Stationary scalar clouds supported by rapidly-rotating acoustic black holes in a photon-fluid model
aa r X i v : . [ g r- q c ] F e b Stationary scalar clouds supported by rapidly-rotating acoustic black holes in aphoton-fluid model
Shahar Hod
The Ruppin Academic Center, Emeq Hefer 40250, IsraelandThe Hadassah Institute, Jerusalem 91010, Israel (Dated: February 5, 2021)It has recently been proved that, in the presence of vortex flows, the fluctuation dynamics of arotating photon-fluid model is governed by the Klein-Gordon equation of an effective massive scalarfield in a (2 + 1)-dimensional acoustic black-hole spacetime. Interestingly, it has been demonstratednumerically that the rotating acoustic black hole, like the familiar Kerr black-hole spacetime, maysupport spatially regular stationary density fluctuations (linearized acoustic scalar ‘clouds’) in itsexterior regions. In particular, it has been shown that the composed rotating-acoustic-black-hole-stationary-scalar-field configurations of the photon-fluid model exist in the narrow dimensionlessregime α ≡ Ω /m Ω H ∈ (1 , α max ) with α max ≃ .
08 [here Ω H is the angular velocity of the black-holehorizon and { Ω , m } are respectively the effective proper mass and the azimuthal harmonic indexof the acoustic scalar field]. In the present paper we use analytical techniques in order to explorethe physical and mathematical properties of the acoustic scalar clouds of the photon-fluid model inthe regime Ω H r H ≫ { Ω (Ω H , m ; n ) } which characterizes the stationary bound-state acoustic scalar clouds of the photon-fluid model.Interestingly, it is proved that the critical (maximal) mass parameter α max , which determines theregime of existence of the composed acoustic-black-hole-stationary-bound-state-massive-scalar-fieldconfigurations, is given by the exact dimensionless relation α max = q . I. INTRODUCTION
Kerr black-hole spacetimes [1] are characterized by the presence of an ergoregion [2], a region outside the black-holehorizon in which matter fields are bound to co-rotate with the central spinning black holes. Interestingly, it has beendemonstrated, both analytically [3] and numerically [4], that this physically intriguing feature of spinning black-holespacetimes allows them to support spatially regular stationary configurations of bosonic (integer-spin) fields thatco-rotate with the central black hole.The stationary hairy scalar field configurations, which in the linearized regime have received the nickname ‘scalarclouds’ in the physics literature [3, 4], are characterized by proper frequencies that are in resonance with the angularvelocity Ω H of the central supporting spinning black hole [3, 4]. In particular, the characteristic proper frequency of astationary scalar cloud with an azimuthal harmonic index m coincides with the critical (marginal) frequency for thesuperradiant scattering phenomenon [5, 6] in the rotating black-hole spacetime [3, 4, 7]: ω = m Ω H . (1)In addition, spatially regular (bounded) bosonic clouds are characterized by the simple upper bound [3, 4] ω < µ , (2)where µ is the proper mass [8] of the supported stationary scalar field. The relation (2) implies that the co-rotatingmassive scalar field configurations are spatially bounded to the central black hole and cannot radiate their energy andangular momentum to infinity.Intriguingly, an analogous physical phenomenon in a rotating photon-fluid system has recently been revealed in thehighly important work [9]. Photon-fluids are nonlinear optical systems whose physical and mathematical propertiescan be described by the hydrodynamic equations of an interacting Bose gas [9–12]. In particular, it has been shown[9, 13] that photon-fluid systems are characterized by the presence of long-wavelength elementary excitations (phonons)that behave as massive scalar fields in an effective acoustic curved spacetime.The dynamics of massive phonons over a draining vortex flow in the photon-fluid model has been investigatedrecently in [9] as the acoustic analogue of the (more familiar) dynamics of massive scalar fields in rotating curved black-hole spacetimes. In particular, it has been explicitly proved [9] that the dynamics of linearized acoustic excitations inthe photon-fluid model with a draining vortex flow are governed by the familiar Klein-Gordon equation of an effectivescalar field of proper mass Ω that propagates in an acoustic (2 + 1)-dimensional spinning black-hole spacetime which,like the familiar Kerr black-hole spacetime, possesses an ergoregion.Intriguingly, using direct numerical techniques, it has been explicitly demonstrated in [9] that the acoustic spinningblack-hole spacetime may support stationary linearized density fluctuations (acoustic scalar ‘clouds’) in its exteriorregions. In particular, it has been revealed [9] that the composed acoustic-black-hole-stationary-massive-scalar-fieldconfigurations of the photon-fluid model are characterized by the narrow regime of existence [9, 14] α ≡ Ω m Ω H ∈ (1 , α max ) with α max ≃ . , (3)where Ω H is the angular velocity that characterizes the acoustic horizon of the central supporting spinning black hole.The main goal of the present paper is to explore, using analytical techniques, the physical and mathematicalproperties of the composed acoustic-spinning-black-hole-stationary-linearized-scalar-field configurations of the photon-fluid model. In particular, we shall derive a remarkably compact analytical formula for the discrete resonance spectrum { Ω (Ω H , m ; n ) } [15] that characterizes the spatially regular stationary acoustic scalar clouds in the dimensionlessregime Ω H r H ≫ α < α max ≃ .
08 [seeEq. (3)] on the regime of existence of the composed acoustic-black-hole-stationary-bound-state-massive-scalar-fieldconfigurations.
II. DESCRIPTION OF THE SYSTEM
We study the physical and mathematical properties of density fluctuations in a rotating photon-fluid model. Intrigu-ingly, a formal equivalence has recently been established in the physically important work [9] between the dynamics oflinearized acoustic phonons that propagate on top of an inhomogeneous photon-fluid and the dynamics of linearizedmassive scalar fields in a spinning curved spacetime. In particular, it has been explicitly proved in [9] that, in thepresence of vortex flows, the dynamics of acoustic density fluctuations in the long-wavelength regime of the photon-fluid model are governed by the Klein-Gordon equation of a massive scalar field in an effective (2 + 1)-dimensionalcurved spacetime.The effective acoustic spacetime of the (2 + 1)-dimensional rotating photon-fluid model can be described, usingpolar coordinates, by the non-trivial curved line element [9, 17] ds = − (cid:16) − r H r − Ω r r (cid:17) dt + (cid:16) − r H r (cid:17) − dr − H r dθdt + r dθ . (4)Here r H is the radius of the acoustic black-hole horizon, which is determined as the circular ring at which the inwardradial velocity v r of the fluid flow equals the speed of sound c s [9, 18, 19]. The physical parameter Ω H in the curvedline element (4) is the angular velocity of the effective acoustic horizon.Interestingly, like the familiar Kerr black-hole solution of the Einstein field equations, the rotating acoustic spacetime(4) of the photon-fluid model is characterized by the presence of an effective ergoregion whose outer boundary [9] r E = 12 r H (cid:16) q r (cid:17) (5)is determined by the condition g tt = 0.As explicitly proved in [9], the spatial behavior of density fluctuations of the form ρ ( t, r, θ ) = ψ ( r ) √ r e imθ − i Ω t (6)in the effective acoustic spacetime (4) of the rotating photon-fluid model are governed by the radial differentialequation h ∆ ddr (cid:16) ∆ ddr (cid:17) − V ( r ; Ω) i ψ ( r ) = 0 , (7)where ∆ ≡ − r H r . (8)The effective radial potential of the photon-fluid system is given by the functional expression V ( r ; Ω) = − (cid:16) Ω − m Ω H r r (cid:17) + ∆ (cid:16) Ω + m r + r H r − ∆4 r (cid:17) . (9)The physical parameter Ω , which plays the role of an effective scalar mass, is the rest energy of the collectiveexcitations (phonons) [9]. The θ -periodicity of the angular function e imθ in the field decomposition (6) implies thatthe azimuthal harmonic index | m | of the scalar perturbation modes is an integer [20].In the next section we shall use analytical techniques in order to derive the discrete resonance spectrum { Ω (Ω H , m ; n ) } n = ∞ n =0 of the composed acoustic-black-hole-stationary-bound-state-linearized-massive-scalar-field con-figurations. The radial eigenfunctions that characterize the stationary scalar clouds of the acoustic curved spacetime(4) are determined by the ordinary differential equation (7) with the physically motivated boundary conditions ofspatially regular (bounded) scalar eigenfunctions at the acoustic black-hole horizon and at spatial infinity [9]: ψ ( r = r H ) < ∞ (10)and ψ ( r → ∞ ) ∼ e − √ Ω − Ω r for Ω < Ω . (11) III. THE DISCRETE RESONANCE SPECTRUM OF THE COMPOSEDACOUSTIC-BLACK-HOLE-SCALAR-CLOUDS CONFIGURATIONS OF THE PHOTON-FLUID MODEL
In the present section we shall analyze the discrete resonance spectrum { Ω (Ω H , m ; n ) } that characterizes thecomposed acoustic-black-hole-stationary-bound-state-linearized-massive-scalar-field configurations of the photon-fluidmodel [9]. The stationary bound-state scalar clouds of the effective rotating black-hole spacetime (4) are characterizedby the resonance condition [9] Ω = m Ω H < Ω . (12)Interestingly, we shall now prove that the resonance spectrum { Ω (Ω H , m ; n ) } of the acoustic scalar clouds can bestudied analytucally in the eikonal large-frequency regime [21]Ω H r H ≫ m (13)of the central supporting spinning black hole.To this end, it is convenient to write the radial differential equation (7), which determines the spatial behaviorof the scalar eigenfunctions in the acoustic black-hole spacetime (4), in the form of the mathematically compactSchr¨odinger-like ordinary differential equation d ψdy − V ( y ) ψ = 0 , (14)where the tortoise radial coordinate y is defined by the differential relation [22] dy = ∆ − dr . (15)Substituting into Eq. (9) the resonant frequency Ω = m Ω H [see Eq. (1)], which characterizes the stationary acousticscalar clouds of the photon-fluid model, one obtains the functional expression V ( r ) = − ( m Ω H ) · (cid:16) − r r (cid:17) + ∆ (cid:16) Ω + m r + r H r − ∆4 r (cid:17) (16)for the effective radial potential V [ r ( y )] of the composed acoustic-black-hole-stationary-bound-state-massive-scalar-field configurations.We shall now show explicitly that the Schr¨odinger-like ordinary differential equation (14) for the spatially regularstationary scalar clouds in the rotating acoustic black-hole spacetime (4) is amenable to a standard WKB analysis[23–27] in the dimensionless regime Ω H r H ≫ V min p V ′′ min = − ( n + 12 ) ; n = 0 , , , ... , (17)where V ′′ ≡ d V /dy . The effective binding potential V min and its second spatial derivative V ′′ min in the WKBresonance condition (17) are evaluated at the minimum point r = r min of the potential (16), where V ′ ≡ dVdy = 0 for r ( y ) = r min . (18)Substituting the effective binding potential (16) of the composed spinning-black-hole-acoustic-scalar-clouds config-urations into Eq. (18), one finds the relationΩ = 4( m Ω H ) ( r − r ) r H r · { O [(Ω H r H ) − ] } for Ω H r H ≫ r H ( r − r ) − ( r min − r H )( r min + r H ) = − r H p r − r m Ω H · (cid:0) n + 12 (cid:1) (20)for the radial location of the minimum r = r min of the effective binding potential (16) that characterizes the composedacoustic-black-hole-stationary-scalar-clouds configurations.As we shall now show, the (rather cumbersome) resonance equation (20) can be solved analytically using an iterationscheme. The zeroth-order solution r (0)min ≡ r min (Ω H r H → ∞ ) of the resonance equation (20) is given by the simpleasymptotic value r (0)min = 3 r H . (21)Next, substituting r min = 3 r H · (cid:2) α (Ω H r H ) − (cid:3) (22)into the resonance equation (20), one finds α = n + √ m · { O [(Ω H r H ) − ] } , (23)which yields the functional expression [see Eq. (22)] r min = 3 r H n √ m (cid:0) n + 12 (cid:1) · (Ω H r H ) − + O [(Ω H r H ) − ] o (24)for the radial location of the minimum of the effective binding potential (16).Finally, substituting (24) into the relation (19), one obtains the discrete resonance spectrumΩ = m Ω H · r n − √ m (cid:0) n + 12 (cid:1) · (Ω H r H ) − + O [(Ω H r H ) − ] o (25)which characterizes the composed acoustic-black-hole-stationary-bound-state-linearized-massive-scalar-field configu-rations of the photon-fluid model. IV. NUMERICAL CONFIRMATION
It is of physical interest to test the accuracy of the analytically derived resonance formula (25) which characterizesthe composed acoustic-black-hole-stationary-scalar-field configurations of the photon-fluid model. The correspondingeffective field masses { Ω (Ω H , m ; n ) } of the acoustic scalar clouds were recently computed numerically in the interestingwork [9].In Table I we display the dimensionless ratios α numerical and α wkb for the fundamental ( n = 0) resonant mode of thestationary bound-state acoustic scalar clouds with m = 1 and for various values of the dimensionless angular velocityΩ H r H of the central supporting spinning black hole. Here { α numerical (Ω H r H ) } are the exact ( numerically computed [9])values of the dimensionless ratio Ω /m Ω H , which characterizes the composed acoustic-black-hole-stationary-bound-state-linearized-massive-scalar-field configurations, and { α wkb (Ω H r H ) } are the corresponding analytically derived val-ues of this dimensionless physical parameter as given by the WKB resonance formula (25).Interestingly, the data presented in Table I reveals an excellent agreement between the numerical data of [9] and theanalytically derived WKB resonance formula (25) of the composed acoustic-black-hole-stationary-massive-scalar-fieldconfigurations [28]. Ω H r H α numerical α wkb H r H , the exact (numerically computed[9]) values of the dimensionless ratio α ≡ Ω /m Ω H for the fundamental ( n = 0) resonant mode of the stationary spatiallyregular massive scalar clouds with m = 1. We also present the corresponding analytically derived values of the dimensionlessratio α ≡ Ω /m Ω H as calculated directly from the WKB resonance formula (25). One finds a remarkably good agreementbetween the analytically derived formula (25) and the numerically computed values [9] of the dimensionless ratio Ω /m Ω H which characterizes the composed acoustic-black-hole-stationary-bound-state-massive-scalar-field configurations [28]. V. SUMMARY
The recently published highly important work [9] has revealed the physically interesting fact that, in the presence ofvortex flows, the dynamics of fluctuations in a rotating photon-fluid model is governed by the Klein-Gordon equationof an effective massive scalar field in a spinning acoustic black-hole spacetime. In particular, it has been demonstratednumerically [9] that co-rotating acoustic scalar clouds, spatially regular bound-state configurations which are madeof stationary linearized massive scalar fields, can be supported by the central spinning (2 + 1)-dimensional acousticblack holes.The important numerical results presented in [9] have nicely demonstrated the fact that, for a given value ofthe horizon angular velocity Ω H of the central supporting black hole, the stationary bound-state acoustic cloudsof the photon-fluid model are characterized by a discrete resonance spectrum { Ω (Ω H , m ; n ) } n = ∞ n =0 for the effectivemass parameter of the supported scalar fields. In particular, it has been revealed that the composed acoustic-black-hole-stationary-bound-state-massive-scalar-field configurations of the photon-fluid model [9] exist in the narrowdimensionless regime α ≡ Ω /m Ω H ∈ (1 , α max ) with α max ≃ . m Ω H = r · n − √ m (cid:0) n + 12 (cid:1) · (Ω H r H ) − + O [(Ω H r H ) − ] o (26)for the discrete resonant spectrum that characterizes the acoustic scalar clouds in the dimensionless regime Ω H r H ≫ m of rapidly-spinning central supporting black holes. The analytically derived formula (26) for the discrete resonant spec-trum of the composed acoustic-spinning-black-hole-massive-scalar-field configurations was shown to agree remarkablywell with direct numerical computations [9] of the corresponding resonant modes of the photon-fluid model.Interestingly, from the resonance formula (26) one finds the asymptotic upper bound (cid:16) Ω m Ω H (cid:17) max = r ACKNOWLEDGMENTS
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2, the acoustic black-hole spacetimes of the photon-fluid model have no strictupper bound on their angular velocities [9].[22] Note that the semi-infinite radial range r ∈ [ r H , ∞ ] of the acoustic black-hole spacetime is mapped into the infinite range y ∈ [ −∞ , + ∞ ] by the radial differential relation (15).[23] B. F. Schutz and C. M. Will, Astrophys. J. , L33 (1985).[24] S. Iyer and C. M. Will, Phys. Rev. D , 3621 (1987).[25] S. Iyer, Phys. Rev. D , 3632 (1987).[26] L. E. Simone and C. M. Will, Class. Quant. Grav. , 963 (1992).[27] S. Hod, Phys. Lett. B , 365 (2015) [arXiv:1506.04148].[28] It is worth noting that the agreement between the numerical data of [9] and the analytically derived WKB resonanceformula (25) of the composed acoustic-black-hole-stationary-bound-state-linearized-massive-scalar-field configurations isgenerally better than 0 .