables in aRM(r) 2 RM(r) a W(r, θ) = , N (r, θ) = RΣ(r) − + , (2b) the Maxwell equation. Σ(r, θ) r r2 The results presented in this paper are mostly relative to 1 h i K2(r, θ) = r2R2 (r) + a2R (r) + a2 cos2 θ N2(r, θ) the Konoplya–Zhidenko black hole [54], which introduces a Σ(r, θ) Σ Σ single extra parameter. Despite its simplicity, this model pre- a W(r, θ) + . (2c) serves a lot of features of the Kerr spacetime: the asymptotic r 3

2 2 2 For further convenience we define ∆ ≡ r N = r RΣ − RMr + being CKerr = M/r+ the compactness of the Kerr black hole, 2 a and we observe that for this class of spacetimes the event while in the extremal case (CKerr = 1) horizon is defined by the largest positive root of ∆ = 0. r Asymptotic flatness and current PPN parameters imply η 3η C − O η3/2 . 2 = 1 3 + 3 + (7) RM → 2M + O (1/r ) as r → ∞. With a suitable change of M 2M the radial coordinate it is possible to set RB or RΣ to 1, so only two of the three radial functions are independent. The Kerr Equation (6) indicates that positive (negative) values of η corresponds to less (more) compact configurations. metric is recovered for RΣ = RB = 1 and RM = 2M. Equa- tions (2) describe a Petrov D spacetime, and as a consequence, For a < M, instead of working with η, deviations from the the Hamilton–Jacobi equation is separable with a generalized Kerr spacetime can be parametrized in terms of the quantity Carter constant [53] — see also AppendixB. In AppendixA, δr, such that the position of the event horizon can be written as r = r + δr — cfr. Eq. (4), although δr can account for we show that the subclass of this spacetime such that RB = 1 0 + 2 large values of η/M3 and it is not limited to a perturbative ex- and RΣ = (1 + ξ/r) also admits separable Maxwell equations for test fields. pansion. This writing is obviously coordinate-dependent but A minimal deformation for the Kerr spacetime was in- since we are using asymptotic Boyer–Lindquist coordinates, troduced by Konoplya and Zhidenko in Ref. [54] and can a significant deviation from Kerr should be similarly acknow- ledged by different observers. be obtained from Eqs. (2) by setting RΣ = RB = 1 and 2 Differently from the Kerr case, in the Konoplya–Zhidenko RM = 2M + η/r . For the rest of the paper we consider this background geometry. spacetime there exists no maximum value for a beyond which the spacetime always describes a naked singularity. As a always enters quadratically in Eq. (3), without loss of generality, A. Event horizons and causal structure in the following we consider positive a. Although this spacetime belongs to a class of metrics which are constructed to describe the spacetime outside the event ho- For the Konoplya–Zhidenko metric the event horizon radius rizon, it is instructive to explore the implications inside the r2 − Mr a2 − is given by the largest positive real root of ∆ = 2 + horizon. This should be taken with great care and interpreted η/r = 0, which in general admits three (possibly complex- prudently, but it might give insights about what a small dif- valued) solutions ference at infinity entails about the structure of the spacetime √ ! inside the horizon. This being said, in what follows we do not 2M 2 2 2 2kπ rk = + 4M − 3a cos β − , (3) limit our analysis to the largest positive real root of ∆ = 0 but 3 3 3 we give a more comprehensive discussion. 1 16M3 − 18Ma2 + 27η The Ricci scalar of the Konoplya–Zhidenko metric is non- β = cos−1 , k = 0, 1, 2 . h i 3 2 4M2 − 3a23/2 vanishing, R = 2η/ r3 r2 + a2 cos2 θ , from which we infer that r = 0 is a physical singularity. We immediately notice that the Kerr limit η → 0 is not con- To classify the solutions of ∆ = 0 it is helpful to introduce tinuous, as in looking for the roots of ∆ = 0 we pass from solving a cubic to a quadratic equation. Nevertheless, for a < M 2 3/2 η = 9Ma2 − 8M3 ± 4M2 − 3a2 , (8) and in the small η/M3 limit, we have ± 27

η η2(2r − r ) and to define three separate parameter regions as (I) η < η−; r = r + − + − + O η3 , (4) 0 + 3 3 (II) η− 6 η 6 η+; (III) η > η+. Then we sort configurations r+(r+ − r−) r+(r+ − r−) according to the value of the spin parameter: below the√ Kerr √ ≡ 2 2 bound, a < M; highly spinning M 6 a < a∗ 2M/ 3; and where r± = M ± M − a are the radii of the event and ultra spinning a > a∗. Cauchy horizon for the Kerr spacetime. For |η|/M3 . 0.07 the difference between r0 calculated as a linear correction to Below the Kerr bound: In region (I), there is only one real r and the exact value as in Eq. (3) is less than 1% for values of + solution given by r2 in Eq. (3) which is always negative and a . 0.9M. Equation (4) does not apply in the extremal limit, hence the spacetime describes a naked singularity. which must be treated separately, as in this case the leading 1/2 In region (II), the equation ∆ = 0 admits three real solutions, order correction is O η and r0 is given by and the event horizon is r0. The root r1 is always positive r while r2 is negative (positive) for η− < η√ < 0 (0 < η < η+). In η η 3/2 2 2 1 r M − O η . particular for η = η−, r0 = (1/3)(2M + 4M − 3a ), while 0 = + 2 + (5) √ M 2M 2 2 for η = η+, r0 = (2/3)(M + 4M − 3a ).

Under these assumptions, the compactness of the spacetime In region (III), r0 is the only positive-deﬁnite real root. for a < M is ! η C = C 1 − + O η2 , (6) Kerr 2 1 r+(r+ − r−) Notice, however, that ∂r0/∂η diverges as η → η−. 4 * * *

t 0 t 0 t 0

0 r1 r0 0 r2 r1 r0 0 r0 r r r

(a) Region (II) with η− < η < 0. (b) Region (II) with 0 < η < η+. (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 > 0 the event horizon is deﬁne advanced coordinates and we plot null rays in Fig.1, h 1 3 i r0 = (2M/3) 1 + cos arccos 27η/2M − 1 . The other where t∗ = t + r − r∗ being r∗ a tortoise coordinate deﬁned by 3 2 2 solutions r1 and r2 are generically complex-valued but for 0 < dr∗/dr = (r + a )/∆. η < 4M3/27 the imaginary part goes to zero and the real part In the external regions, i.e. for r > r0, we observe a peeling is positive. structure, typical of black-hole horizons. In region (II), for η < η < 0, the light-cone structure is nearly similar to that of For M < a < a , η is positive and in the subregion of region − ∗ − a Kerr black hole, there are an outer and an inner horizon and (I) such that 0 < η < η , the only real positive root is r . In − 2 a timelike singularity. In region (II) but for 0 < η < η , a null region (II) the three real roots are positive and the event hori- + trajectory encounters a black-hole horizon, a white-hole-like zon is given by r , while in region (III) the only real solution 0 horizon and then again a black-hole-like horizon to eventually is r . 0 reach a spacelike singularity. In region (III) there is only one Notice that for each value of η in the range 0 < η < 8M3/27, horizon and the light-cone structure looks like the Schwarz- there exists a value of a schild one with a spacelike singularity. −  r 1/2 For further convenience, the angular velocity Ω = gtϕ/gϕϕ M  27η  at the horizon reads a+ = √ 1 + 2 1 + cos β+ , (9) 3  M3  a a Ωk = = , (10) r2 + a2 2Mr + η/r with k k k

6 3 2 where the value of k depends on the specific values of the 1 −1 8M − 540ηM − 729η β+ = cos , black-hole parameters. 3 8 M4 + 27ηM3/2 for which the largest root of ∆ = 0 passes from r0 to r2 dis- B. Ergoregions continuously. Alternatively, for a fixed a, the largest root of ∆ = 0 passes from r to r at η = η . Depending on the spe- 2 0 − An ergosurface is a static limit surface, i.e. no static ob- cific values of the parameters the ratio r /r can be of several 0 2 server is allowed beyond this surface. Ergosurfaces in these orders of magnitude, and the compactness of the black hole black-hole spacetimes are defined as the roots of the equation changes accordingly. 2 2 2 gtt = 0, or equivalently r − 2Mr + a cos θ − η/r = 0, which Ultra spinning: For the particular case a = a∗ with η > 0, read p3 3 r0 = 2M/3 + η − 8M /27 and r1 and r2 are complex-valued √ ! erg 2M 2 2kπ unless η = 8M3/27, for which r = r = r = 2M/3. For r = + 4M2 − 3a2 cos2 θ cos β erg − , (11) 0 1 2 k 3 3 3 a > a∗, η± in Eq. (8) become complex-valued and independ- 1 27η + 16M3 − 18Ma2 cos2 θ ently on the value of η, r0 and r2 are complex-valued, while r1 β erg = cos−1 , k = 0, 1, 2 . is positive for η > 0. 3 2 4M2 − 3a2 cos2 θ3/2 The light-cone structure of these configurations can be For configurations below the Kerr bound in regions (II) and erg richer and significantly different than that of a Kerr black hole. (III), the location of the ergosurface is r0 . As an example, consider a Konoplya–Zhidenko black hole For highly spinning configurations, the ergosurface is again erg below the Kerr bound. Following a standard procedure, we r0 in regions (II) and (III), but it is piecewise and non- 5

erg continuous in region (I): it is given by r0 in the angu- exploits known results to be compared with. In what follows, erg the sign of a is important to distinguish between direct (a > 0) lar interval [θ1, θ2] and r2 in the complementary interval, [0, θ1) ∪ (θ2, π] where θ1,2 (θ2 = π − θ1) are the solutions of and retrograde (a < 0) orbits, so uniquely for the remainder of this subsection we allow a ∈ [−M, M]. 2 3/2 2 2 − 3 − 2 − 2 2 In practice, we expand the light ring radius rc and the im- η = 9Ma cos θ 8M 4M 3a cos θ , (12) 3 27 pact parameter Dc around the Kerr values in powers of η/M . Here we report the leading-order corrections for the most fa- once the values of M, a and η are fixed; the maximum value of −1 + miliar cases, i.e. a = −M, 0, M. When a = −M we find θ1 is cos (M/a), attained for η → 0 . This means that when passing from a configuration in region (I) to one in region (II), 13η η the volume between the ergosurface and the event horizon, the rc ≈ 4M + , Dc ≈ 7M + . (14) 72M2 6M2 ergoregion, can change dramatically. Notice that for configurations below the Kerr bound and In the non-rotating limit, i.e. for a = 0, we get highly spinning and values of η in regions (II) and (III) the √ volume of the ergoregion is maximum for η = η− and it de- 5η √ 3η r ≈ 3M + , D ≈ 3 3M + . (15) creases for larger values. c 18M2 c 6M2 For the particular case a = a∗, the location of the ergos- erg 3 For a = M the leading order correction is milder, urface is r0 as long as η > 8M /27, but piecewise and dis- 3 continuous for 0 < η < 8M /27 as described above. For su- r r 2 perspinning configurations, let θ∗ the smallest root of cos θ = 4η 3η 2 2 3 rc ≈ M + , Dc ≈ 2M + . (16) a∗/a . For 0 < η < 8M /27 the ergoregion is piecewise and 3M M erg erg discontinuous: it is given by r1 for [0, θ∗)∪(π−θ∗, π], r2 for erg For general values of the deformation parameter, and to al- [θ∗, θ1) ∪ (θ2, π − θ∗], and r for [θ1, θ2] where θ1,2 are again 0 low the spin parameter above the Kerr bound, the radius of the the solutions of Eq. (12). For η > 8M3/27 the ergoregion is erg photon orbits and the corresponding impact parameter can be still piecewise but no longer discontinuous: it is given by r0 3 erg determined numerically. For |η|/M . 0.1, r and D have in the interval [θ , θ ] and r in the complementary interval c c 1 2 1 maximum deviations from the Kerr values, respectively, of [0, θ ) ∪ (θ , π]. 1 2 ∼ 3% and ∼ 4% for 0 6 a < 0.9M, which reduce to less The fact that superspinning configuration for some values than 1% for −M 6 a < 0. We have also checked that the light of the deformation parameter can have a piecewise and non- ring is always outside the horizon for η > η− and a 6 a∗. continuous ergoregion, i.e. no longer an ergosurface, poses a serious problem on the viability of these particular configurations as black-hole mimickers. We expect these particular D. The Konoplya–Zhidenko black hole as a solution of general solutions to be dynamically unstable, but this analysis is bey- relativity ond the scope of this paper and is left for future work. Although these parametrized axially symmetric metrics are 2 C. Photon orbits built not to be exact solutions to any gravitational theory, it is an interesting exercise to figure out what kind of matter distribution one would need in general relativity to obtain Photon orbits for the Konoplya–Zhidenko black hole can the Konoplya–Zhidenko black hole as an exact solution, and be studied starting from the geodesic equations derived in Ap- which energy conditions must be violated. pendixB. In particular, the radial null geodesic in the equatorial plane is We start by defining the stress-energy tensor out of the Ein- stein tensor, i.e., Tµν = Gµν/8π, whose non-zero components a2E2 − L2 2M (L − aE)2 η (L − aE)2 are given in AppendixB. r˙2 = E2 + + + , (13) 2 3 5 To characterize the would-be matter content of this space- r r r µ time, a first possibility is to compute the eigenvalues of T ν . where a dot indicates derivative with respect to an affine para- In particular, we identify the energy density with the opposite meter, while E and L are, respectively, the energy and the of the eigenvalue relative to the timelike eigenvector,3 angular momentum of the photon, although it is more con- η venient to characterize the geodesic by the impact parameter ρ = − . (17) D ≡ L/E. 4πr r2 + a2 cos2 θ2 The radius of photon orbits rc and its corresponding impact parameter Dc are determined by Eq. (13) and its derivative evaluated at r = rc = const. The problem is well-known for the Kerr black hole [58], but the term introduced by the de- 2 In Refs. [59, 60] it is shown that the Konoplya–Zhidenko metric is an exact formation parameter η makes the equation no longer amenable solution of a (non-analytical) mixed scalar- f (R) gravitational theory. 3 2 to analytical methods for all values of L and E. Therefore, we Being vt = {a + r /a, 0, 0, 1}, vr = {0, 1, 0, 0}, vθ = {0, 0, 1, 0}, and vϕ = 3 2 µ decide to adopt a small η/M approximation and work below {a sin θ, 0, 0, 1} the eigenvectors of T ν , the timelike vector is vt for ∆ > 0 the Kerr bound. This guarantees some level of analyticity and and vr otherwise. 6

This matter distribution is concentrated close to the singularity III. SUPERRADIANCE FROM THE and mainly along the equatorial plane, but it extends beyond KONOPLYA–ZHIDENKO BLACK HOLE the event horizon although it decays quite fast for large values of the radius. In the Konoplya–Zhidenko background, the scalar (s = 0) Alternatively, the distribution of energy can be character- and electromagnetic (s = ±1) wave equations are separable ized in an observer-dependent way by analysing the contrac- with the angular part described by the spin-weighted spher- tion of the stress-energy tensor with the velocity of a physical oidal harmonics equation and the radial part by µ ν observer, i.e., ρ = T u u . In view of the angular distribution  η µν ! K2 − 2is r − M + K − d dRs  2r2 of Eq. (17), for simplicity, consider a zero-angular-momentum ∆ s ∆s+1 +  observer (ZAMO) in the equatorial plane, whose four-velocity dr dr  ∆ ! in given in AppendixB. It can be verified that s(s + 1)η +4isωr − λ − R = 0 , (20) r3 s η 2r2 + 5a2 where K ≡ (r2 + a2)ω − am and λ ≡ A + a2ω2 − 2maω, being ρZAMO| = − . (18) θ=π/2 8πr7 A the eigenvalue of the angular equation, ω the frequency of the perturbation and m its azimuthal number. The angular eigenvalue is also characterized by the harmonic number l. As Inspection of Eqs. (17) and (18) reveals that the sign of discussed in AppendixA, the physical information contained these energy densities is purely determined by the sign of η: in the solution with spin-weight s is equivalent to that with negative (positive) values of η correspond to a positive (neg- − 3 spin-weight s. This property will be particularly important ative) energy density; assuming a < M and in the small η/M when computing the energy fluxes of electromagnetic waves regime, they also correspond to configurations more (less) at infinity. compact than a Kerr black hole with the same spin — cfr. Eq. (6). These results further imply that, for positive values of η, this matter distribution violates — at least — the weak A. Boundary conditions energy condition. Within this effective description, it is possible to relate the To integrate Eq. (20) we need to supply it with bound- above matter distribution to the flux contribution to the Komar ary conditions. We first introduce a tortoise-like coordin- 2 2 mass [61], ate√ dr∗/dr ≡ (r + a )/∆ and a new radial function Ys(r) = 2 2 s/2 r + a ∆ Rs(r) such that the radial equation becomes ! Z √  η 3 1 µ ν 2 K2 − 2is r − M + K + (4irsω − λ)∆ 2 d y h Tµν − Tgµν n ξ , (19) d Ys  2r2 2 +  Σ 2  2 dr∗  r2 + a2  dG s(s + 1)η∆  where Σ is a spacelike hypersurface that extends from the − − G2 −  Y = 0 , (21) 2  s event horizon to infinity, nµ the unit normal, h the determinant dr∗ r3 r2 + a2 of the induced metric on Σ, T the trace of the stress-energy where G = r∆/(r2 + a2)2 + s∆0/2(r2 + a2). Asymptotically tensor, and ξν the timelike Killing vector. Explicit evaluation (r → ∞), Eq. (21) becomes of Eq. (19) indicates that this contribution can be of the same magnitude of M for some specific values of the black-hole 2 ! d Ys 2 2isω + ω + Ys ≈ 0 , (22) parameters, although for configurations below the Kerr bound dr2 r it is typically of order ±20% of M, where the sign depends on ∗ ±s ∓iωr∗ the sign of η. It would be interesting to explore whether this whose solutions are Ys ∼ r e where the plus (minus) sign amount of putative matter can be used to model “dirty” black refers to outgoing (ingoing) waves. holes as well. Near the event horizon r0 (r∗ → −∞), let k ≡ ω − mΩ0, Ω0 being defined in Eq. (10), then Eq. (21) becomes Configurations on the edge of η = η−, i.e. configurations between regions (I) and (II) — which describe black holes d2Y a2 + r (3r − 4M) s (k − sσ)2 Y ≈ , σ 0 0 , for a > M — seem particularly unstable. As the radius 2 + i s 0 = (23) dr∗ 2r r2 + a2 of the event horizon and the volume of the ergoregion can 0 0 change abruptly and widely, one passes from small to enorm- and the purely ingoing solution at the horizon is Ys ∼ ous violations of the energy conditions. Together with the odd −s/2 −ikr∗ exp [−i (k − isσ) r∗] ∼ ∆ e . piecewise and disconnected ergosurface for some values of the parameter space, this might suggest that not every configuration can mimic actual Kerr black holes. B. Amplification factors Nonetheless, if we drop the assumption that general relativity is the correct gravitational theory, the discussion above The asymptotic solutions to Eq. (22) can be used to define might be extremely different. the energy fluxes of bosonic fields at infinity. Since the 7

0.0 0.0 0.004 0.04 -0.2 0.003 -0.2 0.03 0.002 0.02 -0.4 0.001 -0.4 0.01 0.000 0.00 0 , 1 1 ,

Z -0.6 0.02 0.12 0.22 0.32 0.42 Z -0.6 0.02 0.12 0.22 0.32 0.42 η=0 η=0 -0.8 η=0.04 -0.8 η=-0.01 η=1 η=1 -1.0 η=10 -1.0 η=10 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 ωM ωM

Figure 2. Spectra of the amplification factor for a scalar (left panel) and electromagnetic (right panel) field with l = m = 1 off a Konoplya– Zhidenko black hole with a = 0.99M for selected values of η in units of M3. Inset: Zoom in the superradiant region.

Konoplya–Zhidenko spacetime shares the same asymptotic For each value of the spin-weight s, the quantum numbers behaviour and symmetries of the Kerr spacetime, the de- (l, m) and aω, we first determine the angular eigenvalue using rivation of this section is very similar to what happens for the Leaver method [63]. Second, fixed a value for η, we in- Kerr [62]. tegrate Eq. (21) from the horizon onwards until a sufficiently Consider an incident wave of amplitude I from infinity pro- large radius. Then we compare our numerical solution and ducing a reflected wave of amplitude R, the asymptotic solu- its radial derivative to the asymptotic behaviour in Eq. (24) tion to Eq. (22) can be written as and its derivative to extract the coefficients I and R. Finally, we compute the amplification factor using Eq. (26). To in- −iωr∗ s iωr∗ s Ys ∼ I e r + R e /r . (24) crease the accuracy of this numerical procedure, we consider a higher-order expansion near the horizon and in the asymptotic The total energy flux at infinity per unit solid angle can be region which reduces to those reported in the previous section computed out of the stress-energy tensor of the test fields as at the leading order. The routine is repeated for several values of the frequency (typically) in the interval 0 < ω < 2mΩ . 2 2 0 d E d 2 r Modes with m 6 0 are not superradiant and as a consequence = (Ein + Eout) = lim r T t , (25) dt dΩ dt dΩ r→∞ of the symmetries of the angular and radial equation, the amplification factor is symmetric under Z (ω) = Z (−ω) we where the ingoing and outgoing fluxes dE /dt are propor- s,l,m s,l,−m in/out can consider positive frequencies only. tional, respectively, to |I|2 and |R|2 [62]. When energy is extracted from the black hole, the flux of energy through the ho- We now define our working assumptions for what follows. rizon is negative and energy conservation implies dEin/dt < We allow the deformation parameter in the range η > η− and dEout/dt. It is then possible to define the quantity Zs,l,m = we mainly exclude superspinning configurations from our in- dEout/dEin − 1 which gives the amplification or absorption vestigation, i.e., we focus on black holes below the Kerr bound factor for bosonic waves of spin-weight s and quantum num- and highly spinning in regions (II) and (III) introduced above. This has a practical advantage: the event horizon and the er- bers (l, m) off a black hole. erg In our case of interest, the amplification factors are gosurface are always given by r0 and r0 . Despite the lack of observational evidence for rotating black holes beyond the !±1 |R|2 |R|2 16ω4 Kerr bound [64], it cannot be excluded that some highly spin- Z0,l,m = − 1 , Z±1,l,m = − 1 , (26) ning objects can be produced in high-energy astrophysical |I|2 |I|2 B2 phenomena that dynamically evolve in less spinning config- where B2 = [λ + s(s + 1)]2 + 4maω − 4a2ω2. Notice that urations. Hence it makes sense to explore a bit this parameter the expressions in Eq. (26) are the same as for Kerr as the region. asymptotic behaviour and the symmetries of the Konoplya– Some of our results are presented in Fig.2 and more results Zhidenko black hole are the same. However, the deformation are available online [65]. Both for scalar and electromagnetic parameter η changes the geometry of the near-horizon region fields with quantum numbers l = m = 1, scattered off a black and it is responsible for a different amplification factor, as we hole with spin a = 0.99M, we observe in the insets of Fig.2 show in the next section. that the position of the maximum of the amplification factor is close to the superradiant threshold ω = mΩ0 where the curve becomes very steep, as in the Kerr case. C. Numerical results In absolute values, the maximum amplification factor is about 0.4% and 4.4% for scalar and electromagnetic waves, as For general ω we need to numerical integrate the angular for Kerr. However, in the left panel of Fig.3 we notice that for and radial equations. Our numerical routine works as follows. scalar waves scattering off a Konoplya–Zhidenko black hole 8 with η/M3 ≈ 0.04 the maximum amplification factor is about 6% larger than in the Kerr case, while for electromagnetic 0.001 waves, we observe a maximum amplification factor roughly 10-5 1% larger than in the Kerr case for η/M3 ≈ −0.01. 10-7

0 , l m -9

Z 10 100 106 100 105 l=m=1 104 10-11 80 80 102 100 l=m=2 [ % ]

[ % ] -13 100 10 l=2,m=1 60 60 95

98 Kerr s , 1 max , Kerr s , 1 scalar 0 0.05 0.1 0.15 / I -0.02 0 0.02 / Z 40 40 0.0 0.2 0.4 0.6 0.8 s , 1 I max s , 1 Z electromagnetic 20 scalar ωM 20 electromagnetic 0 0 0 2 4 6 8 10 0 2 4 6 8 10 Figure 4. Typical spectra of the amplification factor Z0,l,m for different η/M3 η/M3 superradiant scalar field modes off a Konoplya–Zhidenko black hole with a = 0.99M and η/M3 = 0.05.

Figure 3. Maximum value of the amplification factor Zs,1,1 (left panel) and integral of the superradiant spectrum Is,1,1 (right panel) for a scalar and electromagnetic field with l = m = 1 as functions though the maximum amplification factor is hierarchically of η, normalized to the maximum value in the Kerr case, i.e. η = 0, smaller than the dominant one. For example, in the range for a = 0.99M. 0.5M . a < M, for both scalar and electromagnetic fields we find Zmax /Zmax ∼ 10−3 while Zmax /Zmax ∼ 0.1 for a & 0.8M. These values of η/M3 are not universal, but depend on the s,2,1 s,2,2 s,2,2 s,1,1 For the l = m = 2 modes, Zmax and I are always smaller value of a/M. For smaller values of a/M, the maximum value s,2,2 s,2,2 than in the Kerr case for positive values of η and a < M, but of Z gets smaller, the position of the peak moves towards s,l,m for negative values the amplification factor can be bigger than smaller values of η/M3 and the frequency range for which in Kerr. Again, this could be interpreted as a consequence of the amplification factor is positive shrinks. For configura- the fact that, for a given a, the ergoregion is larger than the tions with higher spin, say at the Thorne limit a = 0.998M, Kerr ergoregion for negative values of η. On the other hand, the scalar (electromagnetic) amplification factor can be up to the l = 2, m = 1 modes can be more superradiant than in 15% (1%) larger than in the Kerr case. This bigger amplific- the Kerr case, in the sense of Eq. (27), even for positive val- ation factor does not mean that these Kerr-like spacetimes are ues of η when a & 0.8M. For the remaining modes, i.e. with more superradiant than the Kerr spacetime, as the quantity m 6 0, we have verified that the amplification factor is always Z mΩ0 negative, meaning that these modes are not superradiant. Is,l,m = dω Zs,l,m , (27) As previously discussed, the Konoplya–Zhidenko black 0 hole also admits superspinning configurations, i.e. with spin is always smaller than in Kerr, for positive values of η, as parameter a > M. If the rotation parameter is (slightly) above shown in the right panel of Fig.3. However, a bosonic wave the Kerr bound, in principle, such energy extraction could rap- with frequency close to the superradiant threshold can be idly spin down these configurations to produce a black hole significantly more enhanced in a Konoplya–Zhidenko back- with a < M. ground. For negative values of η, which correspond to more compact configurations, Is,l,m is typically bigger than in Kerr and maximal close to η = η−. For large enough positive val- 0.010 a=0.99M ues of the deformation parameter the maximum value of the amplification factor and the range of superradiant frequencies a=1.15M a=1.30M are always smaller than in the Kerr case. The physical explan- 0.005

3 0 , 1 ation to this result is that, typically, for values of η/M , 0 Z the volume of the ergoregion is smaller and hence the energy that can be extracted. In the non-rotating limit, i.e. a = 0, 0.000 superradiance disappears and we recover the recent results on absorption in Schwarzschild-like backgrounds [66, 67]. 0.0 0.1 0.2 0.3 0.4 In the inset of the left panel of Fig.3, we observe that the ωM same maximum value of the amplification factor for a scalar field is obtained for Kerr (η = 0) and for η/M3 ≈ 0.12. This is nothing but an apparent degeneracy, as the spectra and the Figure 5. Spectra of the amplification factor for a scalar field with l m superradiant ranges of frequency are significantly different. = = 1 off a superspinning Konoplya–Zhidenko black hole with η/M3 = 1 for selected values of a/M. In Fig.4 it is evident that the most superradiant mode corresponds to the minimum allowed value of l = m, as in the Kerr case. Modes with different values of (l, m > 0) qualit- For completeness, we consider the scattering of a scalar atively share the same behaviour with the l = m = 1 mode, field off a superspinning black hole. We observe in Fig.5 9 that for η/M3 = 1 and selected values of the black-hole spin, 0.004 the maximum value of the amplification factor can grow (in Mμs =0 principle indefinitely), as well as the range of frequency for 0.003 Mμs = 0.05 which the process is superradiant. But to obtain amplifica- Mμs = 0.1 Mμs = 0.15 tion factors larger than 100% one needs configurations with 0.002 Mμs = 0.2 0,1,1

very large spin parameter or very small positive deforma- Z tion parameter, which are unlikely to describe astrophysical 0.001 black holes. Moreover, as discussed in SectionIIA, one needs to be careful with these conﬁgurations, as in the range 0.000 0 < η < 8M3/27 the position of the event horizon is not always given by r0 for all values of a, and perhaps even more 0.0 0.1 0.2 0.3 0.4 gravely, the ergosurface can be piecewise and non-continuous. ωM 110 105 100 D. Massive scalar ﬁelds 100

[%] 90

95 The extension to a massive scalar ﬁeld with mass µs~ is 80 Mμs =0 quite simple: such mass term in the Klein–Gordon equation max,Kerr 0,1,1 70 0 0.1 0.2 2 / Z Mμs = 0.025 − 2 2 2 2 introduces, after separation, a quantity µs r ∆/ r + a in Mμs = 0.05

max 0,1,1 60 the coefficient of Y0 in Eq. (21) and shifts the frequency of the Z Mμs = 0.075 2 2 2 50 angular equation as ω → ω − µs . Mμs = 0.1 The boundary conditions are slightly modified. In par- 40 ticular, purely ingoing solutions at the horizon still require 0.0 0.2 0.4 0.6 0.8 1.0 3 −ikr∗ η/M Y0 ∼ e , while the asymptotic behaviour at infinity is

2 2 2 q −Mµ /$ $r∗ M µ −2ω /$ $r Figure 6. (Top panel) Spectra of the amplification factor for massive Y ∼ r s e ∼ r ( s ) e , $ = ± µ2 − ω2 . 0 s scalar fields with l = m = 1 off a Konoplya–Zhidenko black hole with (28) a = 0.99M and η/M3 = 0.05, for selected values of the mass parameter. (Bottom panel) Maximum value of the amplification factor Massive waves can be superradiant for frequencies in the Z0,1,1 for a massive scalar field with l = m = 1 as function of η range µs < ω < mΩ0, while they are trapped near the horizon normalized to the maximum value in the Kerr case, i.e. η = 0, for and exponentially suppressed at infinity for ω < µs. a = 0.99M and for selected values of the mass parameter. The numerical routine for the computation of the amplification factor is adapted from that used for massless waves, correcting the asymptotic behaviours accordingly. We limit the massless case, and we also expect a good black-hole mim- this analysis to the l = m = 1 mode for which, in analogy icker not to turn upside-down the Kerr metric. This motiv- with the massless case, we expect the dominant contribution. ates us to investigate the stability of the Konoplya–Zhidenko We repeat the routine for several values of the frequency in spacetime against massive scalar fields in the small η/M3 the interval µs < ω < 2Ω0 − µs. Our results, as those in limit. Remarkably, in this limit, in the low-frequency regime, the top panel of Fig.6, show that superradiance grows with i.e. for ωM 1 and aω 1 and in the small mass approx- the spin parameter a and is less pronounced for more massive imation Mµs 1, the problem can be tackled with analytical fields, as in the Kerr spacetime. The bottom panel of Fig.6 methods — see AppendixC for details. shows that massive waves can be more amplified than in a At leading order, the growth time of instability τ for a Kerr Kerr background with the same spin parameter for some val- black hole perturbed by an axion with mass maxion = µaxion~ = ues of the deformation parameter, analogously to what we 10−20 eV is found for massless fields, though waves with larger masses ! µ M 1 are still less enhanced. Even in this case, for positive values τ . · 6 axion , = 1 58 10 s 8 (29) of η, the Konoplya–Zhidenko black hole is less superradiant µs a (Mµs) than Kerr in the sense of Eq. (27) with the interval of integra- to which the deformation parameter adds the contribution tion adapted to [µs, mΩ0]. (valid as long as η/M3 is small) Kerr black holes develop superradiant instabilities against ! ! massive fields [68] which can be used to constrain the exist- µ η/M3 M 3 1 δτ − . · 3 axion . ence and the mass of ultralight bosons, i.e. using black holes = 7 89 10 s 8 (30) µs 0.01 a (Mµs) as “particle detectors” [69]. In addition, the bosonic cloud can produce long-lasting, monochromatic gravitational-wave sig- Equation (30) implies that for positive (negative) values of nals observable, in principle, in the sensitive band of current the deformation parameter (within a perturbative regime), detectors [47, 48, 70]. We do not expect this picture to be con- the growth time of instability is shorter (longer), i.e. the siderably changed for Kerr-like black holes. Small values of Konoplya–Zhidenko black hole is more (less) unstable than the deformation parameter unveiled an interesting feature in Kerr. For an axion cloud around a supermassive black hole 10

9 −1 with M = 10 M , Mµaxion ≈ 10 , and the growth time of the spin parameter, similarly to what happens for Kerr black instability would be shorter but comparable with the age of holes. Our numerical results show that for large values of the the Universe. Yet, this timescale should also be shorter than deformation parameter and considering the same spin, super- the decay time of the particle for the instability to be really radiance is highly suppressed with respect to the Kerr black effective. hole. This can be interpreted in terms of the volume of the As this preliminary result relies on several assumptions, it is ergoregion: for a Kerr and a Konoplya–Zhidenko black hole to be confirmed by an exhaustive computation of quasi-normal with same a, for positive values of the deformation para- modes and bound states, which is left for future work. In fact, meter the ergoregion is smaller in the latter case as well as this result is valid for slowly rotating black holes hence we the amount of energy that can be extracted, and the effect of cannot conclude whether highly spinning configurations are superradiance is damped. This seems in agreement with the more unstable or not. fact that the proper volume of the ergoregion of slowly rotating black holes in quadratic gravity decreases with respect to the general-relativistic case, suggesting a smaller ampli- IV. DISCUSSION fication factor [73]. Our results for superspinning configurations shown in Fig.5 are compatible with the fact that the energy extraction by the Penrose process for the superspin- In this paper we have studied the superradiant scattering ning Johannsen–Psaltis [24] and Konoplya–Zhidenko metric of scalar and electromagnetic test fields off a Kerr-like black can be significantly larger than for a Kerr black hole [74, 75]. hole. In these spacetimes, the best that we can do is to study Analogously, for negative values of the deformation para- test fields propagating in a fixed background, but often test meter, which correspond to more compact configurations, the fields are a good proxy and the results for scalar and electro- volume of the ergoregion can be larger than that of a Kerr magnetic waves are similar to those for gravitational waves. black hole with the same mass and spin, and as a consequence, However, this is not always true and the case of superradiance the superradiant phenomenon can be enhanced. in general relativity is illustrative: the maximum amplification factors are approximately 0.4% for massless scalar fields with The most interesting feature that we have found is the ex- l = m = 1, 4.4% for electromagnetic waves with l = m = 1 istence of an interval of small values of the deformation para- and 138% for gravitational waves with l = m = 2 [62]. meter for which the maximum of the amplification factor is To guarantee an analytical description of the problem, we larger than in Kerr. This interval contains positive values of have also limited our investigation to a very specific class η for scalar fields when a > 0.97M, while for electromag- of parametrized axially symmetric spacetimes, the Konoplya– netic fields it requires a ≈ M, meaning that there are config- Zhidenko black hole. This does not mean that superradiance urations less compact than a Kerr black hole with the same cannot be present in more general spacetimes, on the contrary, mass and angular momentum for which the superradiant scat- we do expect superradiance to occur in any spacetime rotating tering can be larger. For different values of the spin parameter, sufficiently fast, provided the presence of an ergoregion, but to have more superradiance one needs more compact config- this would most likely require a full numerical simulation. In urations, i.e. with M/r0 > 1. If this trend is respected by this sense, our results represent a first step in the investigation gravitational waves, then a higher amplification factor for less of the phenomenon of superradiance in Kerr-like spacetimes. compact spacetimes would occur for values of a extraordinary Before exploring superradiant scattering around Konoplya– close to the extremal case. Zhidenko black holes, we have studied their structure thor- Besides, we have also presented some initial results on oughly. The simple Konoplya–Zhidenko metric, which shares massive scalar fields. Roughly, their behaviour is similar to with Kerr the same symmetries and asymptotic behaviour, the massless and Kerr cases. Under some approximations — translates into a complicated causal structure. Depending on small frequency, small spin parameter, small scalar mass and the values of the parameters, these configurations can have small deformation parameter — the frequency eigenvalue can from zero up to three horizons. When the spin parameter be determined analytically with asymptotic matching tech- is above the Kerr bound, the ergoregion can be piecewise niques. For the expected most unstable mode l = m = 1, pos- and non-continuous. To consider this model as a valuable itive values of the deformation parameter shorten the growth Kerr black-hole mimicker we might probably need to ex- time of instability, meaning that these spacetimes are more un- clude some regions of the parameter space. Moreover, when stable than Kerr against massive scalar fields. The validity of considered as non-vacuum general-relativistic solutions, these this result is limited: it needs to be taken with great care and configurations require to be sustained by some exotic mat- one should not infer too much information, as it is based on a ter. Yet, in the small-deformation limit and below the Kerr large number of assumptions. A complete numerical investig- bound, we have shown that the horizon and the light ring ation is left for future work. radii are slightly modified with respect to the Kerr values of Within this context, knowledge of superradiant instabilit- only a few percents. Optimistically, future observations of ies can be used to put bounds on the existence and mass of e.g. black-hole shadows could set bounds on the deformation ultralight particles. Nonetheless, if a black hole acts as a parameter [71, 72]. “particle detector”, the presence of (dark) matter around it Regarding superradiance, we have found maximum ampli- might modify the geometry and spoil superradiant effects. In fication for the minimum value allowed of l = m (for scalar SectionIID we interpreted the Konoplya–Zhidenko metric in and electromagnetic fields l = m = 1) and highest values of terms of an exotic matter distribution and show that the mat- 11 ter flux contribution to the Komar mass of the spacetime can Appendix A: The Klein–Gordon and Maxwell equations in be a significant fraction of the black-hole mass M. To have Kerr-like backgrounds this contribution less than, say, 10% of M, and not to suppress superradiant effects, the deformation parameter should take Being s the spin weight of the test field, in linear per- values |η|/M3 . 0.1, indicating once again that the most inter- turbation theory the scalar (s = 0) and Maxwell (s = ±1) esting phenomenology corresponds to small deviations from fields propagate in the background metric. The Klein–Gordon the Kerr geometry. equation for a massless scalar field Φ is easily obtained from The spectra of the amplification factor can look very similar Φ = 0, where the D’Alambert operator is built out of the when comparing a Kerr black hole and a Konoplya–Zhidenko metric (2). To derive the Maxwell equations in such space- black hole with a small deformation parameter, as well as a time, we follow the method proposed in Ref. [76]. massless scalar and a massive scalar with very small mass µ µ µ µ µ First we choose a suitable null tetrad e(a) = {l , n , m , m¯ } parameter. In addition to this fact, there might be a similar that easily reduces to the Kinnersley tetrad [77] in the Kerr “degeneracy” when comparing the spectra of a massless scalar spacetime, i.e. off a slightly deformed Konoplya–Zhidenko black hole with " # µ 1 2 2 ∆ the spectra of a little massive scalar off a Kerr black hole. We l = r RΣ + a , , 0, a , (A1a) have verified that this could actually happen in a number of ∆ RB " # cases. Our criterion for degenerate spectra is when both the µ 1 2 2 ∆ n = r RΣ + a , − , 0, a , (A1b) maximum value of the amplification factor, its corresponding 2Σ RB frequency, its integral as in Eq. (27) and the threshold fre- 1 quency are the same within a tolerance of 5%. Since super- mµ = √ [ia sin θ, 0, 1, i csc θ] , (A1c) radiance is suppressed for massive scalars, we expect this de- 2¯ρ generacy to correspond to positive values of η. This is true for 2 2 2 2 2 where√ ∆ = r RΣ − RM r + a , Σ = r RΣ + a cos θ and ρ = intermediate values of the spin parameter, but for a 0.9M & r RΣ − ia cos θ. The tetrad vectors satisfy the parameter space also include small negative values of 3  0 −1 0 0 η/M . As an example, for a = 0.95M the spectra of a massive   µ −1 0 0 0 scalar field with Mµs ≈ 0.025 and Mµs ≈ 0.05 off a Kerr e e =   . (A2) (a) (b) µ  0 0 0 1 black hole resemble the spectra of a massless scalar field off   a Konoplya–Zhidenko black hole with −0.01 . η/M3 . 0.03 0 0 1 0 3 and 0.02 . η/M . 0.03. The non-vanishing spin coefficients are There are several possible extensions of this work. First 0 2 notice that we considered rotating spacetimes with a horizon, r RΣ ia cos θ meaning that they do not suffer ergoregion instability. The % = − − , (A3a) 2RBΣ Σ 0 presence of the horizon guarantees from the beginning the key  2  ia cos θ  r RΣ  ingredient for superradiance: dissipation. In view of testing −  −  = 2rRB √  , (A3b) the Kerr hypothesis, it could be interesting to use this same 4rRBΣ  RΣ  parametrization and substitute the horizon with a partially re- ∆ h 0 i flective surface and see in which limits superradiance disap- µ − r2R aR θ , = 2 Σ + 2i B cos (A3c) pears. Other possible developments might include consider- 4RBΣ  √  ing non-minimally coupled scalar fields, or use the results in 0  ρ¯ + r RΣ 0  ∆ ∆  2  Ref. [29] to study superradiance of test fields off Kerr–Sen and γ = −  √ r RΣ + 2ia cos θ , (A3d) 4R Σ 8Σ2   Einstein–dilaton–Gauss–Bonnet black holes. B rRB RΣ a sin θ h 2 0i τ = √ 2aRB cos θ − i r RΣ , (A3e) 2 2RBΣρ¯ ACKNOWLEDGMENTS 1 h 2 2 α = √ cot θ Σ − 5a − 5r RΣ 8 2Σρ We are grateful to Vitor Cardoso for several valuable com- 2ia sin θ 2 0 p i ments and a thorough reading of this manuscript, and to Ma- + r RΣ + sin θ 7iar RΣ − 3iaρ , (A3f) RB riano Cadoni for discussions. EF acknowledges partial fin- 0  2  ancial support by CNPq Brazil, process no. 301088/2020-9. ia sin θ  p r RΣ  π = √ r R + − ρ , (A3g) EF and SL acknowledge funding from the Italian Ministry  Σ 2R  of Education and Scientific Research (MIUR) under the grant 2Σρ B  0  PRIN MIUR 2017-MB8AEZ. MO acknowledges partial fin-  2  1   p r RΣ   ancial support by the research project “Theoretical and ex- β = √ ia sin θ r R −  + Σ cot θ . (A3h)   Σ   perimental investigations of accreting neutron stars and black 2 2Σρ¯   2RB   holes”, CUP F71I17000150002, funded by Fondazione di The sourceless decoupled Newman–Penrose equations for Sardegna. The authors thankfully acknowledge Daniele Mura the massless spin-1 field are given by [76] for assistance and computer resources provided by INFN, Sezione di Cagliari. (D − + ¯ − 2% − %¯)(∆ + µ − 2γ) 12 i  − − − − ¯ − 2 0  (δ β α¯ 2τ + π¯) δ + π 2α φ0 = 0 , (A4a) K − iKs∆ dG 2 + − − G Ys = 0, (A8) 2 22  (∆ + γ − γ¯ + 2µ + µ¯)(D − % + 2) (r + ξ) + a √ dr∗  0 2 2 2 2 2 i where G = s∆ /2(r RΣ + a ) + r RΣ∆/(r RΣ + a ) and r is − δ¯ + α + β¯ + 2π − τ¯ (δ − τ + 2β) φ2 = 0 , (A4b) an implicit function of r∗. µ µ µ At infinity (r∗ → ∞), Eq. (A8) can be approximated as where D = l ∇µ, ∆ = n ∇µ and δ = m ∇µ, and the complex µ ν µ ν fields are defined as φ0 = Fµν l m and φ2 = Fµν m¯ n , being ! d2Y 2isω Fµν the electromagnetic field tensor. s ω2 Y , 2 + + s = 0 (A9) Differently from the Kerr case, the spin coefficient is gen- dr∗ r erally non-zero and as a consequence Eqs. (A4) are not separ- ±s ∓iωr∗ able into a radial and angular part. However, one can always from which we see that Ys ∼ r e , where the upper (lower) perform a null rotation of the tetrad to set = 0 [78]. Altern- sign refers to outgoing (ingoing) waves. atively, we can restrict our metric performing a change of the Near the event horizon r0 (r∗ → −∞), Eq. (A8) becomes radial coordinate such that RB = 1 and solving = 0 for RΣ, d2Y ξ 2 s + (k − isσ)2 Y = 0 , (A10) R = 1 + , (A5) 2 s Σ r dr∗ where ξ is a constant parameter. where Under the above assumptions, decomposing the test fields ! as e−iωt eimϕ S (θ) R (r), the scalar and electromagnetic wave ξ(2r + ξ) s k ω 0 − m , = 1 + 2 2 Ω0 (A11) equations separate, with the angular part described by the (r0 + ξ) + a spin-weighted spheroidal harmonics equation 2 0 2 2 r0 1 − RM(r0) − ξ − a ! σ = . (A12) 1 d dS m2 2r (r + ξ)2 + a2 sin θ + a2ω2 cos2 θ − 0 0 sin θ dθ dθ sin2 θ ! The purely ingoing solution at the horizon is given by Ys ∼ 2ms cos θ 2 2 −2aωs cos θ − − s cot θ + s + A S = 0 , (A6) exp [i(k − iσ)r ] ∼ ∆−s/2e−ikr∗ . sin2 θ ∗ Teukolsky and Press showed that one solution of the Teuk- while the radial part by the following equation, olsky equation with spin-weight s contains the same physical information of that with spin-weight −s [62]. This result is a ! " d dR K2 − is∆0K consequence of the fact that the Kerr spacetime is stationary ∆−s ∆s+1 s + + 4isrR ω − λ dr dr ∆ Σ and axisymmetric. This fact holds for this class of metrics too, # s(s + 1) (∆00 − 2) in fact, repeating the same derivation for Eq. (A7) but starting + R = 0 , (A7) with the tetrad 2 s √ 2Σ ∆ r R − ia cos θ 2 2 2 2 ˜µ µ µ µ µ Σ µ where K = r RΣ + a ω − am and λ = A + a ω − 2amω. l = − n , n˜ = − l , m˜ = √ m¯ , ∆ 2Σ r RΣ + ia cos θ The radial functions R0, R1 and R−1 correspond to Φ, φ0 and 2 (A13) φ2/ρ . Equation (A6) together with regular boundary conditions at θ = {0, π} is an eigenvalue problem for the separation con- related to Eqs. (A1) by the simultaneous transformation stant A. For each value of s, m and aω, the eigenvalues are ϕ → −ϕ, t → −t, one finds that, after the separation of the ˜ identified by a number l, whose smallest value is max (|m|, |s|). radial and angular variables, the radial function Rs satisfies The eigenfunctions form a complete and orthonormal set in Eq. (A7) with s → −s and it is related to R−s through θ ∈ [0, π]. For aω = 0, Eq. (A6) reduces to the spin-weighted !s spherical harmonics equation and A = (l − s)(l + s + 1) [79]; 2 R˜ = R− . (A14) for aω 1, Eq. (A6) can be solved perturbatively [80], but in s ∆ s general it must be solved numerically [81]. To integrate Eq. (A7) it is necessary to give boundary conditions at the horizon and at infinity. Therefore, we first introduce a tortoise-like coordinate given by dr /dr ≡ (r2 R + Appendix B: Einstein tensor, geodesic equations and zero ∗ Σ angular momentum observers for the Konoplya–Zhidenko 2 p 2 2 s/2 a )/∆ and the radial function Ys(r) = r RΣ + a ∆ Rs(r). spacetime With these substitutions, Eq. (A7) becomes

2  00 The non-zero components of the Einstein tensor for the d Ys ∆ s(s + 1) (∆ − 2) /2 − λ + 4isω(r + ξ) +  Konoplya–Zhidenko metric read 2  2 dr∗  (r + ξ)2 + a2 13

r2 3 cos2 θ − 5 a2 + 2r −r3 + 2Mr2 + η − a4 cos2 θ sin2 θ Gtt = η , (B1a) r3 r2 + a2 cos2 θ3 a2 r2 + a2 cos2 θ + r 5r3 − 4Mr2 + 5a2r − 2η 2 Gtϕ = aη sin θ , (B1b) r3 r2 + a2 cos2 θ3 2η G = , (B1c) rr r∆ r2 + a2 cos2 θ η 3r2 + a2 cos2 θ G = − , (B1d) θθ r3 r2 + a2 cos2 θ h i a2 a4 − r4 + 4Mr3 + 2rη cos2 θ + r 3r5 + 5a4r − 2a2 (2M − 4r)r2 + η 2 Gϕϕ = −η sin θ . (B1e) r3 r2 + a2 cos2 θ3

  The geodesic equations can be obtained via the Euler– a 2Mr2 + η E − a2L r 1  z r Lz  Lagrange equations from the Lagrangian L = 1 g x˙µ x˙ν, ϕ˙ =  +  , (B5b) 2 µν rΣ  ∆ sin2 θ  where a dot indicates diﬀerentiation with respect to an aﬃne parameter λ. However, it is simpler to use the integrals of 2 2 h 2 2i2 h 2 2 2i Σ r˙ = aLz − E a + r − ∆ (aE − Lz) + Q + r , motion, two of which are related to the obvious symmetries (B5c) of the metric, i.e. stationarity and axisymmetry, that can be 2 ˙2 2 2 2 2 2 2 expressed respectively by Σ θ = a cos θ (E − ) − Lz cot θ + Q . (B5d)

p ≡ g t˙+ g ϕ˙ = −E , p ≡ g ϕ˙ + g t˙ = L , (B2) The four-velocity of a zero-angular-momentum observer in t tt tϕ ϕ ϕϕ tϕ z the equatorial plane is readily obtained, where E and L represent the energy and the angular mo- z r5 + a2 r3 + 2Mr2 + η mentum along the ϕ axis of the particle. Another constant ut = , (B6a) of motion can be obtained observing that the Hamiltonian r3∆ 1 µ ν µ µ r H = gµν p p , where p = ∂L/∂x˙ , is independent of the 2 2 2 2 r a + r η + 2Mr H − 1 2 u = − , (B6b) affine parameter. Therefore we can write = 2 , where r5 2 − is a constant parameter that can be +1, 0, 1, respectively, a 2Mr2 + η for timelike, null and spacelike geodesics. The last integral of uϕ = . (B6c) motion is less obvious and it is related to the separability of r3∆ the Hamilton–Jacobi equation 1 ∂S ∂S Appendix C: Frequency eigeinvalues in the low-frequency, S˙ = gµν , (B3) small-mass and small-deformation limit 2 ∂xµ ∂xν where S is a function of λ and the coordinates. In fact, with In the low-frequency regime, i.e., ωM 1 and aω 1, 2 the ansatz S = − 2 λ − Et + Sθ(θ) + Sr(r) + Lzϕ, Eq. (B3) the amplification factor for waves scattered off a Kerr black separates into an angular and a radial part. The (generalized) hole can be computed analytically [82–84]. The angular equa- 2 Carter constant Q = K − (aE − Lz) is related to the separation tion reduces to the scalar spherical harmonics equation and constant K associated to the hidden symmetry of the metric the angular eigenvalue λ can be approximated as l(l + 1). For generated by a second-order Killing tensor Kµν that satisfies massive scalar field a similar technique can be applied in the ∇(ρKµν) = 0, where the round parentheses denote symmetriz- small mass limit Mµs 1 [44]. We extend this result to the ation with respect to the indices. The explicit form of Kµν is Konoplya–Zhidenko black hole in the limit η/M3 1. The asymptotic matching technique consists in solving the µν (µ ν) 2 µν K = 2Σ l n + r g , (B4) radial equation in the asymptotic and near-horizon regions and relies on the existence of an overlap region in which the two 2 2 2 µ µ where Σ = r +a cos θ while l and n are the vectors defined solutions can be matched. 2 in Eqs. (A1) with RΣ = RB = 1 and RM = 2M + η/r . In the large r limit the radial equation for a massive scalar Using these four integrals of motion it is possible to write field in the Konoplya–Zhidenko background becomes the geodesic equations as 2 ! 00 2 0 l(l + 1) 2Mµs 2 2 2 2 2 R0 (r) + R0(r) + − + + ω − µs R0(r) = 0 . 2Mr + η (r + a )E − aLz r r2 r t˙ = E + , (B5a) r∆Σ (C1) 14

2 2 2 2 l Deﬁning k = µs − ω , ν = Mµs /k, and x = 2kr the above The constant c1 can be determined by comparing the r equation reads terms, ! l − l − n − 00 0 l(l + 1) x (2k) (r+ r−) ( 1) l!(2l + n + 1)!Γ(l + 1 2iq) xR (x) + 2R (x) + − + ν − R (x) = 0 , (C2) c1 = , (C8) 0 0 x 4 0 (2l + 1)!(2l)!

− − i.e., the same equation which governs an electron in the hy- while by comparing the r l 1 terms we get drogen atom. For large x the two independent solutions of !2 ∼ ±(ν+1) ∓x/2 l! Eq. (C2) behave as R0(x) x e . Since we are in- δν = 2iq [2k(r − r )]2l+1 × terested in the unstable modes we take the solution with the + − (2l)!(2l + 1)! upper signs, and the complete solution to Eq. (C2) with such (2l + n + 1)! Yl asymptotic behaviour is j2 + 4q2 . (C9) n! j=1 −x/2 l R0(x) = e x U(l − ν + 1, 2l + 2, x) (C3) Finally, the relation among n, δν and ω = σ + iγ gives being U the confluent hypergeometric function. σ ≈ µs from the real part, while from the imaginary part The regularity of the electron wave-function in x = 0 im- Mµ 3 δν plies that the bound states of the hydrogen atom corresponds iγ = s . (C10) to integer values of ν as ν = l + 1 + n with n positive. As the l + 1 + n M boundary conditions in this case are slightly different from Now we are able to give an estimate for the growth time the quantum mechanics problem (ingoing waves at the hori- of the instability. At zeroth order, combining Eqs. (C9) zon) we guess ν = l + 1 + n + δν where δν is a small complex and (C10) we notice that for m > 0 the imaginary part of number. the frequency is positive and hence the mode is unstable. In In the small x limit, Eq. (C3) is particular, for the most unstable mode, l = m = 1 and n = 0, at leading order Γ(−2l − 1) l Γ(2l + 1) −l−1 R0(x) ≈ x + x . (C4) Γ(−l − ν) Γ(l − ν + 1) 8 a (Mµs) γ = µs , (C11) In terms of the coordinate r and in the small δν limit M 24 (2l + n + 1)! and the growth time, for an axion with mass maxion = µaxion~ = n l n+1 −l−1 −20 R0(r) ≈ (−1) (2kr) + (−1) δν(2l)!n!(2kr) . 10 eV, (2l + 1)! ! (C5) µ M 1 τ ≡ /γ . · 6 axion . 1 = 1 58 10 s 8 (C12) µs a (Mµs) In the near-horizon region we write R0(r) = R˚0(r)+η δR0(r) and we solve order by order in η/M3. We define a new dimen- At first order, the zeroth-order solution enters as a “source sionless coordinate x ≡ (r − r+)/(r+ − r−) and√ the quantity term”, 2 2 q ≡ (am − 2Mr+ω)/(r+ − r−) where r± = M ± M − a are 2 2 00 0 the radial location of the Kerr event and Cauchy horizon. x (x + 1) δR0 (x) + x(2x + 1)(x + 1)δR0(x) At zeroth order, the radial equation reduces to 2 + q − l(l + 1)x(x + 1) δR0(x) = T(x) , (C13) 2 2 ˚00 ˚0 x (x + 1) R0 (x) + x(2x + 1)(x + 1)R0(x) where + q2 − l(l + 1)x(x + 1) R˚ (x) = 0 , (C6) 0 2 ˚0 r+(r+ − r−) T(x) = −R0(x)   2 2 2 2 whose general solution is a combination of the associated Le- 2q q r− − 5r−r+ + 2r+  l l   gendre functions c P (1+2x)+c Q (1+2x) which repres- −  + − l(l + 1) R˚0(x) , (C14) 1 2iq 2 2iq  x Mr  ent, respectively, the ingoing and outgoing waves at the hori- + zon. with R = c Pl (1 + 2x) and c given by Eq. (C8). Now assume there exists an intermediate region in which 0 1 2iq 1 The homogenous problem associated to Eq. (C13) for δR0 the two solutions calculated asymptotically and close to the ˚ horizon overlap. Then the small x limit of the asymptotic solu- is the same as in Eq. (C6) for R0, meaning that its general tion (C5) must be equal to the large x limit of the near-horizon solution is again a combination of the associated Legendre functions, c Pl (1 + 2x) + c Ql (1 + 2x). Again, c can be solution, supplied with the requirement of no outgoing waves 3 2iq 4 2iq 4 set to zero by the request of no outgoing waves at the horizon. at the horizon (c2 = 0). We have The particular solution can be obtained with the method of (2l)! xl (−1)−1−ll! x−l−1 variation of constants, Pl (1 + 2x) ∼ + . 2iq l!Γ(l + 1 − 2iq) (2l + 1)!Γ(−l − 2iq) Z T(z) δR0,2(z) (C7) δR0,p = − δR0,1 dz z2(1 + z)2W(z) 15

Z T(z) δR (z) δR z 0,1 , repeat what we have done for the zeroth-order solution, but + 0,2 d 2 2 (C15) z (1 + z) W(z) matching Eq. (C5) with c1R˚0 + η c3δR0,1 + δR0,p . We ﬁrst solve for c3 and ﬁnd that δν gains a correction proportional where W(x) is the Wronskian associated with δR0,1(x) = Pl (1 + 2x) and δR (x) = Ql (1 + 2x). to η, whose imaginary part sums up to γ computed at zeroth 2iq 0,2 2iq order, As in the zeroth-order calculation, assume that there exists an intermediate overlapping region in which the solution 3 3 ηk Mµs (r+ − r−) h 2 2 i in Eq. (C5) is glued with the large r behaviour of the near- δγ = g Mr + g r + g r−r + g r , 2 2 1 + 2 − 3 + 4 + horizon solution. 48r+q 4q + 1 At this stage, we focus on the l = m = 1 and n = 0 mode (C18) which is, at zeroth order, the most unstable. Using Eq. (C7) with l = 1 and where

h 2 2 i c1 x R1 Mr+ + R2r− + R3r−r+ + R4r+ 2 2 g1 = 8q 2q + 1 4q + 1 =ψ(−2iq) , (C19a) δR0,p ∼ − , 2Mr2 (r − r )2q2(1 + 2iq) (1 − 2iq)3 Γ(1 − 2iq) + + − g − 2 2 2 (C16) 2 = q 4q + 1 28q + 1 , (C19b) 6 4 2 g3 = 2 280q + 130q + 21q + 1 , (C19c) where 6 4 2 g4 = − 224q − 36q − 35q − 2 . (C19d) 2 R1 = 8iq(1 − 2iq) 2q + 1 ψ (−2iq) + γE , (C17a) 2 2 R2 = q (1 − 2iq) 28q − 4iq + 1 , (C17b) We can now evaluate how this correction contributes to the 5 4 3 2 growth time of the instability. At leading order, for an axion, R3 = 280iq − 120q + 70iq − 32q + 5iq − 2 , (C17c) 5 4 3 2 ! ! R4 = −112iq + 20q + 28iq − 25q + 5iq − 2 , (C17d) µ η/M3 M 3 1 δτ − . · 3 axion . = 7 89 10 s 8 (C20) µs 0.01 a (Mµs) being ψ(z) the digamma function, γE the Euler–Mascheroni constant and q is now meant to be computed for m = 1, we

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