Onset of superradiant instabilities in the hydrodynamic vortex model
aa r X i v : . [ g r- q c ] J u l Onset of superradiant instabilities in the hydrodynamic vortex model
Shahar Hod
The Ruppin Academic Center, Emeq Hefer 40250, IsraelandThe Hadassah Institute, Jerusalem 91010, Israel (Dated: September 23, 2018)The hydrodynamic vortex, an effective spacetime geometry for propagating sound waves, isstudied analytically . In contrast with the familiar Kerr black-hole spacetime, the hydrodynamicvortex model is described by an effective acoustic geometry which has no horizons. However, thisacoustic spacetime possesses an ergoregion, a property which it shares with the rotating Kerr space-time. It has recently been shown numerically that this physical system is linearly unstable dueto the superradiant scattering of sound waves in the ergoregion of the effective spacetime. In thepresent study we use analytical tools in order to explore the onset of these superradiant instabilitieswhich characterize the effective spacetime geometry. In particular, we derive a simple analyticalformula which describes the physical properties of the hydrodynamic vortex system in its critical(marginally-stable) state, the state which marks the boundary between stable and unstable fluidconfigurations. The analytically derived formula is shown to agree with the recently publishednumerical data for the hydrodynamic vortex system.
I. INTRODUCTION
One of the most remarkable characteristics of the ro-tating Kerr black-hole spacetime [1] is the existence of an ergoregion [2]: a region in which all physical objects mustco-rotate with the spinning black hole. In particular, nophysical observer inside the ergoregion can remain at restwith respect to inertial asymptotic observes.The presence of the ergoregion in the Kerr black-holespacetime is responsible for the intriguing phenomenonof superradiant wave scattering: it was first realized byZel’dovich [3] (see also [4, 5]) that a co-rotating bosonicfield of the form e imφ e − iωt interacting with a spinningKerr black hole of angular velocity Ω can be amplified(gain energy) if the incident wave field satisfies the su-perradiant condition ω < m Ω . (1)The superradiant scattering of bosonic fields in theKerr black-hole ergoregion (the extraction of rotationalenergy from the black hole) has the potential to destabi-lize the spacetime geometry [6]. However, the Kerr blackhole is known to be stable against massless perturbationfields [5, 7]. The stability of the Kerr spacetime againstthe superradiant scattering of massless bosonic fields inits ergoregion may be attributed to the absorption prop-erties of the black-hole horizon [6]. In particular, theblack-hole horizon acts as a one-way membrane whichabsorbs the (potentially dangerous) perturbation fieldsbefore any instability has the chance to develop in theergoregion [8–11].The reasoning presented above [6] suggests that hori-zonless spacetimes which possess ergoregions may gen-erally be unstable to superradiant scattering of bosonicfields in their ergoregions. This suggestive argument wasraised long ago by Friedman [12]. Most recently, Oliveiraet. al. [6] have explored analogous ergoregion instabili-ties which may develop in fluid flow dynamics. It was first shown by Unruh [13] that the characteristic wave equa-tion for sound waves propagating inside fluids is anal-ogous to the Klein-Gordon wave equation for masslessscalar fields propagating in curved spacetimes [see Eq.(7) below].Oliveira et. al. [6] have studied numerically the hy-drodynamic vortex model, a two-dimensional purely cir-culating flow of a vorticity free ideal fluid. This acousticsystem is analogous to an effective horizonless spacetimewhich nevertheless possesses an ergoregion [14]. In ac-cord with the arguments presented in [12], it has beenestablished in [6] that the effective spacetime geometrywhich corresponds to the hydrodynamic vortex system ischaracterized by unstable acoustic perturbation modes.This is an ergoregion instability [6] which is related tothe absence of an event horizon in the effective rotatingspacetime.A remarkable feature of the hydrodynamic vortexsystem is the existence, for each given value of thesound mode harmonic index m [see Eq. (8) below], of marginally-stable fluid configurations. These stationaryconfigurations mark the boundary between stable andunstable fluid flows. The main goal of the present studyis to obtain an analytical formula which describes thephysical properties of the hydrodynamic vortex systemin its critical (marginally-stable) state. II. DESCRIPTION OF THE SYSTEM
We study the dynamics of a vorticity free barotropicideal fluid. Assuming a two-dimensional purely circulat-ing flow in the xy plane, the background (unperturbed)fluid velocity is characterized by [6] v r = v z = 0 ; v φ = v φ ( r ) . (2)Here r and φ are respectively the radial and azimuthalcoordinates in the xy plane, and z is the coordinate per-pendicular to the plane of flow.Irrotationality of the fluid flow (vorticity free flow) im-plies that the tangential component of the velocity field, v φ , is given by [6] v φ = Cr , (3)where the constant C characterizes the circulationstrength of the fluid. Conservation of angular momen-tum yields ρv φ r = const. (4)which, together with Eq. (3), implies that the fluidbackground density ρ is constant. The assumption ofa barotropic fluid then implies that the background pres-sure P and the speed of sound c are also constants.The two-dimensional circulating fluid flow producesan effective acoustic spacetime, known as the hydrody-namic vortex [6, 14–16], which is characterized by thenon-trivial [17] line element ds = − c (cid:16) − C c r (cid:17) dt + dr − Cdtdφ + r dφ + dz . (5)The rotating acoustic spacetime geometry (5) pos-sesses an ergoregion whose outer boundary is determinedby the circle at which the fluid flow velocity, | C | /r , equalsthe speed of sound c [6, 14–16]: r ergo = | C | c . (6)(We shall henceforth use units in which c = 1. Note thatin these units C has the dimensions of length).We shall now consider small perturbations to the back-ground fluid flow. These perturbations (sound waves)propagate in the acoustic spacetime and their linearizedNavier-Stokes dynamics is governed by the Klein-Gordonwave equation [6, 13, 18] ∇ ν ∇ ν Ψ = 1 p | g | ∂ µ (cid:16)p | g | g µν ∂ ν Ψ (cid:17) = 0 . (7)It proves useful to decompose the perturbation field Ψin the form [19]Ψ( t, r, φ, z ) = 1 √ r ∞ X m = −∞ ψ m ( r ; ω ) e imφ e − iωt , (8)The φ -periodicity of the angular function e imφ enforcesthe azimuthal harmonic index | m | to be an integer. Sub-stituting the decomposition (8) into the Klein-Gordonwave equation (7), one obtains the characteristic radialequation h d dr + (cid:16) ω − Cmr (cid:17) − m − r i ψ m ( r ; ω ) = 0 (9)for each field mode [20]. (We shall henceforth omit theharmonic index m for brevity). III. BOUNDARY CONDITIONS
The background velocity field (3) is singular at theorigin, signaling a breakdown of the physical description.In order to mimic a possible experimental scenario inthe laboratory, Oliveira et. al. [6] have suggested toimpose physically acceptable boundary conditions at a finite radial location, r = r min . In particular, it wasassumed in [6] that an infinitely long cylinder of radius r min made of a certain material with acoustic impedance Z [21] is placed at the center of the fluid system.Oliveira et. al. [6] considered two types of boundaryconditions (BCs) at the surface r = r min of the centralcylinder, characterizing two limiting values of the cylin-der acoustic impedance: Extremely low-Z materials [21]are characterized by the Dirichlet-type boundary condi-tion [6]: ψ ( r = r min ) = 0 , BCI , (10)whereas extremely high-Z materials [21] (that is, a veryrigid boundary cylinder [6]) are characterized by theNeumann-type boundary condition [6]: d Ψ dr ( r = r min ) = d ( ψ/ √ r ) dr ( r = r min ) = 0 , BCII . (11)Following [6], we shall consider purely outgoing waves atlarge distances from the cylinder: ψ ( r → ∞ ) ∼ e iωr . (12) IV. THE ERGOREGION INSTABILITY OF THEHYDRODYNAMIC VORTEX
As emphasized above, the instability of the hydrody-namic vortex system studied in [6] is closely related tothe phenomenon of superradiant scattering [14] of soundwaves in the ergoregion of the effective spacetime geom-etry (5). Thus, the simple inequality [6] r min < C (13)acts as a necessary requirement for the triggering of theergoregion instability. It simply states that the ergore-gion [whose outer boundary is given by r = r ergo = C ,see Eq. (6)] must be part of the physical system.It should be emphasized, however, that not every hy-drodynamic vortex system with r min < r ergo = C is un-stable under perturbations of the m th sound mode [6].In particular, a remarkable feature of the hydrodynamicvortex system is the existence, for each given set ( C, m )of the fluid and field parameters, of a critical cylinderradius, r min = r ∗ min ( C, m ) , (14)which supports stationary ( marginally-stable ) fluid con-figurations.The critical (maximal) cylinder radius (14) marks theboundary between stable and unstable composed fluid-cylinder configurations: composed systems whose cylin-der radius lies in the regime r min > r ∗ min ( C, m ) are sta-ble under perturbations of the m th sound mode (thatis, the sound mode decays in time), whereas composedsystems whose cylinder radius lies in the regime r min The boundary conditions (10)-(12) single out a dis-crete set of complex resonances { ω n ( C, m, r min ) } [22].These quasinormal resonances characterize the temporalresponse of the hydrodynamic system to external (soundmode) perturbations. Note, in particular, that stable (ex-ponentially suppressed) sound modes are characterizedby ℑ ω < 0, whereas unstable (growing in time) soundmodes are characterized by ℑ ω > 0. The stationary(marginally-stable) resonances, which are the solutionswe shall be interested in in this study, are characterizedby ℑ ω = 0.In this section we shall analyze the marginally-stable( ℑ ω = 0) resonances of the hydrodynamic vortex sys-tem. As we shall show below, this hydrodynamic fluidsystem is actually characterized by genuine static reso-nances with ℜ ω = ℑ ω = 0 . (15)In particular, we shall now find the discrete set of crit-ical cylinder radii, { r ∗ min ( C, m ; n ) } , which support thesemarginally-stable fluid configurations.Remarkably, the radial equation (9) can be solved an-alytically for the marginally-stable modes (15). The gen-eral solution of Eq. (9) with ω = 0 can be expressed interms of the Bessel functions of the first and second kinds(see Eq. 9.1.53 of [23]): ψ ( r ; ω = 0) = Ar J m (cid:16) Cmr (cid:17) + Br Y m (cid:16) Cmr (cid:17) , (16)where A and B are normalization constants. The asymp-totic large- r limit ( Cm/r → 0) of Eq. (16) is given by(see Eqs. 9.1.7 and 9.1.9 of [23]) ψ ( r → ∞ ; ω = 0) = Am ! (cid:16) Cm (cid:17) m r − m + − B ( m − π (cid:16) Cm (cid:17) − m r m + . (17)A physically acceptable (finite) solution at infinity re-quires B = 0, which implies ψ ( r ; ω = 0) = Ar J m (cid:16) Cmr (cid:17) . (18) Taking cognizance of the boundary conditions (10)-(11) which characterize the acoustic properties of thecentral cylinder, one finds that the marginally-stable res-onances (15) of the hydrodynamic vortex system corre-spond to the following discrete radii of the cylinder: r ∗ min ( C, m ; n ) = Cmj m,n , BCI , (19)and r ∗ min ( C, m ; n ) = Cmj ′ m,n , BCII , (20)where n = 1 , , , ... . Here j m,n is the n th positive zeroof the Bessel function J m ( x ) and j ′ m,n is the n th positivezero of its first spatial derivative J ′ m ( x ). The real zerosof these functions were studied by many authors, see e.g.[23, 24].For large values of the acoustic harmonic index, m ≫ n , one may use the asymptotic relations (see Eqs. 9.5.14and 9.5.16 of [23]) j m,n = m [1 + b n m − / + O ( m − / )]and j ′ m,n = m [1+ b ′ n m − / + O ( m − / )] [25]. Substitutingthese relations into Eqs. (19) and (20), one finds r ∗ min ( C, m ≫ n ; n ) = C [1 − b n m − / + O ( m − / )] , BCI , (21)and r ∗ min ( C, m ≫ n ; n ) = C [1 − b ′ n m − / + O ( m − / )] , BCII . (22)It is worth emphasizing that Eqs. (21)-(22) provides ananalytical quantitative explanation for the asymptoticlarge- m behavior of the hydrodynamic vortex system asnumerically presented in Fig. 5 of [6].For large overtone numbers, n ≫ m , one may use theasymptotic relations (see Eqs. 9.5.12 and 9.5.13 of [23]) j m,n = ( n + m/ − / π + O ( m /n ) and j ′ m,n = ( n + m/ − / π + O ( m /n ). Substituting these relations intoEqs. (19) and (20), one finds r ∗ min ( C, m ; n ≫ m ) = Cmπn [1 + O ( m/n )] (23)for both types of boundary conditions. The relation (23)implies that, large overtone modes must be supported bysmall radii cylinders in order to be able to trigger super-radiant instabilities in the hydrodynamic vortex system. VI. ANALYTICAL VS. NUMERICAL RESULTS We shall now compare the predictions of the analyti-cally derived formulas (19)-(20) for the critical cylinderradii, r ∗ min ( C, m ; n ), with the corresponding numericaldata recently published by Oliveira et. al. [6]. In Ta-ble I we present such a comparison, from which one findsa remarkably excellent agreement between the analyticalformulas (19)-(20) and the numerical results of [6]. BC m r ∗ min (Numerical) r ∗ min (Analytical)I 1 0.13 0.1305I 2 0.19 0.1947II 1 0.27 0.2716II 2 0.33 0.3274TABLE I: Marginally stable resonances of the hydrodynamicvortex system. We display the critical cylinder radii, r ∗ min ,for the fundamental n = 1 sound mode with fluid circulation C = 0 . VII. SUMMARY AND DISCUSSION In this paper, we have used analytical tools in orderto analyze the marginally-stable resonances of the hydro-dynamic vortex system, an effective spacetime geometryfor sound waves. These resonances are fundamental tothe physics of sound waves in the hydrodynamic acous-tic spacetime: in particular, they mark the onset of thesuperradiant instability in the hydrodynamic vortex sys-tem.In order to mimic a possible experimental scenario inthe laboratory, Oliveira et. al. [6] have recently sug-gested to place a long cylinder of radius r min at the cen-ter of the fluid system, on which physically acceptable boundary conditions [see Eq. (10) and (11)] are imposed.A remarkable feature of the hydrodynamic vortex sys-tem is the existence, for each given set ( C, m ) of thefluid and sound mode parameters, of a critical cylinderradius, r min = r ∗ min ( C, m ), which supports marginally-stable fluid configurations.In the present study we have derived the character-istic resonance conditions [see Eq. (19) and (20)] forthese marginally-stable composed fluid-cylinder config-urations. In particular, it was shown that the criticalcylinder radius r ∗ min (the cylinder radius which, for givenparameters C and m of the system, marks the bound-ary between stable and unstable fluid-cylinder configura-tions) can be expressed in terms of the simple zeros ofthe Bessel function and its first derivative.It was shown that the analytically derived formulas forthe critical cylinder radius [see Eq. (19) and (20)] agreewith the recently published numerical data for the hy-drodynamic vortex system.Finally, it is worth emphasizing that the physical sig-nificance of the critical cylinder radius, r ∗ min , lies in thefact that it is the outermost location of the cylinder whichallows the extraction of rotational energy from the circu-lating fluid (from the ergoregion of the effective spacetimegeometry). ACKNOWLEDGMENTS This research is supported by the Carmel ScienceFoundation. I thank Yael Oren, Arbel M. Ongo andAyelet B. Lata for stimulating discussions. [1] R. P. Kerr, Phys. Rev. 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