Acoustic black-hole bombs and scalar clouds in a photon-fluid model
aa r X i v : . [ g r- q c ] F e b Acoustic black-hole bombs and scalar clouds in a photon-fluid model
Marzena Ciszak and Francesco Marino
1, 2 CNR-Istituto Nazionale di Ottica, Via Sansone 1, I-50019 Sesto Fiorentino (FI), Italy. INFN, Sezione di Firenze, Via Sansone 1, I-50019 Sesto Fiorentino (FI), Italy (Dated: February 10, 2021)Massive bosonic fields in the background of a Kerr black hole can either trigger superradiantinstabilities (black-hole bombs) or form equilibrium configurations corresponding to pure boundstates, known as stationary scalar clouds. Here, similar phenomena are shown to emerge in thefluctuation dynamics of a rotating photon-fluid model. In the presence of suitable vortex flows,the density fluctuations are governed by the massive
Klein-Gordon equation on a (2+1) curvedspacetime, possessing an ergoregion and an event horizon. We report on superradiant instabilitiesoriginating from quasi-bound phonon states trapped by the vortex background and, remarkably,on the existence of stationary modes in synchronous rotation with the horizon. These representthe acoustic counterpart of astrophysical scalar clouds. Our system offers a promising platform foranalogue gravity experiments on superradiant instabilities of massive bosons and black-hole-fieldequilibrium configurations.
I. INTRODUCTION
Superradiant scattering is an effect whereby waves re-flected at the interface of a moving medium are amplified,extracting energy and momentum in the process. For anaxially-symmetric rotating system the amplification oc-curs whenever Ω < m Ω C (1)where Ω is the angular frequency of the incident wave, m its azimuthal wavenumber with respect to the rota-tion axis and Ω C the angular velocity of the rotatingobject. This phenomenon was originally introduced byZel’dovich in 1971 [1] who investigated the scattering ofelectromagnatic waves from a rotating conducting cylin-der. Similarly, a bosonic field impinging upon a KerrBlack Hole (BH) is expected to be amplified if the su-perradiance condition (1) holds, in which case the crit-ical frequency Ω C = Ω H is the angular velocity at theevent horizon [2]. According to quantum mechanics, ro-tating absorbing bodies -including BHs- would supportthe spontaneous emission of superradiant particles, be-ing slowed down by the process [1, 3, 4]. The possibilityto extract energy from a spinning BH was quantified byPenrose some years before [5], and it is related to parti-cles entering the ergosphere with negative energies . Thiswork and the above studies of quantum pair production,can be hystorically considered as precursors to Hawking’sresult on BH evaporation [6].In the presence of a confining mechanism, superradi-ant effects have profound consequences on the stabilityproperties of spinning BHs. An initially small pertur-bation with frequency in the superradiant regime willgrow exponentially in time, if it is prevented from radi-ating its energy to infinity. This is the so-called black-hole in a reference frame compatible with that of a Minkowskian ob-server far from the black hole. bomb by Press and Teukolsky [7]. In their proposal theconfinement is provided by a perfectly reflecting mirrorsurrounding the rotating BH: a field thus undergoes re-peated superradiant scattering between the horizon andthe mirror, resulting in the exponential growth of the fieldamplitude and instability of the system. This is expectede.g. in Kerr–Anti–de Sitter (AdS) spacetimes where theconformal boundary can act as an effective mirror [8].Remarkably, similar phenomena occurs when a KerrBH is simply coupled to a massive scalar field [9–14].The gravitational interaction between the BH and thefield gives rise to a binding potential that prevents low-frequency modes in the regimeΩ < Ω (2)from escaping to spatial infinity, where Ω is the restfrequency of the massive field. The potential gives riseto the formation of a discrete set of quasi-bound (non-stationary) states in the vicinity of the horizon charac-terized by complex eigenfrequencies Ω. Modes furthersatisfying the condition (1), Ω R = Re(Ω) < m Ω H , aresuperradiantly amplified thus triggering the BH bombmechanism. On the other hand, for Ω R > m Ω H themodes are decaying in time, which indicates that the fieldis infalling into the horizon and the system is linearly sta-ble.The boundary between the stable and unstable regimeΩ R = m Ω H is marked by field configurations which arein equilibrium with the BH, i.e. Ω I = Im(Ω)=0. Thesestationary (infinitely long-lived) modes form pure boundstates in synchronous rotation with the horizon, knownas scalar clouds [15, 16].The study of such configurations is of paramount in-terest in BH theory: their existence at the linear levelis intimately linked to hairy black hole solutions of the(nonlinear) Einstein-Klein-Gordon system [17] and theyare considered as viable dark-matter candidates [15]. Up-per bounds to the clouds mass in Kerr spacetime havebeen derived [18], and recently, stationary charged cloudshave been shown to exist around Kerr-Newman BHs[19–21]. On the other hand, superradiant instabilitieshave potential implications in cosmology and high-energyphysics [22], for instance in the discovery of primordialBHs through their coupling with massive bosons or, con-versely, of yet-unknown ultralight particles (e.g. axions)interacting with supermassive BHs [23].Here, we show that similar phenomena can be observedin the excitation dynamics of a rotating photon-fluid inthe presence of both local and nonlocal interactions.Photon-fluids are nonlinear optical systems which canbe described in terms of the hydrodynamic equations ofan interacting Bose gas [24, 25]. Similarly to classical[26, 27] and quantum fluids [28, 29] (for a review see[30]), long-wavelenght excitations (phonons) in an inho-mogeneous flow propagate as a massless scalar field ona curved spacetime endowed with a Lorentzian metric(acoustic metric) [31]. These systems are an ideal plat-form for analogue gravity investigations [32–35]. In par-ticular, superradiant scattering in rotating photon-fluidshas been theoretically demonstrated [36–39], and recentlya rotating BH geometry has been experimentally realized[40].Interestingly, photon-fluids with both local and non-local optical nonlinearities support massive elementaryexcitations which, at lower momenta, correspond to mas-sive phonons with a relativistic energy-momentum rela-tion [41]. In this system, we investigate the phonon dy-namics over a draining vortex flow, as the acoustic ana-logue of a massive scalar field on a rotating BH. We reporton superradiant instabilities, originating from the exis-tence of quasi-bound phonon states trapped in the vicin-ity of the horizon and, remarkably, scalar cloud solutions,equilibrium configurations between the massive phononicfield and the vortex background. Similar states in ana-logue models has been previously discussed by Benone et al. [42], who considered a rotating acoustic BH en-closed in a cylindrical cavity. Unlike ”real” clouds whichoccur for massive fields, phonons in standard fluids haveno mass and a cavity is required to confine them in thevicinity of the horizon. In our model instead, phononicexcitations propagate as a massive Klein-Gordon field,and are naturally trapped by a binding potential in closeanalogy to astrophysical scenarios.The paper is organized as follows. In Sec. II, we intro-duce the photon-fluid model making the connection be-tween hydrodynamic and optical quantities. In Sec. IIIwe review how the massive Klein-Gordon equation gov-erning the propagation of density fluctuations is derivedin the long-wavelength limit. In Sec. IV we decomposethe field to derive the radial Teukolsky equation in thevortex metric, which will be the starting point of ouranalysis. We then show in Sec. V that perturbationswith Ω < m Ω H are superradiantly amplified, producingeither superradiant scattering when Ω > Ω or dynami-cal instabilities when Ω < Ω . In this second case, illus-trated in Sec. VI, quasi-bound states are shown to existfor a discrete set of complex frequencies. Finally, in Sec.VII we focus on scalar clouds solutions and calculate nu- merically the existence lines in the BH-field parametersspace (Ω H ,Ω ). Conclusions and future perspectives arepresented in Sec. VIII. II. PHOTON-FLUID AND COLLECTIVEEXCITATIONS
The propagation of a monochromatic optical beamoscillating at angular frequency ω in a 2D nonlinearmedium can be described within the paraxial approxi-mation in terms of the Nonlinear Schr¨odinger Equation[43] ∂ z E = i k ∇ E − i kn E ∆ n (3)where E is the slowly varying envelope of the electromag-netic field, z is the propagation coordinate, k = 2 πn /λ is the wavenumber, λ the vacuum wavelength and n isthe linear refractive index. The laplacian term ∇ E de-fined with respect to the transverse coordinates r = ( x, y )accounts for diffraction and ∆ n is the nonlinear opti-cal response of the medium. For a purely local (Kerr-defocusing) nonlinearity [45], ∆ n = n | E | with n > x, y ) of the laser beam so that the propa-gation coordinate z plays the role of an effective timevariable t = ( n /c ) z , where c is speed of light in vacuum.Here we consider an optical medium with both localKerr and nonlocal thermo-optical nonlinearities ∆ n = n | E | + n th . The refractive index change n th is thuscoupled to the optical intensity through the stationaryheat equation [46, 47] − ∇ n th = αβκ | E | (4)where κ is the thermal conductivity of the material, α its linear absorption coefficient and β = ∂ n th /∂T > . Nevetheless, a regime infinite-rangenonlocality can be reasonably reproduced by means ofsuitable background optical beams [51]. Similar mod-els arise in semiconductor optical materials [52], nematicliquid crystals [53] and appear also in Bose-Einstein Con-densates (BECs) with simultaneous local and long-range(e.g. dipolar) interactions [54]. Eq. (4) holds in the limit of an infinite medium in the two trans-verse dimensions, and implies an infinite-range for the nonlocalthermo-optical interactions. More realistic models take into ac-count the effects of spatial boundaries by including loss terms inthe stationary heat equation, which make the length-scale of thethermo-optical nonlinearity finite [48–50].
The corresponding hydrodynamic formulation of (3-4)is obtained by means of the Madelung transform E = ρ / e iφ , ∂ t ρ + ∇ · ( ρ v ) = 0 (5) ∂ t ψ + 12 v = − c n n ρ − c n n th + c k n ∇ ρ / ρ / (6)where the optical intensity ρ corresponds to the fluid den-sity and v = ckn ∇ φ ≡ ∇ ψ is the flow velocity. On theright-hand side of (6), the first term provides the local re-pulsive interactions related to the positive bulk pressure P = c n n ρ , n th gives rise to a nonlocal interaction po-tential, while the last term, directly related to diffraction,is the analogue of the Bohm quantum potential. A. Bogoliubov-De Gennes equations anddispersion relation
The evolution of the first-order complex fluctuations ε ( r , t ) of the optical field is obtained by linearizing Eq.(1) around a background solution, E = E (1 + ε + ... )with E = ρ / e iφ . We obtain( ∂ T − i c kn ∂ S ) ε = − i ωn ( n ρ ( ε + ε ∗ ) + n th ) (7)( ∂ T + i c kn ∂ S ) ε ∗ = i ωn ( n ρ ( ε + ε ∗ ) + n th ) (8) −∇ n th = αβκ ρ ( ε + ε ∗ ) (9)in which we introduce the comoving derivative ∂ T = ∂ t + v · ∇ with v = ckn ∇ φ , and the spatial differ-ential operator ∂ S = ρ ∇ · ( ρ ∇ ). Eq. (9) definesthe refractive-index fluctuations due to thermo-optic ef-fect, n th . For n th = 0, Eqs. (7-8) takes the form of thestandard Bogoliubov-De Gennes equations governing thedynamics of elementary excitations in purely local quan-tum fluids [44].In the spatially homogeneous case where both thebackground density ρ and velocity v do not dependon the transverse coordinates, the plane-wave solutionsof Eqs. (7)-(9) satisfy the dispersion relationΩ = Ω + c s K (cid:18) ξ π K (cid:19) (10)where K is the transverse wavevector of the mode ( K isthe wavenumber) Ω = Ω ′ − K · v its angular frequencyin the locally-comoving background frame and Ω = c q αβκn ρ . In analogy to purely local BECs and photon-fluids [31], we define the sound speed as c s ≡ dP ( ρ ) dρ = c n n ρ and the healing length, ξ = λ/ √ n n ρ as thecharacteristic length separating the linear (phononic) andquadratic (single-particle) regime of the dispersion re-lation (10). The length ξ thus determines the critical wavenumber K c = 2 π/ξ associated to the breakdown ofLorentz invariance. Low energy modes with K ≪ K c (phonons) obey the relativistic dispersion relation for amassive particle, where ~ Ω = m ph c s identifies the restenergy of the collective excitations, m ph the rest massand c s plays the role of the light speed. At higher mo-menta K ≫ K c , the terms related to the Bohm quantumpotential become dominant and the excitations propa-gate with a group velocity increasing with K . Simi-lar high-energy (Lorentz-violating) corrections appear inseveral phenomenological approaches to quantum grav-ity, where ~ K c is typically associated to the Planck mo-mentum [55]. III. PHONONS AS A MASSIVEKLEIN-GORDON FIELD
The formal equivalence between phonons propagatingon top of an inhomogeneous photon-fluid and the evo-lution of scalar fields in curved spacetime can be estab-lished starting from the Bogoliubov-de Gennes equations(7)-(8) in the limit K ≪ K c [41]. To this end, we ap-ply the operator ( ∂ T + i c kn ∂ S )( ρ ) to Eq. (7) and weobtain( ∂ T + i c kn ∂ S ) 1 ρ ( ∂ T − i c kn ∂ S ) ε = c s ρ ∂ S ε − i ( ωn ∂ T + i c n ∂ S ) 1 ρ n th (11)The phonon dynamics is obtained by ignoring higher-order spatial derivatives in Eq. (11) ( long-wavelengthlimit ), which indeed are responsible for the Lorentz-breaking, K -terms in the dispersion relation [38, 41].We will turn back on the implications of this approxima-tion at the end of Sect. VII. In this limit and using thefact that the background density ρ satisfies the continu-ity equation (5) with v = v , Eq. (11) can be rewrittenas (cid:3) ε −
12 Ω ( ε + ε ∗ ) = i ωn ( ∂ T + ∇ · v ) n th + c n ∇ · ( n th ∇ ln ρ ) (12)where (cid:3) ≡ − ( ∂ T + ∇ · v ) ∂ T + ∇ · ( c s ∇ ) (13)is the d’Alambertian operator associated with the acous-tic metric g µν = (cid:18) ρ c s (cid:19) (cid:18) − ( c s − v ) − v T0 − v I (cid:19) (14)and I stands for the two-dimensional identity matrix.The complex fluctuations ε can be easily linked to thereal density and phase perturbations through the rela-tions ρ = ρ ( ε + ε ∗ ) and φ = ( i/ ε ∗ − ε ). By meansof these expressions and using ψ = ( c/kn ) φ , Eq. (12)splits into the following system of coupled wave equations (cid:3) (cid:18) ρ ρ (cid:19) − Ω ρ ρ = c n ∇ · ( n th ∇ ln ρ ) (15) (cid:3) ψ = c n ( ∂ T + ∇ · v ) n th (16)for the fluctuations in the velocity-potential ψ and therelative density ρ /ρ . Notably, the above equationsdecouple for a nearly-homogeneous background density ρ ≈ const . On the basis of Eqs. (5)-(6), a nearly-constant density would also imply a nearly-homogeneousflow. Nevertheless, there are specific situations in whichthe simplifying assumption to consider a nearly-constantdensity while letting the flow vary provides an useful andrealistic model of the system. This is indeed the case ofvortex flows in which the horizon and the ergosphere arelocated away from the vortex core in regions of nearlyconstant density [31, 40]. In these conditions Eq. (16)takes the form of the massive Klein-Gordon equation incurved spacetime (cid:3) ρ − Ω ρ = 0 . (17) IV. TEUKOLSKY EQUATION IN THE VORTEXMETRIC
Superradiant instabilities and scalar clouds require (atleast) a rotating Kerr geometry that in acoustic modelscan be simulated by draining vortex flows. In photon-fluids, a vortex arises from the self-trapping of a phasesingularity embedded in a broad optical beam due to thecounterbalanced effects of self-defocusing and diffraction[56]. The resulting pattern is characterized by a darkcore and a helical wave front, E = ρ / ( r ) e iψ , where ρ (0) = 0 and ψ = jθ with the integer j being the topo-logical charge of the vortex. The azimuthal fluid flow isthus v θ = (1 /r ) ∂ θ ψ = cm/ ( kn r ). In order to createan (apparent) horizon an additional radial phase depen-dence is required to provide a non-zero inward radial ve-locity v r . In any region where v r > c s , a sound wave willbe swept inward by the flow and be trapped inside thehorizon, that is formed where v r = c s .Recently, experimental evidence of an ergosphere andhorizon in a photon-fluid with thermo-optical nonlin-earities has been provided using a background beamwith phase ψ = jθ + 2 π p r/r , where r is an ex-perimental parameter controlling the radial phase de-pendence of the beam [40]. Outside the vortex core,the profile ρ ( r ) asymptotes to a constant density andthe flow is well approximated by the above v θ and by v r = ∂ r ψ = − cπ/ ( kn √ r r ). Accordingly, the densityfluctuations are governed by Eq. (17) and the acousticline element in polar coordinates is (up to the conformalfactor ρ /c s ) ds = g µν dx µ dx ν ∼ − c s dt +( dr − v r dt ) +( rdθ − v θ dt ) . The similarity with the equatorial slice of the Kerr ge-ometry becomes clearer through the transformations ofthe time and the azimuthal coordinates dt → dt + | v r | ( c s − v r ) dr ; dθ → dθ + | v r | v θ r ( c s − v r ) dr defined in the exterior region ( r H < r < ∞ ), where r H = ξ /r is the location where v r = c s , i.e. the event horizon.After a rescaling of the time coordinate by c s and definingΩ H = jξπr the metric takes the form ds ∼ − (cid:18) − r H r − r Ω r (cid:19) dt + (cid:16) − r H r (cid:17) − dr − r Ω H dθdt + ( rdθ ) (18)Similarly to the Kerr metric in Boyer—Lindquist coor-dinates, (18) has a coordinate singularity at the eventhorizon r = r H where the radial component g rr goesto infinity. On the other hand, g rr does not containany dependence on the azimuthal flow and thereforethere is no inner horizon. The radius of the ergosphereis given by the vanishing of the temporal component g tt , i.e. r E = r H (1 + p r Ω ), where Ω H is therescaled angular velocity at the horizon. The mixed term g tθ = 2 r Ω H is responsible of the frame dragging due tothe rotating spacetime and disappears when Ω H = 0 (norotation). In this case r E = r H and the metric becomessimilar to Schwarzschild’s.We now consider solutions to the Klein-Gordon equa-tion (17) with metric (18). We decompose the field as ρ ( t, r, θ ) = r − / G ( r ) e − i Ω t + imθ , where the integer m isthe winding number, Ω is normalized to c s and we in-troduce the dimensionless radial coordinate ˆ r = r/r H .We obtain the radial Teukolsky equation on the vortexmetric ∆ dd ˆ r (cid:18) ∆ dd ˆ r (cid:19) G (ˆ r ) + U (ˆ r, ˆΩ) G (ˆ r ) = 0 , (19)where ∆ = (1 − / ˆ r ) and U (ˆ r, ˆΩ) = ˆΩ − m ˆΩ H ˆ r ! +∆ (cid:18) r ∆ − r − m ˆ r − ˆΩ (cid:19) . (20)All angular frequencies in (20) have been rescaled by r H and are thus dimensionless (the frequencies Ω and Ω H,0 are indeed spatial wavenumbers due to the rescaling ofthe time coordinate by c s in (18).In terms of the tortoise coordinate ˆ r ∗ defined as d ˆ r ∗ =∆ − d ˆ r , Eq. (20) reads d G (ˆ r ∗ ) d ˆ r ∗ + U (ˆ r ∗ , ˆΩ) G (ˆ r ∗ ) = 0 . (21)The dimensionless radial equation (21) is the startingpoint of the analysis of the next sections.In Kerr spacetime, superradiance and related phenom-ena depend on the BH angular momentum and, more Ω /m Ω H Ω / m Ω H a) Ω /m Ω H | R | b) Ω /m Ω H | R | c) Trapped States
FIG. 1: a) Reflection coefficients |R (Ω) | in the (Ω ,Ω) parameter space for m = 1 and b) |R (Ω) | for selected values of therest frequency: Ω = 0 (black), for Ω = 0 .
24 Ω H (blue), for Ω = 0 .
42 Ω H (red), for Ω = 0 .
66 Ω H (green). c) |R (Ω) | forΩ = 0 .
12 Ω H and azimuthal numbers m = 1 (black) m = 2 (dashed,blue), m = 3 (dash-dotted,red). In all simulations r H = 1and Ω H = 1 . importantly, on the gravitational coupling given by di-mensionless product (in natural units, G = c = ~ =1) of theblack hole mass M and the field mass µ , M µ ≡ GM µ/ ~ c [14]. The latter corresponds to the ratio of the event hori-zon size to the reduced Compton wavelength of the field,that in our potential (20) is given by the dimensionlessparameter ˆΩ = r H m ph c s / ~ . In the following, we charac-terize superradiant phenomena, quasi-bound states andscalar clouds in our system as ˆΩ is varied. Unless nec-essary, from now on we shall omit hats in all quantities. V. SUPERRADIANT SCATTERING
The occurrence of superradiance is demonstrated bycalculating the Wronskian W of a solution of Eq. (21)and of its complex conjugate near the horizon and atspatial infinity. The tortoise coordinate r ∗ maps r H to r ∗ → −∞ , and r → ∞ to r ∗ → + ∞ . In these limits,Eq. (21) translates into the following harmonic oscillatorequations d G ( r ∗ ) dr ∗ + (Ω − m Ω H ) G ( r ∗ ) , r ∗ → −∞ (22) d G ( r ∗ ) dr ∗ + (Ω − Ω ) G ( r ∗ ) , r ∗ → + ∞ (23)which can be easily solved to get the asymptotic solutions G ( r ∗ ) = T e − i (Ω − m Ω H ) r ∗ , r ∗ → −∞ (24) G ( r ∗ ) = e − i √ Ω − Ω r ∗ + R e i √ Ω − Ω r ∗ , r ∗ → ∞ (25)The solution (24) represents a purely ingoing wave at thehorizon, with is transmission coefficient T , while the firstand second term in (25) correspond respectively to aningoing and a reflected wave with reflection coefficient R .Eqs. (24-25) are limiting approximations of the solution of the full Schr¨odinger equation (21), the Wronskian ofwhich is a constant independent of r ∗ . Moreover, sincethe potential V ( r, Ω) is real (we are now considering onlyreal frequencies Ω), the complex conjugate of any solutionis also a solution. We can thus equate the Wronskian atboth asymptotics W ( G, G ∗ ) | r → r H = W ( G, G ∗ ) | r = →∞ tofind the conservation relation1 − |R| = Ω − m Ω H p Ω − Ω ! |T | . (26)Eq. (26) dictates that modes with frequencies in therange Ω < Ω < m Ω H , are scattered with a reflection co-efficient R > m , i.e., for wavesthat are corotating with the horizon. Due to the presenceof Ω , the frequency range of superradiant scattering isreduced with respect to massless case: waves of frequencyΩ < Ω cannot propagate as they would be either diver-gent or exponentially suppressed at spatial infinity. Thelatter are the quasi-bound states that we will investigatein the next section.The reflection coefficients |R (Ω) | as a function of Ωcan be explicitly calculated by numerical integration ofEq. (21) with asymptotic solutions (22 −
23) and compar-ing the Fourier components of the incident and reflectedwaves. Results for different values of Ω and fixed Ω H are displayed in Fig. 1(a,b). In the superradiant regime,Ω < Ω < m Ω H , the reflection coefficient |R| > m Ω H , beyond which it decays exponentially.The amplification is less pronounced for massive fieldsat all frequencies with respect to the massless case (seeFig. 1b): the larger is Ω , the smaller is |R (Ω) | . In Fig.1(c) we report the reflection coefficient for different val-ues of m showing that as the winding number is increasedthe superresonant effect is continuously decreasing. The r* -10 0 10 20 30 40 50 60 70 80 V -0.3-0.2-0.100.10.20.3 E r go r eg i on Potential WellPotential Barrier
FIG. 2: Real part of V ( r, Ω) as a function of the tortoisecoordinate r ∗ for Ω H = 1 . = 0 . = 0 (blue) and Ω H = 1 . H = 0, Ω = 0 . . . × − i corresponding toa quasi-bound state (see Fig. 3) above phenomenology is similar to what observed in thecase of Kerr spacetime [22, 57, 58]. VI. QUASI-BOUND STATES
The instability of the Kerr spacetime to massive scalarperturbations is a consequence of a binding potential,keeping the superradiant modes confined in the vicinityof the BH.We then start our analysis from the potential term U ( r ∗ , Ω) which determines the behaviour of the radialfunction G . We notice that defining V ( r ∗ , Ω) = Ω − Ω − U ( r ∗ , Ω), Eq. (21) takes the form of the 1D time-independent Schr¨odinger equation − d G ( r ∗ ) dr ∗ + V ( r ∗ , Ω) G ( r ∗ ) = (Ω − Ω ) G ( r ∗ ) (27)for a particle of ”energy” (Ω − Ω ) moving in a poten-tial V ( r, Ω). Unlike the standard Schr¨odinger eigenvalueproblem, the potential depends on the (generally com-plex) eigenfrequency Ω of the modes. Nevertheless, thephysical mechanism behind the superradiant amplifica-tion of trapped modes can be still interpreted in termsof the properties of V ( r, Ω). A necessary condition forthe existence of the quasi-bound states is the presence ofa local potential minimum. An example of the potential V is displayed in Fig. 1(a), where we used the complexeigenfrequency of one of the quasi-bound states shown inFig. 2(b) (black curve). Since Ω I ≪ Ω R , the above pic-ture is barely affected by Ω I . Outside the ergoregion, the curve has the typical shape of a bond interatomic poten-tial [59], repulsive at short distances and attractive oth-erwise, and consists of a finite centrifugal barrier locatedin the vicinity of r E and of a neighboring potential well.The former is mainly related to rotation while the lattercannot form in the massless limit (see dashed curves conΩ H = 0 e Ω = 0 in Fig. 1(a)). Trapped states in the po-tential well can tunnel into the ergoregion and partiallyenter the horizon. For Ω R < m Ω H , the transmitted wavecarries negative energy into the black hole and thus thereflected wave will be reflected with a larger amplitude,while remaining localized in the potential well. This leadsto multiple superradiant scattering in the vicinity of theBH causing the dynamical instability. TABLE I: Complex eigenfrequencies of quasi-bound statesdisplayed in Fig. 3b (from down to up)Ω Ω R Ω I ( × − )0.35202527 0.35034321 0.0780.43203101 0.42909303 0.0800.52803790 0.52298580 0.2300.62582270 0.61784984 0.5000.73338598 0.72119458 1.0800.83917135 0.82167984 2.0400.93517824 0.91176974 3.0901.0498531 1.0178151 3.6401.1583054 1.1163115 -0.462 The above states are described by quasi-bound (non-stationary) modes behaving as purely ingoing waves atthe horizon (as measured by a comoving observer) andexponentially-decaying (bounded) solutions at spatial in-finity. In terms of the tortoise coordinate, their asymp-totic radial profiles are written as G ( r ∗ ) ∼ e − i (Ω − m Ω H ) r ∗ , r ∗ → −∞ (28) G ( r ∗ ) ∼ e − √ Ω − Ω r ∗ , r ∗ → + ∞ (29)For a given Ω , the radial equation (27) [or, equivalentlyEq. (21)] together with the above boundary conditionsgives rise to a discrete spectrum of complex eigenfrequen-cies { Ω(Ω ) } = Ω R + i Ω I . The imaginary part Ω I setsthe growth (Ω I >
0) or decay (Ω I <
0) rate of the am-plitude of the scalar field. We remind that in additionto bound-states, the problem admits also a discrete setof resonances corresponding to outgoing waves at spatialinfinity ∼ e √ Ω − Ω r ∗ , which are the acoustic analogue ofthe so-called quasi-normal modes [60–62]. In the follow-ing we focus on quasi- and purely-bound state solutions.Fixed the black hole angular frequency Ω H and therest frequency of the massive field Ω H , quasi-bound statesare uniquely identified by two “quantum” numbers: thewinding number of the field m and the non-negative n corresponding to the node number of G ( r ∗ ). Notice thatalthough the topological charge of vortex j is also quan-tized, it does not uniquely determine the state, as Ω H can be continuosly varied e.g. by changing the locationof the horizon. r* -5 50 105 160-0.2-0.100.10.2 b) R e ( G ) a) Ω /m Ω H - Ω I -2024 c) Ω R - m Ω H -0.6-0.4-0.200.21.03 1.035 1.04 1.045 0 FIG. 3: a) Radial profiles of fundamental quasi-bound states ( n = 0, m = 1) for different values of Ω and Ω H = 1 . and the corresponding eigenfrequencies are reported in Table I. b) Radial profiles of fundamental n = 0 (black)and excited states n = 1 (blue) and n = 2 (red) for Ω = 0 . H = 1 . . . × − i , Ω = 0 . . × − i and Ω = 0 . . i , respectively.c) Bound state spectrum of fundamental modes ( n = 0, m = 1) as a function of Ω and angular velocities Ω H = 1 . H = 1 . H = 1 . To calculate the quasi-bound state spectrum we nu-merically solve Eq. (21) implementing a Numerov in-tegration scheme and one-parameter shooting methodbased on bisection procedure to find the eigenvalues. Inour calculations we keep Ω H fixed and treat Ω R as theeigenvalue to be determined, whereas the parameters Ω I and Ω are continuously scanned. First we perform for-ward integration, starting from r = r H + δr until a suit-able chosen matching point r m , and then a backwardone, starting from r = r max until that point. The trialvalue of Ω R is properly adjusted in order to get the pre-determined number n of nodes and the matching betweenthe forward and backward solutions. The latter is evalu-ated by comparing their logarithmic derivative at r m .The radial profiles of quasi-bound modes with n = 0and m = 1 for different values of Ω are illustrated inFig. 3(a). The solutions are strongly reminiscent of theground-state radial wavefunctions in hydrogenic poten-tials. We observe that the position of their maxima in-creases with Ω and, contextually, the correspondent pro-files are less localized (i.e. they extend over a larger por-tion of space). This behaviour is consistent in the frame-work of a quantum mechanical interpretation of G as anatomic orbital, in which the mode ”position” related tothe Bohr radius of the massive particle is proportional toits Compton wavelength, thus scaling as 1 / Ω .In Fig. 3(c) we report the spectrum of the fundamen-tal quasi-bound state for ( n , m )=(0,1). The plot clearlyshows that a dynamical instability (Ω I >
0) occurs withinthe superradiant regime, Ω R < Ω < m Ω H . Similarly tothe curves of the reflection coefficient R (see Fig. 1),the growth rate displays a clear maximum close the crit- ical frequency (Ω R . Ω ∼ m Ω H ) and increases with theBH angular velocity. Instead, modes with Ω R > m Ω H are decaying (Ω I < R = m Ω H the growth rate isequal to 1. In these conditions the trapped modes areneither growing nor vanishing. This is precisely the equi-librium condition for the existence of stationary cloudsthat we will discuss in the next section. At lower val-ues of Ω the growth rate is continuosly decreasing, ap-proaching the value Ω I = 0 as Ω R → Ω . For Ω > Ω quasi-bound states no longer exist and the system entersthe regime of supperradiant scattering (see Sec. V). Theabove spectrum is fully consistent to what observed formassive scalar fields on Kerr spacetime [14, 63, 64].In Fig. 3(b) we show quasi-bound states for m = 1and n = 0 , ,
2. We find that increasing the numberof nodes the profiles are less localized and their center-of-mass shifts to higher values of r ∗ . These solutionshave been found for three complex eigenfrequencies, thereal part of which increases with n (see caption of Fig.3). Therefore, when all the other parameters are fixed,modes with a higher number of nodes are more energeticwith respect to the fundamental and can be actually in-terpreted as excited states of the field. VII. SCALAR CLOUDS
As discussed in the previous section, at the superradi-ant instability threshold the phonon field is in equilibriumwith the BH forming a pure bound state, i.e. Ω I = 0. Wethus solve numerically Eq. (21) fixing Ω R = Ω = m Ω H . Ω /m Ω H Ω = m Ω H c) G -0.0500.05 a) r* 0 10 20 30 G -0.0500.05 b) FIG. 4: Radial profiles of fundamental n = 0 (black) and excited scalar clouds n = 1 (blue) and n = 2 (red) for m = 1 and a)Ω = 0 . = 7 . . n = 0), Ω = 0 . n = 1), Ω = 0 . n = 2) and b) Ω = 6 . n = 0), Ω = 6 . n = 1) and Ω = 6 . n = 2. Existence lines in the (Ω H , Ω ) parameter space for clouds: ( n , m )=(0,1) (solid-black);( n , m )=(0,2) (dashed-black);( n , m )=(0,3) (solid-black);( n , m )=(1,1) (solid-blue);( n , m )=(1,2) (dashed-blue);( n , m )=(1,3) (dotted-blue);( n , m )=(2,1)(solid-red);( n , m )=(2,2) (dashed-red);( n , m )=(1,3) (dotted-red). In this case, the radial function admits a power seriesexpansion close to the horizon G ∼ c ( r −
1) + c ( r − + ..., (30)where the coefficients can be found replacing (30) intoEq. (21) and solving it order by order in terms of powersof (r-1). We adopt a numerical method similar to the oneused to calculate quasi-bound states: starting with thenear horizon expansion (30), we fix all quantum numbersand scan Ω to find the eigenfrequencies Ω for which G remains bounded, as prescribed by the condition (29).In our scheme we have truncated the expansion at thesecond-order. Solutions with the right asymptotic be-havior exist only for a discrete set of Ω corresponding tonumber of nodes n of the radial function G .In Fig. 4(a-b) we plot the radial dependence of fun-damental and excited clouds for two different rest fre-quencies. The profile is always bounded at the hori-zon and decrease exponentially as r ∗ → ∞ . Similarlyto quasi-bound states, scalar clouds with larger Ω ex-hibit a smaller orbital radius and a smaller width of theassociated wave profile.The above simulations allow to construct the existenceline of clouds in the (Ω H , Ω ) space. Such lines for thefundamental n = 0 and excited modes n = 1 , m are displayed in Fig. 4(c).Since these lines demarcate the threshold of the superra-diant instability, the area below (above) a given line con-tains unstable (stable) solutions against the correspond-ing mode.We observe that the existence lines shift towards highervalues of Ω H for higher n and m , which is in agreement with the interpretation that excited configurations re-quire a larger background rotation for equilibrium. Allthe lines are converging as Ω → Ω . However, this is justan asymptotic limit since in that point Ω = m Ω H andaccordingly to conditions (28-29) the radial function G would become a fully delocalized (plane) wave.We now comment on how close to the horizon theclouds can be localized. We first notice that for anyset of parameters the radial profile of the fundamentalmode n = 0 attains its maximum in correspondence ofthe minimum of the potential well. A rough estimateof its position can be thus obtained by finding the localminimum of V ( r, Ω).In order to obtain more elucidating expressions, we ap-proximate the potential considering only the first leadingterms in 1/r, i.e. V ( r, Ω) = − Ω r + wr (31)where we used Ω = m Ω H and we defined w = m +2Ω − /
4. Eq. (31) closely resemble the Kratzer molec-ular potential with an attractive Coulombian and a re-pulsive inverse-square term [65]. The minimum is foundat r cl = 2 w/ Ω . The dependence of r cl from BH and fieldparameters can be inferred with the help of the existencelines in Fig. 4(c). As Ω → m Ω H , Ω → r cl = 2Ω − m Ω ≥ ≡ r min (32)We can see that r cl increases as the angular frequencydescreases, diverging as Ω H →
0. This behaviour is in Ω H r m a x FIG. 5: Effective positions of the fundamental cloud solution( n = 0 , m = 1) (dashed-green) and the first two excited states( n = 1 , m = 1) (dotted-blue) and ( n = 2 , m = 1) (dashed-dotted-red) as a function of Ω H . The solid-black curve is thefunction r min (Ω H ) as defined in (32). agreement to what observed in the case of Kerr BHs andis consistent with the fact that clouds are supported onlyin the presence of rotation. Similar considerations applyto the limit case Ω →
0, keeping Ω H fixed: without anyother confinement mechanism, clouds can form only inthe presence of massive scalar fields.In Fig. 5 we plot the effective position of cloud solu-tions, calculated as the radial coordinate for which thefunction 2 πr | G ( r ) | attains its maximum value [68], as afunction of Ω H . As expected, the excited states n = 1 , n = 0 (solutions withhigher azimuthal numbers m are even farther). All curvesare in agreement with the inequality (32) that thus pro-vide a lower bound for the clouds position.Unlike the Kerr metric, acoustic BHs have no mathe-matical upper limits on the angular velocity [36, 66]. Onthe other hand, a soft bound is placed by K ≪ K c , i.e.the long-wavelength regime in which the massive Klein-Gordon equation well describe the phonon dynamics.This condition implies an upper bound for the (dimen-sionless) angular frequency Ω H = r H K c /m that wouldcorrespond to an absolute lower limit for the position ofcloud solutions. Incidentally, for Ω H ∼
1, Eq. (32) isin perfect agreement with the “no-short hair” conjecture[67], stating that for static and spherically symmetricBHs the region of non-asymptotic behavior of the BHhair (hairosphere), of which the scalar clouds are the lin-ear seeds, must extend beyond 3 / H = r H K c /m only separates two differ-ent regimes, namely the Lorentz-invariant phonon regimeand the high-energy regime of Bogoliubov excitations.Recent works in local photon-fluids have demonstratedthe occurrence of a resonant amplification phenomenonfor Bogoliubov excitations, which reduces to the standardphonon superradiance in the long-wavelength limit [38].Remarkably, the superradiant regime Ω < m Ω H remainsunaffected independently of the energy of the excitation,and the effects of the breakdown of the long-wavelegthapproximation manifest only in a reduced amplificationfactor. While the generalization of these results to thecase of our photon-fluid is not straightforward, we wouldnot expect strong qualitative differences, in particular forscalar clouds solutions, which exist right at the instabil-ity thereshold Ω = m Ω H . All this will be the subject ofa forthcoming investigation. VIII. CONCLUSIONS AND PERSPECTIVES
Quantum fluids of light such as exciton-polaritonsBECs and more recently photon-fluids have offered al-ternative platforms for analogue gravity investigations.Recent experiments in these systems provided evidenceof acoustic horizons in 1D flows [33, 70] and rotating BHgeometries [40, 71] in (2+1)-dimensions.Here, we have theoretically investigated a photon-fluidwith both local and nonlocal thermo-optical interactions,the fluctuations of which behave as a massive Klein-Gordon field. In the presence of suitable vortex flows,superradiant instabilities are shown to arise due to theexistence of phonon states which are prevented from radi-ating their energy to infinity. At the superradiant insta-bility thereshold, these modes give rise to purely boundstates in synchronous rotation with the horizon, whichexactly represent the acoustic counterpart of stationaryscalar clouds around Kerr BHs. This generalizes pre-vious studies to the case of massive phonons and pavethe way for observing superradiant instabilities and mas-sive scalar clouds configurations in the laboratory. Sincemost analogue gravity models are dealing with masslessexcitations, this is one of the very few systems in whichthese phenomena could be investigated and, possibly inthe near future, experimentally observed.Interestingly, scalar clouds have been related to theexistence of Kerr BHs with scalar hair [17]: these solu-tions of the linear Klein-Gordon equation are actuallythe seeds of a new family of solutions of the fully non-linear Einstein-Klein-Gordon system, which correspondto Kerr-like geometries deformed by the cloud backre-action [72]. As also remarked in [42], the specific form0of the equations ruling the background is thus crucialto relate linear clouds around Kerr BHs to (nonlinear)scalar hairy BHs. As in all analog models, our backreac-tion is encoded in a set of nonlinear equations [cf. 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